algebrite/sources/integral.coffee

Computer Algebra System in Coffeescript

###
 Table of integrals

The symbol f is just a dummy symbol for creating a list f(A,B,C,C,...) where

  A  is the template expression

  B  is the result expression

  C  is an optional list of conditional expressions
###

itab = [
  # 1
  "f(a,a*x)",
  # 9 (need a caveat for 7 so we can put 9 after 7)
  "f(1/x,log(x))",
  # 7
  "f(x^a,x^(a+1)/(a+1))",
  # five specialisations of case 7 for speed.
  # Covers often-occurring exponents: each of
  # these case ends up in a dedicated entry, so we
  # only have to do one sure-shot match.
  "f(x^(-2),-x^(-1))",
  "f(x^(-1/2),2*x^(1/2))",
  "f(x^(1/2),2/3*x^(3/2))",
  "f(x,x^2/2)",
  "f(x^2,x^3/3)",
  # 12
  "f(exp(a*x),1/a*exp(a*x))",
  "f(exp(a*x+b),1/a*exp(a*x+b))",
  "f(x*exp(a*x^2),exp(a*x^2)/(2*a))",
  "f(x*exp(a*x^2+b),exp(a*x^2+b)/(2*a))",
  # 14
  "f(log(a*x),x*log(a*x)-x)",
  # 15
  "f(a^x,a^x/log(a),or(not(number(a)),a>0))",
  # 16
  "f(1/(a+x^2),1/sqrt(a)*arctan(x/sqrt(a)),or(not(number(a)),a>0))",
  # 17
  "f(1/(a-x^2),1/sqrt(a)*arctanh(x/sqrt(a)))",
  # 19
  "f(1/sqrt(a-x^2),arcsin(x/(sqrt(a))))",
  # 20
  "f(1/sqrt(a+x^2),log(x+sqrt(a+x^2)))",
  # 27
  "f(1/(a+b*x),1/b*log(a+b*x))",
  # 28
  "f(1/(a+b*x)^2,-1/(b*(a+b*x)))",
  # 29
  "f(1/(a+b*x)^3,-1/(2*b)*1/(a+b*x)^2)",
  # 30
  "f(x/(a+b*x),x/b-a*log(a+b*x)/b/b)",
  # 31
  "f(x/(a+b*x)^2,1/b^2*(log(a+b*x)+a/(a+b*x)))",
  # 33
  "f(x^2/(a+b*x),1/b^2*(1/2*(a+b*x)^2-2*a*(a+b*x)+a^2*log(a+b*x)))",
  # 34
  "f(x^2/(a+b*x)^2,1/b^3*(a+b*x-2*a*log(a+b*x)-a^2/(a+b*x)))",
  # 35
  "f(x^2/(a+b*x)^3,1/b^3*(log(a+b*x)+2*a/(a+b*x)-1/2*a^2/(a+b*x)^2))",
  # 37
  "f(1/x*1/(a+b*x),-1/a*log((a+b*x)/x))",
  # 38
  "f(1/x*1/(a+b*x)^2,1/a*1/(a+b*x)-1/a^2*log((a+b*x)/x))",
  # 39
  "f(1/x*1/(a+b*x)^3,1/a^3*(1/2*((2*a+b*x)/(a+b*x))^2+log(x/(a+b*x))))",
  # 40
  "f(1/x^2*1/(a+b*x),-1/(a*x)+b/a^2*log((a+b*x)/x))",
  # 41
  "f(1/x^3*1/(a+b*x),(2*b*x-a)/(2*a^2*x^2)+b^2/a^3*log(x/(a+b*x)))",
  # 42
  "f(1/x^2*1/(a+b*x)^2,-(a+2*b*x)/(a^2*x*(a+b*x))+2*b/a^3*log((a+b*x)/x))",
  # 60
  "f(1/(a+b*x^2),1/sqrt(a*b)*arctan(x*sqrt(a*b)/a),or(not(number(a*b)),a*b>0))",
  # 61
  "f(1/(a+b*x^2),1/(2*sqrt(-a*b))*log((a+x*sqrt(-a*b))/(a-x*sqrt(-a*b))),or(not(number(a*b)),a*b<0))",
  # 62 is the same as 60
  # 63
  "f(x/(a+b*x^2),1/2*1/b*log(a+b*x^2))",
  #64
  "f(x^2/(a+b*x^2),x/b-a/b*integral(1/(a+b*x^2),x))",
  #65
  "f(1/(a+b*x^2)^2,x/(2*a*(a+b*x^2))+1/2*1/a*integral(1/(a+b*x^2),x))",
  #66 is covered by 61
  #70
  "f(1/x*1/(a+b*x^2),1/2*1/a*log(x^2/(a+b*x^2)))",
  #71
  "f(1/x^2*1/(a+b*x^2),-1/(a*x)-b/a*integral(1/(a+b*x^2),x))",
  #74
  "f(1/(a+b*x^3),1/3*1/a*(a/b)^(1/3)*(1/2*log(((a/b)^(1/3)+x)^3/(a+b*x^3))+sqrt(3)*arctan((2*x-(a/b)^(1/3))*(a/b)^(-1/3)/sqrt(3))))",
  #76
  "f(x^2/(a+b*x^3),1/3*1/b*log(a+b*x^3))",
  # float(defint(1/(2+3*X^4),X,0,pi)) gave wrong result.
  # Also, the tests related to the indefinite integral
  # fail since we rationalise expressions "better", so I'm thinking
  # to take this out completely as it seemed to give the
  # wrong results in the first place.
  #77
  #"f(1/(a+b*x^4),1/2*1/a*(a/b/4)^(1/4)*(1/2*log((x^2+2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2))/(x^2-2*(a/b/4)^(1/4)*x+2*(a/b/4)^(1/2)))+arctan(2*(a/b/4)^(1/4)*x/(2*(a/b/4)^(1/2)-x^2))),or(not(number(a*b)),a*b>0))",
  #78
  #"f(1/(a+b*x^4),1/2*(-a/b)^(1/4)/a*(1/2*log((x+(-a/b)^(1/4))/(x-(-a/b)^(1/4)))+arctan(x*(-a/b)^(-1/4))),or(not(number(a*b)),a*b<0))",
  #79
  "f(x/(a+b*x^4),1/2*sqrt(b/a)/b*arctan(x^2*sqrt(b/a)),or(not(number(a*b)),a*b>0))",
  #80
  "f(x/(a+b*x^4),1/4*sqrt(-b/a)/b*log((x^2-sqrt(-a/b))/(x^2+sqrt(-a/b))),or(not(number(a*b)),a*b<0))",

  # float(defint(X^2/(2+3*X^4),X,0,pi)) gave wrong result.
  # Also, the tests related to the indefinite integral
  # fail since we rationalise expressions "better", so I'm thinking
  # to take this out completely as it seemed to give the
  # wrong results in the first place.
  #81
  #"f(x^2/(a+b*x^4),1/4*1/b*(a/b/4)^(-1/4)*(1/2*log((x^2-2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4))/(x^2+2*(a/b/4)^(1/4)*x+2*sqrt(a/b/4)))+arctan(2*(a/b/4)^(1/4)*x/(2*sqrt(a/b/4)-x^2))),or(not(number(a*b)),a*b>0))",
  #82
  #"f(x^2/(a+b*x^4),1/4*1/b*(-a/b)^(-1/4)*(log((x-(-a/b)^(1/4))/(x+(-a/b)^(1/4)))+2*arctan(x*(-a/b)^(-1/4))),or(not(number(a*b)),a*b<0))",
  #83
  "f(x^3/(a+b*x^4),1/4*1/b*log(a+b*x^4))",
  #124
  "f(sqrt(a+b*x),2/3*1/b*sqrt((a+b*x)^3))",
  #125
  "f(x*sqrt(a+b*x),-2*(2*a-3*b*x)*sqrt((a+b*x)^3)/15/b^2)",
  #126
  "f(x^2*sqrt(a+b*x),2*(8*a^2-12*a*b*x+15*b^2*x^2)*sqrt((a+b*x)^3)/105/b^3)",
  #128
  "f(sqrt(a+b*x)/x,2*sqrt(a+b*x)+a*integral(1/x*1/sqrt(a+b*x),x))",
  #129
  "f(sqrt(a+b*x)/x^2,-sqrt(a+b*x)/x+b/2*integral(1/x*1/sqrt(a+b*x),x))",
  #131
  "f(1/sqrt(a+b*x),2*sqrt(a+b*x)/b)",
  #132
  "f(x/sqrt(a+b*x),-2/3*(2*a-b*x)*sqrt(a+b*x)/b^2)",
  #133
  "f(x^2/sqrt(a+b*x),2/15*(8*a^2-4*a*b*x+3*b^2*x^2)*sqrt(a+b*x)/b^3)",
  #135
  "f(1/x*1/sqrt(a+b*x),1/sqrt(a)*log((sqrt(a+b*x)-sqrt(a))/(sqrt(a+b*x)+sqrt(a))),or(not(number(a)),a>0))",
  #136
  "f(1/x*1/sqrt(a+b*x),2/sqrt(-a)*arctan(sqrt(-(a+b*x)/a)),or(not(number(a)),a<0))",
  #137
  "f(1/x^2*1/sqrt(a+b*x),-sqrt(a+b*x)/a/x-1/2*b/a*integral(1/x*1/sqrt(a+b*x),x))",
  #156
  "f(sqrt(x^2+a),1/2*(x*sqrt(x^2+a)+a*log(x+sqrt(x^2+a))))",
  #157
  "f(1/sqrt(x^2+a),log(x+sqrt(x^2+a)))",
  #158
  "f(1/x*1/sqrt(x^2+a),arcsec(x/sqrt(-a))/sqrt(-a),or(not(number(a)),a<0))",
  #159
  "f(1/x*1/sqrt(x^2+a),-1/sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
  #160
  "f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
  #161
  "f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(-a)*arcsec(x/sqrt(-a)),or(not(number(a)),a<0))",
  #162
  "f(x/sqrt(x^2+a),sqrt(x^2+a))",
  #163
  "f(x*sqrt(x^2+a),1/3*sqrt((x^2+a)^3))",
  #164 need an unexpanded version?
  "f(sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2+a^(1/3))^3)+3/2*a^(1/3)*x*sqrt(x^2+a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2+a^(1/3)))))",
    # match doesn't work for the following
  "f(sqrt(-a+x^6-3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2-a^(1/3))^3)-3/2*a^(1/3)*x*sqrt(x^2-a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2-a^(1/3)))))",
  #165
  "f(1/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),x/a^(1/3)/sqrt(x^2+a^(1/3)))",
  #166
  "f(x/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),-1/sqrt(x^2+a^(1/3)))",
  #167
  "f(x*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/5*sqrt((x^2+a^(1/3))^5))",
  #168
  "f(x^2*sqrt(x^2+a),1/4*x*sqrt((x^2+a)^3)-1/8*a*x*sqrt(x^2+a)-1/8*a^2*log(x+sqrt(x^2+a)))",
  #169
  "f(x^3*sqrt(x^2+a),(1/5*x^2-2/15*a)*sqrt((x^2+a)^3),and(number(a),a>0))",
  #170
  "f(x^3*sqrt(x^2+a),sqrt((x^2+a)^5)/5-a*sqrt((x^2+a)^3)/3,and(number(a),a<0))",
  #171
  "f(x^2/sqrt(x^2+a),1/2*x*sqrt(x^2+a)-1/2*a*log(x+sqrt(x^2+a)))",
  #172
  "f(x^3/sqrt(x^2+a),1/3*sqrt((x^2+a)^3)-a*sqrt(x^2+a))",
  #173
  "f(1/x^2*1/sqrt(x^2+a),-sqrt(x^2+a)/a/x)",
  #174
  "f(1/x^3*1/sqrt(x^2+a),-1/2*sqrt(x^2+a)/a/x^2+1/2*log((sqrt(a)+sqrt(x^2+a))/x)/a^(3/2),or(not(number(a)),a>0))",
  #175
  "f(1/x^3*1/sqrt(x^2-a),1/2*sqrt(x^2-a)/a/x^2+1/2*1/(a^(3/2))*arcsec(x/(a^(1/2))),or(not(number(a)),a>0))",
  #176+
  "f(x^2*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/6*x*sqrt((x^2+a^(1/3))^5)-1/24*a^(1/3)*x*sqrt((x^2+a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2+a^(1/3))-1/16*a*log(x+sqrt(x^2+a^(1/3))),or(not(number(a)),a>0))",
  #176-
  "f(x^2*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/6*x*sqrt((x^2-a^(1/3))^5)+1/24*a^(1/3)*x*sqrt((x^2-a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2-a^(1/3))+1/16*a*log(x+sqrt(x^2-a^(1/3))),or(not(number(a)),a>0))",
  #177+
  "f(x^3*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/7*sqrt((x^2+a^(1/3))^7)-1/5*a^(1/3)*sqrt((x^2+a^(1/3))^5),or(not(number(a)),a>0))",
  #177-
  "f(x^3*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/7*sqrt((x^2-a^(1/3))^7)+1/5*a^(1/3)*sqrt((x^2-a^(1/3))^5),or(not(number(a)),a>0))",
  #196
  "f(1/(x-a)/sqrt(x^2-a^2),-sqrt(x^2-a^2)/a/(x-a))",
  #197
  "f(1/(x+a)/sqrt(x^2-a^2),sqrt(x^2-a^2)/a/(x+a))",
  #200+
  "f(sqrt(a-x^2),1/2*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(abs(a)))))",
  #201    (seems to be handled somewhere else)
  #202
  "f(1/x*1/sqrt(a-x^2),-1/sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",
  #203
  "f(sqrt(a-x^2)/x,sqrt(a-x^2)-sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))",
  #204
  "f(x/sqrt(a-x^2),-sqrt(a-x^2))",
  #205
  "f(x*sqrt(a-x^2),-1/3*sqrt((a-x^2)^3))",
  #210
  "f(x^2*sqrt(a-x^2),-x/4*sqrt((a-x^2)^3)+1/8*a*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(a))),or(not(number(a)),a>0))",
  #211
  "f(x^3*sqrt(a-x^2),(-1/5*x^2-2/15*a)*sqrt((a-x^2)^3),or(not(number(a)),a>0))",
  #214
  "f(x^2/sqrt(a-x^2),-x/2*sqrt(a-x^2)+a/2*arcsin(x/sqrt(a)),or(not(number(a)),a>0))",
  #215
  "f(1/x^2*1/sqrt(a-x^2),-sqrt(a-x^2)/a/x,or(not(number(a)),a>0))",
  #216
  "f(sqrt(a-x^2)/x^2,-sqrt(a-x^2)/x-arcsin(x/sqrt(a)),or(not(number(a)),a>0))",
  #217
  "f(sqrt(a-x^2)/x^3,-1/2*sqrt(a-x^2)/x^2+1/2*log((sqrt(a)+sqrt(a-x^2))/x)/sqrt(a),or(not(number(a)),a>0))",
  #218
  "f(sqrt(a-x^2)/x^4,-1/3*sqrt((a-x^2)^3)/a/x^3,or(not(number(a)),a>0))",
  # 273
  "f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*log(x*sqrt(a)+sqrt(a*x^2+b))/2/sqrt(a),and(number(a),a>0))",
  # 274
  "f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*arcsin(x*sqrt(-a/b))/2/sqrt(-a),and(number(a),a<0))",
  # 290
  "f(sin(a*x),-cos(a*x)/a)",
  # 291
  "f(cos(a*x),sin(a*x)/a)",
  # 292
  "f(tan(a*x),-log(cos(a*x))/a)",
  # 293
  "f(1/tan(a*x),log(sin(a*x))/a)",
  # 294
  "f(1/cos(a*x),log(tan(pi/4+a*x/2))/a)",
  # 295
  "f(1/sin(a*x),log(tan(a*x/2))/a)",
  # 296
  "f(sin(a*x)^2,x/2-sin(2*a*x)/(4*a))",
  # 297
  "f(sin(a*x)^3,-cos(a*x)*(sin(a*x)^2+2)/(3*a))",
  # 298
  "f(sin(a*x)^4,3/8*x-sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a))",
  # 302
  "f(cos(a*x)^2,x/2+sin(2*a*x)/(4*a))",
  # 303
  "f(cos(a*x)^3,sin(a*x)*(cos(a*x)^2+2)/(3*a))",
  # 304
  "f(cos(a*x)^4,3/8*x+sin(2*a*x)/(4*a)+sin(4*a*x)/(32*a))",
  # 308
  "f(1/sin(a*x)^2,-1/(a*tan(a*x)))",
  # 312
  "f(1/cos(a*x)^2,tan(a*x)/a)",
  # 318
  "f(sin(a*x)*cos(a*x),sin(a*x)^2/(2*a))",
  # 320
  "f(sin(a*x)^2*cos(a*x)^2,-sin(4*a*x)/(32*a)+x/8)",
  # 326
  "f(sin(a*x)/cos(a*x)^2,1/(a*cos(a*x)))",
  # 327
  "f(sin(a*x)^2/cos(a*x),(log(tan(pi/4+a*x/2))-sin(a*x))/a)",
  # 328
  "f(cos(a*x)/sin(a*x)^2,-1/(a*sin(a*x)))",
  # 329
  "f(1/(sin(a*x)*cos(a*x)),log(tan(a*x))/a)",
  # 330
  "f(1/(sin(a*x)*cos(a*x)^2),(1/cos(a*x)+log(tan(a*x/2)))/a)",
  # 331
  "f(1/(sin(a*x)^2*cos(a*x)),(log(tan(pi/4+a*x/2))-1/sin(a*x))/a)",
  # 333
  "f(1/(sin(a*x)^2*cos(a*x)^2),-2/(a*tan(2*a*x)))",
  # 335
  "f(sin(a+b*x),-cos(a+b*x)/b)",
  # 336
  "f(cos(a+b*x),sin(a+b*x)/b)",
  # 337+ (with the addition of b)
  "f(1/(b+b*sin(a*x)),-tan(pi/4-a*x/2)/a/b)",
  # 337- (with the addition of b)
  "f(1/(b-b*sin(a*x)),tan(pi/4+a*x/2)/a/b)",
  # 338 (with the addition of b)
  "f(1/(b+b*cos(a*x)),tan(a*x/2)/a/b)",
  # 339 (with the addition of b)
  "f(1/(b-b*cos(a*x)),-1/tan(a*x/2)/a/b)",
  # 340
  "f(1/(a+b*sin(x)),1/sqrt(b^2-a^2)*log((a*tan(x/2)+b-sqrt(b^2-a^2))/(a*tan(x/2)+b+sqrt(b^2-a^2))),b^2-a^2)",   # check that b^2-a^2 is not zero
  # 341
  "f(1/(a+b*cos(x)),1/sqrt(b^2-a^2)*log((sqrt(b^2-a^2)*tan(x/2)+a+b)/(sqrt(b^2-a^2)*tan(x/2)-a-b)),b^2-a^2)",   # check that b^2-a^2 is not zero
  # 389
  "f(x*sin(a*x),sin(a*x)/a^2-x*cos(a*x)/a)",
  # 390
  "f(x^2*sin(a*x),2*x*sin(a*x)/a^2-(a^2*x^2-2)*cos(a*x)/a^3)",
  # 393
  "f(x*cos(a*x),cos(a*x)/a^2+x*sin(a*x)/a)",
  # 394
  "f(x^2*cos(a*x),2*x*cos(a*x)/a^2+(a^2*x^2-2)*sin(a*x)/a^3)",
  # 441
  "f(arcsin(a*x),x*arcsin(a*x)+sqrt(1-a^2*x^2)/a)",
  # 442
  "f(arccos(a*x),x*arccos(a*x)-sqrt(1-a^2*x^2)/a)",
  # 443
  "f(arctan(a*x),x*arctan(a*x)-1/2*log(1+a^2*x^2)/a)",
  # 485 (with addition of a)
  # however commenting out since it's a duplicate of 14
  # "f(log(a*x),x*log(a*x)-x)",
  # 486 (with addition of a)
  "f(x*log(a*x),x^2*log(a*x)/2-x^2/4)",
  # 487 (with addition of a)
  "f(x^2*log(a*x),x^3*log(a*x)/3-1/9*x^3)",
  # 489
  "f(log(x)^2,x*log(x)^2-2*x*log(x)+2*x)",
  # 493 (with addition of a)
  "f(1/x*1/(a+log(x)),log(a+log(x)))",
  # 499
  "f(log(a*x+b),(a*x+b)*log(a*x+b)/a-x)",
  # 500
  "f(log(a*x+b)/x^2,a/b*log(x)-(a*x+b)*log(a*x+b)/b/x)",
  # 554
  "f(sinh(x),cosh(x))",
  # 555
  "f(cosh(x),sinh(x))",
  # 556
  "f(tanh(x),log(cosh(x)))",
  # 560
  "f(x*sinh(x),x*cosh(x)-sinh(x))",
  # 562
  "f(x*cosh(x),x*sinh(x)-cosh(x))",
  # 566
  "f(sinh(x)^2,sinh(2*x)/4-x/2)",
  # 569
  "f(tanh(x)^2,x-tanh(x))",
  # 572
  "f(cosh(x)^2,sinh(2*x)/4+x/2)",
  # ?
  "f(x^3*exp(a*x^2),exp(a*x^2)*(x^2/a-1/(a^2))/2)",
  # ?
  "f(x^3*exp(a*x^2+b),exp(a*x^2)*exp(b)*(x^2/a-1/(a^2))/2)",
  # ?
  "f(exp(a*x^2),-i*sqrt(pi)*erf(i*sqrt(a)*x)/sqrt(a)/2)",
  # ?
  "f(erf(a*x),x*erf(a*x)+exp(-a^2*x^2)/a/sqrt(pi))",

  # these are needed for the surface integral in the manual

  "f(x^2*(1-x^2)^(3/2),(x*sqrt(1-x^2)*(-8*x^4+14*x^2-3)+3*arcsin(x))/48)",
  "f(x^2*(1-x^2)^(5/2),(x*sqrt(1-x^2)*(48*x^6-136*x^4+118*x^2-15)+15*arcsin(x))/384)",
  "f(x^4*(1-x^2)^(3/2),(-x*sqrt(1-x^2)*(16*x^6-24*x^4+2*x^2+3)+3*arcsin(x))/128)",

  "f(x*exp(a*x),exp(a*x)*(a*x-1)/(a^2))",
  "f(x*exp(a*x+b),exp(a*x+b)*(a*x-1)/(a^2))",

  "f(x^2*exp(a*x),exp(a*x)*(a^2*x^2-2*a*x+2)/(a^3))",
  "f(x^2*exp(a*x+b),exp(a*x+b)*(a^2*x^2-2*a*x+2)/(a^3))",

  "f(x^3*exp(a*x),exp(a*x)*x^3/a-3/a*integral(x^2*exp(a*x),x))",
  "f(x^3*exp(a*x+b),exp(a*x+b)*x^3/a-3/a*integral(x^2*exp(a*x+b),x))",
  0

]


#define F p3
#define X p4
#define N p5

Eval_integral = ->
  i = 0
  n = 0

  # evaluate 1st arg to get function F

  p1 = cdr(p1)
  push(car(p1))
  Eval()

  # evaluate 2nd arg and then...

  # example    result of 2nd arg  what to do
  #
  # integral(f)    nil      guess X, N = nil
  # integral(f,2)  2      guess X, N = 2
  # integral(f,x)  x      X = x, N = nil
  # integral(f,x,2)  x      X = x, N = 2
  # integral(f,x,y)  x      X = x, N = y

  p1 = cdr(p1)
  push(car(p1))
  Eval()

  p2 = pop()
  if (p2 == symbol(NIL))
    guess()
    push(symbol(NIL))
  else if (isNumericAtom(p2))
    guess()
    push(p2)
  else
    push(p2)
    p1 = cdr(p1)
    push(car(p1))
    Eval()

  p5 = pop()
  p4 = pop()
  p3 = pop()

  while (1)

    # N might be a symbol instead of a number

    if (isNumericAtom(p5))
      push(p5)
      n = pop_integer()
      if (isNaN(n))
        stop("nth integral: check n")
    else
      n = 1

    push(p3)

    if (n >= 0)
      for i in [0...n]
        push(p4)
        integral()
    else
      n = -n
      for i in [0...n]
        push(p4)
        derivative()

    p3 = pop()

    # if N is nil then arglist is exhausted

    if (p5 == symbol(NIL))
      break

    # otherwise...

    # N    arg1    what to do
    #
    # number  nil    break
    # number  number    N = arg1, continue
    # number  symbol    X = arg1, N = arg2, continue
    #
    # symbol  nil    X = N, N = nil, continue
    # symbol  number    X = N, N = arg1, continue
    # symbol  symbol    X = N, N = arg1, continue

    if (isNumericAtom(p5))
      p1 = cdr(p1)
      push(car(p1))
      Eval()
      p5 = pop()
      if (p5 == symbol(NIL))
        break;    # arglist exhausted
      if (isNumericAtom(p5))
        doNothing = 1    # N = arg1
      else
        p4 = p5;    # X = arg1
        p1 = cdr(p1)
        push(car(p1))
        Eval()
        p5 = pop();  # N = arg2
    else
      p4 = p5;      # X = N
      p1 = cdr(p1)
      push(car(p1))
      Eval()
      p5 = pop();    # N = arg1

  push(p3);  # final result

integral = ->
  save()
  p2 = pop()
  p1 = pop()
  if (car(p1) == symbol(ADD))
    integral_of_sum()
  else if (car(p1) == symbol(MULTIPLY))
    integral_of_product()
  else
    integral_of_form()
  p1 = pop()
  if (Find(p1, symbol(INTEGRAL)))
    stop("integral: sorry, could not find a solution")
  push(p1)
  simplify();  # polish the result
  Eval();    # normalize the result
  restore()

integral_of_sum = ->
  p1 = cdr(p1)
  push(car(p1))
  push(p2)
  integral()
  p1 = cdr(p1)
  while (iscons(p1))
    push(car(p1))
    push(p2)
    integral()
    add()
    p1 = cdr(p1)

integral_of_product = ->
  push(p1)
  push(p2)
  partition()
  p1 = pop();      # pop variable part
  integral_of_form()
  multiply();      # multiply constant part

integral_of_form = ->
  hc = italu_hashcode(p1, p2).toFixed(6)
  tab = hashed_itab[hc]
  # console.log('hashcode('+p1+', '+p2+') = ' + hc + ' '+!!tab);
  if (!tab)
    # debugger
    # italu_hashcode(p1, p2)
    push_symbol(INTEGRAL)
    push(p1)
    push(p2)
    list(3)
    return
  push(p1) # free variable
  push(p2) # input expression
  transform(tab, false)
  p3 = pop()
  if (p3 == symbol(NIL))
    push_symbol(INTEGRAL)
    push(p1)
    push(p2)
    list(3)
  else
    push(p3)


# Implementation of hash codes based on ITALU (An Integral Table Look-Up)
# https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680004891.pdf
# see Appendix A, page 153

# The first two values are from the ITALU paper.
# The others are just arbitrary constants.
hashcode_values =
  'x':0.95532,
  'constexp':1.43762,
  'constant':1.14416593629414332,
  'constbase':1.20364122304218824,
  'sin': 1.73305482518303221,
  'arcsin': 1.6483368529465804,
  'cos': 1.058672123686340116,
  'arccos': 1.8405225918106694,
  'tan': 1.12249437762925064,
  'arctan': 1.1297397925394962,
  'sinh':1.8176164926060078,
  'cosh':1.9404934661708022,
  'tanh':1.6421307715103121,
  'log':1.47744370135492387,
  'erf':1.0825269225702916,

italu_hashcode = (u, x) ->
  if (issymbol(u))
    if (equal(u, x))
      return hashcode_values.x
    else
      return hashcode_values.constant
  else if (iscons(u))
    switch (symnum(car(u)))
      when ADD then return hash_addition(cdr(u),x)
      when MULTIPLY then return hash_multiplication(cdr(u), x)
      when POWER then return hash_power(cadr(u), caddr(u), x)
      when EXP then return hash_power(symbol(E), cadr(u), x)
      when SQRT
        push_double(0.5)
        half = pop()
        return hash_power(cadr(u), half, x)
      else return hash_function(u, x)
    
  return hashcode_values.constant

hash_function = (u, x) ->
  if !Find(cadr(u), x)
    return hashcode_values.constant
  name = car(u)
  arg_hash = italu_hashcode(cadr(u), x)
  base = hashcode_values[name.printname]
  if (!base)
    throw new Error('Unsupported function ' + name.printname)
  return Math.pow(base, arg_hash)

hash_addition = (terms, x) ->
  term_set = {}
  while (iscons(terms))
    term = car(terms)
    terms = cdr(terms)
    term_hash = 0
    if Find(term, x)
      term_hash = italu_hashcode(term, x)
    else
      # The original algorithm would skip this,
      # but recording that it was present helps
      # prevent collisions.
      term_hash = hashcode_values.constant
    term_set[term_hash.toFixed(6)] = true
  sum = 0
  for own k,v of term_set
    sum = sum + parseFloat(k,10)
  return sum

hash_multiplication = (terms, x) ->
  product = 1
  while (iscons(terms))
    term = car(terms)
    terms = cdr(terms)
    if (Find(term, x))
      product = product * italu_hashcode(term, x)
  return product

hash_power = (base, power, x) ->
  base_hash = hashcode_values.constant
  exp_hash = hashcode_values.constexp
  if (Find(base, x))
    base_hash = italu_hashcode(base, x)
  if (Find(power, x))
    exp_hash = italu_hashcode(power, x)
  else
    # constant to constant = constant
    if base_hash == hashcode_values.constant
      return hashcode_values.constant
    if isminusone(power)
      exp_hash = -1
    else if isoneovertwo(power)
      exp_hash = 0.5
    else if isminusoneovertwo(power)
      exp_hash = -0.5
    else if equalq(power, 2, 1)
      exp_hash = 2
    else if equalq(power, -2, 1)
      exp_hash = -2
  return Math.pow(base_hash, exp_hash)

make_hashed_itab = () ->
  tab = {}
  for s in itab
    if !s
      break
    scan_meta(s)
    f = pop()
    u = cadr(f)
    h = italu_hashcode(u, symbol(METAX))
    key = h.toFixed(6)
    if (!tab[key])
      tab[key]=[]
    tab[key].push(s)
  console.log('hashed_itab = '+JSON.stringify(tab, null, 2))
  return tab

$.make_hashed_itab = make_hashed_itab

# pre-calculated hashed integral table.
# in case the integral table is changed, use this
#   Algebrite.make_hashed_itab()
# and copy the resulting JSON in here.
hashed_itab = {
  "1.144166": [
    "f(a,a*x)"
  ],
  "1.046770": [
    "f(1/x,log(x))"
  ],
  "0.936400": [
    "f(x^a,x^(a+1)/(a+1))"
  ],
  "1.095727": [
    "f(x^(-2),-x^(-1))"
  ],
  "1.023118": [
    "f(x^(-1/2),2*x^(1/2))"
  ],
  "0.977405": [
    "f(x^(1/2),2/3*x^(3/2))"
  ],
  "0.955320": [
    "f(x,x^2/2)"
  ],
  "0.912636": [
    "f(x^2,x^3/3)"
  ],
  "1.137302": [
    "f(exp(a*x),1/a*exp(a*x))",
    "f(a^x,a^x/log(a),or(not(number(a)),a>0))"
  ],
  "1.326774": [
    "f(exp(a*x+b),1/a*exp(a*x+b))"
  ],
  "1.080259": [
    "f(x*exp(a*x^2),exp(a*x^2)/(2*a))"
  ],
  "1.260228": [
    "f(x*exp(a*x^2+b),exp(a*x^2+b)/(2*a))"
  ],
  "1.451902": [
    "f(log(a*x),x*log(a*x)-x)"
  ],
  "0.486192": [
    "f(1/(a+x^2),1/sqrt(a)*arctan(x/sqrt(a)),or(not(number(a)),a>0))",
    "f(1/(a-x^2),1/sqrt(a)*arctanh(x/sqrt(a)))",
    "f(1/(a+b*x^2),1/sqrt(a*b)*arctan(x*sqrt(a*b)/a),or(not(number(a*b)),a*b>0))",
    "f(1/(a+b*x^2),1/(2*sqrt(-a*b))*log((a+x*sqrt(-a*b))/(a-x*sqrt(-a*b))),or(not(number(a*b)),a*b<0))"
  ],
  "0.697274": [
    "f(1/sqrt(a-x^2),arcsin(x/(sqrt(a))))",
    "f(1/sqrt(a+x^2),log(x+sqrt(a+x^2)))",
    "f(1/sqrt(x^2+a),log(x+sqrt(x^2+a)))"
  ],
  "0.476307": [
    "f(1/(a+b*x),1/b*log(a+b*x))"
  ],
  "0.226868": [
    "f(1/(a+b*x)^2,-1/(b*(a+b*x)))"
  ],
  "2.904531": [
    "f(1/(a+b*x)^3,-1/(2*b)*1/(a+b*x)^2)"
  ],
  "0.455026": [
    "f(x/(a+b*x),x/b-a*log(a+b*x)/b/b)"
  ],
  "0.216732": [
    "f(x/(a+b*x)^2,1/b^2*(log(a+b*x)+a/(a+b*x)))"
  ],
  "0.434695": [
    "f(x^2/(a+b*x),1/b^2*(1/2*(a+b*x)^2-2*a*(a+b*x)+a^2*log(a+b*x)))"
  ],
  "0.207048": [
    "f(x^2/(a+b*x)^2,1/b^3*(a+b*x-2*a*log(a+b*x)-a^2/(a+b*x)))"
  ],
  "2.650781": [
    "f(x^2/(a+b*x)^3,1/b^3*(log(a+b*x)+2*a/(a+b*x)-1/2*a^2/(a+b*x)^2))"
  ],
  "0.498584": [
    "f(1/x*1/(a+b*x),-1/a*log((a+b*x)/x))"
  ],
  "0.237479": [
    "f(1/x*1/(a+b*x)^2,1/a*1/(a+b*x)-1/a^2*log((a+b*x)/x))"
  ],
  "3.040375": [
    "f(1/x*1/(a+b*x)^3,1/a^3*(1/2*((2*a+b*x)/(a+b*x))^2+log(x/(a+b*x))))"
  ],
  "0.521902": [
    "f(1/x^2*1/(a+b*x),-1/(a*x)+b/a^2*log((a+b*x)/x))"
  ],
  "0.446014": [
    "f(1/x^3*1/(a+b*x),(2*b*x-a)/(2*a^2*x^2)+b^2/a^3*log(x/(a+b*x)))"
  ],
  "0.248586": [
    "f(1/x^2*1/(a+b*x)^2,-(a+2*b*x)/(a^2*x*(a+b*x))+2*b/a^3*log((a+b*x)/x))"
  ],
  "0.464469": [
    "f(x/(a+b*x^2),1/2*1/b*log(a+b*x^2))"
  ],
  "0.443716": [
    "f(x^2/(a+b*x^2),x/b-a/b*integral(1/(a+b*x^2),x))"
  ],
  "0.236382": [
    "f(1/(a+b*x^2)^2,x/(2*a*(a+b*x^2))+1/2*1/a*integral(1/(a+b*x^2),x))"
  ],
  "0.508931": [
    "f(1/x*1/(a+b*x^2),1/2*1/a*log(x^2/(a+b*x^2)))"
  ],
  "0.532733": [
    "f(1/x^2*1/(a+b*x^2),-1/(a*x)-b/a*integral(1/(a+b*x^2),x))"
  ],
  "0.480638": [
    "f(1/(a+b*x^3),1/3*1/a*(a/b)^(1/3)*(1/2*log(((a/b)^(1/3)+x)^3/(a+b*x^3))+sqrt(3)*arctan((2*x-(a/b)^(1/3))*(a/b)^(-1/3)/sqrt(3))))"
  ],
  "0.438648": [
    "f(x^2/(a+b*x^3),1/3*1/b*log(a+b*x^3))"
  ],
  "0.459164": [
    "f(x/(a+b*x^4),1/2*sqrt(b/a)/b*arctan(x^2*sqrt(b/a)),or(not(number(a*b)),a*b>0))",
    "f(x/(a+b*x^4),1/4*sqrt(-b/a)/b*log((x^2-sqrt(-a/b))/(x^2+sqrt(-a/b))),or(not(number(a*b)),a*b<0))"
  ],
  "0.450070": [
    "f(x^3/(a+b*x^4),1/4*1/b*log(a+b*x^4))"
  ],
  "1.448960": [
    "f(sqrt(a+b*x),2/3*1/b*sqrt((a+b*x)^3))"
  ],
  "1.384221": [
    "f(x*sqrt(a+b*x),-2*(2*a-3*b*x)*sqrt((a+b*x)^3)/15/b^2)"
  ],
  "1.322374": [
    "f(x^2*sqrt(a+b*x),2*(8*a^2-12*a*b*x+15*b^2*x^2)*sqrt((a+b*x)^3)/105/b^3)"
  ],
  "1.516728": [
    "f(sqrt(a+b*x)/x,2*sqrt(a+b*x)+a*integral(1/x*1/sqrt(a+b*x),x))"
  ],
  "1.587665": [
    "f(sqrt(a+b*x)/x^2,-sqrt(a+b*x)/x+b/2*integral(1/x*1/sqrt(a+b*x),x))"
  ],
  "0.690150": [
    "f(1/sqrt(a+b*x),2*sqrt(a+b*x)/b)"
  ],
  "0.659314": [
    "f(x/sqrt(a+b*x),-2/3*(2*a-b*x)*sqrt(a+b*x)/b^2)"
  ],
  "0.629856": [
    "f(x^2/sqrt(a+b*x),2/15*(8*a^2-4*a*b*x+3*b^2*x^2)*sqrt(a+b*x)/b^3)"
  ],
  "0.722428": [
    "f(1/x*1/sqrt(a+b*x),1/sqrt(a)*log((sqrt(a+b*x)-sqrt(a))/(sqrt(a+b*x)+sqrt(a))),or(not(number(a)),a>0))",
    "f(1/x*1/sqrt(a+b*x),2/sqrt(-a)*arctan(sqrt(-(a+b*x)/a)),or(not(number(a)),a<0))"
  ],
  "0.756216": [
    "f(1/x^2*1/sqrt(a+b*x),-sqrt(a+b*x)/a/x-1/2*b/a*integral(1/x*1/sqrt(a+b*x),x))"
  ],
  "1.434156": [
    "f(sqrt(x^2+a),1/2*(x*sqrt(x^2+a)+a*log(x+sqrt(x^2+a))))",
    "f(sqrt(a-x^2),1/2*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(abs(a)))))",
    "f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*log(x*sqrt(a)+sqrt(a*x^2+b))/2/sqrt(a),and(number(a),a>0))",
    "f(sqrt(a*x^2+b),x*sqrt(a*x^2+b)/2+b*arcsin(x*sqrt(-a/b))/2/sqrt(-a),and(number(a),a<0))"
  ],
  "0.729886": [
    "f(1/x*1/sqrt(x^2+a),arcsec(x/sqrt(-a))/sqrt(-a),or(not(number(a)),a<0))",
    "f(1/x*1/sqrt(x^2+a),-1/sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
    "f(1/x*1/sqrt(a-x^2),-1/sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))"
  ],
  "1.501230": [
    "f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(a)*log((sqrt(a)+sqrt(x^2+a))/x),or(not(number(a)),a>0))",
    "f(sqrt(x^2+a)/x,sqrt(x^2+a)-sqrt(-a)*arcsec(x/sqrt(-a)),or(not(number(a)),a<0))",
    "f(sqrt(a-x^2)/x,sqrt(a-x^2)-sqrt(a)*log((sqrt(a)+sqrt(a-x^2))/x),or(not(number(a)),a>0))"
  ],
  "0.666120": [
    "f(x/sqrt(x^2+a),sqrt(x^2+a))",
    "f(x/sqrt(a-x^2),-sqrt(a-x^2))"
  ],
  "1.370077": [
    "f(x*sqrt(x^2+a),1/3*sqrt((x^2+a)^3))",
    "f(x*sqrt(a-x^2),-1/3*sqrt((a-x^2)^3))"
  ],
  "1.730087": [
    "f(sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2+a^(1/3))^3)+3/2*a^(1/3)*x*sqrt(x^2+a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2+a^(1/3)))))",
    "f(sqrt(-a+x^6-3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/4*(x*sqrt((x^2-a^(1/3))^3)-3/2*a^(1/3)*x*sqrt(x^2-a^(1/3))+3/2*a^(2/3)*log(x+sqrt(x^2-a^(1/3)))))"
  ],
  "0.578006": [
    "f(1/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),x/a^(1/3)/sqrt(x^2+a^(1/3)))"
  ],
  "0.552180": [
    "f(x/sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),-1/sqrt(x^2+a^(1/3)))"
  ],
  "1.652787": [
    "f(x*sqrt(a+x^6+3*a^(1/3)*x^4+3*a^(2/3)*x^2),1/5*sqrt((x^2+a^(1/3))^5))"
  ],
  "1.308862": [
    "f(x^2*sqrt(x^2+a),1/4*x*sqrt((x^2+a)^3)-1/8*a*x*sqrt(x^2+a)-1/8*a^2*log(x+sqrt(x^2+a)))",
    "f(x^2*sqrt(a-x^2),-x/4*sqrt((a-x^2)^3)+1/8*a*(x*sqrt(a-x^2)+a*arcsin(x/sqrt(a))),or(not(number(a)),a>0))"
  ],
  "1.342944": [
    "f(x^3*sqrt(x^2+a),(1/5*x^2-2/15*a)*sqrt((x^2+a)^3),and(number(a),a>0))",
    "f(x^3*sqrt(x^2+a),sqrt((x^2+a)^5)/5-a*sqrt((x^2+a)^3)/3,and(number(a),a<0))",
    "f(x^3*sqrt(a-x^2),(-1/5*x^2-2/15*a)*sqrt((a-x^2)^3),or(not(number(a)),a>0))",
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    "f(x^2*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/6*x*sqrt((x^2-a^(1/3))^5)+1/24*a^(1/3)*x*sqrt((x^2-a^(1/3))^3)-1/16*a^(2/3)*x*sqrt(x^2-a^(1/3))+1/16*a*log(x+sqrt(x^2-a^(1/3))),or(not(number(a)),a>0))"
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    "f(x^3*sqrt(-a-3*a^(1/3)*x^4+3*a^(2/3)*x^2+x^6),1/7*sqrt((x^2-a^(1/3))^7)+1/5*a^(1/3)*sqrt((x^2-a^(1/3))^5),or(not(number(a)),a>0))"
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    "f(1/(b-b*sin(a*x)),tan(pi/4+a*x/2)/a/b)",
    "f(1/(a+b*sin(x)),1/sqrt(b^2-a^2)*log((a*tan(x/2)+b-sqrt(b^2-a^2))/(a*tan(x/2)+b+sqrt(b^2-a^2))),b^2-a^2)"
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    "f(1/(b-b*cos(a*x)),-1/tan(a*x/2)/a/b)",
    "f(1/(a+b*cos(x)),1/sqrt(b^2-a^2)*log((sqrt(b^2-a^2)*tan(x/2)+a+b)/(sqrt(b^2-a^2)*tan(x/2)-a-b)),b^2-a^2)"
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  ],
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  ],
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    "f(1/x*1/(a+log(x)),log(a+log(x)))"
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    "f(log(a*x+b)/x^2,a/b*log(x)-(a*x+b)*log(a*x+b)/b/x)"
  ],
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    "f(tanh(x),log(cosh(x)))"
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  ],
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  ],
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    "f(sinh(x)^2,sinh(2*x)/4-x/2)"
  ],
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    "f(tanh(x)^2,x-tanh(x))"
  ],
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  ],
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  ],
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    "f(x^3*exp(a*x^2+b),exp(a*x^2)*exp(b)*(x^2/a-1/(a^2))/2)"
  ],
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    "f(exp(a*x^2),-i*sqrt(pi)*erf(i*sqrt(a)*x)/sqrt(a)/2)"
  ],
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    "f(erf(a*x),x*erf(a*x)+exp(-a^2*x^2)/a/sqrt(pi))"
  ],
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    "f(x^2*(1-x^2)^(5/2),(x*sqrt(1-x^2)*(48*x^6-136*x^4+118*x^2-15)+15*arcsin(x))/384)"
  ],
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    "f(x^4*(1-x^2)^(3/2),(-x*sqrt(1-x^2)*(16*x^6-24*x^4+2*x^2+3)+3*arcsin(x))/128)"
  ],
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    "f(x*exp(a*x),exp(a*x)*(a*x-1)/(a^2))"
  ],
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  ],
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  ],
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  ],
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    "f(x^3*exp(a*x+b),exp(a*x+b)*x^3/a-3/a*integral(x^2*exp(a*x+b),x))"
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}