# Maple integration test file: "7 Inverse hyperbolic functions\7.1 Inverse hyperbolic sine\7.1.2 (d x)^m (a+b arcsinh(c x))^n.txt"

lst:=[

# Integrands of the form (d x)^m (a+b ArcSinh[c x])^n

# Integrands of the form (b x)^m ArcSinh[a x])^n

# Integrands of the form x^m ArcSinh[a x]^n

# n>0
[x^4*arcsinh(a*x),x,4,2/15*(1+a^2*x^2)^(3/2)/a^5-1/25*(1+a^2*x^2)^(5/2)/a^5+1/5*x^5*arcsinh(a*x)-1/5*sqrt(1+a^2*x^2)/a^5],
[x^3*arcsinh(a*x),x,4,-3/32*arcsinh(a*x)/a^4+1/4*x^4*arcsinh(a*x)+3/32*x*sqrt(1+a^2*x^2)/a^3-1/16*x^3*sqrt(1+a^2*x^2)/a],
[x^2*arcsinh(a*x),x,4,-1/9*(1+a^2*x^2)^(3/2)/a^3+1/3*x^3*arcsinh(a*x)+1/3*sqrt(1+a^2*x^2)/a^3],
[x*arcsinh(a*x),x,3,1/4*arcsinh(a*x)/a^2+1/2*x^2*arcsinh(a*x)-1/4*x*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x),x,2,x*arcsinh(a*x)-sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)/x,x,5,-1/2*arcsinh(a*x)^2+arcsinh(a*x)*log(1-exp(2*arcsinh(a*x)))+1/2*polylog(2,exp(2*arcsinh(a*x)))],
[arcsinh(a*x)/x^2,x,4,-arcsinh(a*x)/x-a*arctanh(sqrt(1+a^2*x^2))],
[arcsinh(a*x)/x^3,x,2,-1/2*arcsinh(a*x)/x^2-1/2*a*sqrt(1+a^2*x^2)/x],
[arcsinh(a*x)/x^4,x,5,-1/3*arcsinh(a*x)/x^3+1/6*a^3*arctanh(sqrt(1+a^2*x^2))-1/6*a*sqrt(1+a^2*x^2)/x^2],
[arcsinh(a*x)/x^5,x,3,-1/4*arcsinh(a*x)/x^4-1/12*a*sqrt(1+a^2*x^2)/x^3+1/6*a^3*sqrt(1+a^2*x^2)/x],
[arcsinh(a*x)/x^6,x,6,-1/5*arcsinh(a*x)/x^5-3/40*a^5*arctanh(sqrt(1+a^2*x^2))-1/20*a*sqrt(1+a^2*x^2)/x^4+3/40*a^3*sqrt(1+a^2*x^2)/x^2],
[x^4*arcsinh(a*x)^2,x,7,16/75*x/a^4-8/225*x^3/a^2+2/125*x^5+1/5*x^5*arcsinh(a*x)^2-16/75*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^5+8/75*x^2*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-2/25*x^4*arcsinh(a*x)*sqrt(1+a^2*x^2)/a],
[x^3*arcsinh(a*x)^2,x,6,-3/32*x^2/a^2+1/32*x^4-3/32*arcsinh(a*x)^2/a^4+1/4*x^4*arcsinh(a*x)^2+3/16*x*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-1/8*x^3*arcsinh(a*x)*sqrt(1+a^2*x^2)/a],
[x^2*arcsinh(a*x)^2,x,5,-4/9*x/a^2+2/27*x^3+1/3*x^3*arcsinh(a*x)^2+4/9*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-2/9*x^2*arcsinh(a*x)*sqrt(1+a^2*x^2)/a],
[x*arcsinh(a*x)^2,x,4,1/4*x^2+1/4*arcsinh(a*x)^2/a^2+1/2*x^2*arcsinh(a*x)^2-1/2*x*arcsinh(a*x)*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)^2,x,3,2*x+x*arcsinh(a*x)^2-2*arcsinh(a*x)*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)^2/x,x,6,-1/3*arcsinh(a*x)^3+arcsinh(a*x)^2*log(1-exp(2*arcsinh(a*x)))+arcsinh(a*x)*polylog(2,exp(2*arcsinh(a*x)))-1/2*polylog(3,exp(2*arcsinh(a*x)))],
[arcsinh(a*x)^2/x^2,x,7,-arcsinh(a*x)^2/x-4*a*arcsinh(a*x)*arctanh(exp(arcsinh(a*x)))-2*a*polylog(2,-exp(arcsinh(a*x)))+2*a*polylog(2,exp(arcsinh(a*x)))],
[arcsinh(a*x)^2/x^3,x,3,-1/2*arcsinh(a*x)^2/x^2+a^2*log(x)-a*arcsinh(a*x)*sqrt(1+a^2*x^2)/x],
[arcsinh(a*x)^2/x^4,x,9,-1/3*a^2/x-1/3*arcsinh(a*x)^2/x^3+2/3*a^3*arcsinh(a*x)*arctanh(exp(arcsinh(a*x)))+1/3*a^3*polylog(2,-exp(arcsinh(a*x)))-1/3*a^3*polylog(2,exp(arcsinh(a*x)))-1/3*a*arcsinh(a*x)*sqrt(1+a^2*x^2)/x^2],
[arcsinh(a*x)^2/x^5,x,5,-1/12*a^2/x^2-1/4*arcsinh(a*x)^2/x^4-1/3*a^4*log(x)-1/6*a*arcsinh(a*x)*sqrt(1+a^2*x^2)/x^3+1/3*a^3*arcsinh(a*x)*sqrt(1+a^2*x^2)/x],
[x^4*arcsinh(a*x)^3,x,14,76/1125*(1+a^2*x^2)^(3/2)/a^5-6/625*(1+a^2*x^2)^(5/2)/a^5+16/25*x*arcsinh(a*x)/a^4-8/75*x^3*arcsinh(a*x)/a^2+6/125*x^5*arcsinh(a*x)+1/5*x^5*arcsinh(a*x)^3-298/375*sqrt(1+a^2*x^2)/a^5-8/25*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a^5+4/25*x^2*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a^3-3/25*x^4*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a],
[x^3*arcsinh(a*x)^3,x,11,-45/256*arcsinh(a*x)/a^4-9/32*x^2*arcsinh(a*x)/a^2+3/32*x^4*arcsinh(a*x)-3/32*arcsinh(a*x)^3/a^4+1/4*x^4*arcsinh(a*x)^3+45/256*x*sqrt(1+a^2*x^2)/a^3-3/128*x^3*sqrt(1+a^2*x^2)/a+9/32*x*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a^3-3/16*x^3*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a],
[x^2*arcsinh(a*x)^3,x,9,-2/27*(1+a^2*x^2)^(3/2)/a^3-4/3*x*arcsinh(a*x)/a^2+2/9*x^3*arcsinh(a*x)+1/3*x^3*arcsinh(a*x)^3+14/9*sqrt(1+a^2*x^2)/a^3+2/3*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a^3-1/3*x^2*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a],
[x*arcsinh(a*x)^3,x,6,3/8*arcsinh(a*x)/a^2+3/4*x^2*arcsinh(a*x)+1/4*arcsinh(a*x)^3/a^2+1/2*x^2*arcsinh(a*x)^3-3/8*x*sqrt(1+a^2*x^2)/a-3/4*x*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)^3,x,4,6*x*arcsinh(a*x)+x*arcsinh(a*x)^3-6*sqrt(1+a^2*x^2)/a-3*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)^3/x,x,7,-1/4*arcsinh(a*x)^4+arcsinh(a*x)^3*log(1-exp(2*arcsinh(a*x)))+3/2*arcsinh(a*x)^2*polylog(2,exp(2*arcsinh(a*x)))-3/2*arcsinh(a*x)*polylog(3,exp(2*arcsinh(a*x)))+3/4*polylog(4,exp(2*arcsinh(a*x)))],
[arcsinh(a*x)^3/x^2,x,9,-arcsinh(a*x)^3/x-6*a*arcsinh(a*x)^2*arctanh(exp(arcsinh(a*x)))-6*a*arcsinh(a*x)*polylog(2,-exp(arcsinh(a*x)))+6*a*arcsinh(a*x)*polylog(2,exp(arcsinh(a*x)))+6*a*polylog(3,-exp(arcsinh(a*x)))-6*a*polylog(3,exp(arcsinh(a*x)))],
[arcsinh(a*x)^3/x^3,x,7,-3/2*a^2*arcsinh(a*x)^2-1/2*arcsinh(a*x)^3/x^2+3*a^2*arcsinh(a*x)*log(1-exp(2*arcsinh(a*x)))+3/2*a^2*polylog(2,exp(2*arcsinh(a*x)))-3/2*a*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/x],
[arcsinh(a*x)^3/x^4,x,14,-a^2*arcsinh(a*x)/x-1/3*arcsinh(a*x)^3/x^3+a^3*arcsinh(a*x)^2*arctanh(exp(arcsinh(a*x)))-a^3*arctanh(sqrt(1+a^2*x^2))+a^3*arcsinh(a*x)*polylog(2,-exp(arcsinh(a*x)))-a^3*arcsinh(a*x)*polylog(2,exp(arcsinh(a*x)))-a^3*polylog(3,-exp(arcsinh(a*x)))+a^3*polylog(3,exp(arcsinh(a*x)))-1/2*a*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/x^2],
[arcsinh(a*x)^3/x^5,x,10,-1/4*a^2*arcsinh(a*x)/x^2+1/2*a^4*arcsinh(a*x)^2-1/4*arcsinh(a*x)^3/x^4-a^4*arcsinh(a*x)*log(1-exp(2*arcsinh(a*x)))-1/2*a^4*polylog(2,exp(2*arcsinh(a*x)))-1/4*a^3*sqrt(1+a^2*x^2)/x-1/4*a*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/x^3+1/2*a^3*arcsinh(a*x)^2*sqrt(1+a^2*x^2)/x],
[x^5*arcsinh(a*x)^4,x,23,245/1152*x^2/a^4-65/3456*x^4/a^2+1/324*x^6+245/1152*arcsinh(a*x)^2/a^6+5/16*x^2*arcsinh(a*x)^2/a^4-5/48*x^4*arcsinh(a*x)^2/a^2+1/18*x^6*arcsinh(a*x)^2+5/96*arcsinh(a*x)^4/a^6+1/6*x^6*arcsinh(a*x)^4-245/576*x*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^5+65/864*x^3*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-1/54*x^5*arcsinh(a*x)*sqrt(1+a^2*x^2)/a-5/24*x*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a^5+5/36*x^3*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a^3-1/9*x^5*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a],
[x^4*arcsinh(a*x)^4,x,19,16576/5625*x/a^4-1088/16875*x^3/a^2+24/3125*x^5+32/25*x*arcsinh(a*x)^2/a^4-16/75*x^3*arcsinh(a*x)^2/a^2+12/125*x^5*arcsinh(a*x)^2+1/5*x^5*arcsinh(a*x)^4-16576/5625*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^5+1088/5625*x^2*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-24/625*x^4*arcsinh(a*x)*sqrt(1+a^2*x^2)/a-32/75*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a^5+16/75*x^2*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a^3-4/25*x^4*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a],
[x^3*arcsinh(a*x)^4,x,14,-45/128*x^2/a^2+3/128*x^4-45/128*arcsinh(a*x)^2/a^4-9/16*x^2*arcsinh(a*x)^2/a^2+3/16*x^4*arcsinh(a*x)^2-3/32*arcsinh(a*x)^4/a^4+1/4*x^4*arcsinh(a*x)^4+45/64*x*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-3/32*x^3*arcsinh(a*x)*sqrt(1+a^2*x^2)/a+3/8*x*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a^3-1/4*x^3*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a],
[x^2*arcsinh(a*x)^4,x,11,-160/27*x/a^2+8/81*x^3-8/3*x*arcsinh(a*x)^2/a^2+4/9*x^3*arcsinh(a*x)^2+1/3*x^3*arcsinh(a*x)^4+160/27*arcsinh(a*x)*sqrt(1+a^2*x^2)/a^3-8/27*x^2*arcsinh(a*x)*sqrt(1+a^2*x^2)/a+8/9*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a^3-4/9*x^2*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a],
[x*arcsinh(a*x)^4,x,7,3/4*x^2+3/4*arcsinh(a*x)^2/a^2+3/2*x^2*arcsinh(a*x)^2+1/4*arcsinh(a*x)^4/a^2+1/2*x^2*arcsinh(a*x)^4-3/2*x*arcsinh(a*x)*sqrt(1+a^2*x^2)/a-x*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)^4,x,5,24*x+12*x*arcsinh(a*x)^2+x*arcsinh(a*x)^4-24*arcsinh(a*x)*sqrt(1+a^2*x^2)/a-4*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/a],
[arcsinh(a*x)^4/x,x,8,-1/5*arcsinh(a*x)^5+arcsinh(a*x)^4*log(1-exp(2*arcsinh(a*x)))+2*arcsinh(a*x)^3*polylog(2,exp(2*arcsinh(a*x)))-3*arcsinh(a*x)^2*polylog(3,exp(2*arcsinh(a*x)))+3*arcsinh(a*x)*polylog(4,exp(2*arcsinh(a*x)))-3/2*polylog(5,exp(2*arcsinh(a*x)))],
[arcsinh(a*x)^4/x^2,x,11,-arcsinh(a*x)^4/x-8*a*arcsinh(a*x)^3*arctanh(exp(arcsinh(a*x)))-12*a*arcsinh(a*x)^2*polylog(2,-exp(arcsinh(a*x)))+12*a*arcsinh(a*x)^2*polylog(2,exp(arcsinh(a*x)))+24*a*arcsinh(a*x)*polylog(3,-exp(arcsinh(a*x)))-24*a*arcsinh(a*x)*polylog(3,exp(arcsinh(a*x)))-24*a*polylog(4,-exp(arcsinh(a*x)))+24*a*polylog(4,exp(arcsinh(a*x)))],
[arcsinh(a*x)^4/x^3,x,8,-2*a^2*arcsinh(a*x)^3-1/2*arcsinh(a*x)^4/x^2+6*a^2*arcsinh(a*x)^2*log(1-exp(2*arcsinh(a*x)))+6*a^2*arcsinh(a*x)*polylog(2,exp(2*arcsinh(a*x)))-3*a^2*polylog(3,exp(2*arcsinh(a*x)))-2*a*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/x],
[arcsinh(a*x)^4/x^4,x,19,-2*a^2*arcsinh(a*x)^2/x-1/3*arcsinh(a*x)^4/x^3-8*a^3*arcsinh(a*x)*arctanh(exp(arcsinh(a*x)))+4/3*a^3*arcsinh(a*x)^3*arctanh(exp(arcsinh(a*x)))-4*a^3*polylog(2,-exp(arcsinh(a*x)))+2*a^3*arcsinh(a*x)^2*polylog(2,-exp(arcsinh(a*x)))+4*a^3*polylog(2,exp(arcsinh(a*x)))-2*a^3*arcsinh(a*x)^2*polylog(2,exp(arcsinh(a*x)))-4*a^3*arcsinh(a*x)*polylog(3,-exp(arcsinh(a*x)))+4*a^3*arcsinh(a*x)*polylog(3,exp(arcsinh(a*x)))+4*a^3*polylog(4,-exp(arcsinh(a*x)))-4*a^3*polylog(4,exp(arcsinh(a*x)))-2/3*a*arcsinh(a*x)^3*sqrt(1+a^2*x^2)/x^2],

# n<0
[x^6/arcsinh(a*x),x,7,-5/64*Chi(arcsinh(a*x))/a^7+9/64*Chi(3*arcsinh(a*x))/a^7-5/64*Chi(5*arcsinh(a*x))/a^7+1/64*Chi(7*arcsinh(a*x))/a^7],
[x^5/arcsinh(a*x),x,6,5/32*Shi(2*arcsinh(a*x))/a^6-1/8*Shi(4*arcsinh(a*x))/a^6+1/32*Shi(6*arcsinh(a*x))/a^6],
[x^4/arcsinh(a*x),x,6,1/8*Chi(arcsinh(a*x))/a^5-3/16*Chi(3*arcsinh(a*x))/a^5+1/16*Chi(5*arcsinh(a*x))/a^5],
[x^3/arcsinh(a*x),x,5,-1/4*Shi(2*arcsinh(a*x))/a^4+1/8*Shi(4*arcsinh(a*x))/a^4],
[x^2/arcsinh(a*x),x,5,-1/4*Chi(arcsinh(a*x))/a^3+1/4*Chi(3*arcsinh(a*x))/a^3],
[x/arcsinh(a*x),x,4,1/2*Shi(2*arcsinh(a*x))/a^2],
[1/arcsinh(a*x),x,2,Chi(arcsinh(a*x))/a],
[1/(x*arcsinh(a*x)),x,0,Unintegrable(1/(x*arcsinh(a*x)),x)],
[1/(x^2*arcsinh(a*x)),x,0,Unintegrable(1/(x^2*arcsinh(a*x)),x)],
[x^6/arcsinh(a*x)^2,x,6,-5/64*Shi(arcsinh(a*x))/a^7+27/64*Shi(3*arcsinh(a*x))/a^7-25/64*Shi(5*arcsinh(a*x))/a^7+7/64*Shi(7*arcsinh(a*x))/a^7-x^6*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x^5/arcsinh(a*x)^2,x,5,5/16*Chi(2*arcsinh(a*x))/a^6-1/2*Chi(4*arcsinh(a*x))/a^6+3/16*Chi(6*arcsinh(a*x))/a^6-x^5*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x^4/arcsinh(a*x)^2,x,5,1/8*Shi(arcsinh(a*x))/a^5-9/16*Shi(3*arcsinh(a*x))/a^5+5/16*Shi(5*arcsinh(a*x))/a^5-x^4*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x^3/arcsinh(a*x)^2,x,4,-1/2*Chi(2*arcsinh(a*x))/a^4+1/2*Chi(4*arcsinh(a*x))/a^4-x^3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x^2/arcsinh(a*x)^2,x,4,-1/4*Shi(arcsinh(a*x))/a^3+3/4*Shi(3*arcsinh(a*x))/a^3-x^2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x/arcsinh(a*x)^2,x,2,Chi(2*arcsinh(a*x))/a^2-x*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[1/arcsinh(a*x)^2,x,3,Shi(arcsinh(a*x))/a-sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[1/(x*arcsinh(a*x)^2),x,0,Unintegrable(1/(x*arcsinh(a*x)^2),x)],
[1/(x^2*arcsinh(a*x)^2),x,0,Unintegrable(1/(x^2*arcsinh(a*x)^2),x)],
[x^4/arcsinh(a*x)^3,x,14,-2*x^3/(a^2*arcsinh(a*x))-5/2*x^5/arcsinh(a*x)+1/16*Chi(arcsinh(a*x))/a^5-27/32*Chi(3*arcsinh(a*x))/a^5+25/32*Chi(5*arcsinh(a*x))/a^5-1/2*x^4*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^2)],
[x^3/arcsinh(a*x)^3,x,12,-3/2*x^2/(a^2*arcsinh(a*x))-2*x^4/arcsinh(a*x)-1/2*Shi(2*arcsinh(a*x))/a^4+Shi(4*arcsinh(a*x))/a^4-1/2*x^3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^2)],
[x^2/arcsinh(a*x)^3,x,10,-x/(a^2*arcsinh(a*x))-3/2*x^3/arcsinh(a*x)-1/8*Chi(arcsinh(a*x))/a^3+9/8*Chi(3*arcsinh(a*x))/a^3-1/2*x^2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^2)],
[x/arcsinh(a*x)^3,x,7,(-1/2)/(a^2*arcsinh(a*x))-x^2/arcsinh(a*x)+Shi(2*arcsinh(a*x))/a^2-1/2*x*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^2)],
[1/arcsinh(a*x)^3,x,4,-1/2*x/arcsinh(a*x)+1/2*Chi(arcsinh(a*x))/a-1/2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^2)],
[1/(x*arcsinh(a*x)^3),x,0,Unintegrable(1/(x*arcsinh(a*x)^3),x)],
[1/(x^2*arcsinh(a*x)^3),x,0,Unintegrable(1/(x^2*arcsinh(a*x)^3),x)],
[x^4/arcsinh(a*x)^4,x,12,-2/3*x^3/(a^2*arcsinh(a*x)^2)-5/6*x^5/arcsinh(a*x)^2+1/48*Shi(arcsinh(a*x))/a^5-27/32*Shi(3*arcsinh(a*x))/a^5+125/96*Shi(5*arcsinh(a*x))/a^5-1/3*x^4*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^3)-2*x^2*sqrt(1+a^2*x^2)/(a^3*arcsinh(a*x))-25/6*x^4*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x^3/arcsinh(a*x)^4,x,9,-1/2*x^2/(a^2*arcsinh(a*x)^2)-2/3*x^4/arcsinh(a*x)^2-1/3*Chi(2*arcsinh(a*x))/a^4+4/3*Chi(4*arcsinh(a*x))/a^4-1/3*x^3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^3)-x*sqrt(1+a^2*x^2)/(a^3*arcsinh(a*x))-8/3*x^3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x^2/arcsinh(a*x)^4,x,10,-1/3*x/(a^2*arcsinh(a*x)^2)-1/2*x^3/arcsinh(a*x)^2-1/24*Shi(arcsinh(a*x))/a^3+9/8*Shi(3*arcsinh(a*x))/a^3-1/3*x^2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^3)-1/3*sqrt(1+a^2*x^2)/(a^3*arcsinh(a*x))-3/2*x^2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[x/arcsinh(a*x)^4,x,5,(-1/6)/(a^2*arcsinh(a*x)^2)-1/3*x^2/arcsinh(a*x)^2+2/3*Chi(2*arcsinh(a*x))/a^2-1/3*x*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^3)-2/3*x*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[1/arcsinh(a*x)^4,x,5,-1/6*x/arcsinh(a*x)^2+1/6*Shi(arcsinh(a*x))/a-1/3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^3)-1/6*sqrt(1+a^2*x^2)/(a*arcsinh(a*x))],
[1/(x*arcsinh(a*x)^4),x,0,Unintegrable(1/(x*arcsinh(a*x)^4),x)],
[1/(x^2*arcsinh(a*x)^4),x,0,Unintegrable(1/(x^2*arcsinh(a*x)^4),x)],

# Integrands of the form x^m ArcSinh[a x]^(n/2)

# n>0
[x^4*sqrt(arcsinh(a*x)),x,19,1/320*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5-1/320*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5-1/64*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^5+1/64*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^5+1/32*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5-1/32*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+1/5*x^5*sqrt(arcsinh(a*x))],
[x^3*sqrt(arcsinh(a*x)),x,14,1/32*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4+1/32*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-1/256*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4-1/256*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4-3/32*sqrt(arcsinh(a*x))/a^4+1/4*x^4*sqrt(arcsinh(a*x))],
[x^2*sqrt(arcsinh(a*x)),x,14,1/48*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3-1/48*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3-1/16*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3+1/16*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3+1/3*x^3*sqrt(arcsinh(a*x))],
[x*sqrt(arcsinh(a*x)),x,9,-1/16*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2-1/16*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2+1/4*sqrt(arcsinh(a*x))/a^2+1/2*x^2*sqrt(arcsinh(a*x))],
[sqrt(arcsinh(a*x)),x,7,1/4*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-1/4*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a+x*sqrt(arcsinh(a*x))],
[sqrt(arcsinh(a*x))/x,x,0,Unintegrable(sqrt(arcsinh(a*x))/x,x)],
[x^4*arcsinh(a*x)^(3/2),x,41,1/5*x^5*arcsinh(a*x)^(3/2)+3/3200*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5+3/3200*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5-1/200*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^5-1/200*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^5+3/64*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+3/64*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5-3/3200*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-3/3200*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-4/25*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a^5+2/25*x^2*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a^3-3/50*x^4*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a],
[x^3*arcsinh(a*x)^(3/2),x,25,-3/32*arcsinh(a*x)^(3/2)/a^4+1/4*x^4*arcsinh(a*x)^(3/2)+3/128*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-3/128*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-3/2048*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+3/2048*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+9/64*x*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a^3-3/32*x^3*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a],
[x^2*arcsinh(a*x)^(3/2),x,22,1/3*x^3*arcsinh(a*x)^(3/2)+1/96*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3+1/96*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3-3/32*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-3/32*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3+1/3*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a^3-1/6*x^2*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a],
[x*arcsinh(a*x)^(3/2),x,11,1/4*arcsinh(a*x)^(3/2)/a^2+1/2*x^2*arcsinh(a*x)^(3/2)-3/64*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2+3/64*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2-3/8*x*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a],
[arcsinh(a*x)^(3/2),x,8,x*arcsinh(a*x)^(3/2)+3/8*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a+3/8*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-3/2*sqrt(1+a^2*x^2)*sqrt(arcsinh(a*x))/a],
[arcsinh(a*x)^(3/2)/x,x,0,Unintegrable(arcsinh(a*x)^(3/2)/x,x)],
[x^4*arcsinh(a*x)^(5/2),x,44,1/5*x^5*arcsinh(a*x)^(5/2)+3/6400*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5-3/6400*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5-1/240*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^5+1/240*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^5+15/128*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5-15/128*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5-1/1280*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5+1/1280*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-4/15*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a^5+2/15*x^2*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a^3-1/10*x^4*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a+2/5*x*sqrt(arcsinh(a*x))/a^4-1/15*x^3*sqrt(arcsinh(a*x))/a^2+3/100*x^5*sqrt(arcsinh(a*x))],
[x^3*arcsinh(a*x)^(5/2),x,27,-3/32*arcsinh(a*x)^(5/2)/a^4+1/4*x^4*arcsinh(a*x)^(5/2)+15/512*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4+15/512*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-15/16384*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4-15/16384*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+15/64*x*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a^3-5/32*x^3*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a-225/2048*sqrt(arcsinh(a*x))/a^4-45/256*x^2*sqrt(arcsinh(a*x))/a^2+15/256*x^4*sqrt(arcsinh(a*x))],
[x^2*arcsinh(a*x)^(5/2),x,24,1/3*x^3*arcsinh(a*x)^(5/2)+5/576*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3-5/576*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3-15/64*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3+15/64*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3+5/9*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a^3-5/18*x^2*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a-5/6*x*sqrt(arcsinh(a*x))/a^2+5/36*x^3*sqrt(arcsinh(a*x))],
[x*arcsinh(a*x)^(5/2),x,12,1/4*arcsinh(a*x)^(5/2)/a^2+1/2*x^2*arcsinh(a*x)^(5/2)-15/256*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2-15/256*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2-5/8*x*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a+15/64*sqrt(arcsinh(a*x))/a^2+15/32*x^2*sqrt(arcsinh(a*x))],
[arcsinh(a*x)^(5/2),x,9,x*arcsinh(a*x)^(5/2)+15/16*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-15/16*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-5/2*arcsinh(a*x)^(3/2)*sqrt(1+a^2*x^2)/a+15/4*x*sqrt(arcsinh(a*x))],
[arcsinh(a*x)^(5/2)/x,x,0,Unintegrable(arcsinh(a*x)^(5/2)/x,x)],

# n<0
[x^4/sqrt(arcsinh(a*x)),x,18,1/32*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5+1/32*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(1/5*Pi)/a^5+1/16*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+1/16*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5-1/32*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-1/32*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5],
[x^3/sqrt(arcsinh(a*x)),x,13,1/8*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-1/8*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-1/32*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+1/32*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4],
[x^2/sqrt(arcsinh(a*x)),x,13,1/8*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3+1/8*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(1/3*Pi)/a^3-1/8*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-1/8*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3],
[x/sqrt(arcsinh(a*x)),x,8,-1/4*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2+1/4*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2],
[1/sqrt(arcsinh(a*x)),x,6,1/2*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a+1/2*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a],
[1/(x*sqrt(arcsinh(a*x))),x,0,Unintegrable(1/(x*sqrt(arcsinh(a*x))),x)],
[1/(x^2*sqrt(arcsinh(a*x))),x,0,Unintegrable(1/(x^2*sqrt(arcsinh(a*x))),x)],
[x^4/arcsinh(a*x)^(3/2),x,17,-1/8*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+1/8*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+3/16*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-3/16*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-1/16*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(5*Pi)/a^5+1/16*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(5*Pi)/a^5-2*x^4*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[x^3/arcsinh(a*x)^(3/2),x,12,-1/2*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4-1/2*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^4+1/4*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+1/4*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4-2*x^3*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[x^2/arcsinh(a*x)^(3/2),x,12,1/4*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-1/4*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-1/4*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^3+1/4*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^3-2*x^2*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[x/arcsinh(a*x)^(3/2),x,6,erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2+erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(1/2*Pi)/a^2-2*x*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[1/arcsinh(a*x)^(3/2),x,7,-erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a+erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-2*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[1/(x*arcsinh(a*x)^(3/2)),x,0,Unintegrable(1/(x*arcsinh(a*x)^(3/2)),x)],
[x^4/arcsinh(a*x)^(5/2),x,34,1/12*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+1/12*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5-3/8*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-3/8*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5+5/24*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(5*Pi)/a^5+5/24*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(5*Pi)/a^5-2/3*x^4*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(3/2))-16/3*x^3/(a^2*sqrt(arcsinh(a*x)))-20/3*x^5/sqrt(arcsinh(a*x))],
[x^3/arcsinh(a*x)^(5/2),x,24,-2/3*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+2/3*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+1/3*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^4-1/3*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^4-2/3*x^3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(3/2))-4*x^2/(a^2*sqrt(arcsinh(a*x)))-16/3*x^4/sqrt(arcsinh(a*x))],
[x^2/arcsinh(a*x)^(5/2),x,22,-1/6*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-1/6*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3+1/2*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^3+1/2*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^3-2/3*x^2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(3/2))-8/3*x/(a^2*sqrt(arcsinh(a*x)))-4*x^3/sqrt(arcsinh(a*x))],
[x/arcsinh(a*x)^(5/2),x,11,-2/3*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^2+2/3*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^2-2/3*x*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(3/2))+(-4/3)/(a^2*sqrt(arcsinh(a*x)))-8/3*x^2/sqrt(arcsinh(a*x))],
[1/arcsinh(a*x)^(5/2),x,8,2/3*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a+2/3*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-2/3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(3/2))-4/3*x/sqrt(arcsinh(a*x))],
[1/(x*arcsinh(a*x)^(5/2)),x,0,Unintegrable(1/(x*arcsinh(a*x)^(5/2)),x)],
[x^4/arcsinh(a*x)^(7/2),x,32,-16/15*x^3/(a^2*arcsinh(a*x)^(3/2))-4/3*x^5/arcsinh(a*x)^(3/2)-1/30*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+1/30*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^5+9/20*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-9/20*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^5-5/12*erf(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(5*Pi)/a^5+5/12*erfi(sqrt(5)*sqrt(arcsinh(a*x)))*sqrt(5*Pi)/a^5-2/5*x^4*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(5/2))-32/5*x^2*sqrt(1+a^2*x^2)/(a^3*sqrt(arcsinh(a*x)))-40/3*x^4*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[x^3/arcsinh(a*x)^(7/2),x,21,-4/5*x^2/(a^2*arcsinh(a*x)^(3/2))-16/15*x^4/arcsinh(a*x)^(3/2)+16/15*erf(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4+16/15*erfi(2*sqrt(arcsinh(a*x)))*sqrt(Pi)/a^4-4/15*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^4-4/15*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^4-2/5*x^3*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(5/2))-16/5*x*sqrt(1+a^2*x^2)/(a^3*sqrt(arcsinh(a*x)))-128/15*x^3*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[x^2/arcsinh(a*x)^(7/2),x,22,-8/15*x/(a^2*arcsinh(a*x)^(3/2))-4/5*x^3/arcsinh(a*x)^(3/2)+1/15*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-1/15*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a^3-3/5*erf(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^3+3/5*erfi(sqrt(3)*sqrt(arcsinh(a*x)))*sqrt(3*Pi)/a^3-2/5*x^2*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(5/2))-16/15*sqrt(1+a^2*x^2)/(a^3*sqrt(arcsinh(a*x)))-24/5*x^2*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[x/arcsinh(a*x)^(7/2),x,9,(-4/15)/(a^2*arcsinh(a*x)^(3/2))-8/15*x^2/arcsinh(a*x)^(3/2)+8/15*erf(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^2+8/15*erfi(sqrt(2)*sqrt(arcsinh(a*x)))*sqrt(2*Pi)/a^2-2/5*x*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(5/2))-32/15*x*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[1/arcsinh(a*x)^(7/2),x,9,-4/15*x/arcsinh(a*x)^(3/2)-4/15*erf(sqrt(arcsinh(a*x)))*sqrt(Pi)/a+4/15*erfi(sqrt(arcsinh(a*x)))*sqrt(Pi)/a-2/5*sqrt(1+a^2*x^2)/(a*arcsinh(a*x)^(5/2))-8/15*sqrt(1+a^2*x^2)/(a*sqrt(arcsinh(a*x)))],
[1/(x*arcsinh(a*x)^(7/2)),x,0,Unintegrable(1/(x*arcsinh(a*x)^(7/2)),x)],

# Integrands of the form (b x)^m ArcSin[a x]^n with m symbolic
[x^m*arcsinh(a*x)^4,x,1,x^(1+m)*arcsinh(a*x)^4/(1+m)-4*a*Unintegrable(x^(1+m)*arcsinh(a*x)^3/sqrt(1+a^2*x^2),x)/(1+m)],
[x^m*arcsinh(a*x)^3,x,1,x^(1+m)*arcsinh(a*x)^3/(1+m)-3*a*Unintegrable(x^(1+m)*arcsinh(a*x)^2/sqrt(1+a^2*x^2),x)/(1+m)],
[x^m*arcsinh(a*x)^2,x,2,x^(1+m)*arcsinh(a*x)^2/(1+m)-2*a*x^(2+m)*arcsinh(a*x)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],-a^2*x^2)/(2+3*m+m^2)+2*a^2*x^(3+m)*HypergeometricPFQ([1,3/2+1/2*m,3/2+1/2*m],[2+1/2*m,5/2+1/2*m],-a^2*x^2)/(6+11*m+6*m^2+m^3)],
[x^m*arcsinh(a*x),x,2,x^(1+m)*arcsinh(a*x)/(1+m)-a*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],-a^2*x^2)/(2+3*m+m^2)],
[x^m/arcsinh(a*x),x,0,Unintegrable(x^m/arcsinh(a*x),x)],
[x^m/arcsinh(a*x)^2,x,0,Unintegrable(x^m/arcsinh(a*x)^2,x)],
[x^m*arcsinh(a*x)^(5/2),x,0,Unintegrable(x^m*arcsinh(a*x)^(5/2),x)],
[x^m*arcsinh(a*x)^(3/2),x,0,Unintegrable(x^m*arcsinh(a*x)^(3/2),x)],
[x^m*sqrt(arcsinh(a*x)),x,0,Unintegrable(x^m*sqrt(arcsinh(a*x)),x)],
[x^m/sqrt(arcsinh(a*x)),x,0,Unintegrable(x^m/sqrt(arcsinh(a*x)),x)],
[x^m/arcsinh(a*x)^(3/2),x,0,Unintegrable(x^m/arcsinh(a*x)^(3/2),x)],

# Integrands of the form (b x)^m ArcSin[a x]^n with n symbolic
[(b*x)^m*arcsinh(a*x)^n,x,0,Unintegrable((b*x)^m*arcsinh(a*x)^n,x)],
[x^4*arcsinh(a*x)^n,x,12,1/32*5^(-1-n)*arcsinh(a*x)^n*GAMMA(1+n,-5*arcsinh(a*x))/(a^5*(-arcsinh(a*x))^n)-1/32*arcsinh(a*x)^n*GAMMA(1+n,-3*arcsinh(a*x))/(3^n*a^5*(-arcsinh(a*x))^n)+1/16*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/(a^5*(-arcsinh(a*x))^n)-1/16*GAMMA(1+n,arcsinh(a*x))/a^5+1/32*GAMMA(1+n,3*arcsinh(a*x))/(3^n*a^5)-1/32*5^(-1-n)*GAMMA(1+n,5*arcsinh(a*x))/a^5],
[x^3*arcsinh(a*x)^n,x,9,arcsinh(a*x)^n*GAMMA(1+n,-4*arcsinh(a*x))/(2^(2*(3+n))*a^4*(-arcsinh(a*x))^n)-2^(-4-n)*arcsinh(a*x)^n*GAMMA(1+n,-2*arcsinh(a*x))/(a^4*(-arcsinh(a*x))^n)-2^(-4-n)*GAMMA(1+n,2*arcsinh(a*x))/a^4+GAMMA(1+n,4*arcsinh(a*x))/(2^(2*(3+n))*a^4)],
[x^2*arcsinh(a*x)^n,x,9,1/8*3^(-1-n)*arcsinh(a*x)^n*GAMMA(1+n,-3*arcsinh(a*x))/(a^3*(-arcsinh(a*x))^n)-1/8*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/(a^3*(-arcsinh(a*x))^n)+1/8*GAMMA(1+n,arcsinh(a*x))/a^3-1/8*3^(-1-n)*GAMMA(1+n,3*arcsinh(a*x))/a^3],
[x*arcsinh(a*x)^n,x,6,2^(-3-n)*arcsinh(a*x)^n*GAMMA(1+n,-2*arcsinh(a*x))/(a^2*(-arcsinh(a*x))^n)+2^(-3-n)*GAMMA(1+n,2*arcsinh(a*x))/a^2],
[arcsinh(a*x)^n,x,4,1/2*arcsinh(a*x)^n*GAMMA(1+n,-arcsinh(a*x))/(a*(-arcsinh(a*x))^n)-1/2*GAMMA(1+n,arcsinh(a*x))/a],
[arcsinh(a*x)^n/x,x,0,Unintegrable(arcsinh(a*x)^n/x,x)],
[arcsinh(a*x)^n/x^2,x,0,Unintegrable(arcsinh(a*x)^n/x^2,x)],

# Integrands of the form (d x)^m (a+b ArcSinh[c x]^p

# Integrands of the form x^m (a+b ArcSinh[c x])^(n/2)

# n>0
[x^2*(a+b*arcsinh(c*x))^(1/2),x,14,1/48*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/c^3-1/48*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(1/3*Pi)/(c^3*exp(3*a/b))-1/16*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/c^3+1/16*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(c^3*exp(a/b))+1/3*x^3*sqrt(a+b*arcsinh(c*x))],
[x*(a+b*arcsinh(c*x))^(1/2),x,9,-1/16*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/c^2-1/16*erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(1/2*Pi)/(c^2*exp(2*a/b))+1/4*sqrt(a+b*arcsinh(c*x))/c^2+1/2*x^2*sqrt(a+b*arcsinh(c*x))],
[(a+b*arcsinh(c*x))^(1/2),x,7,1/4*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/c-1/4*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(b)*sqrt(Pi)/(c*exp(a/b))+x*sqrt(a+b*arcsinh(c*x))],
[x^2*(a+b*arcsinh(c*x))^(3/2),x,22,1/3*x^3*(a+b*arcsinh(c*x))^(3/2)+1/96*b^(3/2)*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/3*Pi)/c^3+1/96*b^(3/2)*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/3*Pi)/(c^3*exp(3*a/b))-3/32*b^(3/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/c^3-3/32*b^(3/2)*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c^3*exp(a/b))+1/3*b*sqrt(1+c^2*x^2)*sqrt(a+b*arcsinh(c*x))/c^3-1/6*b*x^2*sqrt(1+c^2*x^2)*sqrt(a+b*arcsinh(c*x))/c],
[x*(a+b*arcsinh(c*x))^(3/2),x,11,1/4*(a+b*arcsinh(c*x))^(3/2)/c^2+1/2*x^2*(a+b*arcsinh(c*x))^(3/2)-3/64*b^(3/2)*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/c^2+3/64*b^(3/2)*erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/(c^2*exp(2*a/b))-3/8*b*x*sqrt(1+c^2*x^2)*sqrt(a+b*arcsinh(c*x))/c],
[(a+b*arcsinh(c*x))^(3/2),x,8,x*(a+b*arcsinh(c*x))^(3/2)+3/8*b^(3/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/c+3/8*b^(3/2)*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c*exp(a/b))-3/2*b*sqrt(1+c^2*x^2)*sqrt(a+b*arcsinh(c*x))/c],
[x^2*(a+b*arcsinh(c*x))^(5/2),x,24,1/3*x^3*(a+b*arcsinh(c*x))^(5/2)+5/576*b^(5/2)*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/3*Pi)/c^3-5/576*b^(5/2)*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/3*Pi)/(c^3*exp(3*a/b))-15/64*b^(5/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/c^3+15/64*b^(5/2)*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c^3*exp(a/b))+5/9*b*(a+b*arcsinh(c*x))^(3/2)*sqrt(1+c^2*x^2)/c^3-5/18*b*x^2*(a+b*arcsinh(c*x))^(3/2)*sqrt(1+c^2*x^2)/c-5/6*b^2*x*sqrt(a+b*arcsinh(c*x))/c^2+5/36*b^2*x^3*sqrt(a+b*arcsinh(c*x))],
[x*(a+b*arcsinh(c*x))^(5/2),x,12,1/4*(a+b*arcsinh(c*x))^(5/2)/c^2+1/2*x^2*(a+b*arcsinh(c*x))^(5/2)-15/256*b^(5/2)*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/c^2-15/256*b^(5/2)*erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/(c^2*exp(2*a/b))-5/8*b*x*(a+b*arcsinh(c*x))^(3/2)*sqrt(1+c^2*x^2)/c+15/64*b^2*sqrt(a+b*arcsinh(c*x))/c^2+15/32*b^2*x^2*sqrt(a+b*arcsinh(c*x))],
[(a+b*arcsinh(c*x))^(5/2),x,9,x*(a+b*arcsinh(c*x))^(5/2)+15/16*b^(5/2)*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/c-15/16*b^(5/2)*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c*exp(a/b))-5/2*b*(a+b*arcsinh(c*x))^(3/2)*sqrt(1+c^2*x^2)/c+15/4*b^2*x*sqrt(a+b*arcsinh(c*x))],

# n<0
[x^2/(a+b*arcsinh(c*x))^(1/2),x,13,1/8*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/3*Pi)/(c^3*sqrt(b))+1/8*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/3*Pi)/(c^3*exp(3*a/b)*sqrt(b))-1/8*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c^3*sqrt(b))-1/8*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c^3*exp(a/b)*sqrt(b))],
[x/(a+b*arcsinh(c*x))^(1/2),x,8,-1/4*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/(c^2*sqrt(b))+1/4*erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/(c^2*exp(2*a/b)*sqrt(b))],
[1/(a+b*arcsinh(c*x))^(1/2),x,6,1/2*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c*sqrt(b))+1/2*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(c*exp(a/b)*sqrt(b))],
[x^2/(a+b*arcsinh(c*x))^(3/2),x,12,1/4*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*c^3)-1/4*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*c^3*exp(a/b))-1/4*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(3*Pi)/(b^(3/2)*c^3)+1/4*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(3*Pi)/(b^(3/2)*c^3*exp(3*a/b))-2*x^2*sqrt(1+c^2*x^2)/(b*c*sqrt(a+b*arcsinh(c*x)))],
[x/(a+b*arcsinh(c*x))^(3/2),x,6,exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*c^2)+erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(1/2*Pi)/(b^(3/2)*c^2*exp(2*a/b))-2*x*sqrt(1+c^2*x^2)/(b*c*sqrt(a+b*arcsinh(c*x)))],
[1/(a+b*arcsinh(c*x))^(3/2),x,7,-exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*c)+erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(3/2)*c*exp(a/b))-2*sqrt(1+c^2*x^2)/(b*c*sqrt(a+b*arcsinh(c*x)))],
[x^2/(a+b*arcsinh(c*x))^(5/2),x,22,-1/6*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*c^3)-1/6*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*c^3*exp(a/b))+1/2*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(3*Pi)/(b^(5/2)*c^3)+1/2*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(3*Pi)/(b^(5/2)*c^3*exp(3*a/b))-2/3*x^2*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x))^(3/2))-8/3*x/(b^2*c^2*sqrt(a+b*arcsinh(c*x)))-4*x^3/(b^2*sqrt(a+b*arcsinh(c*x)))],
[x/(a+b*arcsinh(c*x))^(5/2),x,11,-2/3*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*c^2)+2/3*erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(2*Pi)/(b^(5/2)*c^2*exp(2*a/b))-2/3*x*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x))^(3/2))+(-4/3)/(b^2*c^2*sqrt(a+b*arcsinh(c*x)))-8/3*x^2/(b^2*sqrt(a+b*arcsinh(c*x)))],
[1/(a+b*arcsinh(c*x))^(5/2),x,8,2/3*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*c)+2/3*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(5/2)*c*exp(a/b))-2/3*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x))^(3/2))-4/3*x/(b^2*sqrt(a+b*arcsinh(c*x)))],
[x^2/(a+b*arcsinh(c*x))^(7/2),x,22,-8/15*x/(b^2*c^2*(a+b*arcsinh(c*x))^(3/2))-4/5*x^3/(b^2*(a+b*arcsinh(c*x))^(3/2))+1/15*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*c^3)-1/15*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*c^3*exp(a/b))-3/5*exp(3*a/b)*erf(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(3*Pi)/(b^(7/2)*c^3)+3/5*erfi(sqrt(3)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(3*Pi)/(b^(7/2)*c^3*exp(3*a/b))-2/5*x^2*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x))^(5/2))-16/15*sqrt(1+c^2*x^2)/(b^3*c^3*sqrt(a+b*arcsinh(c*x)))-24/5*x^2*sqrt(1+c^2*x^2)/(b^3*c*sqrt(a+b*arcsinh(c*x)))],
[x/(a+b*arcsinh(c*x))^(7/2),x,9,(-4/15)/(b^2*c^2*(a+b*arcsinh(c*x))^(3/2))-8/15*x^2/(b^2*(a+b*arcsinh(c*x))^(3/2))+8/15*exp(2*a/b)*erf(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*c^2)+8/15*erfi(sqrt(2)*sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(2*Pi)/(b^(7/2)*c^2*exp(2*a/b))-2/5*x*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x))^(5/2))-32/15*x*sqrt(1+c^2*x^2)/(b^3*c*sqrt(a+b*arcsinh(c*x)))],
[1/(a+b*arcsinh(c*x))^(7/2),x,9,-4/15*x/(b^2*(a+b*arcsinh(c*x))^(3/2))-4/15*exp(a/b)*erf(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*c)+4/15*erfi(sqrt(a+b*arcsinh(c*x))/sqrt(b))*sqrt(Pi)/(b^(7/2)*c*exp(a/b))-2/5*sqrt(1+c^2*x^2)/(b*c*(a+b*arcsinh(c*x))^(5/2))-8/15*sqrt(1+c^2*x^2)/(b^3*c*sqrt(a+b*arcsinh(c*x)))]]:
