# Maple integration test file: "7 Inverse hyperbolic functions\7.3 Inverse hyperbolic tangent\7.3.6 Exponentials of inverse hyperbolic tangent functions.txt"

lst:=[

# Integrands of the form u E^(n ArcTanh[a x])

# Integrands of the form x^m E^(n ArcTanh[a x])

# Integrands of the form x^m E^(n ArcTanh[a x])

# n>0
[exp(arctanh(a*x))*x^4,x,6,3/8*arcsin(a*x)/a^5-4/15*x^2*sqrt(1-a^2*x^2)/a^3-1/4*x^3*sqrt(1-a^2*x^2)/a^2-1/5*x^4*sqrt(1-a^2*x^2)/a-1/120*(64+45*a*x)*sqrt(1-a^2*x^2)/a^5],
[exp(arctanh(a*x))*x^3,x,5,3/8*arcsin(a*x)/a^4-1/3*x^2*sqrt(1-a^2*x^2)/a^2-1/4*x^3*sqrt(1-a^2*x^2)/a-1/24*(16+9*a*x)*sqrt(1-a^2*x^2)/a^4],
[exp(arctanh(a*x))*x^2,x,7,1/3*(1-a^2*x^2)^(3/2)/a^3+1/2*arcsin(a*x)/a^3-sqrt(1-a^2*x^2)/a^3-1/2*x*sqrt(1-a^2*x^2)/a^2],
[exp(arctanh(a*x))*x,x,3,1/2*arcsin(a*x)/a^2-1/2*(2+a*x)*sqrt(1-a^2*x^2)/a^2],
[exp(arctanh(a*x)),x,3,arcsin(a*x)/a-sqrt(1-a^2*x^2)/a],
[exp(arctanh(a*x))/x,x,6,arcsin(a*x)-arctanh(sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/x^2,x,5,-a*arctanh(sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))/x^3,x,6,-1/2*a^2*arctanh(sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/x^2-a*sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))/x^4,x,7,-1/2*a^3*arctanh(sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/x^3-1/2*a*sqrt(1-a^2*x^2)/x^2-2/3*a^2*sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))/x^5,x,8,-3/8*a^4*arctanh(sqrt(1-a^2*x^2))-1/4*sqrt(1-a^2*x^2)/x^4-1/3*a*sqrt(1-a^2*x^2)/x^3-3/8*a^2*sqrt(1-a^2*x^2)/x^2-2/3*a^3*sqrt(1-a^2*x^2)/x],
[exp(2*arctanh(a*x))*x^3,x,3,-2*x/a^3-x^2/a^2-2/3*x^3/a-1/4*x^4-2*log(1-a*x)/a^4],
[exp(2*arctanh(a*x))*x^2,x,3,-2*x/a^2-x^2/a-1/3*x^3-2*log(1-a*x)/a^3],
[exp(2*arctanh(a*x))*x,x,3,-2*x/a-1/2*x^2-2*log(1-a*x)/a^2],
[exp(2*arctanh(a*x)),x,3,-x-2*log(1-a*x)/a],
[exp(2*arctanh(a*x))/x,x,3,log(x)-2*log(1-a*x)],
[exp(2*arctanh(a*x))/x^2,x,3,(-1)/x+2*a*log(x)-2*a*log(1-a*x)],
[exp(2*arctanh(a*x))/x^3,x,3,(-1/2)/x^2-2*a/x+2*a^2*log(x)-2*a^2*log(1-a*x)],
[exp(2*arctanh(a*x))/x^4,x,3,(-1/3)/x^3-a/x^2-2*a^2/x+2*a^3*log(x)-2*a^3*log(1-a*x)],
[exp(3*arctanh(a*x))*x^2,x,10,-11/2*arcsin(a*x)/a^3+(1+a*x)^3/(a^3*sqrt(1-a^2*x^2))+1/3*(3+a*x)^2*sqrt(1-a^2*x^2)/a^3+1/6*(28+3*a*x)*sqrt(1-a^2*x^2)/a^3],
[exp(3*arctanh(a*x))*x,x,9,3/2*(1-a^2*x^2)^(3/2)/(a^2*(1-a*x))+(1-a^2*x^2)^(5/2)/(a^2*(1-a*x)^3)-9/2*arcsin(a*x)/a^2+9/2*sqrt(1-a^2*x^2)/a^2],
[exp(3*arctanh(a*x)),x,5,-3*arcsin(a*x)/a+2*(1+a*x)^2/(a*sqrt(1-a^2*x^2))+3*sqrt(1-a^2*x^2)/a],
[exp(3*arctanh(a*x))/x,x,8,-arcsin(a*x)-arctanh(sqrt(1-a^2*x^2))+4*sqrt(1-a^2*x^2)/(1-a*x)],
[exp(3*arctanh(a*x))/x^2,x,8,-3*a*arctanh(sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/x+4*a*sqrt(1-a^2*x^2)/(1-a*x)],
[exp(3*arctanh(a*x))/x^3,x,12,-9/2*a^2*arctanh(sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/x^2-3*a*sqrt(1-a^2*x^2)/x+4*a^2*sqrt(1-a^2*x^2)/(1-a*x)],
[exp(3*arctanh(a*x))/x^4,x,14,-11/2*a^3*arctanh(sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/x^3-3/2*a*sqrt(1-a^2*x^2)/x^2-14/3*a^2*sqrt(1-a^2*x^2)/x+4*a^3*sqrt(1-a^2*x^2)/(1-a*x)],
[exp(4*arctanh(a*x))*x^3,x,3,12*x/a^3+4*x^2/a^2+4/3*x^3/a+1/4*x^4+4/(a^4*(1-a*x))+16*log(1-a*x)/a^4],
[exp(4*arctanh(a*x))*x^2,x,3,8*x/a^2+2*x^2/a+1/3*x^3+4/(a^3*(1-a*x))+12*log(1-a*x)/a^3],
[exp(4*arctanh(a*x))*x,x,3,4*x/a+1/2*x^2+4/(a^2*(1-a*x))+8*log(1-a*x)/a^2],
[exp(4*arctanh(a*x)),x,3,x+4/(a*(1-a*x))+4*log(1-a*x)/a],
[exp(4*arctanh(a*x))/x,x,3,4/(1-a*x)+log(x)],
[exp(4*arctanh(a*x))/x^2,x,3,(-1)/x+4*a/(1-a*x)+4*a*log(x)-4*a*log(1-a*x)],
[exp(4*arctanh(a*x))/x^3,x,3,(-1/2)/x^2-4*a/x+4*a^2/(1-a*x)+8*a^2*log(x)-8*a^2*log(1-a*x)],
[exp(4*arctanh(a*x))/x^4,x,3,(-1/3)/x^3-2*a/x^2-8*a^2/x+4*a^3/(1-a*x)+12*a^3*log(x)-12*a^3*log(1-a*x)],

# n<0
[x^3/exp(arctanh(a*x)),x,5,-3/8*arcsin(a*x)/a^4-1/3*x^2*sqrt(1-a^2*x^2)/a^2+1/4*x^3*sqrt(1-a^2*x^2)/a-1/24*(16-9*a*x)*sqrt(1-a^2*x^2)/a^4],
[x^2/exp(arctanh(a*x)),x,7,-1/3*(1-a^2*x^2)^(3/2)/a^3+1/2*arcsin(a*x)/a^3+sqrt(1-a^2*x^2)/a^3-1/2*x*sqrt(1-a^2*x^2)/a^2],
[x/exp(arctanh(a*x)),x,3,-1/2*arcsin(a*x)/a^2-1/2*(2-a*x)*sqrt(1-a^2*x^2)/a^2],
[exp(-arctanh(a*x)),x,3,arcsin(a*x)/a+sqrt(1-a^2*x^2)/a],
[1/(exp(arctanh(a*x))*x),x,6,-arcsin(a*x)-arctanh(sqrt(1-a^2*x^2))],
[1/(exp(arctanh(a*x))*x^2),x,5,a*arctanh(sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/x],
[1/(exp(arctanh(a*x))*x^3),x,6,-1/2*a^2*arctanh(sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/x^2+a*sqrt(1-a^2*x^2)/x],
[1/(exp(arctanh(a*x))*x^4),x,7,1/2*a^3*arctanh(sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/x^3+1/2*a*sqrt(1-a^2*x^2)/x^2-2/3*a^2*sqrt(1-a^2*x^2)/x],
[1/(exp(arctanh(a*x))*x^5),x,8,-3/8*a^4*arctanh(sqrt(1-a^2*x^2))-1/4*sqrt(1-a^2*x^2)/x^4+1/3*a*sqrt(1-a^2*x^2)/x^3-3/8*a^2*sqrt(1-a^2*x^2)/x^2+2/3*a^3*sqrt(1-a^2*x^2)/x],
[x^3/exp(2*arctanh(a*x)),x,3,2*x/a^3-x^2/a^2+2/3*x^3/a-1/4*x^4-2*log(1+a*x)/a^4],
[x^2/exp(2*arctanh(a*x)),x,3,-2*x/a^2+x^2/a-1/3*x^3+2*log(1+a*x)/a^3],
[x/exp(2*arctanh(a*x)),x,3,2*x/a-1/2*x^2-2*log(1+a*x)/a^2],
[exp(-2*arctanh(a*x)),x,3,-x+2*log(1+a*x)/a],
[1/(exp(2*arctanh(a*x))*x),x,3,log(x)-2*log(1+a*x)],
[1/(exp(2*arctanh(a*x))*x^2),x,3,(-1)/x-2*a*log(x)+2*a*log(1+a*x)],
[1/(exp(2*arctanh(a*x))*x^3),x,3,(-1/2)/x^2+2*a/x+2*a^2*log(x)-2*a^2*log(1+a*x)],
[1/(exp(2*arctanh(a*x))*x^4),x,3,(-1/3)/x^3+a/x^2-2*a^2/x-2*a^3*log(x)+2*a^3*log(1+a*x)],
[x^3/exp(3*arctanh(a*x)),x,14,51/8*arcsin(a*x)/a^4+(1-a*x)^3/(a^4*sqrt(1-a^2*x^2))+27/4*sqrt(1-a^2*x^2)/a^4+x^2*sqrt(1-a^2*x^2)/a^2-1/4*x^3*sqrt(1-a^2*x^2)/a+9/8*(2-3*a*x)*sqrt(1-a^2*x^2)/a^4],
[x^2/exp(3*arctanh(a*x)),x,10,-11/2*arcsin(a*x)/a^3-(1-a*x)^3/(a^3*sqrt(1-a^2*x^2))-1/6*(28-3*a*x)*sqrt(1-a^2*x^2)/a^3-1/3*(3-a*x)^2*sqrt(1-a^2*x^2)/a^3],
[x/exp(3*arctanh(a*x)),x,9,3/2*(1-a^2*x^2)^(3/2)/(a^2*(1+a*x))+(1-a^2*x^2)^(5/2)/(a^2*(1+a*x)^3)+9/2*arcsin(a*x)/a^2+9/2*sqrt(1-a^2*x^2)/a^2],
[exp(-3*arctanh(a*x)),x,5,-3*arcsin(a*x)/a-2*(1-a*x)^2/(a*sqrt(1-a^2*x^2))-3*sqrt(1-a^2*x^2)/a],
[1/(exp(3*arctanh(a*x))*x),x,8,arcsin(a*x)-arctanh(sqrt(1-a^2*x^2))+4*sqrt(1-a^2*x^2)/(1+a*x)],
[1/(exp(3*arctanh(a*x))*x^2),x,8,3*a*arctanh(sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/x-4*a*sqrt(1-a^2*x^2)/(1+a*x)],
[1/(exp(3*arctanh(a*x))*x^3),x,12,-9/2*a^2*arctanh(sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/x^2+3*a*sqrt(1-a^2*x^2)/x+4*a^2*sqrt(1-a^2*x^2)/(1+a*x)],
[1/(exp(3*arctanh(a*x))*x^4),x,14,11/2*a^3*arctanh(sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/x^3+3/2*a*sqrt(1-a^2*x^2)/x^2-14/3*a^2*sqrt(1-a^2*x^2)/x-4*a^3*sqrt(1-a^2*x^2)/(1+a*x)],
[1/(exp(3*arctanh(a*x))*x^5),x,19,-51/8*a^4*arctanh(sqrt(1-a^2*x^2))-1/4*sqrt(1-a^2*x^2)/x^4+a*sqrt(1-a^2*x^2)/x^3-19/8*a^2*sqrt(1-a^2*x^2)/x^2+6*a^3*sqrt(1-a^2*x^2)/x+4*a^4*sqrt(1-a^2*x^2)/(1+a*x)],

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

# n>0
[exp(1/2*arctanh(a*x))*x^m,x,2,x^(1+m)*AppellF1(1+m,1/4,-1/4,2+m,a*x,-a*x)/(1+m)],
[exp(1/2*arctanh(a*x))*x^2,x,15,-3/8*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^3-1/12*(1-a*x)^(3/4)*(1+a*x)^(5/4)/a^3-1/3*x*(1-a*x)^(3/4)*(1+a*x)^(5/4)/a^2+3/8*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-3/8*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-3/16*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))+3/16*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))],
[exp(1/2*arctanh(a*x))*x,x,14,-1/4*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^2-1/2*(1-a*x)^(3/4)*(1+a*x)^(5/4)/a^2+1/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))-1/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))-1/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))+1/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))],
[exp(1/2*arctanh(a*x)),x,13,-(1-a*x)^(3/4)*(1+a*x)^(1/4)/a+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-1/2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))+1/2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[exp(1/2*arctanh(a*x))/x,x,17,-2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))-log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)+log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)],
[exp(1/2*arctanh(a*x))/x^2,x,6,-(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-a*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-a*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(1/2*arctanh(a*x))/x^3,x,7,-1/4*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-1/2*(1-a*x)^(3/4)*(1+a*x)^(5/4)/x^2-1/4*a^2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-1/4*a^2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(1/2*arctanh(a*x))/x^4,x,9,-1/3*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^3-5/12*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^2-11/24*a^2*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-3/8*a^3*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-3/8*a^3*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(1/2*arctanh(a*x))/x^5,x,10,-1/4*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^4-7/24*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^3-29/96*a^2*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^2-83/192*a^3*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-11/64*a^4*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-11/64*a^4*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(1/2*arctanh(a*x))/x^6,x,11,-1/5*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^5-9/40*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^4-11/48*a^2*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^3-269/960*a^3*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^2-611/1920*a^4*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-31/128*a^5*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-31/128*a^5*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(3/2*arctanh(a*x))*x^m,x,2,x^(1+m)*AppellF1(1+m,3/4,-3/4,2+m,a*x,-a*x)/(1+m)],
[exp(3/2*arctanh(a*x))*x^3,x,15,-41/64*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^4-1/4*x^2*(1-a*x)^(1/4)*(1+a*x)^(7/4)/a^2-1/32*(1-a*x)^(1/4)*(1+a*x)^(7/4)*(11+4*a*x)/a^4+123/64*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))-123/64*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+123/128*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))-123/128*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))],
[exp(3/2*arctanh(a*x))*x^2,x,15,-17/24*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^3-1/4*(1-a*x)^(1/4)*(1+a*x)^(7/4)/a^3-1/3*x*(1-a*x)^(1/4)*(1+a*x)^(7/4)/a^2+17/8*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-17/8*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))+17/16*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))-17/16*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))],
[exp(3/2*arctanh(a*x))*x,x,14,-3/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^2-1/2*(1-a*x)^(1/4)*(1+a*x)^(7/4)/a^2+9/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))-9/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+9/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))-9/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))],
[exp(3/2*arctanh(a*x)),x,13,-(1-a*x)^(1/4)*(1+a*x)^(3/4)/a+3*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-3*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+3/2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-3/2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[exp(3/2*arctanh(a*x))/x,x,17,2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))+log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)],
[exp(3/2*arctanh(a*x))/x^2,x,6,-(1-a*x)^(1/4)*(1+a*x)^(3/4)/x+3*a*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-3*a*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(3/2*arctanh(a*x))/x^3,x,7,-3/4*a*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x-1/2*(1-a*x)^(1/4)*(1+a*x)^(7/4)/x^2+9/4*a^2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-9/4*a^2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(3/2*arctanh(a*x))/x^4,x,9,-1/3*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^3-7/12*a*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^2-23/24*a^2*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x+17/8*a^3*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-17/8*a^3*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(3/2*arctanh(a*x))/x^5,x,10,-1/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^4-3/8*a*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^3-15/32*a^2*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^2-63/64*a^3*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x+123/64*a^4*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-123/64*a^4*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(5/2*arctanh(a*x))*x^m,x,2,x^(1+m)*AppellF1(1+m,5/4,-5/4,2+m,a*x,-a*x)/(1+m)],
[exp(5/2*arctanh(a*x))*x^3,x,16,475/64*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^4+4*x^3*(1+a*x)^(5/4)/(a*(1-a*x)^(1/4))+17/4*x^2*(1-a*x)^(3/4)*(1+a*x)^(5/4)/a^2+1/96*(1-a*x)^(3/4)*(1+a*x)^(5/4)*(521+452*a*x)/a^4-475/64*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+475/64*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+475/128*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))-475/128*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))],
[exp(5/2*arctanh(a*x))*x^2,x,16,55/8*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^3+11/4*(1-a*x)^(3/4)*(1+a*x)^(5/4)/a^3+2*(1+a*x)^(9/4)/(a^3*(1-a*x)^(1/4))+1/3*(1-a*x)^(3/4)*(1+a*x)^(9/4)/a^3-55/8*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))+55/8*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))+55/16*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))-55/16*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))],
[exp(5/2*arctanh(a*x))*x,x,15,25/4*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^2+5/2*(1-a*x)^(3/4)*(1+a*x)^(5/4)/a^2+2*(1+a*x)^(9/4)/(a^2*(1-a*x)^(1/4))-25/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+25/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+25/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))-25/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))],
[exp(5/2*arctanh(a*x)),x,14,5*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a+4*(1+a*x)^(5/4)/(a*(1-a*x)^(1/4))-5*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+5*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+5/2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-5/2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[exp(5/2*arctanh(a*x))/x,x,19,8*(1+a*x)^(1/4)/(1-a*x)^(1/4)-2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))+log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)+arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)],
[exp(5/2*arctanh(a*x))/x^2,x,7,10*a*(1+a*x)^(1/4)/(1-a*x)^(1/4)-(1+a*x)^(5/4)/(x*(1-a*x)^(1/4))-5*a*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-5*a*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(5/2*arctanh(a*x))/x^3,x,8,25/2*a^2*(1+a*x)^(1/4)/(1-a*x)^(1/4)-5/4*a*(1+a*x)^(5/4)/(x*(1-a*x)^(1/4))-1/2*(1+a*x)^(9/4)/(x^2*(1-a*x)^(1/4))-25/4*a^2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-25/4*a^2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(5/2*arctanh(a*x))/x^4,x,10,287/24*a^3*(1+a*x)^(1/4)/(1-a*x)^(1/4)-1/3*(1+a*x)^(1/4)/(x^3*(1-a*x)^(1/4))-13/12*a*(1+a*x)^(1/4)/(x^2*(1-a*x)^(1/4))-61/24*a^2*(1+a*x)^(1/4)/(x*(1-a*x)^(1/4))-55/8*a^3*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-55/8*a^3*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[exp(5/2*arctanh(a*x))/x^5,x,11,2467/192*a^4*(1+a*x)^(1/4)/(1-a*x)^(1/4)-1/4*(1+a*x)^(1/4)/(x^4*(1-a*x)^(1/4))-17/24*a*(1+a*x)^(1/4)/(x^3*(1-a*x)^(1/4))-113/96*a^2*(1+a*x)^(1/4)/(x^2*(1-a*x)^(1/4))-521/192*a^3*(1+a*x)^(1/4)/(x*(1-a*x)^(1/4))-475/64*a^4*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-475/64*a^4*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],

# n<0
[x^m/exp(1/2*arctanh(a*x)),x,2,x^(1+m)*AppellF1(1+m,-1/4,1/4,2+m,a*x,-a*x)/(1+m)],
[x^3/exp(1/2*arctanh(a*x)),x,15,-11/64*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^4-1/4*x^2*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^2-1/96*(25-4*a*x)*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^4-11/64*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+11/64*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))-11/128*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))+11/128*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))],
[x^2/exp(1/2*arctanh(a*x)),x,15,3/8*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^3+1/12*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^3-1/3*x*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^2+3/8*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-3/8*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))+3/16*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))-3/16*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))],
[x/exp(1/2*arctanh(a*x)),x,14,-1/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^2-1/2*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^2-1/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+1/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))-1/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))+1/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))],
[exp(-1/2*arctanh(a*x)),x,13,(1-a*x)^(1/4)*(1+a*x)^(3/4)/a+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+1/2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-1/2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[1/(exp(1/2*arctanh(a*x))*x),x,17,2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))-log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)+log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)+arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)],
[1/(exp(1/2*arctanh(a*x))*x^2),x,6,-(1-a*x)^(1/4)*(1+a*x)^(3/4)/x-a*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))+a*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(1/2*arctanh(a*x))*x^3),x,7,1/4*a*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x-1/2*(1-a*x)^(5/4)*(1+a*x)^(3/4)/x^2+1/4*a^2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-1/4*a^2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(1/2*arctanh(a*x))*x^4),x,9,-1/3*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^3+5/12*a*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^2-11/24*a^2*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x-3/8*a^3*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))+3/8*a^3*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(1/2*arctanh(a*x))*x^5),x,10,-1/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^4+7/24*a*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^3-29/96*a^2*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x^2+83/192*a^3*(1-a*x)^(1/4)*(1+a*x)^(3/4)/x+11/64*a^4*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-11/64*a^4*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[x^m/exp(3/2*arctanh(a*x)),x,2,x^(1+m)*AppellF1(1+m,-3/4,3/4,2+m,a*x,-a*x)/(1+m)],
[x^3/exp(3/2*arctanh(a*x)),x,15,-41/64*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^4-1/4*x^2*(1-a*x)^(7/4)*(1+a*x)^(1/4)/a^2-1/32*(11-4*a*x)*(1-a*x)^(7/4)*(1+a*x)^(1/4)/a^4-123/64*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+123/64*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+123/128*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))-123/128*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))],
[x^2/exp(3/2*arctanh(a*x)),x,15,17/24*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^3+1/4*(1-a*x)^(7/4)*(1+a*x)^(1/4)/a^3-1/3*x*(1-a*x)^(7/4)*(1+a*x)^(1/4)/a^2+17/8*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-17/8*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-17/16*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))+17/16*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))],
[x/exp(3/2*arctanh(a*x)),x,14,-3/4*(1-a*x)^(3/4)*(1+a*x)^(1/4)/a^2-1/2*(1-a*x)^(7/4)*(1+a*x)^(1/4)/a^2-9/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+9/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+9/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))-9/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))],
[exp(-3/2*arctanh(a*x)),x,13,(1-a*x)^(3/4)*(1+a*x)^(1/4)/a+3*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-3*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-3/2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))+3/2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[1/(exp(3/2*arctanh(a*x))*x),x,17,-2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))+log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)+arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)],
[1/(exp(3/2*arctanh(a*x))*x^2),x,6,-(1-a*x)^(3/4)*(1+a*x)^(1/4)/x+3*a*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))+3*a*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(3/2*arctanh(a*x))*x^3),x,7,3/4*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-1/2*(1-a*x)^(7/4)*(1+a*x)^(1/4)/x^2-9/4*a^2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-9/4*a^2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(3/2*arctanh(a*x))*x^4),x,9,-1/3*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^3+7/12*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^2-23/24*a^2*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x+17/8*a^3*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))+17/8*a^3*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(3/2*arctanh(a*x))*x^5),x,10,-1/4*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^4+3/8*a*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^3-15/32*a^2*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x^2+63/64*a^3*(1-a*x)^(3/4)*(1+a*x)^(1/4)/x-123/64*a^4*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-123/64*a^4*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[x^m/exp(5/2*arctanh(a*x)),x,2,x^(1+m)*AppellF1(1+m,-5/4,5/4,2+m,a*x,-a*x)/(1+m)],
[x^3/exp(5/2*arctanh(a*x)),x,16,-4*x^3*(1-a*x)^(5/4)/(a*(1+a*x)^(1/4))+475/64*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^4+17/4*x^2*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^2+1/96*(521-452*a*x)*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^4+475/64*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))-475/64*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^4*sqrt(2))+475/128*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))-475/128*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^4*sqrt(2))],
[x^2/exp(5/2*arctanh(a*x)),x,16,-2*(1-a*x)^(9/4)/(a^3*(1+a*x)^(1/4))-55/8*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^3-11/4*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^3-1/3*(1-a*x)^(9/4)*(1+a*x)^(3/4)/a^3-55/8*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))+55/8*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^3*sqrt(2))-55/16*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))+55/16*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^3*sqrt(2))],
[x/exp(5/2*arctanh(a*x)),x,15,2*(1-a*x)^(9/4)/(a^2*(1+a*x)^(1/4))+25/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a^2+5/2*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a^2+25/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))-25/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a^2*sqrt(2))+25/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))-25/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a^2*sqrt(2))],
[exp(-5/2*arctanh(a*x)),x,14,-4*(1-a*x)^(5/4)/(a*(1+a*x)^(1/4))-5*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a-5*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+5*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-5/2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))+5/2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[1/(exp(5/2*arctanh(a*x))*x),x,19,8*(1-a*x)^(1/4)/(1+a*x)^(1/4)+2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))+log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)-log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/sqrt(2)+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)],
[1/(exp(5/2*arctanh(a*x))*x^2),x,7,-10*a*(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(5/4)/(x*(1+a*x)^(1/4))-5*a*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))+5*a*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(5/2*arctanh(a*x))*x^3),x,8,25/2*a^2*(1-a*x)^(1/4)/(1+a*x)^(1/4)+5/4*a*(1-a*x)^(5/4)/(x*(1+a*x)^(1/4))-1/2*(1-a*x)^(9/4)/(x^2*(1+a*x)^(1/4))+25/4*a^2*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-25/4*a^2*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(5/2*arctanh(a*x))*x^4),x,10,-287/24*a^3*(1-a*x)^(1/4)/(1+a*x)^(1/4)-1/3*(1-a*x)^(1/4)/(x^3*(1+a*x)^(1/4))+13/12*a*(1-a*x)^(1/4)/(x^2*(1+a*x)^(1/4))-61/24*a^2*(1-a*x)^(1/4)/(x*(1+a*x)^(1/4))-55/8*a^3*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))+55/8*a^3*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],
[1/(exp(5/2*arctanh(a*x))*x^5),x,11,2467/192*a^4*(1-a*x)^(1/4)/(1+a*x)^(1/4)-1/4*(1-a*x)^(1/4)/(x^4*(1+a*x)^(1/4))+17/24*a*(1-a*x)^(1/4)/(x^3*(1+a*x)^(1/4))-113/96*a^2*(1-a*x)^(1/4)/(x^2*(1+a*x)^(1/4))+521/192*a^3*(1-a*x)^(1/4)/(x*(1+a*x)^(1/4))+475/64*a^4*arctan((1+a*x)^(1/4)/(1-a*x)^(1/4))-475/64*a^4*arctanh((1+a*x)^(1/4)/(1-a*x)^(1/4))],

# Integrands of the form x^m E^(n/3 ArcTanh[a x])

# n>0
[exp(1/3*arctanh(x))*x^m,x,2,x^(1+m)*AppellF1(1+m,1/6,-1/6,2+m,x,-x)/(1+m)],
[exp(1/3*arctanh(x))*x^2,x,16,-19/54*(1-x)^(5/6)*(1+x)^(1/6)-1/18*(1-x)^(5/6)*(1+x)^(7/6)-1/3*(1-x)^(5/6)*x*(1+x)^(7/6)-19/81*arctan((1-x)^(1/6)/(1+x)^(1/6))+19/162*arctan(-2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-19/162*arctan(2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-19/108*log(1+(1-x)^(1/3)/(1+x)^(1/3)-(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))/sqrt(3)+19/108*log(1+(1-x)^(1/3)/(1+x)^(1/3)+(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))/sqrt(3)],
[exp(1/3*arctanh(x))*x,x,15,-1/6*(1-x)^(5/6)*(1+x)^(1/6)-1/2*(1-x)^(5/6)*(1+x)^(7/6)-1/9*arctan((1-x)^(1/6)/(1+x)^(1/6))+1/18*arctan(-2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-1/18*arctan(2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-1/12*log(1+(1-x)^(1/3)/(1+x)^(1/3)-(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))/sqrt(3)+1/12*log(1+(1-x)^(1/3)/(1+x)^(1/3)+(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))/sqrt(3)],
[exp(1/3*arctanh(x)),x,14,-(1-x)^(5/6)*(1+x)^(1/6)-2/3*arctan((1-x)^(1/6)/(1+x)^(1/6))+1/3*arctan(-2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-1/3*arctan(2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-1/2*log(1+(1-x)^(1/3)/(1+x)^(1/3)-(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))/sqrt(3)+1/2*log(1+(1-x)^(1/3)/(1+x)^(1/3)+(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))/sqrt(3)],
[exp(1/3*arctanh(x))/x,x,25,-2*arctan((1-x)^(1/6)/(1+x)^(1/6))+arctan(-2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-arctan(2*(1-x)^(1/6)/(1+x)^(1/6)+sqrt(3))-2*arctanh((1+x)^(1/6)/(1-x)^(1/6))+1/2*log(1-(1+x)^(1/6)/(1-x)^(1/6)+(1+x)^(1/3)/(1-x)^(1/3))-1/2*log(1+(1+x)^(1/6)/(1-x)^(1/6)+(1+x)^(1/3)/(1-x)^(1/3))+arctan((1-2*(1+x)^(1/6)/(1-x)^(1/6))/sqrt(3))*sqrt(3)-arctan((1+2*(1+x)^(1/6)/(1-x)^(1/6))/sqrt(3))*sqrt(3)-1/2*log(1+(1-x)^(1/3)/(1+x)^(1/3)-(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))*sqrt(3)+1/2*log(1+(1-x)^(1/3)/(1+x)^(1/3)+(1-x)^(1/6)*sqrt(3)/(1+x)^(1/6))*sqrt(3)],
[exp(1/3*arctanh(x))/x^2,x,13,-(1-x)^(5/6)*(1+x)^(1/6)/x-2/3*arctanh((1+x)^(1/6)/(1-x)^(1/6))+1/6*log(1-(1+x)^(1/6)/(1-x)^(1/6)+(1+x)^(1/3)/(1-x)^(1/3))-1/6*log(1+(1+x)^(1/6)/(1-x)^(1/6)+(1+x)^(1/3)/(1-x)^(1/3))+arctan((1-2*(1+x)^(1/6)/(1-x)^(1/6))/sqrt(3))/sqrt(3)-arctan((1+2*(1+x)^(1/6)/(1-x)^(1/6))/sqrt(3))/sqrt(3)],
[exp(1/3*arctanh(x))/x^3,x,14,-1/6*(1-x)^(5/6)*(1+x)^(1/6)/x-1/2*(1-x)^(5/6)*(1+x)^(7/6)/x^2-1/9*arctanh((1+x)^(1/6)/(1-x)^(1/6))+1/36*log(1-(1+x)^(1/6)/(1-x)^(1/6)+(1+x)^(1/3)/(1-x)^(1/3))-1/36*log(1+(1+x)^(1/6)/(1-x)^(1/6)+(1+x)^(1/3)/(1-x)^(1/3))+1/6*arctan((1-2*(1+x)^(1/6)/(1-x)^(1/6))/sqrt(3))/sqrt(3)-1/6*arctan((1+2*(1+x)^(1/6)/(1-x)^(1/6))/sqrt(3))/sqrt(3)],
[exp(2/3*arctanh(x))*x^m,x,2,x^(1+m)*AppellF1(1+m,1/3,-1/3,2+m,x,-x)/(1+m)],
[exp(2/3*arctanh(x))*x^2,x,5,-11/27*(1-x)^(2/3)*(1+x)^(1/3)-1/9*(1-x)^(2/3)*(1+x)^(4/3)-1/3*(1-x)^(2/3)*x*(1+x)^(4/3)+11/81*log(1+x)+11/27*log(1+(1-x)^(1/3)/(1+x)^(1/3))+22/27*arctan(1/sqrt(3)-2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arctanh(x))*x,x,4,-1/3*(1-x)^(2/3)*(1+x)^(1/3)-1/2*(1-x)^(2/3)*(1+x)^(4/3)+1/9*log(1+x)+1/3*log(1+(1-x)^(1/3)/(1+x)^(1/3))+2/3*arctan(1/sqrt(3)-2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arctanh(x)),x,3,-(1-x)^(2/3)*(1+x)^(1/3)+1/3*log(1+x)+log(1+(1-x)^(1/3)/(1+x)^(1/3))+2*arctan(1/sqrt(3)-2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arctanh(x))/x,x,4,-1/2*log(x)+1/2*log(1+x)+3/2*log(1+(1-x)^(1/3)/(1+x)^(1/3))+3/2*log((1-x)^(1/3)-(1+x)^(1/3))+arctan(1/sqrt(3)-2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))*sqrt(3)+arctan(1/sqrt(3)+2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))*sqrt(3)],
[exp(2/3*arctanh(x))/x^2,x,3,-(1-x)^(2/3)*(1+x)^(1/3)/x-1/3*log(x)+log((1-x)^(1/3)-(1+x)^(1/3))+2*arctan(1/sqrt(3)+2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arctanh(x))/x^3,x,4,-1/3*(1-x)^(2/3)*(1+x)^(1/3)/x-1/2*(1-x)^(2/3)*(1+x)^(4/3)/x^2-1/9*log(x)+1/3*log((1-x)^(1/3)-(1+x)^(1/3))+2/3*arctan(1/sqrt(3)+2*(1-x)^(1/3)/((1+x)^(1/3)*sqrt(3)))/sqrt(3)],

# n<0

# Integrands of the form x^m E^(n/4 ArcTanh[a x])

# n>0
[exp(1/4*arctanh(a*x))*x^m,x,2,x^(1+m)*AppellF1(1+m,1/8,-1/8,2+m,a*x,-a*x)/(1+m)],
[exp(1/4*arctanh(a*x))*x^2,x,27,-11/32*(1-a*x)^(7/8)*(1+a*x)^(1/8)/a^3-1/24*(1-a*x)^(7/8)*(1+a*x)^(9/8)/a^3-1/3*x*(1-a*x)^(7/8)*(1+a*x)^(9/8)/a^2+11/128*arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))/a^3-11/128*arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))/a^3-11/256*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))/a^3+11/256*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))/a^3+11/128*arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))/a^3-11/128*arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))/a^3-11/256*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))/a^3+11/256*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))/a^3],
[exp(1/4*arctanh(a*x))*x,x,26,-1/8*(1-a*x)^(7/8)*(1+a*x)^(1/8)/a^2-1/2*(1-a*x)^(7/8)*(1+a*x)^(9/8)/a^2+1/32*arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))/a^2-1/32*arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))/a^2-1/64*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))/a^2+1/64*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))/a^2+1/32*arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))/a^2-1/32*arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))/a^2-1/64*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))/a^2+1/64*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))/a^2],
[exp(1/4*arctanh(a*x)),x,25,-(1-a*x)^(7/8)*(1+a*x)^(1/8)/a+1/4*arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))/a-1/4*arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))/a-1/8*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))/a+1/8*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))/a+1/4*arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))/a-1/4*arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))/a-1/8*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))/a+1/8*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))/a],
[exp(1/4*arctanh(a*x))/x,x,39,-2*arctan((1+a*x)^(1/8)/(1-a*x)^(1/8))-2*arctanh((1+a*x)^(1/8)/(1-a*x)^(1/8))+log(1+(1+a*x)^(1/4)/(1-a*x)^(1/4)-(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)-log(1+(1+a*x)^(1/4)/(1-a*x)^(1/4)+(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)+arctan(1-(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))*sqrt(2)-arctan(1+(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))*sqrt(2)+arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))-1/2*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))+1/2*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2-sqrt(2))/(1+a*x)^(1/8))*sqrt(2-sqrt(2))+arctan((-2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-arctan((2*(1-a*x)^(1/8)/(1+a*x)^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))-1/2*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)-(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))+1/2*log(1+(1-a*x)^(1/4)/(1+a*x)^(1/4)+(1-a*x)^(1/8)*sqrt(2+sqrt(2))/(1+a*x)^(1/8))*sqrt(2+sqrt(2))],
[exp(1/4*arctanh(a*x))/x^2,x,16,-(1-a*x)^(7/8)*(1+a*x)^(1/8)/x-1/2*a*arctan((1+a*x)^(1/8)/(1-a*x)^(1/8))-1/2*a*arctanh((1+a*x)^(1/8)/(1-a*x)^(1/8))+1/2*a*arctan(1-(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)-1/2*a*arctan(1+(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)+1/4*a*log(1+(1+a*x)^(1/4)/(1-a*x)^(1/4)-(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)-1/4*a*log(1+(1+a*x)^(1/4)/(1-a*x)^(1/4)+(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)],
[exp(1/4*arctanh(a*x))/x^3,x,17,-1/8*a*(1-a*x)^(7/8)*(1+a*x)^(1/8)/x-1/2*(1-a*x)^(7/8)*(1+a*x)^(9/8)/x^2-1/16*a^2*arctan((1+a*x)^(1/8)/(1-a*x)^(1/8))-1/16*a^2*arctanh((1+a*x)^(1/8)/(1-a*x)^(1/8))+1/16*a^2*arctan(1-(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)-1/16*a^2*arctan(1+(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)+1/32*a^2*log(1+(1+a*x)^(1/4)/(1-a*x)^(1/4)-(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)-1/32*a^2*log(1+(1+a*x)^(1/4)/(1-a*x)^(1/4)+(1+a*x)^(1/8)*sqrt(2)/(1-a*x)^(1/8))/sqrt(2)],

# n<0

# Integrands of the form x^m E^(n ArcTanh[a x]) with m symbolic
[exp(4*arctanh(a*x))*x^m,x,4,x^(1+m)/(1+m)+4*x^(1+m)/(1-a*x)-4*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)],
[exp(3*arctanh(a*x))*x^m,x,9,-3*x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)-a*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)+4*x^(1+m)*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+4*a*x^(2+m)*hypergeom([3/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[exp(2*arctanh(a*x))*x^m,x,3,-x^(1+m)/(1+m)+2*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)/(1+m)],
[exp(arctanh(a*x))*x^m,x,4,x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+a*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[x^m/exp(arctanh(a*x)),x,4,x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)-a*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[x^m/exp(2*arctanh(a*x)),x,3,-x^(1+m)/(1+m)+2*x^(1+m)*hypergeom([1,1+m],[2+m],-a*x)/(1+m)],
[x^m/exp(3*arctanh(a*x)),x,9,-3*x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+a*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)+4*x^(1+m)*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)-4*a*x^(2+m)*hypergeom([3/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],

# Integrands of the form x^m E^(n ArcTanh[a x]) with n symbolic
[exp(n*arctanh(a*x))*x^m,x,2,x^(1+m)*AppellF1(1+m,1/2*n,-1/2*n,2+m,a*x,-a*x)/(1+m)],
[exp(n*arctanh(a*x))*x^3,x,4,-1/4*x^2*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/a^2-1/24*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))*(6+n^2+2*a*n*x)/a^4-1/3*2^(-2+1/2*n)*n*(8+n^2)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^4*(2-n))],
[exp(n*arctanh(a*x))*x^2,x,4,-1/6*n*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/a^3-1/3*x*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/a^2-1/3*2^(1/2*n)*(2+n^2)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^3*(2-n))],
[exp(n*arctanh(a*x))*x,x,3,-1/2*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/a^2-2^(1/2*n)*n*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^2*(2-n))],
[exp(n*arctanh(a*x)),x,2,-2^(1+1/2*n)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a*(2-n))],
[exp(n*arctanh(a*x))/x,x,4,2*(1+a*x)^(1/2*n)*hypergeom([1,-1/2*n],[1-1/2*n],(1-a*x)/(1+a*x))/(n*(1-a*x)^(1/2*n))-2^(1+1/2*n)*hypergeom([-1/2*n,-1/2*n],[1-1/2*n],1/2*(1-a*x))/(n*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/x^2,x,2,-4*a*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(1-a*x)/(1+a*x))/(2-n)],
[exp(n*arctanh(a*x))/x^3,x,3,-1/2*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/x^2-2*a^2*n*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(1-a*x)/(1+a*x))/(2-n)],
[exp(n*arctanh(a*x))/x^4,x,5,-1/3*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/x^3-1/6*a*n*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(2+n))/x^2-2/3*a^3*(2+n^2)*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(1-a*x)/(1+a*x))/(2-n)],

# Integrands of the form u E^(n ArcTanh[a x]) (c-a c x)^p

# Integrands of the form E^(n ArcTanh[a x]) (c-c a x)^p

# n>0
[exp(arctanh(a*x))*(c-a*c*x)^p,x,3,-2*(c-a*c*x)^(1+p)*hypergeom([-1/2,1/2+p],[3/2+p],1/2*(1-a*x))*sqrt(2)/(a*c*(1+2*p)*sqrt(1-a*x))],
[exp(arctanh(a*x))*(c-a*c*x)^4,x,6,7/12*c^4*(1-a^2*x^2)^(3/2)/a+7/20*c^4*(1-a*x)*(1-a^2*x^2)^(3/2)/a+1/5*c^4*(1-a*x)^2*(1-a^2*x^2)^(3/2)/a+7/8*c^4*arcsin(a*x)/a+7/8*c^4*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^3,x,5,5/12*c^3*(1-a^2*x^2)^(3/2)/a+1/4*c^3*(1-a*x)*(1-a^2*x^2)^(3/2)/a+5/8*c^3*arcsin(a*x)/a+5/8*c^3*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^2,x,4,1/3*c^2*(1-a^2*x^2)^(3/2)/a+1/2*c^2*arcsin(a*x)/a+1/2*c^2*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x),x,3,1/2*c*arcsin(a*x)/a+1/2*c*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))/(c-a*c*x),x,3,-arcsin(a*x)/(a*c)+2*sqrt(1-a^2*x^2)/(a*c*(1-a*x))],
[exp(arctanh(a*x))/(c-a*c*x)^2,x,2,1/3*(1-a^2*x^2)^(3/2)/(a*c^2*(1-a*x)^3)],
[exp(arctanh(a*x))/(c-a*c*x)^3,x,3,1/5*(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^4)+1/15*(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^3)],
[exp(arctanh(a*x))/(c-a*c*x)^4,x,4,1/7*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^5)+2/35*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^4)+2/105*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^3)],
[exp(arctanh(a*x))/(c-a*c*x)^5,x,5,1/9*(1-a^2*x^2)^(3/2)/(a*c^5*(1-a*x)^6)+1/21*(1-a^2*x^2)^(3/2)/(a*c^5*(1-a*x)^5)+2/105*(1-a^2*x^2)^(3/2)/(a*c^5*(1-a*x)^4)+2/315*(1-a^2*x^2)^(3/2)/(a*c^5*(1-a*x)^3)],
[exp(2*arctanh(a*x))*(c-a*c*x)^p,x,4,-2*(c-a*c*x)^p/(a*p)+(c-a*c*x)^(1+p)/(a*c*(1+p))],
[exp(2*arctanh(a*x))*(c-a*c*x)^5,x,3,-2/5*c^5*(1-a*x)^5/a+1/6*c^5*(1-a*x)^6/a],
[exp(2*arctanh(a*x))*(c-a*c*x)^4,x,3,-1/2*c^4*(1-a*x)^4/a+1/5*c^4*(1-a*x)^5/a],
[exp(2*arctanh(a*x))*(c-a*c*x)^3,x,3,-2/3*c^3*(1-a*x)^3/a+1/4*c^3*(1-a*x)^4/a],
[exp(2*arctanh(a*x))*(c-a*c*x)^2,x,3,c^2*x-1/3*a^2*c^2*x^3],
[exp(2*arctanh(a*x))*(c-a*c*x),x,1,c*x+1/2*a*c*x^2,1/2*c*exp(2*arctanh(a*x))*(1-a^2*x^2)/a],
[exp(2*arctanh(a*x))/(c-a*c*x),x,3,2/(a*c*(1-a*x))+log(1-a*x)/(a*c)],
[exp(2*arctanh(a*x))/(c-a*c*x)^2,x,2,x/(c^2*(1-a*x)^2)],
[exp(2*arctanh(a*x))/(c-a*c*x)^3,x,3,2/3/(a*c^3*(1-a*x)^3)+(-1/2)/(a*c^3*(1-a*x)^2)],
[exp(2*arctanh(a*x))/(c-a*c*x)^4,x,3,1/2/(a*c^4*(1-a*x)^4)+(-1/3)/(a*c^4*(1-a*x)^3)],
[exp(3*arctanh(a*x))*(c-a*c*x)^p,x,3,4*(c-a*c*x)^(1+p)*hypergeom([-3/2,-1/2+p],[1/2+p],1/2*(1-a*x))*sqrt(2)/(a*c*(1-2*p)*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*(c-a*c*x)^4,x,5,1/4*c^4*x*(1-a^2*x^2)^(3/2)+1/5*c^4*(1-a^2*x^2)^(5/2)/a+3/8*c^4*arcsin(a*x)/a+3/8*c^4*x*sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a*c*x)^3,x,4,1/4*c^3*x*(1-a^2*x^2)^(3/2)+3/8*c^3*arcsin(a*x)/a+3/8*c^3*x*sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a*c*x)^2,x,4,-1/3*c^2*(1-a^2*x^2)^(3/2)/a+1/2*c^2*arcsin(a*x)/a+1/2*c^2*x*sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a*c*x),x,4,-1/2*c*(1-a^2*x^2)^(3/2)/(a*(1-a*x))+3/2*c*arcsin(a*x)/a-3/2*c*sqrt(1-a^2*x^2)/a],
[exp(3*arctanh(a*x))/(c-a*c*x),x,4,2/3*(1-a^2*x^2)^(3/2)/(a*c*(1-a*x)^3)+arcsin(a*x)/(a*c)-2*sqrt(1-a^2*x^2)/(a*c*(1-a*x))],
[exp(3*arctanh(a*x))/(c-a*c*x)^2,x,2,1/5*(1-a^2*x^2)^(5/2)/(a*c^2*(1-a*x)^5)],
[exp(3*arctanh(a*x))/(c-a*c*x)^3,x,3,1/7*(1-a^2*x^2)^(5/2)/(a*c^3*(1-a*x)^6)+1/35*(1-a^2*x^2)^(5/2)/(a*c^3*(1-a*x)^5)],
[exp(3*arctanh(a*x))/(c-a*c*x)^4,x,4,1/9*(1-a^2*x^2)^(5/2)/(a*c^4*(1-a*x)^7)+2/63*(1-a^2*x^2)^(5/2)/(a*c^4*(1-a*x)^6)+2/315*(1-a^2*x^2)^(5/2)/(a*c^4*(1-a*x)^5)],
[exp(3*arctanh(a*x))/(c-a*c*x)^5,x,5,1/11*(1-a^2*x^2)^(5/2)/(a*c^5*(1-a*x)^8)+1/33*(1-a^2*x^2)^(5/2)/(a*c^5*(1-a*x)^7)+2/231*(1-a^2*x^2)^(5/2)/(a*c^5*(1-a*x)^6)+2/1155*(1-a^2*x^2)^(5/2)/(a*c^5*(1-a*x)^5)],
[exp(4*arctanh(a*x))*(c-a*c*x)^p,x,4,4*c*(c-a*c*x)^(-1+p)/(a*(1-p))+4*(c-a*c*x)^p/(a*p)-(c-a*c*x)^(1+p)/(a*c*(1+p))],
[exp(4*arctanh(a*x))*(c-a*c*x)^5,x,3,-c^5*(1-a*x)^4/a+4/5*c^5*(1-a*x)^5/a-1/6*c^5*(1-a*x)^6/a],
[exp(4*arctanh(a*x))*(c-a*c*x)^4,x,4,c^4*x-2/3*a^2*c^4*x^3+1/5*a^4*c^4*x^5],
[exp(4*arctanh(a*x))*(c-a*c*x)^3,x,3,2/3*c^3*(1+a*x)^3/a-1/4*c^3*(1+a*x)^4/a],
[exp(4*arctanh(a*x))*(c-a*c*x)^2,x,2,1/3*c^2*(1+a*x)^3/a],
[exp(4*arctanh(a*x))*(c-a*c*x),x,3,-3*c*x-1/2*a*c*x^2-4*c*log(1-a*x)/a],
[exp(4*arctanh(a*x))/(c-a*c*x),x,3,2/(a*c*(1-a*x)^2)+(-4)/(a*c*(1-a*x))-log(1-a*x)/(a*c)],
[exp(4*arctanh(a*x))/(c-a*c*x)^2,x,2,1/6*(1+a*x)^3/(a*c^2*(1-a*x)^3)],
[exp(4*arctanh(a*x))/(c-a*c*x)^3,x,3,1/(a*c^3*(1-a*x)^4)+(-4/3)/(a*c^3*(1-a*x)^3)+1/2/(a*c^3*(1-a*x)^2)],
[exp(4*arctanh(a*x))/(c-a*c*x)^4,x,3,4/5/(a*c^4*(1-a*x)^5)+(-1)/(a*c^4*(1-a*x)^4)+1/3/(a*c^4*(1-a*x)^3)],

# n<0
[(c-a*c*x)^p/exp(arctanh(a*x)),x,3,-(c-a*c*x)^(1+p)*hypergeom([1/2,3/2+p],[5/2+p],1/2*(1-a*x))*sqrt(2)*sqrt(1-a*x)/(a*c*(3+2*p))],
[(c-a*c*x)^3/exp(arctanh(a*x)),x,6,35/8*c^3*arcsin(a*x)/a+35/8*c^3*sqrt(1-a^2*x^2)/a+35/24*c^3*(1-a*x)*sqrt(1-a^2*x^2)/a+7/12*c^3*(1-a*x)^2*sqrt(1-a^2*x^2)/a+1/4*c^3*(1-a*x)^3*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)^2/exp(arctanh(a*x)),x,5,5/2*c^2*arcsin(a*x)/a+5/2*c^2*sqrt(1-a^2*x^2)/a+5/6*c^2*(1-a*x)*sqrt(1-a^2*x^2)/a+1/3*c^2*(1-a*x)^2*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)/exp(arctanh(a*x)),x,4,3/2*c*arcsin(a*x)/a+3/2*c*sqrt(1-a^2*x^2)/a+1/2*c*(1-a*x)*sqrt(1-a^2*x^2)/a],
[1/(exp(arctanh(a*x))*(c-a*c*x)),x,2,arcsin(a*x)/(a*c)],
[1/(exp(arctanh(a*x))*(c-a*c*x)^2),x,2,sqrt(1-a^2*x^2)/(a*c^2*(1-a*x))],
[1/(exp(arctanh(a*x))*(c-a*c*x)^3),x,3,1/3*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^2)+1/3*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x))],
[1/(exp(arctanh(a*x))*(c-a*c*x)^4),x,4,1/5*sqrt(1-a^2*x^2)/(a*c^4*(1-a*x)^3)+2/15*sqrt(1-a^2*x^2)/(a*c^4*(1-a*x)^2)+2/15*sqrt(1-a^2*x^2)/(a*c^4*(1-a*x))],
[1/(exp(arctanh(a*x))*(c-a*c*x)^5),x,5,1/7*sqrt(1-a^2*x^2)/(a*c^5*(1-a*x)^4)+3/35*sqrt(1-a^2*x^2)/(a*c^5*(1-a*x)^3)+2/35*sqrt(1-a^2*x^2)/(a*c^5*(1-a*x)^2)+2/35*sqrt(1-a^2*x^2)/(a*c^5*(1-a*x))],
[(c-a*c*x)^p/exp(2*arctanh(a*x)),x,3,-1/2*(c-a*c*x)^(2+p)*hypergeom([1,2+p],[3+p],1/2*(1-a*x))/(a*c^2*(2+p))],
[(c-a*c*x)^4/exp(2*arctanh(a*x)),x,3,-16*c^4*x+4*c^4*(1-a*x)^2/a+4/3*c^4*(1-a*x)^3/a+1/2*c^4*(1-a*x)^4/a+1/5*c^4*(1-a*x)^5/a+32*c^4*log(1+a*x)/a],
[(c-a*c*x)^3/exp(2*arctanh(a*x)),x,3,-8*c^3*x+2*c^3*(1-a*x)^2/a+2/3*c^3*(1-a*x)^3/a+1/4*c^3*(1-a*x)^4/a+16*c^3*log(1+a*x)/a],
[(c-a*c*x)^2/exp(2*arctanh(a*x)),x,3,-4*c^2*x+c^2*(1-a*x)^2/a+1/3*c^2*(1-a*x)^3/a+8*c^2*log(1+a*x)/a],
[(c-a*c*x)/exp(2*arctanh(a*x)),x,3,-3*c*x+1/2*a*c*x^2+4*c*log(1+a*x)/a],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)),x,2,log(1+a*x)/(a*c)],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^2),x,3,arctanh(a*x)/(a*c^2)],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^3),x,4,1/2/(a*c^3*(1-a*x))+1/2*arctanh(a*x)/(a*c^3)],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^4),x,4,1/4/(a*c^4*(1-a*x)^2)+1/4/(a*c^4*(1-a*x))+1/4*arctanh(a*x)/(a*c^4)],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^5),x,4,1/6/(a*c^5*(1-a*x)^3)+1/8/(a*c^5*(1-a*x)^2)+1/8/(a*c^5*(1-a*x))+1/8*arctanh(a*x)/(a*c^5)],
[(c-a*c*x)^p/exp(3*arctanh(a*x)),x,3,-(1-a*x)^(3/2)*(c-a*c*x)^(1+p)*hypergeom([3/2,5/2+p],[7/2+p],1/2*(1-a*x))/(a*c*(5+2*p)*sqrt(2))],
[(c-a*c*x)^3/exp(3*arctanh(a*x)),x,7,-315/8*c^3*arcsin(a*x)/a-2*c^3*(1-a*x)^5/(a*sqrt(1-a^2*x^2))-315/8*c^3*sqrt(1-a^2*x^2)/a-105/8*c^3*(1-a*x)*sqrt(1-a^2*x^2)/a-21/4*c^3*(1-a*x)^2*sqrt(1-a^2*x^2)/a-9/4*c^3*(1-a*x)^3*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)^2/exp(3*arctanh(a*x)),x,6,-35/2*c^2*arcsin(a*x)/a-2*c^2*(1-a*x)^4/(a*sqrt(1-a^2*x^2))-35/2*c^2*sqrt(1-a^2*x^2)/a-35/6*c^2*(1-a*x)*sqrt(1-a^2*x^2)/a-7/3*c^2*(1-a*x)^2*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)/exp(3*arctanh(a*x)),x,5,-15/2*c*arcsin(a*x)/a-2*c*(1-a*x)^3/(a*sqrt(1-a^2*x^2))-15/2*c*sqrt(1-a^2*x^2)/a-5/2*c*(1-a*x)*sqrt(1-a^2*x^2)/a],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)),x,3,-arcsin(a*x)/(a*c)-2*(1-a*x)/(a*c*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^2),x,2,(-1+a*x)/(a*c^2*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^3),x,2,x/(c^3*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^4),x,3,2/3*x/(c^4*sqrt(1-a^2*x^2))+1/3/(a*c^4*(1-a*x)*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^5),x,4,2/5*x/(c^5*sqrt(1-a^2*x^2))+1/5/(a*c^5*(1-a*x)^2*sqrt(1-a^2*x^2))+1/5/(a*c^5*(1-a*x)*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^6),x,5,8/35*x/(c^6*sqrt(1-a^2*x^2))+1/7/(a*c^6*(1-a*x)^3*sqrt(1-a^2*x^2))+4/35/(a*c^6*(1-a*x)^2*sqrt(1-a^2*x^2))+4/35/(a*c^6*(1-a*x)*sqrt(1-a^2*x^2))],

# Integrands of the form E^(n ArcTanh[a x])  (c-c a x)^(p/2)

# n>0
[exp(arctanh(a*x))*(c-a*c*x)^(9/2),x,6,4096/3465*c^6*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+32/99*c^3*(c-a*c*x)^(3/2)*(1-a^2*x^2)^(3/2)/a+2/11*c^2*(c-a*c*x)^(5/2)*(1-a^2*x^2)^(3/2)/a+1024/1155*c^5*(1-a^2*x^2)^(3/2)/(a*sqrt(c-a*c*x))+128/231*c^4*(1-a^2*x^2)^(3/2)*sqrt(c-a*c*x)/a],
[exp(arctanh(a*x))*(c-a*c*x)^(7/2),x,5,256/315*c^5*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+2/9*c^2*(c-a*c*x)^(3/2)*(1-a^2*x^2)^(3/2)/a+64/105*c^4*(1-a^2*x^2)^(3/2)/(a*sqrt(c-a*c*x))+8/21*c^3*(1-a^2*x^2)^(3/2)*sqrt(c-a*c*x)/a],
[exp(arctanh(a*x))*(c-a*c*x)^(5/2),x,4,64/105*c^4*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+16/35*c^3*(1-a^2*x^2)^(3/2)/(a*sqrt(c-a*c*x))+2/7*c^2*(1-a^2*x^2)^(3/2)*sqrt(c-a*c*x)/a],
[exp(arctanh(a*x))*(c-a*c*x)^(3/2),x,3,8/15*c^3*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+2/5*c^2*(1-a^2*x^2)^(3/2)/(a*sqrt(c-a*c*x))],
[exp(arctanh(a*x))*sqrt(c-a*c*x),x,2,2/3*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))],
[exp(arctanh(a*x))/sqrt(c-a*c*x),x,4,2*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))*sqrt(2)/(a*sqrt(c))-2*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))],
[exp(arctanh(a*x))/(c-a*c*x)^(3/2),x,4,-arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(3/2)*sqrt(2))+sqrt(1-a^2*x^2)/(a*(c-a*c*x)^(3/2))],
[exp(arctanh(a*x))/(c-a*c*x)^(5/2),x,5,-1/8*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(5/2)*sqrt(2))+1/2*sqrt(1-a^2*x^2)/(a*(c-a*c*x)^(5/2))-1/8*sqrt(1-a^2*x^2)/(a*c*(c-a*c*x)^(3/2))],
[exp(arctanh(a*x))/(c-a*c*x)^(7/2),x,6,-1/32*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(7/2)*sqrt(2))+1/3*sqrt(1-a^2*x^2)/(a*(c-a*c*x)^(7/2))-1/24*sqrt(1-a^2*x^2)/(a*c*(c-a*c*x)^(5/2))-1/32*sqrt(1-a^2*x^2)/(a*c^2*(c-a*c*x)^(3/2))],
[exp(2*arctanh(a*x))*(c-a*c*x)^(7/2),x,4,-4/7*(c-a*c*x)^(7/2)/a+2/9*(c-a*c*x)^(9/2)/(a*c)],
[exp(2*arctanh(a*x))*(c-a*c*x)^(5/2),x,4,-4/5*(c-a*c*x)^(5/2)/a+2/7*(c-a*c*x)^(7/2)/(a*c)],
[exp(2*arctanh(a*x))*(c-a*c*x)^(3/2),x,4,-4/3*(c-a*c*x)^(3/2)/a+2/5*(c-a*c*x)^(5/2)/(a*c)],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x),x,4,2/3*(c-a*c*x)^(3/2)/(a*c)-4*sqrt(c-a*c*x)/a],
[exp(2*arctanh(a*x))/sqrt(c-a*c*x),x,4,4/(a*sqrt(c-a*c*x))+2*sqrt(c-a*c*x)/(a*c)],
[exp(2*arctanh(a*x))/(c-a*c*x)^(3/2),x,4,4/3/(a*(c-a*c*x)^(3/2))+(-2)/(a*c*sqrt(c-a*c*x))],
[exp(2*arctanh(a*x))/(c-a*c*x)^(5/2),x,4,4/5/(a*(c-a*c*x)^(5/2))+(-2/3)/(a*c*(c-a*c*x)^(3/2))],
[exp(2*arctanh(a*x))/(c-a*c*x)^(7/2),x,4,4/7/(a*(c-a*c*x)^(7/2))+(-2/5)/(a*c*(c-a*c*x)^(5/2))],
[exp(3*arctanh(a*x))*(c-a*c*x)^(9/2),x,5,256/1155*c^7*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(5/2))+64/231*c^6*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(3/2))+8/33*c^5*(1-a^2*x^2)^(5/2)/(a*sqrt(c-a*c*x))+2/11*c^4*(1-a^2*x^2)^(5/2)*sqrt(c-a*c*x)/a],
[exp(3*arctanh(a*x))*(c-a*c*x)^(7/2),x,4,64/315*c^6*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(5/2))+16/63*c^5*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(3/2))+2/9*c^4*(1-a^2*x^2)^(5/2)/(a*sqrt(c-a*c*x))],
[exp(3*arctanh(a*x))*(c-a*c*x)^(5/2),x,3,8/35*c^5*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(5/2))+2/7*c^4*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(3/2))],
[exp(3*arctanh(a*x))*(c-a*c*x)^(3/2),x,2,2/5*c^4*(1-a^2*x^2)^(5/2)/(a*(c-a*c*x)^(5/2))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x),x,5,-2/3*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+4*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))*sqrt(2)*sqrt(c)/a-4*c*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))],
[exp(3*arctanh(a*x))/sqrt(c-a*c*x),x,5,c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(5/2))-3*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))*sqrt(2)/(a*sqrt(c))+3*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))],
[exp(3*arctanh(a*x))/(c-a*c*x)^(3/2),x,5,1/2*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(7/2))+3/4*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(3/2)*sqrt(2))-3/4*sqrt(1-a^2*x^2)/(a*(c-a*c*x)^(3/2))],
[exp(3*arctanh(a*x))/(c-a*c*x)^(5/2),x,6,1/3*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(9/2))+1/16*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(5/2)*sqrt(2))-1/4*sqrt(1-a^2*x^2)/(a*(c-a*c*x)^(5/2))+1/16*sqrt(1-a^2*x^2)/(a*c*(c-a*c*x)^(3/2))],
[exp(3*arctanh(a*x))/(c-a*c*x)^(7/2),x,7,1/4*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(11/2))+3/256*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(7/2)*sqrt(2))-1/8*sqrt(1-a^2*x^2)/(a*(c-a*c*x)^(7/2))+1/64*sqrt(1-a^2*x^2)/(a*c*(c-a*c*x)^(5/2))+3/256*sqrt(1-a^2*x^2)/(a*c^2*(c-a*c*x)^(3/2))],

# n<0
[(c-a*c*x)^(9/2)/exp(arctanh(a*x)),x,7,512/231*c^3*(c-a*c*x)^(3/2)*sqrt(1-a^2*x^2)/a+640/693*c^2*(c-a*c*x)^(5/2)*sqrt(1-a^2*x^2)/a+40/99*c*(c-a*c*x)^(7/2)*sqrt(1-a^2*x^2)/a+2/11*(c-a*c*x)^(9/2)*sqrt(1-a^2*x^2)/a+16384/693*c^5*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))+4096/693*c^4*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)^(7/2)/exp(arctanh(a*x)),x,6,128/105*c^2*(c-a*c*x)^(3/2)*sqrt(1-a^2*x^2)/a+32/63*c*(c-a*c*x)^(5/2)*sqrt(1-a^2*x^2)/a+2/9*(c-a*c*x)^(7/2)*sqrt(1-a^2*x^2)/a+4096/315*c^4*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))+1024/315*c^3*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)^(5/2)/exp(arctanh(a*x)),x,5,24/35*c*(c-a*c*x)^(3/2)*sqrt(1-a^2*x^2)/a+2/7*(c-a*c*x)^(5/2)*sqrt(1-a^2*x^2)/a+256/35*c^3*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))+64/35*c^2*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a],
[(c-a*c*x)^(3/2)/exp(arctanh(a*x)),x,4,2/5*(c-a*c*x)^(3/2)*sqrt(1-a^2*x^2)/a+64/15*c^2*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))+16/15*c*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a],
[sqrt(c-a*c*x)/exp(arctanh(a*x)),x,3,8/3*c*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))+2/3*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a],
[1/(exp(arctanh(a*x))*sqrt(c-a*c*x)),x,2,2*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))],
[1/(exp(arctanh(a*x))*(c-a*c*x)^(3/2)),x,3,arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))*sqrt(2)/(a*c^(3/2))],
[1/(exp(arctanh(a*x))*(c-a*c*x)^(5/2)),x,4,1/2*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(5/2)*sqrt(2))+1/2*sqrt(1-a^2*x^2)/(a*c*(c-a*c*x)^(3/2))],
[1/(exp(arctanh(a*x))*(c-a*c*x)^(7/2)),x,5,3/16*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(7/2)*sqrt(2))+1/4*sqrt(1-a^2*x^2)/(a*c*(c-a*c*x)^(5/2))+3/16*sqrt(1-a^2*x^2)/(a*c^2*(c-a*c*x)^(3/2))],
[(c-a*c*x)^(7/2)/exp(2*arctanh(a*x)),x,9,16/3*c^2*(c-a*c*x)^(3/2)/a+8/5*c*(c-a*c*x)^(5/2)/a+4/7*(c-a*c*x)^(7/2)/a+2/9*(c-a*c*x)^(9/2)/(a*c)-32*c^(7/2)*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/a+32*c^3*sqrt(c-a*c*x)/a],
[(c-a*c*x)^(5/2)/exp(2*arctanh(a*x)),x,8,8/3*c*(c-a*c*x)^(3/2)/a+4/5*(c-a*c*x)^(5/2)/a+2/7*(c-a*c*x)^(7/2)/(a*c)-16*c^(5/2)*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/a+16*c^2*sqrt(c-a*c*x)/a],
[(c-a*c*x)^(3/2)/exp(2*arctanh(a*x)),x,7,4/3*(c-a*c*x)^(3/2)/a+2/5*(c-a*c*x)^(5/2)/(a*c)-8*c^(3/2)*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/a+8*c*sqrt(c-a*c*x)/a],
[sqrt(c-a*c*x)/exp(2*arctanh(a*x)),x,6,2/3*(c-a*c*x)^(3/2)/(a*c)-4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a+4*sqrt(c-a*c*x)/a],
[1/(exp(2*arctanh(a*x))*sqrt(c-a*c*x)),x,5,-2*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*sqrt(c))+2*sqrt(c-a*c*x)/(a*c)],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^(3/2)),x,4,-arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*c^(3/2))],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^(5/2)),x,5,-arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))/(a*c^(5/2)*sqrt(2))+1/(a*c^2*sqrt(c-a*c*x))],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^(7/2)),x,6,1/3/(a*c^2*(c-a*c*x)^(3/2))-1/2*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))/(a*c^(7/2)*sqrt(2))+1/2/(a*c^3*sqrt(c-a*c*x))],
[1/(exp(2*arctanh(a*x))*(c-a*c*x)^(9/2)),x,7,1/5/(a*c^2*(c-a*c*x)^(5/2))+1/6/(a*c^3*(c-a*c*x)^(3/2))-1/4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))/(a*c^(9/2)*sqrt(2))+1/4/(a*c^4*sqrt(c-a*c*x))],
[(c-a*c*x)^(5/2)/exp(3*arctanh(a*x)),x,6,1024/35*c*(c-a*c*x)^(3/2)/(a*sqrt(1-a^2*x^2))+128/35*(c-a*c*x)^(5/2)/(a*sqrt(1-a^2*x^2))+32/35*(c-a*c*x)^(7/2)/(a*c*sqrt(1-a^2*x^2))+2/7*(c-a*c*x)^(9/2)/(a*c^2*sqrt(1-a^2*x^2))-4096/35*c^2*sqrt(c-a*c*x)/(a*sqrt(1-a^2*x^2))],
[(c-a*c*x)^(3/2)/exp(3*arctanh(a*x)),x,5,64/5*(c-a*c*x)^(3/2)/(a*sqrt(1-a^2*x^2))+8/5*(c-a*c*x)^(5/2)/(a*c*sqrt(1-a^2*x^2))+2/5*(c-a*c*x)^(7/2)/(a*c^2*sqrt(1-a^2*x^2))-256/5*c*sqrt(c-a*c*x)/(a*sqrt(1-a^2*x^2))],
[sqrt(c-a*c*x)/exp(3*arctanh(a*x)),x,4,16/3*(c-a*c*x)^(3/2)/(a*c*sqrt(1-a^2*x^2))+2/3*(c-a*c*x)^(5/2)/(a*c^2*sqrt(1-a^2*x^2))-64/3*sqrt(c-a*c*x)/(a*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*sqrt(c-a*c*x)),x,3,2*(c-a*c*x)^(3/2)/(a*c^2*sqrt(1-a^2*x^2))-8*sqrt(c-a*c*x)/(a*c*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^(3/2)),x,2,-2*sqrt(c-a*c*x)/(a*c^2*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^(5/2)),x,4,arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(5/2)*sqrt(2))-sqrt(c-a*c*x)/(a*c^3*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^(7/2)),x,5,3/4*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(7/2)*sqrt(2))+1/2/(a*c^3*sqrt(c-a*c*x)*sqrt(1-a^2*x^2))-3/4*sqrt(c-a*c*x)/(a*c^4*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a*c*x)^(9/2)),x,6,15/32*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))/(a*c^(9/2)*sqrt(2))+1/4/(a*c^3*(c-a*c*x)^(3/2)*sqrt(1-a^2*x^2))+5/16/(a*c^4*sqrt(c-a*c*x)*sqrt(1-a^2*x^2))-15/32*sqrt(c-a*c*x)/(a*c^5*sqrt(1-a^2*x^2))],

# n symbolic
[exp(n*arctanh(a*x))*(c-a*c*x)^(7/2),x,3,-2^(1+1/2*n)*(c-a*c*x)^(9/2)*hypergeom([1/2*(9-n),-1/2*n],[1/2*(11-n)],1/2*(1-a*x))/(a*c*(9-n)*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*(c-a*c*x)^(5/2),x,3,-2^(1+1/2*n)*(c-a*c*x)^(7/2)*hypergeom([1/2*(7-n),-1/2*n],[1/2*(9-n)],1/2*(1-a*x))/(a*c*(7-n)*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*(c-a*c*x)^(3/2),x,3,-2^(1+1/2*n)*(c-a*c*x)^(5/2)*hypergeom([1/2*(5-n),-1/2*n],[1/2*(7-n)],1/2*(1-a*x))/(a*c*(5-n)*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*sqrt(c-a*c*x),x,3,-2^(1+1/2*n)*(c-a*c*x)^(3/2)*hypergeom([1/2*(3-n),-1/2*n],[1/2*(5-n)],1/2*(1-a*x))/(a*c*(3-n)*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/sqrt(c-a*c*x),x,3,-2^(1+1/2*n)*hypergeom([1/2*(1-n),-1/2*n],[1/2*(3-n)],1/2*(1-a*x))*sqrt(c-a*c*x)/(a*c*(1-n)*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(c-a*c*x)^(3/2),x,3,2^(1+1/2*n)*hypergeom([1/2*(-1-n),-1/2*n],[1/2*(1-n)],1/2*(1-a*x))/(a*c*(1+n)*(1-a*x)^(1/2*n)*sqrt(c-a*c*x))],
[exp(n*arctanh(a*x))/(c-a*c*x)^(5/2),x,3,2^(1+1/2*n)*hypergeom([1/2*(-3-n),-1/2*n],[1/2*(-1-n)],1/2*(1-a*x))/(a*c*(3+n)*(1-a*x)^(1/2*n)*(c-a*c*x)^(3/2))],
[exp(n*arctanh(a*x))/(c-a*c*x)^(7/2),x,3,2^(1+1/2*n)*hypergeom([1/2*(-5-n),-1/2*n],[1/2*(-3-n)],1/2*(1-a*x))/(a*c*(5+n)*(1-a*x)^(1/2*n)*(c-a*c*x)^(5/2))],

# Integrands of the form x^m E^ArcTanh[a x] (c-c a x)^p

# p>0
[exp(arctanh(a*x))*x^4*(c-a*c*x),x,5,1/16*c*arcsin(a*x)/a^5-1/16*c*x*sqrt(1-a^2*x^2)/a^4-1/24*c*x^3*sqrt(1-a^2*x^2)/a^2+1/6*c*x^5*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*x^3*(c-a*c*x),x,4,-1/3*c*(1-a^2*x^2)^(3/2)/a^4+1/5*c*(1-a^2*x^2)^(5/2)/a^4],
[exp(arctanh(a*x))*x^2*(c-a*c*x),x,4,1/8*c*arcsin(a*x)/a^3-1/8*c*x*sqrt(1-a^2*x^2)/a^2+1/4*c*x^3*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*x*(c-a*c*x),x,2,-1/3*c*(1-a^2*x^2)^(3/2)/a^2],
[exp(arctanh(a*x))*(c-a*c*x),x,3,1/2*c*arcsin(a*x)/a+1/2*c*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)/x,x,5,-c*arctanh(sqrt(1-a^2*x^2))+c*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)/x^2,x,3,-a*c*arcsin(a*x)-c*sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))*(c-a*c*x)/x^3,x,5,1/2*a^2*c*arctanh(sqrt(1-a^2*x^2))-1/2*c*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)/x^4,x,2,-1/3*c*(1-a^2*x^2)^(3/2)/x^3],
[exp(arctanh(a*x))*x^3*(c-a*c*x)^2,x,6,-1/5*c^2*x^2*(1-a^2*x^2)^(3/2)/a^2+1/6*c^2*x^3*(1-a^2*x^2)^(3/2)/a-1/120*c^2*(16-15*a*x)*(1-a^2*x^2)^(3/2)/a^4-1/16*c^2*arcsin(a*x)/a^4-1/16*c^2*x*sqrt(1-a^2*x^2)/a^3],
[exp(arctanh(a*x))*x^2*(c-a*c*x)^2,x,9,1/3*c^2*(1-a^2*x^2)^(3/2)/a^3-1/4*c^2*x*(1-a^2*x^2)^(3/2)/a^2-1/5*c^2*(1-a^2*x^2)^(5/2)/a^3+1/8*c^2*arcsin(a*x)/a^3+1/8*c^2*x*sqrt(1-a^2*x^2)/a^2],
[exp(arctanh(a*x))*x*(c-a*c*x)^2,x,4,-1/12*c^2*(4-3*a*x)*(1-a^2*x^2)^(3/2)/a^2-1/8*c^2*arcsin(a*x)/a^2-1/8*c^2*x*sqrt(1-a^2*x^2)/a],
[exp(arctanh(a*x))*(c-a*c*x)^2,x,4,1/3*c^2*(1-a^2*x^2)^(3/2)/a+1/2*c^2*arcsin(a*x)/a+1/2*c^2*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^2/x,x,7,-1/2*c^2*arcsin(a*x)-c^2*arctanh(sqrt(1-a^2*x^2))+1/2*c^2*(2-a*x)*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^2/x^2,x,7,-a*c^2*arcsin(a*x)+a*c^2*arctanh(sqrt(1-a^2*x^2))-c^2*(1+a*x)*sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))*(c-a*c*x)^2/x^3,x,7,a^2*c^2*arcsin(a*x)+1/2*a^2*c^2*arctanh(sqrt(1-a^2*x^2))-1/2*c^2*(1-2*a*x)*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^2/x^4,x,6,-1/3*c^2*(1-a^2*x^2)^(3/2)/x^3-1/2*a^3*c^2*arctanh(sqrt(1-a^2*x^2))+1/2*a*c^2*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^2/x^5,x,7,-1/4*c^2*(1-a^2*x^2)^(3/2)/x^4+1/3*a*c^2*(1-a^2*x^2)^(3/2)/x^3+1/8*a^4*c^2*arctanh(sqrt(1-a^2*x^2))-1/8*a^2*c^2*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^2/x^6,x,8,-1/5*c^2*(1-a^2*x^2)^(3/2)/x^5+1/4*a*c^2*(1-a^2*x^2)^(3/2)/x^4-2/15*a^2*c^2*(1-a^2*x^2)^(3/2)/x^3-1/8*a^5*c^2*arctanh(sqrt(1-a^2*x^2))+1/8*a^3*c^2*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^2/x^7,x,9,-1/6*c^2*(1-a^2*x^2)^(3/2)/x^6+1/5*a*c^2*(1-a^2*x^2)^(3/2)/x^5-1/8*a^2*c^2*(1-a^2*x^2)^(3/2)/x^4+2/15*a^3*c^2*(1-a^2*x^2)^(3/2)/x^3+1/16*a^6*c^2*arctanh(sqrt(1-a^2*x^2))-1/16*a^4*c^2*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*x^3*(c-a*c*x)^3,x,7,-11/35*c^3*x^2*(1-a^2*x^2)^(3/2)/a^2+1/3*c^3*x^3*(1-a^2*x^2)^(3/2)/a-1/7*c^3*x^4*(1-a^2*x^2)^(3/2)-1/420*c^3*(88-105*a*x)*(1-a^2*x^2)^(3/2)/a^4-1/8*c^3*arcsin(a*x)/a^4-1/8*c^3*x*sqrt(1-a^2*x^2)/a^3],
[exp(arctanh(a*x))*x^2*(c-a*c*x)^3,x,6,2/5*c^3*x^2*(1-a^2*x^2)^(3/2)/a-1/6*c^3*x^3*(1-a^2*x^2)^(3/2)+1/120*c^3*(32-45*a*x)*(1-a^2*x^2)^(3/2)/a^3+3/16*c^3*arcsin(a*x)/a^3+3/16*c^3*x*sqrt(1-a^2*x^2)/a^2],
[exp(arctanh(a*x))*x*(c-a*c*x)^3,x,5,-1/5*c^3*x^2*(1-a^2*x^2)^(3/2)-1/30*c^3*(14-15*a*x)*(1-a^2*x^2)^(3/2)/a^2-1/4*c^3*arcsin(a*x)/a^2-1/4*c^3*x*sqrt(1-a^2*x^2)/a],
[exp(arctanh(a*x))*(c-a*c*x)^3,x,5,5/12*c^3*(1-a^2*x^2)^(3/2)/a+1/4*c^3*(1-a*x)*(1-a^2*x^2)^(3/2)/a+5/8*c^3*arcsin(a*x)/a+5/8*c^3*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^3/x,x,8,-1/3*c^3*(1-a^2*x^2)^(3/2)-c^3*arcsin(a*x)-c^3*arctanh(sqrt(1-a^2*x^2))+c^3*(1-a*x)*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^3/x^2,x,8,-c^3*(1-a^2*x^2)^(3/2)/x-1/2*a*c^3*arcsin(a*x)+2*a*c^3*arctanh(sqrt(1-a^2*x^2))-1/2*a*c^3*(4+a*x)*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^3/x^3,x,8,-1/2*c^3*(1-a^2*x^2)^(3/2)/x^2+2*a^2*c^3*arcsin(a*x)-1/2*a^2*c^3*arctanh(sqrt(1-a^2*x^2))+1/2*a*c^3*(4+a*x)*sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))*(c-a*c*x)^3/x^4,x,8,-1/3*c^3*(1-a^2*x^2)^(3/2)/x^3-a^3*c^3*arcsin(a*x)-a^3*c^3*arctanh(sqrt(1-a^2*x^2))+a*c^3*(1-a*x)*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^3/x^5,x,7,-1/4*c^3*(1-a^2*x^2)^(3/2)/x^4+2/3*a*c^3*(1-a^2*x^2)^(3/2)/x^3+5/8*a^4*c^3*arctanh(sqrt(1-a^2*x^2))-5/8*a^2*c^3*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^3/x^6,x,8,-1/5*c^3*(1-a^2*x^2)^(3/2)/x^5+1/2*a*c^3*(1-a^2*x^2)^(3/2)/x^4-7/15*a^2*c^3*(1-a^2*x^2)^(3/2)/x^3-1/4*a^5*c^3*arctanh(sqrt(1-a^2*x^2))+1/4*a^3*c^3*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*x^3*(c-a*c*x)^4,x,8,-19/35*c^4*x^2*(1-a^2*x^2)^(3/2)/a^2+29/48*c^4*x^3*(1-a^2*x^2)^(3/2)/a-3/7*c^4*x^4*(1-a^2*x^2)^(3/2)+1/8*a*c^4*x^5*(1-a^2*x^2)^(3/2)-1/6720*c^4*(2432-3045*a*x)*(1-a^2*x^2)^(3/2)/a^4-29/128*c^4*arcsin(a*x)/a^4-29/128*c^4*x*sqrt(1-a^2*x^2)/a^3],
[exp(arctanh(a*x))*x^2*(c-a*c*x)^4,x,7,5/7*c^4*x^2*(1-a^2*x^2)^(3/2)/a-1/2*c^4*x^3*(1-a^2*x^2)^(3/2)+1/7*a*c^4*x^4*(1-a^2*x^2)^(3/2)+5/168*c^4*(16-21*a*x)*(1-a^2*x^2)^(3/2)/a^3+5/16*c^4*arcsin(a*x)/a^3+5/16*c^4*x*sqrt(1-a^2*x^2)/a^2],
[exp(arctanh(a*x))*x*(c-a*c*x)^4,x,7,-7/24*c^4*(1-a^2*x^2)^(3/2)/a^2-7/40*c^4*(1-a*x)*(1-a^2*x^2)^(3/2)/a^2-1/10*c^4*(1-a*x)^2*(1-a^2*x^2)^(3/2)/a^2-1/6*c^4*(1-a*x)^3*(1-a^2*x^2)^(3/2)/a^2-7/16*c^4*arcsin(a*x)/a^2-7/16*c^4*x*sqrt(1-a^2*x^2)/a],
[exp(arctanh(a*x))*(c-a*c*x)^4,x,6,7/12*c^4*(1-a^2*x^2)^(3/2)/a+7/20*c^4*(1-a*x)*(1-a^2*x^2)^(3/2)/a+1/5*c^4*(1-a*x)^2*(1-a^2*x^2)^(3/2)/a+7/8*c^4*arcsin(a*x)/a+7/8*c^4*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^4/x,x,9,-c^4*(1-a^2*x^2)^(3/2)+1/4*a*c^4*x*(1-a^2*x^2)^(3/2)-13/8*c^4*arcsin(a*x)-c^4*arctanh(sqrt(1-a^2*x^2))+1/8*c^4*(8-13*a*x)*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^4/x^2,x,9,1/3*a*c^4*(1-a^2*x^2)^(3/2)-c^4*(1-a^2*x^2)^(3/2)/x+1/2*a*c^4*arcsin(a*x)+3*a*c^4*arctanh(sqrt(1-a^2*x^2))-1/2*a*c^4*(6-a*x)*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^4/x^3,x,9,-1/2*c^4*(1-a^2*x^2)^(3/2)/x^2+3*a*c^4*(1-a^2*x^2)^(3/2)/x+5/2*a^2*c^4*arcsin(a*x)-5/2*a^2*c^4*arctanh(sqrt(1-a^2*x^2))+5/2*a^2*c^4*(1+a*x)*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a*c*x)^4/x^4,x,9,-1/3*c^4*(1-a^2*x^2)^(3/2)/x^3+3/2*a*c^4*(1-a^2*x^2)^(3/2)/x^2-3*a^3*c^4*arcsin(a*x)-1/2*a^3*c^4*arctanh(sqrt(1-a^2*x^2))-1/2*a^2*c^4*(6-a*x)*sqrt(1-a^2*x^2)/x],
[exp(arctanh(a*x))*(c-a*c*x)^4/x^5,x,9,a^4*c^4*arcsin(a*x)+13/8*a^4*c^4*arctanh(sqrt(1-a^2*x^2))-1/4*c^4*sqrt(1-a^2*x^2)/x^4+a*c^4*sqrt(1-a^2*x^2)/x^3-11/8*a^2*c^4*sqrt(1-a^2*x^2)/x^2,-1/4*c^4*(1-a^2*x^2)^(3/2)/x^4+a*c^4*(1-a^2*x^2)^(3/2)/x^3+a^4*c^4*arcsin(a*x)+13/8*a^4*c^4*arctanh(sqrt(1-a^2*x^2))-1/8*a^2*c^4*(13-8*a*x)*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^4/x^6,x,8,-1/5*c^4*(1-a^2*x^2)^(3/2)/x^5+3/4*a*c^4*(1-a^2*x^2)^(3/2)/x^4-17/15*a^2*c^4*(1-a^2*x^2)^(3/2)/x^3-7/8*a^5*c^4*arctanh(sqrt(1-a^2*x^2))+7/8*a^3*c^4*sqrt(1-a^2*x^2)/x^2],
[exp(arctanh(a*x))*(c-a*c*x)^4/x^7,x,9,-1/6*c^4*(1-a^2*x^2)^(3/2)/x^6+3/5*a*c^4*(1-a^2*x^2)^(3/2)/x^5-7/8*a^2*c^4*(1-a^2*x^2)^(3/2)/x^4+11/15*a^3*c^4*(1-a^2*x^2)^(3/2)/x^3+7/16*a^6*c^4*arctanh(sqrt(1-a^2*x^2))-7/16*a^4*c^4*sqrt(1-a^2*x^2)/x^2],

# p<0
[exp(arctanh(a*x))*x^4/(c-a*c*x),x,8,-27/8*arcsin(a*x)/(a^5*c)+(1+a*x)^2/(a^5*c*sqrt(1-a^2*x^2))+13/3*sqrt(1-a^2*x^2)/(a^5*c)+11/8*x*sqrt(1-a^2*x^2)/(a^4*c)+2/3*x^2*sqrt(1-a^2*x^2)/(a^3*c)+1/4*x^3*sqrt(1-a^2*x^2)/(a^2*c)],
[exp(arctanh(a*x))*x^3/(c-a*c*x),x,7,-3*arcsin(a*x)/(a^4*c)+(1+a*x)^2/(a^4*c*sqrt(1-a^2*x^2))+11/3*sqrt(1-a^2*x^2)/(a^4*c)+x*sqrt(1-a^2*x^2)/(a^3*c)+1/3*x^2*sqrt(1-a^2*x^2)/(a^2*c)],
[exp(arctanh(a*x))*x^2/(c-a*c*x),x,5,-5/2*arcsin(a*x)/(a^3*c)+(1+a*x)^2/(a^3*c*sqrt(1-a^2*x^2))+1/2*(6+a*x)*sqrt(1-a^2*x^2)/(a^3*c)],
[exp(arctanh(a*x))*x/(c-a*c*x),x,4,(1-a^2*x^2)^(3/2)/(a^2*c*(1-a*x)^2)-2*arcsin(a*x)/(a^2*c)+2*sqrt(1-a^2*x^2)/(a^2*c)],
[exp(arctanh(a*x))/(c-a*c*x),x,3,-arcsin(a*x)/(a*c)+2*sqrt(1-a^2*x^2)/(a*c*(1-a*x))],
[exp(arctanh(a*x))/(x*(c-a*c*x)),x,7,-arctanh(sqrt(1-a^2*x^2))/c+2*(1+a*x)/(c*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a*c*x)),x,7,-2*a*arctanh(sqrt(1-a^2*x^2))/c+2*a*(1+a*x)/(c*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(c*x)],
[exp(arctanh(a*x))/(x^3*(c-a*c*x)),x,8,-5/2*a^2*arctanh(sqrt(1-a^2*x^2))/c+2*a^2*(1+a*x)/(c*sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/(c*x^2)-2*a*sqrt(1-a^2*x^2)/(c*x)],
[exp(arctanh(a*x))/(x^4*(c-a*c*x)),x,9,-3*a^3*arctanh(sqrt(1-a^2*x^2))/c+2*a^3*(1+a*x)/(c*sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/(c*x^3)-a*sqrt(1-a^2*x^2)/(c*x^2)-8/3*a^2*sqrt(1-a^2*x^2)/(c*x)],
[exp(arctanh(a*x))*x^4/(c-a*c*x)^2,x,10,1/3*(1+a*x)^3/(a^5*c^2*(1-a^2*x^2)^(3/2))+17/2*arcsin(a*x)/(a^5*c^2)-2*(1+a*x)^3/(a^5*c^2*sqrt(1-a^2*x^2))-5/2*sqrt(1-a^2*x^2)/(a^5*c^2)-1/6*(5+a*x)*sqrt(1-a^2*x^2)/(a^5*c^2)-1/3*(5+a*x)^2*sqrt(1-a^2*x^2)/(a^5*c^2)],
[exp(arctanh(a*x))*x^3/(c-a*c*x)^2,x,6,1/3*(1+a*x)^3/(a^4*c^2*(1-a^2*x^2)^(3/2))+11/2*arcsin(a*x)/(a^4*c^2)-3*(1+a*x)^2/(a^4*c^2*sqrt(1-a^2*x^2))-1/2*(12+a*x)*sqrt(1-a^2*x^2)/(a^4*c^2)],
[exp(arctanh(a*x))*x^2/(c-a*c*x)^2,x,5,1/3*(1-a^2*x^2)^(3/2)/(a^3*c^2*(1-a*x)^3)+(1-a^2*x^2)^(3/2)/(a^3*c^2*(1-a*x)^2)+3*arcsin(a*x)/(a^3*c^2)-6*sqrt(1-a^2*x^2)/(a^3*c^2*(1-a*x))],
[exp(arctanh(a*x))*x/(c-a*c*x)^2,x,4,1/3*(1-a^2*x^2)^(3/2)/(a^2*c^2*(1-a*x)^3)+arcsin(a*x)/(a^2*c^2)-2*sqrt(1-a^2*x^2)/(a^2*c^2*(1-a*x))],
[exp(arctanh(a*x))/(c-a*c*x)^2,x,2,1/3*(1-a^2*x^2)^(3/2)/(a*c^2*(1-a*x)^3)],
[exp(arctanh(a*x))/(x*(c-a*c*x)^2),x,8,4/3*(1+a*x)/(c^2*(1-a^2*x^2)^(3/2))-arctanh(sqrt(1-a^2*x^2))/c^2+1/3*(3+5*a*x)/(c^2*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a*c*x)^2),x,8,4/3*a*(1+a*x)/(c^2*(1-a^2*x^2)^(3/2))-3*a*arctanh(sqrt(1-a^2*x^2))/c^2+1/3*a*(9+11*a*x)/(c^2*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(c^2*x)],
[exp(arctanh(a*x))/(x^3*(c-a*c*x)^2),x,9,4/3*a^2*(1+a*x)/(c^2*(1-a^2*x^2)^(3/2))-11/2*a^2*arctanh(sqrt(1-a^2*x^2))/c^2+1/3*a^2*(15+17*a*x)/(c^2*sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/(c^2*x^2)-3*a*sqrt(1-a^2*x^2)/(c^2*x)],
[exp(arctanh(a*x))/(x^4*(c-a*c*x)^2),x,10,4/3*a^3*(1+a*x)/(c^2*(1-a^2*x^2)^(3/2))-17/2*a^3*arctanh(sqrt(1-a^2*x^2))/c^2+1/3*a^3*(21+23*a*x)/(c^2*sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/(c^2*x^3)-3/2*a*sqrt(1-a^2*x^2)/(c^2*x^2)-17/3*a^2*sqrt(1-a^2*x^2)/(c^2*x)],
[exp(arctanh(a*x))*x^4/(c-a*c*x)^3,x,7,1/5*(1+a*x)^4/(a^5*c^3*(1-a^2*x^2)^(5/2))-19/15*(1+a*x)^3/(a^5*c^3*(1-a^2*x^2)^(3/2))-19/2*arcsin(a*x)/(a^5*c^3)+6*(1+a*x)^2/(a^5*c^3*sqrt(1-a^2*x^2))+1/2*(20+a*x)*sqrt(1-a^2*x^2)/(a^5*c^3)],
[exp(arctanh(a*x))*x^3/(c-a*c*x)^3,x,9,1/5*(1-a^2*x^2)^(3/2)/(a^4*c^3*(1-a*x)^4)-14/15*(1-a^2*x^2)^(3/2)/(a^4*c^3*(1-a*x)^3)-(1-a^2*x^2)^(3/2)/(a^4*c^3*(1-a*x)^2)-4*arcsin(a*x)/(a^4*c^3)+8*sqrt(1-a^2*x^2)/(a^4*c^3*(1-a*x))],
[exp(arctanh(a*x))*x^2/(c-a*c*x)^3,x,8,1/5*(1-a^2*x^2)^(3/2)/(a^3*c^3*(1-a*x)^4)-3/5*(1-a^2*x^2)^(3/2)/(a^3*c^3*(1-a*x)^3)-arcsin(a*x)/(a^3*c^3)+2*sqrt(1-a^2*x^2)/(a^3*c^3*(1-a*x))],
[exp(arctanh(a*x))*x/(c-a*c*x)^3,x,3,1/5*(1-a^2*x^2)^(3/2)/(a^2*c^3*(1-a*x)^4)-4/15*(1-a^2*x^2)^(3/2)/(a^2*c^3*(1-a*x)^3)],
[exp(arctanh(a*x))/(c-a*c*x)^3,x,3,1/5*(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^4)+1/15*(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^3)],
[exp(arctanh(a*x))/(x*(c-a*c*x)^3),x,9,8/5*(1+a*x)/(c^3*(1-a^2*x^2)^(5/2))+4/5*a*x/(c^3*(1-a^2*x^2)^(3/2))-arctanh(sqrt(1-a^2*x^2))/c^3+1/5*(5+8*a*x)/(c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a*c*x)^3),x,9,8/5*a*(1+a*x)/(c^3*(1-a^2*x^2)^(5/2))+4/15*a*(5+8*a*x)/(c^3*(1-a^2*x^2)^(3/2))-4*a*arctanh(sqrt(1-a^2*x^2))/c^3+1/15*a*(60+79*a*x)/(c^3*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(c^3*x)],
[exp(arctanh(a*x))/(x^3*(c-a*c*x)^3),x,10,8/5*a^2*(1+a*x)/(c^3*(1-a^2*x^2)^(5/2))+4/15*a^2*(10+13*a*x)/(c^3*(1-a^2*x^2)^(3/2))-19/2*a^2*arctanh(sqrt(1-a^2*x^2))/c^3+1/15*a^2*(135+164*a*x)/(c^3*sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/(c^3*x^2)-4*a*sqrt(1-a^2*x^2)/(c^3*x)],
[exp(arctanh(a*x))/(x^4*(c-a*c*x)^3),x,11,8/5*a^3*(1+a*x)/(c^3*(1-a^2*x^2)^(5/2))+4/5*a^3*(5+6*a*x)/(c^3*(1-a^2*x^2)^(3/2))-18*a^3*arctanh(sqrt(1-a^2*x^2))/c^3+1/5*a^3*(80+93*a*x)/(c^3*sqrt(1-a^2*x^2))-1/3*sqrt(1-a^2*x^2)/(c^3*x^3)-2*a*sqrt(1-a^2*x^2)/(c^3*x^2)-29/3*a^2*sqrt(1-a^2*x^2)/(c^3*x)],
[exp(arctanh(a*x))*x^5/(c-a*c*x)^4,x,8,1/7*(1+a*x)^5/(a^6*c^4*(1-a^2*x^2)^(7/2))-33/35*(1+a*x)^4/(a^6*c^4*(1-a^2*x^2)^(5/2))+317/105*(1+a*x)^3/(a^6*c^4*(1-a^2*x^2)^(3/2))+29/2*arcsin(a*x)/(a^6*c^4)-10*(1+a*x)^2/(a^6*c^4*sqrt(1-a^2*x^2))-1/2*(30+a*x)*sqrt(1-a^2*x^2)/(a^6*c^4)],
[exp(arctanh(a*x))*x^4/(c-a*c*x)^4,x,12,1/7*(1-a^2*x^2)^(3/2)/(a^5*c^4*(1-a*x)^5)-26/35*(1-a^2*x^2)^(3/2)/(a^5*c^4*(1-a*x)^4)+184/105*(1-a^2*x^2)^(3/2)/(a^5*c^4*(1-a*x)^3)+(1-a^2*x^2)^(3/2)/(a^5*c^4*(1-a*x)^2)+5*arcsin(a*x)/(a^5*c^4)-10*sqrt(1-a^2*x^2)/(a^5*c^4*(1-a*x))],
[exp(arctanh(a*x))*x^3/(c-a*c*x)^4,x,11,1/7*(1-a^2*x^2)^(3/2)/(a^4*c^4*(1-a*x)^5)-19/35*(1-a^2*x^2)^(3/2)/(a^4*c^4*(1-a*x)^4)+86/105*(1-a^2*x^2)^(3/2)/(a^4*c^4*(1-a*x)^3)+arcsin(a*x)/(a^4*c^4)-2*sqrt(1-a^2*x^2)/(a^4*c^4*(1-a*x))],
[exp(arctanh(a*x))*x^2/(c-a*c*x)^4,x,5,1/7*(1-a^2*x^2)^(3/2)/(a^3*c^4*(1-a*x)^5)-12/35*(1-a^2*x^2)^(3/2)/(a^3*c^4*(1-a*x)^4)+23/105*(1-a^2*x^2)^(3/2)/(a^3*c^4*(1-a*x)^3)],
[exp(arctanh(a*x))*x/(c-a*c*x)^4,x,4,1/7*(1-a^2*x^2)^(3/2)/(a^2*c^4*(1-a*x)^5)-1/7*(1-a^2*x^2)^(3/2)/(a^2*c^4*(1-a*x)^4)-1/21*(1-a^2*x^2)^(3/2)/(a^2*c^4*(1-a*x)^3)],
[exp(arctanh(a*x))/(c-a*c*x)^4,x,4,1/7*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^5)+2/35*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^4)+2/105*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^3)],
[exp(arctanh(a*x))/(x*(c-a*c*x)^4),x,10,16/7*(1+a*x)/(c^4*(1-a^2*x^2)^(7/2))-4/35*(7-3*a*x)/(c^4*(1-a^2*x^2)^(5/2))+1/105*(35+83*a*x)/(c^4*(1-a^2*x^2)^(3/2))-arctanh(sqrt(1-a^2*x^2))/c^4+1/105*(105+166*a*x)/(c^4*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a*c*x)^4),x,10,16/7*a*(1+a*x)/(c^4*(1-a^2*x^2)^(7/2))+4/35*a*(7+17*a*x)/(c^4*(1-a^2*x^2)^(5/2))+1/105*a*(175+307*a*x)/(c^4*(1-a^2*x^2)^(3/2))-5*a*arctanh(sqrt(1-a^2*x^2))/c^4+1/105*a*(525+719*a*x)/(c^4*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(c^4*x)],
[exp(arctanh(a*x))/(x^3*(c-a*c*x)^4),x,11,16/7*a^2*(1+a*x)/(c^4*(1-a^2*x^2)^(7/2))+4/35*a^2*(21+31*a*x)/(c^4*(1-a^2*x^2)^(5/2))+1/105*a^2*(455+671*a*x)/(c^4*(1-a^2*x^2)^(3/2))-29/2*a^2*arctanh(sqrt(1-a^2*x^2))/c^4+1/105*a^2*(1470+1867*a*x)/(c^4*sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/(c^4*x^2)-5*a*sqrt(1-a^2*x^2)/(c^4*x)],
[exp(arctanh(x))*x*(1+x),x,6,arcsin(x)-1/3*(1+x)^(3/2)*sqrt(1-x)-1/3*(1+x)^(5/2)*sqrt(1-x)-sqrt(1-x)*sqrt(1+x)],
[exp(arctanh(x))*(1+x),x,5,3/2*arcsin(x)-1/2*(1+x)^(3/2)*sqrt(1-x)-3/2*sqrt(1-x)*sqrt(1+x)],
[exp(arctanh(x))*x*(1+x)^2,x,7,15/8*arcsin(x)-5/8*(1+x)^(3/2)*sqrt(1-x)-1/4*(1+x)^(5/2)*sqrt(1-x)-1/4*(1+x)^(7/2)*sqrt(1-x)-15/8*sqrt(1-x)*sqrt(1+x)],
[exp(arctanh(x))*(1+x)^2,x,6,5/2*arcsin(x)-5/6*(1+x)^(3/2)*sqrt(1-x)-1/3*(1+x)^(5/2)*sqrt(1-x)-5/2*sqrt(1-x)*sqrt(1+x)],
[exp(arctanh(x))*x/(1+x),x,2,-sqrt(1-x)*sqrt(1+x)],
[exp(arctanh(x))/(1+x),x,3,arcsin(x)],
[exp(arctanh(x))*x/(1+x)^2,x,4,arcsin(x)+sqrt(1-x)/sqrt(1+x)],
[exp(arctanh(x))/(1+x)^2,x,2,-sqrt(1-x)/sqrt(1+x)],

# p/2>0
[exp(arctanh(x))*x*(1+x)^(3/2),x,3,16/3*(1-x)^(3/2)-2*(1-x)^(5/2)+2/7*(1-x)^(7/2)-8*sqrt(1-x)],
[exp(arctanh(x))*(1+x)^(3/2),x,3,8/3*(1-x)^(3/2)-2/5*(1-x)^(5/2)-8*sqrt(1-x)],
[exp(arctanh(x))*(1-x)^(3/2)*x,x,5,-4/3*(1+x)^(3/2)+6/5*(1+x)^(5/2)-2/7*(1+x)^(7/2),-4/21*(1+x)^(3/2)+2/35*(1+x)^(5/2)-2/7*(1-x^2)^(3/2)*sqrt(1-x)],
[exp(arctanh(x))*(1-x)^(3/2),x,4,4/3*(1+x)^(3/2)-2/5*(1+x)^(5/2)],
[exp(arctanh(x))*x*(1+x)^(1/2),x,3,2*(1-x)^(3/2)-2/5*(1-x)^(5/2)-4*sqrt(1-x)],
[exp(arctanh(x))*(1+x)^(1/2),x,3,2/3*(1-x)^(3/2)-4*sqrt(1-x)],
[exp(arctanh(x))*(1-x)^(1/2)*x,x,4,-2/3*(1+x)^(3/2)+2/5*(1+x)^(5/2)],
[exp(arctanh(x))*(1-x)^(1/2),x,3,2/3*(1+x)^(3/2)],

# p/2<0
[exp(arctanh(x))*x/(1+x)^(1/2),x,3,2/3*(1-x)^(3/2)-2*sqrt(1-x)],
[exp(arctanh(x))/(1+x)^(1/2),x,2,-2*sqrt(1-x)],
[exp(arctanh(x))*x/(1-x)^(1/2),x,5,-2/3*(1+x)^(3/2)+2*arctanh(sqrt(1+x)/sqrt(2))*sqrt(2)-2*sqrt(1+x)],
[exp(arctanh(x))/(1-x)^(1/2),x,5,2*arctanh(sqrt(1+x)/sqrt(2))*sqrt(2)-2*sqrt(1+x)],
[exp(arctanh(x))*x/(1+x)^(3/2),x,4,arctanh(sqrt(1-x)/sqrt(2))*sqrt(2)-2*sqrt(1-x)],
[exp(arctanh(x))/(1+x)^(3/2),x,3,-arctanh(sqrt(1-x)/sqrt(2))*sqrt(2)],
[exp(arctanh(x))*x/(1-x)^(3/2),x,5,1/2*(1+x)^(3/2)/(1-x)-5*arctanh(sqrt(1+x)/sqrt(2))/sqrt(2)+5/2*sqrt(1+x)],
[exp(arctanh(x))/(1-x)^(3/2),x,5,-arctanh(sqrt(1+x)/sqrt(2))/sqrt(2)+sqrt(1+x)/(1-x)],

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-c a x)^(p/2)

# n>0
[exp(arctanh(a*x))*x^m*sqrt(c-a*c*x),x,4,2/3*c*x^m*(1+a*x)*hypergeom([3/2,-m],[5/2],1+a*x)*sqrt(1-a^2*x^2)/(a*(-a*x)^m*sqrt(c-a*c*x))],
[exp(arctanh(a*x))*x^2*sqrt(c-a*c*x),x,4,-8/105*c^2*(1-a^2*x^2)^(3/2)/(a^3*(c-a*c*x)^(3/2))+2/7*c^2*x^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+8/35*c*(1-a^2*x^2)^(3/2)/(a^3*sqrt(c-a*c*x))],
[exp(arctanh(a*x))*x*sqrt(c-a*c*x),x,3,2/15*c^2*(1-a^2*x^2)^(3/2)/(a^2*(c-a*c*x)^(3/2))-2/5*c*(1-a^2*x^2)^(3/2)/(a^2*sqrt(c-a*c*x))],
[exp(arctanh(a*x))*sqrt(c-a*c*x),x,2,2/3*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))],
[exp(arctanh(a*x))*sqrt(c-a*c*x)/x,x,4,-2*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)+2*c*sqrt(1-a^2*x^2)/sqrt(c-a*c*x)],
[exp(arctanh(a*x))*sqrt(c-a*c*x)/x^2,x,4,-a*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)-c*sqrt(1-a^2*x^2)/(x*sqrt(c-a*c*x))],
[exp(2*arctanh(a*x))*x^3*sqrt(c-a*c*x),x,4,14/3*(c-a*c*x)^(3/2)/(a^4*c)-18/5*(c-a*c*x)^(5/2)/(a^4*c^2)+10/7*(c-a*c*x)^(7/2)/(a^4*c^3)-2/9*(c-a*c*x)^(9/2)/(a^4*c^4)-4*sqrt(c-a*c*x)/a^4],
[exp(2*arctanh(a*x))*x^2*sqrt(c-a*c*x),x,4,10/3*(c-a*c*x)^(3/2)/(a^3*c)-8/5*(c-a*c*x)^(5/2)/(a^3*c^2)+2/7*(c-a*c*x)^(7/2)/(a^3*c^3)-4*sqrt(c-a*c*x)/a^3],
[exp(2*arctanh(a*x))*x*sqrt(c-a*c*x),x,4,2*(c-a*c*x)^(3/2)/(a^2*c)-2/5*(c-a*c*x)^(5/2)/(a^2*c^2)-4*sqrt(c-a*c*x)/a^2],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x),x,4,2/3*(c-a*c*x)^(3/2)/(a*c)-4*sqrt(c-a*c*x)/a],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x)/x,x,5,-2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-2*sqrt(c-a*c*x)],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x)/x^2,x,5,-3*a*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-sqrt(c-a*c*x)/x],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x)/x^3,x,6,-7/4*a^2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-1/2*sqrt(c-a*c*x)/x^2-7/4*a*sqrt(c-a*c*x)/x],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x)/x^4,x,7,-11/8*a^3*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-1/3*sqrt(c-a*c*x)/x^3-11/12*a*sqrt(c-a*c*x)/x^2-11/8*a^2*sqrt(c-a*c*x)/x],
[exp(2*arctanh(a*x))*sqrt(c-a*c*x)/x^5,x,8,-75/64*a^4*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-1/4*sqrt(c-a*c*x)/x^4-5/8*a*sqrt(c-a*c*x)/x^3-25/32*a^2*sqrt(c-a*c*x)/x^2-75/64*a^3*sqrt(c-a*c*x)/x],
[exp(3*arctanh(a*x))*x^3*sqrt(c-a*c*x),x,8,-2/3*(1+a*x)^(3/2)*(c-a*c*x)^(3/2)/(a^4*c*(1-a*x)^(3/2))-2/5*(1+a*x)^(5/2)*(c-a*c*x)^(3/2)/(a^4*c*(1-a*x)^(3/2))+2/7*(1+a*x)^(7/2)*(c-a*c*x)^(3/2)/(a^4*c*(1-a*x)^(3/2))-2/9*(1+a*x)^(9/2)*(c-a*c*x)^(3/2)/(a^4*c*(1-a*x)^(3/2))+4*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(a^4*c*(1-a*x)^(3/2))-4*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(a^4*c*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*x^2*sqrt(c-a*c*x),x,8,-2/3*(1+a*x)^(3/2)*(c-a*c*x)^(3/2)/(a^3*c*(1-a*x)^(3/2))-2/7*(1+a*x)^(7/2)*(c-a*c*x)^(3/2)/(a^3*c*(1-a*x)^(3/2))+4*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(a^3*c*(1-a*x)^(3/2))-4*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(a^3*c*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*x*sqrt(c-a*c*x),x,7,-2/3*(1+a*x)^(3/2)*(c-a*c*x)^(3/2)/(a^2*c*(1-a*x)^(3/2))-2/5*(1+a*x)^(5/2)*(c-a*c*x)^(3/2)/(a^2*c*(1-a*x)^(3/2))+4*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(a^2*c*(1-a*x)^(3/2))-4*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(a^2*c*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x),x,5,-2/3*c^2*(1-a^2*x^2)^(3/2)/(a*(c-a*c*x)^(3/2))+4*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/(sqrt(2)*sqrt(c-a*c*x)))*sqrt(2)*sqrt(c)/a-4*c*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x)/x,x,8,-2*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x))/(c*(1-a*x)^(3/2))+4*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(c*(1-a*x)^(3/2))-2*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x)/x^2,x,8,-5*a*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x))/(c*(1-a*x)^(3/2))+4*a*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(c*(1-a*x)^(3/2))-(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x)/x^3,x,9,-23/4*a^2*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x))/(c*(1-a*x)^(3/2))+4*a^2*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(c*(1-a*x)^(3/2))-1/2*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x^2*(1-a*x)^(3/2))-9/4*a*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x)/x^4,x,10,-45/8*a^3*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x))/(c*(1-a*x)^(3/2))+4*a^3*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(c*(1-a*x)^(3/2))-1/3*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x^3*(1-a*x)^(3/2))-13/12*a*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x^2*(1-a*x)^(3/2))-19/8*a^2*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x*(1-a*x)^(3/2))],
[exp(3*arctanh(a*x))*sqrt(c-a*c*x)/x^5,x,11,-363/64*a^4*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x))/(c*(1-a*x)^(3/2))+4*a^4*(c-a*c*x)^(3/2)*arctanh(sqrt(1+a*x)/sqrt(2))*sqrt(2)/(c*(1-a*x)^(3/2))-1/4*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x^4*(1-a*x)^(3/2))-17/24*a*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x^3*(1-a*x)^(3/2))-107/96*a^2*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x^2*(1-a*x)^(3/2))-149/64*a^3*(c-a*c*x)^(3/2)*sqrt(1+a*x)/(c*x*(1-a*x)^(3/2))],

# n<0
[x^m*sqrt(c-a*c*x)/exp(arctanh(a*x)),x,5,2*(5+4*m)*x^m*(1+a*x)*hypergeom([1/2,-m],[3/2],1+a*x)*sqrt(c-a*c*x)/(a*(3+2*m)*(-a*x)^m*sqrt(1-a^2*x^2))-2*c*x^(1+m)*sqrt(1-a^2*x^2)/((3+2*m)*sqrt(c-a*c*x))],
[x^2*sqrt(c-a*c*x)/exp(arctanh(a*x)),x,5,104/105*c*sqrt(1-a^2*x^2)/(a^3*sqrt(c-a*c*x))+26/35*c*x^2*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))-2/7*c*x^3*sqrt(1-a^2*x^2)/sqrt(c-a*c*x)+104/105*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a^3],
[x*sqrt(c-a*c*x)/exp(arctanh(a*x)),x,4,-2/5*(c-a*c*x)^(3/2)*sqrt(1-a^2*x^2)/(a^2*c)-8/5*c*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a*c*x))-2/5*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a^2],
[sqrt(c-a*c*x)/exp(arctanh(a*x)),x,3,8/3*c*sqrt(1-a^2*x^2)/(a*sqrt(c-a*c*x))+2/3*sqrt(c-a*c*x)*sqrt(1-a^2*x^2)/a],
[sqrt(c-a*c*x)/(exp(arctanh(a*x))*x),x,4,-2*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)-2*c*sqrt(1-a^2*x^2)/sqrt(c-a*c*x)],
[sqrt(c-a*c*x)/(exp(arctanh(a*x))*x^2),x,4,3*a*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)-c*sqrt(1-a^2*x^2)/(x*sqrt(c-a*c*x))],
[sqrt(c-a*c*x)/(exp(arctanh(a*x))*x^3),x,5,-7/4*a^2*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)-1/2*c*sqrt(1-a^2*x^2)/(x^2*sqrt(c-a*c*x))+7/4*a*c*sqrt(1-a^2*x^2)/(x*sqrt(c-a*c*x))],
[sqrt(c-a*c*x)/(exp(arctanh(a*x))*x^4),x,6,11/8*a^3*arctanh(sqrt(c)*sqrt(1-a^2*x^2)/sqrt(c-a*c*x))*sqrt(c)-1/3*c*sqrt(1-a^2*x^2)/(x^3*sqrt(c-a*c*x))+11/12*a*c*sqrt(1-a^2*x^2)/(x^2*sqrt(c-a*c*x))-11/8*a^2*c*sqrt(1-a^2*x^2)/(x*sqrt(c-a*c*x))],
[x^3*sqrt(c-a*c*x)/exp(2*arctanh(a*x)),x,8,-2/3*(c-a*c*x)^(3/2)/(a^4*c)-2/5*(c-a*c*x)^(5/2)/(a^4*c^2)+2/7*(c-a*c*x)^(7/2)/(a^4*c^3)-2/9*(c-a*c*x)^(9/2)/(a^4*c^4)+4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^4-4*sqrt(c-a*c*x)/a^4],
[x^2*sqrt(c-a*c*x)/exp(2*arctanh(a*x)),x,8,2/3*(c-a*c*x)^(3/2)/(a^3*c)+2/7*(c-a*c*x)^(7/2)/(a^3*c^3)-4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^3+4*sqrt(c-a*c*x)/a^3],
[x*sqrt(c-a*c*x)/exp(2*arctanh(a*x)),x,7,-2/3*(c-a*c*x)^(3/2)/(a^2*c)-2/5*(c-a*c*x)^(5/2)/(a^2*c^2)+4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^2-4*sqrt(c-a*c*x)/a^2],
[sqrt(c-a*c*x)/exp(2*arctanh(a*x)),x,6,2/3*(c-a*c*x)^(3/2)/(a*c)-4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a+4*sqrt(c-a*c*x)/a],
[sqrt(c-a*c*x)/(exp(2*arctanh(a*x))*x),x,8,-2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-2*sqrt(c-a*c*x)],
[sqrt(c-a*c*x)/(exp(2*arctanh(a*x))*x^2),x,8,5*a*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-4*a*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-sqrt(c-a*c*x)/x],
[sqrt(c-a*c*x)/(exp(2*arctanh(a*x))*x^3),x,9,-23/4*a^2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+4*a^2*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-1/2*sqrt(c-a*c*x)/x^2+9/4*a*sqrt(c-a*c*x)/x],
[sqrt(c-a*c*x)/(exp(2*arctanh(a*x))*x^4),x,10,45/8*a^3*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-4*a^3*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-1/3*sqrt(c-a*c*x)/x^3+13/12*a*sqrt(c-a*c*x)/x^2-19/8*a^2*sqrt(c-a*c*x)/x],
[sqrt(c-a*c*x)/(exp(2*arctanh(a*x))*x^5),x,11,-363/64*a^4*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+4*a^4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-1/4*sqrt(c-a*c*x)/x^4+17/24*a*sqrt(c-a*c*x)/x^3-107/96*a^2*sqrt(c-a*c*x)/x^2+149/64*a^3*sqrt(c-a*c*x)/x],
[x^3*sqrt(c-a*c*x)/exp(3*arctanh(a*x)),x,4,-50/3*c^2*(1-a*x)^(3/2)*(1+a*x)^(3/2)/(a^4*(c-a*c*x)^(3/2))+38/5*c^2*(1-a*x)^(3/2)*(1+a*x)^(5/2)/(a^4*(c-a*c*x)^(3/2))-2*c^2*(1-a*x)^(3/2)*(1+a*x)^(7/2)/(a^4*(c-a*c*x)^(3/2))+2/9*c^2*(1-a*x)^(3/2)*(1+a*x)^(9/2)/(a^4*(c-a*c*x)^(3/2))+8*c^2*(1-a*x)^(3/2)/(a^4*(c-a*c*x)^(3/2)*sqrt(1+a*x))+32*c^2*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^4*(c-a*c*x)^(3/2))],
[x^2*sqrt(c-a*c*x)/exp(3*arctanh(a*x)),x,4,26/3*c^2*(1-a*x)^(3/2)*(1+a*x)^(3/2)/(a^3*(c-a*c*x)^(3/2))-12/5*c^2*(1-a*x)^(3/2)*(1+a*x)^(5/2)/(a^3*(c-a*c*x)^(3/2))+2/7*c^2*(1-a*x)^(3/2)*(1+a*x)^(7/2)/(a^3*(c-a*c*x)^(3/2))-8*c^2*(1-a*x)^(3/2)/(a^3*(c-a*c*x)^(3/2)*sqrt(1+a*x))-24*c^2*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^3*(c-a*c*x)^(3/2))],
[x*sqrt(c-a*c*x)/exp(3*arctanh(a*x)),x,4,-10/3*c^2*(1-a*x)^(3/2)*(1+a*x)^(3/2)/(a^2*(c-a*c*x)^(3/2))+2/5*c^2*(1-a*x)^(3/2)*(1+a*x)^(5/2)/(a^2*(c-a*c*x)^(3/2))+8*c^2*(1-a*x)^(3/2)/(a^2*(c-a*c*x)^(3/2)*sqrt(1+a*x))+16*c^2*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-a*c*x)^(3/2))],
[sqrt(c-a*c*x)/exp(3*arctanh(a*x)),x,4,16/3*(c-a*c*x)^(3/2)/(a*c*sqrt(1-a^2*x^2))+2/3*(c-a*c*x)^(5/2)/(a*c^2*sqrt(1-a^2*x^2))-64/3*sqrt(c-a*c*x)/(a*sqrt(1-a^2*x^2))],
[sqrt(c-a*c*x)/(exp(3*arctanh(a*x))*x),x,6,-2*c^2*(1-a*x)^(3/2)*arctanh(sqrt(1+a*x))/(c-a*c*x)^(3/2)+8*c^2*(1-a*x)^(3/2)/((c-a*c*x)^(3/2)*sqrt(1+a*x))+2*c^2*(1-a*x)^(3/2)*sqrt(1+a*x)/(c-a*c*x)^(3/2)],
[sqrt(c-a*c*x)/(exp(3*arctanh(a*x))*x^2),x,6,7*a*c^2*(1-a*x)^(3/2)*arctanh(sqrt(1+a*x))/(c-a*c*x)^(3/2)-9*a*c^2*(1-a*x)^(3/2)/((c-a*c*x)^(3/2)*sqrt(1+a*x))-c^2*(1-a*x)^(3/2)/(x*(c-a*c*x)^(3/2)*sqrt(1+a*x))],
[sqrt(c-a*c*x)/(exp(3*arctanh(a*x))*x^3),x,7,-47/4*a^2*c^2*(1-a*x)^(3/2)*arctanh(sqrt(1+a*x))/(c-a*c*x)^(3/2)+47/4*a^2*c^2*(1-a*x)^(3/2)/((c-a*c*x)^(3/2)*sqrt(1+a*x))-1/2*c^2*(1-a*x)^(3/2)/(x^2*(c-a*c*x)^(3/2)*sqrt(1+a*x))+13/4*a*c^2*(1-a*x)^(3/2)/(x*(c-a*c*x)^(3/2)*sqrt(1+a*x))],
[sqrt(c-a*c*x)/(exp(3*arctanh(a*x))*x^4),x,8,119/8*a^3*c^2*(1-a*x)^(3/2)*arctanh(sqrt(1+a*x))/(c-a*c*x)^(3/2)-119/8*a^3*c^2*(1-a*x)^(3/2)/((c-a*c*x)^(3/2)*sqrt(1+a*x))-1/3*c^2*(1-a*x)^(3/2)/(x^3*(c-a*c*x)^(3/2)*sqrt(1+a*x))+19/12*a*c^2*(1-a*x)^(3/2)/(x^2*(c-a*c*x)^(3/2)*sqrt(1+a*x))-119/24*a^2*c^2*(1-a*x)^(3/2)/(x*(c-a*c*x)^(3/2)*sqrt(1+a*x))],
[sqrt(c-a*c*x)/(exp(3*arctanh(a*x))*x^5),x,9,-1115/64*a^4*c^2*(1-a*x)^(3/2)*arctanh(sqrt(1+a*x))/(c-a*c*x)^(3/2)+1115/64*a^4*c^2*(1-a*x)^(3/2)/((c-a*c*x)^(3/2)*sqrt(1+a*x))-1/4*c^2*(1-a*x)^(3/2)/(x^4*(c-a*c*x)^(3/2)*sqrt(1+a*x))+25/24*a*c^2*(1-a*x)^(3/2)/(x^3*(c-a*c*x)^(3/2)*sqrt(1+a*x))-223/96*a^2*c^2*(1-a*x)^(3/2)/(x^2*(c-a*c*x)^(3/2)*sqrt(1+a*x))+1115/192*a^3*c^2*(1-a*x)^(3/2)/(x*(c-a*c*x)^(3/2)*sqrt(1+a*x))],

# Integrands of the form E^(n ArcTanh[a x]) (c-c a x)^p with n symbolic
[(c-a*c*x)^p/exp(2*p*arctanh(a*x)),x,3,-(1-a*x)^p*(c-a*c*x)^(1+p)*hypergeom([p,1+2*p],[2*(1+p)],1/2*(1-a*x))/(2^p*a*c*(1+2*p))],
[exp(2*p*arctanh(a*x))*(c-a*c*x)^p,x,3,(1+a*x)^(1+p)*(c-a*c*x)^p/(a*(1+p)*(1-a*x)^p)],
[exp(n*arctanh(a*x))*(c-a*c*x)^p,x,3,-2^(1+1/2*n)*(c-a*c*x)^(1+p)*hypergeom([-1/2*n,1-1/2*n+p],[2-1/2*n+p],1/2*(1-a*x))/(a*c*(2-n+2*p)*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*(c-a*c*x)^3,x,2,-2^(1+1/2*n)*c^3*(1-a*x)^(4-1/2*n)*hypergeom([4-1/2*n,-1/2*n],[5-1/2*n],1/2*(1-a*x))/(a*(8-n))],
[exp(n*arctanh(a*x))*(c-a*c*x)^2,x,2,-2^(1+1/2*n)*c^2*(1-a*x)^(3-1/2*n)*hypergeom([3-1/2*n,-1/2*n],[4-1/2*n],1/2*(1-a*x))/(a*(6-n))],
[exp(n*arctanh(a*x))*(c-a*c*x),x,2,-2^(1+1/2*n)*c*(1-a*x)^(2-1/2*n)*hypergeom([2-1/2*n,-1/2*n],[3-1/2*n],1/2*(1-a*x))/(a*(4-n))],
[exp(n*arctanh(a*x))/(c-a*c*x),x,2,2^(1+1/2*n)*hypergeom([-1/2*n,-1/2*n],[1-1/2*n],1/2*(1-a*x))/(a*c*n*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(c-a*c*x)^2,x,2,(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(2+n))/(a*c^2*(2+n))],
[exp(n*arctanh(a*x))/(c-a*c*x)^3,x,3,(1-a*x)^(1/2*(-2-n))*(1+a*x)^(1/2*(2+n))/(a*c^3*(8+6*n+n^2))+(1-a*x)^(-2-1/2*n)*(1+a*x)^(1/2*(2+n))/(a*c^3*(4+n))],
[exp(n*arctanh(a*x))/(c-a*c*x)^4,x,4,2*(1-a*x)^(1/2*(-4-n))*(1+a*x)^(1/2*(2+n))/(a*c^4*(24+10*n+n^2))+2*(1-a*x)^(1/2*(-2-n))*(1+a*x)^(1/2*(2+n))/(a*c^4*(48+44*n+12*n^2+n^3))+(1-a*x)^(-3-1/2*n)*(1+a*x)^(1/2*(2+n))/(a*c^4*(6+n))],

# Integrands of the form u E^(n ArcTanh[a x]) (c-c/(a x))^p

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a x))^p

# n>0
[exp(arctanh(a*x))*(c-c/(a*x))^p,x,3,(c-c/(a*x))^p*x*AppellF1(1-p,1/2-p,-1/2,2-p,a*x,-a*x)/((1-p)*(1-a*x)^p)],
[exp(arctanh(a*x))*(c-c/(a*x))^4,x,10,-1/3*c^4*(1-a^2*x^2)^(3/2)/(a^4*x^3)+3/2*c^4*(1-a^2*x^2)^(3/2)/(a^3*x^2)-3*c^4*arcsin(a*x)/a-1/2*c^4*arctanh(sqrt(1-a^2*x^2))/a-1/2*c^4*(6-a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))*(c-c/(a*x))^3,x,9,1/2*c^3*(1-a^2*x^2)^(3/2)/(a^3*x^2)-2*c^3*arcsin(a*x)/a+1/2*c^3*arctanh(sqrt(1-a^2*x^2))/a-1/2*c^3*(4+a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))*(c-c/(a*x))^2,x,8,-c^2*arcsin(a*x)/a+c^2*arctanh(sqrt(1-a^2*x^2))/a-c^2*(1+a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))*(c-c/(a*x)),x,6,c*arctanh(sqrt(1-a^2*x^2))/a-c*sqrt(1-a^2*x^2)/a],
[exp(arctanh(a*x))/(c-c/(a*x)),x,5,-(1-a^2*x^2)^(3/2)/(a*c*(1-a*x)^2)+2*arcsin(a*x)/(a*c)-2*sqrt(1-a^2*x^2)/(a*c)],
[exp(arctanh(a*x))/(c-c/(a*x))^2,x,6,1/3*(1-a^2*x^2)^(3/2)/(a*c^2*(1-a*x)^3)+(1-a^2*x^2)^(3/2)/(a*c^2*(1-a*x)^2)+3*arcsin(a*x)/(a*c^2)-6*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x))],
[exp(arctanh(a*x))/(c-c/(a*x))^3,x,10,-1/5*(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^4)+14/15*(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^3)+(1-a^2*x^2)^(3/2)/(a*c^3*(1-a*x)^2)+4*arcsin(a*x)/(a*c^3)-8*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x))],
[exp(arctanh(a*x))/(c-c/(a*x))^4,x,13,1/7*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^5)-26/35*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^4)+184/105*(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^3)+(1-a^2*x^2)^(3/2)/(a*c^4*(1-a*x)^2)+5*arcsin(a*x)/(a*c^4)-10*sqrt(1-a^2*x^2)/(a*c^4*(1-a*x))],
[exp(2*arctanh(a*x))*(c-c/(a*x))^p,x,6,-(c-c/(a*x))^p*x-(2-p)*(c-c/(a*x))^p*hypergeom([1,p],[1+p],1+(-1)/(a*x))/(a*p)],
[exp(2*arctanh(a*x))*(c-c/(a*x))^5,x,4,1/4*c^5/(a^5*x^4)-c^5/(a^4*x^3)+c^5/(a^3*x^2)+2*c^5/(a^2*x)-c^5*x+3*c^5*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a*x))^4,x,4,-1/3*c^4/(a^4*x^3)+c^4/(a^3*x^2)-c^4*x+2*c^4*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a*x))^3,x,4,1/2*c^3/(a^3*x^2)-c^3/(a^2*x)-c^3*x+c^3*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a*x))^2,x,5,-c^2/(a^2*x)-c^2*x],
[exp(2*arctanh(a*x))*(c-c/(a*x)),x,4,-c*x-c*log(x)/a],
[exp(2*arctanh(a*x))/(c-c/(a*x)),x,4,-x/c+(-2)/(a*c*(1-a*x))-3*log(1-a*x)/(a*c)],
[exp(2*arctanh(a*x))/(c-c/(a*x))^2,x,4,-x/c^2+1/(a*c^2*(1-a*x)^2)+(-5)/(a*c^2*(1-a*x))-4*log(1-a*x)/(a*c^2)],
[exp(2*arctanh(a*x))/(c-c/(a*x))^3,x,4,-x/c^3+(-2/3)/(a*c^3*(1-a*x)^3)+7/2/(a*c^3*(1-a*x)^2)+(-9)/(a*c^3*(1-a*x))-5*log(1-a*x)/(a*c^3)],
[exp(2*arctanh(a*x))/(c-c/(a*x))^4,x,4,-x/c^4+1/2/(a*c^4*(1-a*x)^4)+(-3)/(a*c^4*(1-a*x)^3)+8/(a*c^4*(1-a*x)^2)+(-14)/(a*c^4*(1-a*x))-6*log(1-a*x)/(a*c^4)],
[exp(3*arctanh(a*x))*(c-c/(a*x))^4,x,9,-1/6*c^4*(2-3*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+c^4*arcsin(a*x)/a-3/2*c^4*arctanh(sqrt(1-a^2*x^2))/a+1/2*c^4*(2+3*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(3*arctanh(a*x))*(c-c/(a*x))^3,x,7,1/2*c^3*(1-a^2*x^2)^(3/2)/(a^3*x^2)-3/2*c^3*arctanh(sqrt(1-a^2*x^2))/a+3/2*c^3*sqrt(1-a^2*x^2)/a],
[exp(3*arctanh(a*x))*(c-c/(a*x))^2,x,9,-c^2*arcsin(a*x)/a-c^2*arctanh(sqrt(1-a^2*x^2))/a-c^2*(1-a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(3*arctanh(a*x))*(c-c/(a*x)),x,9,-2*c*arcsin(a*x)/a+c*arctanh(sqrt(1-a^2*x^2))/a+c*sqrt(1-a^2*x^2)/a],
[exp(3*arctanh(a*x))/(c-c/(a*x)),x,6,8/3*(1-a^2*x^2)^(3/2)/(a*c*(1-a*x)^2)-1/3*(1-a^2*x^2)^(5/2)/(a*c*(1-a*x)^4)-4*arcsin(a*x)/(a*c)+4*sqrt(1-a^2*x^2)/(a*c)],
[exp(3*arctanh(a*x))/(c-c/(a*x))^2,x,9,1/5*(1+a*x)^5/(a*c^2*(1-a^2*x^2)^(5/2))-2/3*(1+a*x)^4/(a*c^2*(1-a^2*x^2)^(3/2))-5*arcsin(a*x)/(a*c^2)+10/3*(1+a*x)^2/(a*c^2*sqrt(1-a^2*x^2))+5*sqrt(1-a^2*x^2)/(a*c^2)],
[exp(3*arctanh(a*x))/(c-c/(a*x))^3,x,9,-1/7*(1+a*x)^6/(a*c^3*(1-a^2*x^2)^(7/2))+4/7*(1+a*x)^5/(a*c^3*(1-a^2*x^2)^(5/2))-(1+a*x)^4/(a*c^3*(1-a^2*x^2)^(3/2))-6*arcsin(a*x)/(a*c^3)+4*(1+a*x)^2/(a*c^3*sqrt(1-a^2*x^2))+6*sqrt(1-a^2*x^2)/(a*c^3)],
[exp(3*arctanh(a*x))/(c-c/(a*x))^4,x,10,1/9*(1+a*x)^7/(a*c^4*(1-a^2*x^2)^(9/2))-34/63*(1+a*x)^6/(a*c^4*(1-a^2*x^2)^(7/2))+344/315*(1+a*x)^5/(a*c^4*(1-a^2*x^2)^(5/2))-4/3*(1+a*x)^4/(a*c^4*(1-a^2*x^2)^(3/2))-7*arcsin(a*x)/(a*c^4)+14/3*(1+a*x)^2/(a*c^4*sqrt(1-a^2*x^2))+7*sqrt(1-a^2*x^2)/(a*c^4)],
[exp(4*arctanh(a*x))*(c-c/(a*x))^p,x,7,-c*(5-p)*(c-c/(a*x))^(-1+p)/(a*(1-p))+c*(c-c/(a*x))^(-1+p)*x+(4-p)*(c-c/(a*x))^p*hypergeom([1,p],[1+p],1+(-1)/(a*x))/(a*p)],
[exp(4*arctanh(a*x))*(c-c/(a*x))^5,x,4,1/4*c^5/(a^5*x^4)-1/3*c^5/(a^4*x^3)-c^5/(a^3*x^2)+2*c^5/(a^2*x)+c^5*x-c^5*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a*x))^4,x,5,-1/3*c^4/(a^4*x^3)+2*c^4/(a^2*x)+c^4*x],
[exp(4*arctanh(a*x))*(c-c/(a*x))^3,x,4,1/2*c^3/(a^3*x^2)+c^3/(a^2*x)+c^3*x+c^3*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a*x))^2,x,4,-c^2/(a^2*x)+c^2*x+2*c^2*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a*x)),x,4,c*x-c*log(x)/a+4*c*log(1-a*x)/a],
[exp(4*arctanh(a*x))/(c-c/(a*x)),x,4,x/c+(-2)/(a*c*(1-a*x)^2)+8/(a*c*(1-a*x))+5*log(1-a*x)/(a*c)],
[exp(4*arctanh(a*x))/(c-c/(a*x))^2,x,4,x/c^2+4/3/(a*c^2*(1-a*x)^3)+(-6)/(a*c^2*(1-a*x)^2)+13/(a*c^2*(1-a*x))+6*log(1-a*x)/(a*c^2)],
[exp(4*arctanh(a*x))/(c-c/(a*x))^3,x,4,x/c^3+(-1)/(a*c^3*(1-a*x)^4)+16/3/(a*c^3*(1-a*x)^3)+(-25/2)/(a*c^3*(1-a*x)^2)+19/(a*c^3*(1-a*x))+7*log(1-a*x)/(a*c^3)],
[exp(4*arctanh(a*x))/(c-c/(a*x))^4,x,4,x/c^4+4/5/(a*c^4*(1-a*x)^5)+(-5)/(a*c^4*(1-a*x)^4)+41/3/(a*c^4*(1-a*x)^3)+(-22)/(a*c^4*(1-a*x)^2)+26/(a*c^4*(1-a*x))+8*log(1-a*x)/(a*c^4)],

# n<0
[(c-c/(a*x))^p/exp(arctanh(a*x)),x,3,(c-c/(a*x))^p*x*AppellF1(1-p,-1/2-p,1/2,2-p,a*x,-a*x)/((1-p)*(1-a*x)^p)],
[(c-c/(a*x))^4/exp(arctanh(a*x)),x,11,5*c^4*arcsin(a*x)/a+25/2*c^4*arctanh(sqrt(1-a^2*x^2))/a+c^4*sqrt(1-a^2*x^2)/a-1/3*c^4*sqrt(1-a^2*x^2)/(a^4*x^3)+5/2*c^4*sqrt(1-a^2*x^2)/(a^3*x^2)-32/3*c^4*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a*x))^3/exp(arctanh(a*x)),x,10,4*c^3*arcsin(a*x)/a+13/2*c^3*arctanh(sqrt(1-a^2*x^2))/a+c^3*sqrt(1-a^2*x^2)/a+1/2*c^3*sqrt(1-a^2*x^2)/(a^3*x^2)-4*c^3*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a*x))^2/exp(arctanh(a*x)),x,9,3*c^2*arcsin(a*x)/a+3*c^2*arctanh(sqrt(1-a^2*x^2))/a+c^2*sqrt(1-a^2*x^2)/a-c^2*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a*x))/exp(arctanh(a*x)),x,8,2*c*arcsin(a*x)/a+c*arctanh(sqrt(1-a^2*x^2))/a+c*sqrt(1-a^2*x^2)/a],
[1/(exp(arctanh(a*x))*(c-c/(a*x))),x,3,sqrt(1-a^2*x^2)/(a*c)],
[1/(exp(arctanh(a*x))*(c-c/(a*x))^2),x,6,-arcsin(a*x)/(a*c^2)+sqrt(1-a^2*x^2)/(a*c^2)+sqrt(1-a^2*x^2)/(a*c^2*(1-a*x))],
[1/(exp(arctanh(a*x))*(c-c/(a*x))^3),x,7,-1/3*(1+a*x)^2/(a*c^3*(1-a^2*x^2)^(3/2))-2*arcsin(a*x)/(a*c^3)+8/3*(1+a*x)/(a*c^3*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a*c^3)],
[1/(exp(arctanh(a*x))*(c-c/(a*x))^4),x,8,1/5*(1+a*x)^3/(a*c^4*(1-a^2*x^2)^(5/2))-6/5*(1+a*x)^2/(a*c^4*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^4)+24/5*(1+a*x)/(a*c^4*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a*c^4)],
[(c-c/(a*x))^p/exp(2*arctanh(a*x)),x,8,-(c-c/(a*x))^(2+p)*x/c^2-1/2*(c-c/(a*x))^(2+p)*hypergeom([1,2+p],[3+p],1/2*(a+(-1)/x)/a)/(a*c^2*(2+p))+(c-c/(a*x))^(2+p)*hypergeom([1,2+p],[3+p],1+(-1)/(a*x))/(a*c^2)],
[(c-c/(a*x))^4/exp(2*arctanh(a*x)),x,4,-1/3*c^4/(a^4*x^3)+3*c^4/(a^3*x^2)-16*c^4/(a^2*x)-c^4*x-26*c^4*log(x)/a+32*c^4*log(1+a*x)/a],
[(c-c/(a*x))^3/exp(2*arctanh(a*x)),x,4,1/2*c^3/(a^3*x^2)-5*c^3/(a^2*x)-c^3*x-11*c^3*log(x)/a+16*c^3*log(1+a*x)/a],
[(c-c/(a*x))^2/exp(2*arctanh(a*x)),x,4,-c^2/(a^2*x)-c^2*x-4*c^2*log(x)/a+8*c^2*log(1+a*x)/a],
[(c-c/(a*x))/exp(2*arctanh(a*x)),x,4,-c*x-c*log(x)/a+4*c*log(1+a*x)/a],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))),x,4,-x/c+log(1+a*x)/(a*c)],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^2),x,5,-x/c^2+arctanh(a*x)/(a*c^2)],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^3),x,4,-x/c^3+(-1/2)/(a*c^3*(1-a*x))-5/4*log(1-a*x)/(a*c^3)+1/4*log(1+a*x)/(a*c^3)],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^4),x,4,-x/c^4+1/4/(a*c^4*(1-a*x)^2)+(-7/4)/(a*c^4*(1-a*x))-17/8*log(1-a*x)/(a*c^4)+1/8*log(1+a*x)/(a*c^4)],
[(c-c/(a*x))^3/exp(3*arctanh(a*x)),x,11,-6*c^3*arcsin(a*x)/a+33/2*c^3*arctanh(sqrt(1-a^2*x^2))/a-32*c^3*(1-a*x)/(a*sqrt(1-a^2*x^2))-c^3*sqrt(1-a^2*x^2)/a+1/2*c^3*sqrt(1-a^2*x^2)/(a^3*x^2)-6*c^3*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a*x))^2/exp(3*arctanh(a*x)),x,10,-5*c^2*arcsin(a*x)/a+5*c^2*arctanh(sqrt(1-a^2*x^2))/a-16*c^2*(1-a*x)/(a*sqrt(1-a^2*x^2))-c^2*sqrt(1-a^2*x^2)/a-c^2*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a*x))/exp(3*arctanh(a*x)),x,9,-4*c*arcsin(a*x)/a+c*arctanh(sqrt(1-a^2*x^2))/a-8*c*(1-a*x)/(a*sqrt(1-a^2*x^2))-c*sqrt(1-a^2*x^2)/a],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))),x,5,-2*arcsin(a*x)/(a*c)-(1-a*x)^2/(a*c*sqrt(1-a^2*x^2))-2*sqrt(1-a^2*x^2)/(a*c)],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^2),x,6,-arcsin(a*x)/(a*c^2)+(-1+a*x)/(a*c^2*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c^2)],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^3),x,5,(-1)/(a*c^3*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c^3)],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^4),x,7,1/3*a^2*x^3*(1+a*x)/(c^4*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^4)-1/3*x*(3+4*a*x)/(c^4*sqrt(1-a^2*x^2))-8/3*sqrt(1-a^2*x^2)/(a*c^4)],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^5),x,8,-1/5*(1+a*x)^2/(a*c^5*(1-a^2*x^2)^(5/2))+22/15*(1+a*x)/(a*c^5*(1-a^2*x^2)^(3/2))+2*arcsin(a*x)/(a*c^5)-2/15*(30+23*a*x)/(a*c^5*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c^5)],

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a x))^(p/2)

# n>0
[exp(arctanh(a*x))*(c-c/(a*x))^(9/2),x,8,-7*a^(7/2)*(c-c/(a*x))^(9/2)*x^(9/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(9/2)-1/105*a^3*(c-c/(a*x))^(9/2)*x^4*(54-227*a*x)*sqrt(1+a*x)/(1-a*x)^(9/2)-10/21*a^2*(c-c/(a*x))^(9/2)*x^3*sqrt(1+a*x)/(1-a*x)^(5/2)+2/5*a*(c-c/(a*x))^(9/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)-2/7*(c-c/(a*x))^(9/2)*x*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))*(c-c/(a*x))^(7/2),x,7,5*a^(5/2)*(c-c/(a*x))^(7/2)*x^(7/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(7/2)+2/3*a*(c-c/(a*x))^(7/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)-1/15*a^2*(c-c/(a*x))^(7/2)*x^3*(18+31*a*x)*sqrt(1+a*x)/(1-a*x)^(7/2)-2/5*(c-c/(a*x))^(7/2)*x*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))*(c-c/(a*x))^(5/2),x,7,-2/3*(c-c/(a*x))^(5/2)*x*(1+a*x)^(3/2)/(1-a*x)^(5/2)+4*a*(c-c/(a*x))^(5/2)*x^2*(1+a*x)^(3/2)/(1-a*x)^(5/2)-3*a^(3/2)*(c-c/(a*x))^(5/2)*x^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(5/2)-3*a^2*(c-c/(a*x))^(5/2)*x^3*sqrt(1+a*x)/(1-a*x)^(5/2)],
[exp(arctanh(a*x))*(c-c/(a*x))^(3/2),x,7,-2*(c-c/(a*x))^(3/2)*x*(1-a^2*x^2)^(3/2)/(1-a*x)^3+(c-c/(a*x))^(3/2)*x^(3/2)*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)/(1-a*x)^(3/2)+a*(c-c/(a*x))^(3/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)],
[exp(arctanh(a*x))*sqrt(c-c/(a*x)),x,6,arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))-c*sqrt(1-a^2*x^2)/(a*sqrt(c-c/(a*x))),arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))+x*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))/sqrt(c-c/(a*x)),x,8,-3*arcsinh(sqrt(a)*sqrt(x))*sqrt(1-a*x)/(a^(3/2)*sqrt(c-c/(a*x))*sqrt(x))+2*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(1-a*x)/(a^(3/2)*sqrt(c-c/(a*x))*sqrt(x))-sqrt(1-a*x)*sqrt(1+a*x)/(a*sqrt(c-c/(a*x)))],
[exp(arctanh(a*x))/(c-c/(a*x))^(3/2),x,9,5*(1-a*x)^(3/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2))-7*(1-a*x)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2)*sqrt(2))+2*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(3/2)*x)+sqrt(1-a*x)*sqrt(1+a*x)/(a*(c-c/(a*x))^(3/2))],
[exp(arctanh(a*x))/(c-c/(a*x))^(5/2),x,10,-7*(1-a*x)^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2))+79/8*(1-a*x)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2)*sqrt(2))-11/8*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(5/2)*x)-23/8*(1-a*x)^(5/2)*sqrt(1+a*x)/(a^3*(c-c/(a*x))^(5/2)*x^2)+1/2*sqrt(1-a*x)*sqrt(1+a*x)/(a*(c-c/(a*x))^(5/2))],
[exp(2*arctanh(a*x))*(c-c/(a*x))^(9/2),x,11,-5/3*c^3*(c-c/(a*x))^(3/2)/a-c^2*(c-c/(a*x))^(5/2)/a-5/7*c*(c-c/(a*x))^(7/2)/a-(c-c/(a*x))^(9/2)*x+5*c^(9/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-5*c^4*sqrt(c-c/(a*x))/a],
[exp(2*arctanh(a*x))*(c-c/(a*x))^(7/2),x,10,-c^2*(c-c/(a*x))^(3/2)/a-3/5*c*(c-c/(a*x))^(5/2)/a-(c-c/(a*x))^(7/2)*x+3*c^(7/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-3*c^3*sqrt(c-c/(a*x))/a],
[exp(2*arctanh(a*x))*(c-c/(a*x))^(5/2),x,9,-1/3*c*(c-c/(a*x))^(3/2)/a-(c-c/(a*x))^(5/2)*x+c^(5/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-c^2*sqrt(c-c/(a*x))/a],
[exp(2*arctanh(a*x))*(c-c/(a*x))^(3/2),x,8,-(c-c/(a*x))^(3/2)*x-c^(3/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+c*sqrt(c-c/(a*x))/a],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x)),x,7,-3*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a-x*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))/sqrt(c-c/(a*x)),x,8,-5*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*sqrt(c))+5/(a*sqrt(c-c/(a*x)))-x/sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))/(c-c/(a*x))^(3/2),x,9,7/3/(a*(c-c/(a*x))^(3/2))-x/(c-c/(a*x))^(3/2)-7*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(3/2))+7/(a*c*sqrt(c-c/(a*x)))],
[exp(2*arctanh(a*x))/(c-c/(a*x))^(5/2),x,10,9/5/(a*(c-c/(a*x))^(5/2))+3/(a*c*(c-c/(a*x))^(3/2))-x/(c-c/(a*x))^(5/2)-9*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(5/2))+9/(a*c^2*sqrt(c-c/(a*x)))],
[exp(2*arctanh(a*x))/(c-c/(a*x))^(7/2),x,11,11/7/(a*(c-c/(a*x))^(7/2))+11/5/(a*c*(c-c/(a*x))^(5/2))+11/3/(a*c^2*(c-c/(a*x))^(3/2))-x/(c-c/(a*x))^(7/2)-11*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(7/2))+11/(a*c^3*sqrt(c-c/(a*x)))],
[exp(3*arctanh(a*x))*(c-c/(a*x))^(9/2),x,8,3/35*a^2*(c-c/(a*x))^(9/2)*x^3*(6-17*a*x)*(1+a*x)^(3/2)/(1-a*x)^(9/2)+6/35*a*(c-c/(a*x))^(9/2)*x^2*(1+a*x)^(3/2)/(1-a*x)^(5/2)-2/7*(c-c/(a*x))^(9/2)*x*(1+a*x)^(3/2)/(1-a*x)^(3/2)+3*a^(7/2)*(c-c/(a*x))^(9/2)*x^(9/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(9/2)-3*a^3*(c-c/(a*x))^(9/2)*x^4*sqrt(1+a*x)/(1-a*x)^(9/2)],
[exp(3*arctanh(a*x))*(c-c/(a*x))^(7/2),x,8,2/3*a^2*(c-c/(a*x))^(7/2)*x^3*(1+a*x)^(3/2)/(1-a*x)^(7/2)-2/5*(c-c/(a*x))^(7/2)*x*(1+a*x)^(5/2)/(1-a*x)^(7/2)+4/3*a*(c-c/(a*x))^(7/2)*x^2*(1+a*x)^(5/2)/(1-a*x)^(7/2)-a^(5/2)*(c-c/(a*x))^(7/2)*x^(7/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(7/2)-a^3*(c-c/(a*x))^(7/2)*x^4*sqrt(1+a*x)/(1-a*x)^(7/2)],
[exp(3*arctanh(a*x))*(c-c/(a*x))^(5/2),x,8,2/3*a*(c-c/(a*x))^(5/2)*x^2*(1+a*x)^(3/2)/(1-a*x)^(5/2)-2/3*(c-c/(a*x))^(5/2)*x*(1-a^2*x^2)^(5/2)/(1-a*x)^5-a^(3/2)*(c-c/(a*x))^(5/2)*x^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(5/2)-a^2*(c-c/(a*x))^(5/2)*x^3*sqrt(1+a*x)/(1-a*x)^(5/2)],
[exp(3*arctanh(a*x))*(c-c/(a*x))^(3/2),x,7,-2*(c-c/(a*x))^(3/2)*x*(1+a*x)^(3/2)/(1-a*x)^(3/2)+3*(c-c/(a*x))^(3/2)*x^(3/2)*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)/(1-a*x)^(3/2)+3*a*(c-c/(a*x))^(3/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x)),x,8,-5*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))+4*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))-x*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(3*arctanh(a*x))/sqrt(c-c/(a*x)),x,9,(1+a*x)^(3/2)/(a*sqrt(c-c/(a*x))*sqrt(1-a*x))+7*arcsinh(sqrt(a)*sqrt(x))*sqrt(1-a*x)/(a^(3/2)*sqrt(c-c/(a*x))*sqrt(x))-5*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(1-a*x)/(a^(3/2)*sqrt(c-c/(a*x))*sqrt(x))+2*sqrt(1-a*x)*sqrt(1+a*x)/(a*sqrt(c-c/(a*x)))],
[exp(3*arctanh(a*x))/(c-c/(a*x))^(3/2),x,10,-9*(1-a*x)^(3/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2))+51/4*(1-a*x)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2)*sqrt(2))+1/2*(1+a*x)^(3/2)/(a*(c-c/(a*x))^(3/2)*sqrt(1-a*x))-9/8*(1+a*x)^(3/2)*sqrt(1-a*x)/(a^2*(c-c/(a*x))^(3/2)*x)-21/8*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(3/2)*x)],
[exp(3*arctanh(a*x))/(c-c/(a*x))^(5/2),x,11,43/32*(1-a*x)^(3/2)*(1+a*x)^(3/2)/(a^3*(c-c/(a*x))^(5/2)*x^2)+11*(1-a*x)^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2))-249/16*(1-a*x)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2)*sqrt(2))+1/3*(1+a*x)^(3/2)/(a*(c-c/(a*x))^(5/2)*sqrt(1-a*x))-13/24*(1+a*x)^(3/2)*sqrt(1-a*x)/(a^2*(c-c/(a*x))^(5/2)*x)+103/32*(1-a*x)^(5/2)*sqrt(1+a*x)/(a^3*(c-c/(a*x))^(5/2)*x^2)],

# n<0
[(c-c/(a*x))^(9/2)/exp(arctanh(a*x)),x,8,11*a^(7/2)*(c-c/(a*x))^(9/2)*x^(9/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(9/2)-94/21*a^2*(c-c/(a*x))^(9/2)*x^3*sqrt(1+a*x)/(1-a*x)^(5/2)+6/5*a*(c-c/(a*x))^(9/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)+1/105*a^3*(c-c/(a*x))^(9/2)*x^4*(2718+521*a*x)*sqrt(1+a*x)/(1-a*x)^(9/2)-2/7*(c-c/(a*x))^(9/2)*x*sqrt(1+a*x)/sqrt(1-a*x)],
[(c-c/(a*x))^(7/2)/exp(arctanh(a*x)),x,7,-9*a^(5/2)*(c-c/(a*x))^(7/2)*x^(7/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(7/2)+2*a*(c-c/(a*x))^(7/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)-1/5*a^2*(c-c/(a*x))^(7/2)*x^3*(66+7*a*x)*sqrt(1+a*x)/(1-a*x)^(7/2)-2/5*(c-c/(a*x))^(7/2)*x*sqrt(1+a*x)/sqrt(1-a*x)],
[(c-c/(a*x))^(5/2)/exp(arctanh(a*x)),x,6,7*a^(3/2)*(c-c/(a*x))^(5/2)*x^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(5/2)+1/3*a*(c-c/(a*x))^(5/2)*x^2*(18-a*x)*sqrt(1+a*x)/(1-a*x)^(5/2)-2/3*(c-c/(a*x))^(5/2)*x*sqrt(1+a*x)/sqrt(1-a*x)],
[(c-c/(a*x))^(3/2)/exp(arctanh(a*x)),x,6,-5*(c-c/(a*x))^(3/2)*x^(3/2)*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)/(1-a*x)^(3/2)-2*(c-c/(a*x))^(3/2)*x*sqrt(1+a*x)/(1-a*x)^(3/2)+a*(c-c/(a*x))^(3/2)*x^2*sqrt(1+a*x)/(1-a*x)^(3/2)],
[sqrt(c-c/(a*x))/exp(arctanh(a*x)),x,6,3*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))-x*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(1-a*x)],
[1/(exp(arctanh(a*x))*sqrt(c-c/(a*x))),x,6,-arcsinh(sqrt(a)*sqrt(x))*sqrt(1-a*x)/(a^(3/2)*sqrt(c-c/(a*x))*sqrt(x))+sqrt(1-a*x)*sqrt(1+a*x)/(a*sqrt(c-c/(a*x)))],
[1/(exp(arctanh(a*x))*(c-c/(a*x))^(3/2)),x,9,-(1-a*x)^(3/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2))+(1-a*x)^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2))-(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(3/2)*x)],
[1/(exp(arctanh(a*x))*(c-c/(a*x))^(5/2)),x,9,3*(1-a*x)^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2))-9/2*(1-a*x)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2)*sqrt(2))+1/2*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(5/2)*x)+3/2*(1-a*x)^(5/2)*sqrt(1+a*x)/(a^3*(c-c/(a*x))^(5/2)*x^2)],
[1/(exp(arctanh(a*x))*(c-c/(a*x))^(7/2)),x,10,-5*(1-a*x)^(7/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(9/2)*(c-c/(a*x))^(7/2)*x^(7/2))+115/16*(1-a*x)^(7/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(9/2)*(c-c/(a*x))^(7/2)*x^(7/2)*sqrt(2))+1/4*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(7/2)*x)-15/16*(1-a*x)^(5/2)*sqrt(1+a*x)/(a^3*(c-c/(a*x))^(7/2)*x^2)-35/16*(1-a*x)^(7/2)*sqrt(1+a*x)/(a^4*(c-c/(a*x))^(7/2)*x^3)],
[(c-c/(a*x))^(9/2)/exp(2*arctanh(a*x)),x,14,19/3*c^3*(c-c/(a*x))^(3/2)/a+3/5*c^2*(c-c/(a*x))^(5/2)/a-5/7*c*(c-c/(a*x))^(7/2)/a-(c-c/(a*x))^(9/2)*x+13*c^(9/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-64*c^(9/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a+51*c^4*sqrt(c-c/(a*x))/a],
[(c-c/(a*x))^(7/2)/exp(2*arctanh(a*x)),x,13,5/3*c^2*(c-c/(a*x))^(3/2)/a-3/5*c*(c-c/(a*x))^(5/2)/a-(c-c/(a*x))^(7/2)*x+11*c^(7/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-32*c^(7/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a+21*c^3*sqrt(c-c/(a*x))/a],
[(c-c/(a*x))^(5/2)/exp(2*arctanh(a*x)),x,12,-1/3*c*(c-c/(a*x))^(3/2)/a-(c-c/(a*x))^(5/2)*x+9*c^(5/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-16*c^(5/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a+7*c^2*sqrt(c-c/(a*x))/a],
[(c-c/(a*x))^(3/2)/exp(2*arctanh(a*x)),x,11,-(c-c/(a*x))^(3/2)*x+7*c^(3/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-8*c^(3/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a+c*sqrt(c-c/(a*x))/a],
[sqrt(c-c/(a*x))/exp(2*arctanh(a*x)),x,10,5*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a-4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a-x*sqrt(c-c/(a*x))],
[1/(exp(2*arctanh(a*x))*sqrt(c-c/(a*x))),x,10,3*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*sqrt(c))-2*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*sqrt(c))-x*sqrt(c-c/(a*x))/c],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^(3/2)),x,11,arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(3/2))-arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*c^(3/2))-x*sqrt(c-c/(a*x))/c^2],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^(5/2)),x,11,-arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(5/2))-arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))/(a*c^(5/2)*sqrt(2))+2/(a*c^2*sqrt(c-c/(a*x)))-x/(c^2*sqrt(c-c/(a*x)))],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^(7/2)),x,12,4/3/(a*c^2*(c-c/(a*x))^(3/2))-x/(c^2*(c-c/(a*x))^(3/2))-3*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(7/2))-1/2*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))/(a*c^(7/2)*sqrt(2))+7/2/(a*c^3*sqrt(c-c/(a*x)))],
[1/(exp(2*arctanh(a*x))*(c-c/(a*x))^(9/2)),x,13,6/5/(a*c^2*(c-c/(a*x))^(5/2))+11/6/(a*c^3*(c-c/(a*x))^(3/2))-x/(c^2*(c-c/(a*x))^(5/2))-5*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(9/2))-1/4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))/(a*c^(9/2)*sqrt(2))+21/4/(a*c^4*sqrt(c-c/(a*x)))],
[(c-c/(a*x))^(9/2)/exp(3*arctanh(a*x)),x,9,-15*a^(7/2)*(c-c/(a*x))^(9/2)*x^(9/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(9/2)+5/7*a^4*(c-c/(a*x))^(9/2)*x^5*(587-109*a*x)/((1-a*x)^(9/2)*sqrt(1+a*x))+70*a^3*(c-c/(a*x))^(9/2)*x^4/((1-a*x)^(5/2)*sqrt(1+a*x))-50/7*a^2*(c-c/(a*x))^(9/2)*x^3/((1-a*x)^(3/2)*sqrt(1+a*x))+10/7*a*(c-c/(a*x))^(9/2)*x^2/(sqrt(1-a*x)*sqrt(1+a*x))-2/7*(c-c/(a*x))^(9/2)*x*sqrt(1-a*x)/sqrt(1+a*x)],
[(c-c/(a*x))^(7/2)/exp(3*arctanh(a*x)),x,8,13*a^(5/2)*(c-c/(a*x))^(7/2)*x^(7/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(7/2)-1/15*a^3*(c-c/(a*x))^(7/2)*x^4*(2525-427*a*x)/((1-a*x)^(7/2)*sqrt(1+a*x))-398/15*a^2*(c-c/(a*x))^(7/2)*x^3/((1-a*x)^(3/2)*sqrt(1+a*x))+38/15*a*(c-c/(a*x))^(7/2)*x^2/(sqrt(1-a*x)*sqrt(1+a*x))-2/5*(c-c/(a*x))^(7/2)*x*sqrt(1-a*x)/sqrt(1+a*x)],
[(c-c/(a*x))^(5/2)/exp(3*arctanh(a*x)),x,7,-11*a^(3/2)*(c-c/(a*x))^(5/2)*x^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(1-a*x)^(5/2)+1/3*a^2*(c-c/(a*x))^(5/2)*x^3*(191-25*a*x)/((1-a*x)^(5/2)*sqrt(1+a*x))+26/3*a*(c-c/(a*x))^(5/2)*x^2/(sqrt(1-a*x)*sqrt(1+a*x))-2/3*(c-c/(a*x))^(5/2)*x*sqrt(1-a*x)/sqrt(1+a*x)],
[(c-c/(a*x))^(3/2)/exp(3*arctanh(a*x)),x,6,9*(c-c/(a*x))^(3/2)*x^(3/2)*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)/(1-a*x)^(3/2)-a*(c-c/(a*x))^(3/2)*x^2*(23-a*x)/((1-a*x)^(3/2)*sqrt(1+a*x))-2*(c-c/(a*x))^(3/2)*x*sqrt(1-a*x)/sqrt(1+a*x)],
[sqrt(c-c/(a*x))/exp(3*arctanh(a*x)),x,6,-7*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))+8*x*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))+x*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[1/(exp(3*arctanh(a*x))*sqrt(c-c/(a*x))),x,7,5*arcsinh(sqrt(a)*sqrt(x))*sqrt(1-a*x)/(a^(3/2)*sqrt(c-c/(a*x))*sqrt(x))-5*sqrt(1-a*x)/(a*sqrt(c-c/(a*x))*sqrt(1+a*x))-x*(1-a*x)/(sqrt(c-c/(a*x))*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^(3/2)),x,7,-3*(1-a*x)^(3/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(5/2)*(c-c/(a*x))^(3/2)*x^(3/2))-2*(1-a*x)^(3/2)/(a*(c-c/(a*x))^(3/2)*sqrt(1+a*x))+3*(1-a*x)^(3/2)*sqrt(1+a*x)/(a^2*(c-c/(a*x))^(3/2)*x)],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^(5/2)),x,9,(1-a*x)^(5/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2))+(1-a*x)^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(7/2)*(c-c/(a*x))^(5/2)*x^(5/2)*sqrt(2))+(1-a*x)^(5/2)/(a^2*(c-c/(a*x))^(5/2)*x*sqrt(1+a*x))-2*(1-a*x)^(5/2)*sqrt(1+a*x)/(a^3*(c-c/(a*x))^(5/2)*x^2)],
[1/(exp(3*arctanh(a*x))*(c-c/(a*x))^(7/2)),x,10,(1-a*x)^(7/2)*arcsinh(sqrt(a)*sqrt(x))/(a^(9/2)*(c-c/(a*x))^(7/2)*x^(7/2))-11/4*(1-a*x)^(7/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))/(a^(9/2)*(c-c/(a*x))^(7/2)*x^(7/2)*sqrt(2))+1/2*(1-a*x)^(5/2)/(a^2*(c-c/(a*x))^(7/2)*x*sqrt(1+a*x))-1/4*(1-a*x)^(7/2)/(a^3*(c-c/(a*x))^(7/2)*x^2*sqrt(1+a*x))+7/4*(1-a*x)^(7/2)*sqrt(1+a*x)/(a^4*(c-c/(a*x))^(7/2)*x^3)],

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-c/(a x))^p

# n>0
[exp(3*arctanh(a*x))/((c-c/(a*x))*x^3),x,9,-8/3*a^2*(1+a*x)/(c*(1-a^2*x^2)^(3/2))+4*a^2*arctanh(sqrt(1-a^2*x^2))/c-4/3*a^2*(3+4*a*x)/(c*sqrt(1-a^2*x^2))+a*sqrt(1-a^2*x^2)/(c*x)],

# n<0

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-c/(a x))^(p/2)

# n>0
[exp(arctanh(a*x))*x^m*sqrt(c-c/(a*x)),x,4,2*x^(1+m)*hypergeom([-1/2,1/2+m],[3/2+m],-a*x)*sqrt(c-c/(a*x))/((1+2*m)*sqrt(1-a*x))],
[exp(arctanh(a*x))*x^2*sqrt(c-c/(a*x)),x,8,1/8*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(5/2)*sqrt(1-a*x))-1/8*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^2*sqrt(1-a*x))+1/12*x^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))+1/3*x^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))*x*sqrt(c-c/(a*x)),x,7,-1/4*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(3/2)*sqrt(1-a*x))+1/4*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))+1/2*x^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))*sqrt(c-c/(a*x)),x,6,arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))+x*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))*sqrt(c-c/(a*x))/x,x,6,2*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(arctanh(a*x))*sqrt(c-c/(a*x))/x^2,x,4,-2/3*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x*sqrt(1-a*x))],
[exp(arctanh(a*x))*sqrt(c-c/(a*x))/x^3,x,5,-2/5*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x))+4/15*a*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x*sqrt(1-a*x))],
[exp(arctanh(a*x))*sqrt(c-c/(a*x))/x^4,x,6,-2/7*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x^3*sqrt(1-a*x))+8/35*a*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x))-16/105*a^2*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x*sqrt(1-a*x))],
[exp(arctanh(a*x))*sqrt(c-c/(a*x))/x^5,x,7,-2/9*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x^4*sqrt(1-a*x))+4/21*a*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x^3*sqrt(1-a*x))-16/105*a^2*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x))+32/315*a^3*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x*sqrt(1-a*x))],
[exp(2*arctanh(a*x))*x^3*sqrt(c-c/(a*x)),x,10,-75/64*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^4-75/64*x*sqrt(c-c/(a*x))/a^3-25/32*x^2*sqrt(c-c/(a*x))/a^2-5/8*x^3*sqrt(c-c/(a*x))/a-1/4*x^4*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*x^2*sqrt(c-c/(a*x)),x,9,-11/8*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^3-11/8*x*sqrt(c-c/(a*x))/a^2-11/12*x^2*sqrt(c-c/(a*x))/a-1/3*x^3*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*x*sqrt(c-c/(a*x)),x,8,-7/4*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^2-7/4*x*sqrt(c-c/(a*x))/a-1/2*x^2*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x)),x,7,-3*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a-x*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x))/x,x,7,-2*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)-2*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x))/x^2,x,6,2/3*a*(c-c/(a*x))^(3/2)/c-4*a*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x))/x^3,x,6,2*a^2*(c-c/(a*x))^(3/2)/c-2/5*a^2*(c-c/(a*x))^(5/2)/c^2-4*a^2*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x))/x^4,x,6,10/3*a^3*(c-c/(a*x))^(3/2)/c-8/5*a^3*(c-c/(a*x))^(5/2)/c^2+2/7*a^3*(c-c/(a*x))^(7/2)/c^3-4*a^3*sqrt(c-c/(a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a*x))/x^5,x,6,14/3*a^4*(c-c/(a*x))^(3/2)/c-18/5*a^4*(c-c/(a*x))^(5/2)/c^2+10/7*a^4*(c-c/(a*x))^(7/2)/c^3-2/9*a^4*(c-c/(a*x))^(9/2)/c^4-4*a^4*sqrt(c-c/(a*x))],
[exp(3*arctanh(a*x))*x^3*sqrt(c-c/(a*x)),x,11,-21/32*x*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(a^3*sqrt(1-a*x))-11/24*x^2*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(a^2*sqrt(1-a*x))-1/4*x^3*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(a*sqrt(1-a*x))-363/64*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(7/2)*sqrt(1-a*x))+4*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/(a^(7/2)*sqrt(1-a*x))-107/64*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^3*sqrt(1-a*x))],
[exp(3*arctanh(a*x))*x^2*sqrt(c-c/(a*x)),x,10,-3/4*x*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(a^2*sqrt(1-a*x))-1/3*x^2*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(a*sqrt(1-a*x))-45/8*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(5/2)*sqrt(1-a*x))+4*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/(a^(5/2)*sqrt(1-a*x))-13/8*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^2*sqrt(1-a*x))],
[exp(3*arctanh(a*x))*x*sqrt(c-c/(a*x)),x,9,-1/2*x*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(a*sqrt(1-a*x))-23/4*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(3/2)*sqrt(1-a*x))+4*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/(a^(3/2)*sqrt(1-a*x))-7/4*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x)),x,8,-5*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))+4*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))-x*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x))/x,x,8,-2*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)+4*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(a)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x))/x^2,x,6,-2/3*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x*sqrt(1-a*x))+4*a^(3/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-4*a*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x))/x^3,x,7,-2/3*a*(1+a*x)^(3/2)*sqrt(c-c/(a*x))/(x*sqrt(1-a*x))-2/5*(1+a*x)^(5/2)*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x))+4*a^(5/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-4*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x))/x^4,x,9,4*a^(7/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-104/21*a^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)-2/7*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^3*sqrt(1-a*x))-6/7*a*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^2*sqrt(1-a*x))-32/21*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x*sqrt(1-a*x))],
[exp(3*arctanh(a*x))*sqrt(c-c/(a*x))/x^5,x,10,4*a^(9/2)*arctanh(sqrt(2)*sqrt(a)*sqrt(x)/sqrt(1+a*x))*sqrt(2)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-1576/315*a^4*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)-2/9*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^4*sqrt(1-a*x))-38/63*a*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^3*sqrt(1-a*x))-92/105*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^2*sqrt(1-a*x))-472/315*a^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x*sqrt(1-a*x))],

# n<0
[x^m*sqrt(c-c/(a*x))/exp(arctanh(a*x)),x,5,(3+4*m)*x^(1+m)*hypergeom([1/2,1/2+m],[3/2+m],-a*x)*sqrt(c-c/(a*x))/((1+3*m+2*m^2)*sqrt(1-a*x))-x^(1+m)*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/((1+m)*(1-a*x))],
[x^2*sqrt(c-c/(a*x))/exp(arctanh(a*x)),x,8,11/8*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(5/2)*sqrt(1-a*x))-11/8*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^2*sqrt(1-a*x))+11/12*x^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))-1/3*x^3*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(1-a*x)],
[x*sqrt(c-c/(a*x))/exp(arctanh(a*x)),x,7,-7/4*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(3/2)*sqrt(1-a*x))+7/4*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))-1/2*x^2*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(1-a*x)],
[sqrt(c-c/(a*x))/exp(arctanh(a*x)),x,6,3*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))-x*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(1-a*x)],
[sqrt(c-c/(a*x))/(exp(arctanh(a*x))*x),x,6,-2*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-2*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(1-a*x)],
[sqrt(c-c/(a*x))/(exp(arctanh(a*x))*x^2),x,5,10/3*a*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)-2/3*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(x*(1-a*x))],
[sqrt(c-c/(a*x))/(exp(arctanh(a*x))*x^3),x,6,-12/5*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)+6/5*a*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x*sqrt(1-a*x))-2/5*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(x^2*(1-a*x))],
[sqrt(c-c/(a*x))/(exp(arctanh(a*x))*x^4),x,7,208/105*a^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)+26/35*a*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^2*sqrt(1-a*x))-104/105*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x*sqrt(1-a*x))-2/7*sqrt(c-c/(a*x))*sqrt(1-a^2*x^2)/(x^3*(1-a*x))],
[x^3*sqrt(c-c/(a*x))/exp(2*arctanh(a*x)),x,13,-363/64*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^4+4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^4+149/64*x*sqrt(c-c/(a*x))/a^3-107/96*x^2*sqrt(c-c/(a*x))/a^2+17/24*x^3*sqrt(c-c/(a*x))/a-1/4*x^4*sqrt(c-c/(a*x))],
[x^2*sqrt(c-c/(a*x))/exp(2*arctanh(a*x)),x,12,45/8*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^3-4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^3-19/8*x*sqrt(c-c/(a*x))/a^2+13/12*x^2*sqrt(c-c/(a*x))/a-1/3*x^3*sqrt(c-c/(a*x))],
[x*sqrt(c-c/(a*x))/exp(2*arctanh(a*x)),x,11,-23/4*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^2+4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^2+9/4*x*sqrt(c-c/(a*x))/a-1/2*x^2*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/exp(2*arctanh(a*x)),x,10,5*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a-4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a-x*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arctanh(a*x))*x),x,10,-2*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)+4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-2*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arctanh(a*x))*x^2),x,8,2/3*a*(c-c/(a*x))^(3/2)/c-4*a*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+4*a*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arctanh(a*x))*x^3),x,9,-2/3*a^2*(c-c/(a*x))^(3/2)/c-2/5*a^2*(c-c/(a*x))^(5/2)/c^2+4*a^2*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-4*a^2*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arctanh(a*x))*x^4),x,10,2/3*a^3*(c-c/(a*x))^(3/2)/c+2/7*a^3*(c-c/(a*x))^(7/2)/c^3-4*a^3*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+4*a^3*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arctanh(a*x))*x^5),x,10,-2/3*a^4*(c-c/(a*x))^(3/2)/c-2/5*a^4*(c-c/(a*x))^(5/2)/c^2+2/7*a^4*(c-c/(a*x))^(7/2)/c^3-2/9*a^4*(c-c/(a*x))^(9/2)/c^4+4*a^4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-4*a^4*sqrt(c-c/(a*x))],
[x^3*sqrt(c-c/(a*x))/exp(3*arctanh(a*x)),x,9,1115/64*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(7/2)*sqrt(1-a*x))+8*x^4*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))-1115/64*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^3*sqrt(1-a*x))+1115/96*x^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^2*sqrt(1-a*x))-223/24*x^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))+1/4*x^4*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[x^2*sqrt(c-c/(a*x))/exp(3*arctanh(a*x)),x,8,-119/8*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(5/2)*sqrt(1-a*x))+8*x^3*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))+119/8*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a^2*sqrt(1-a*x))-119/12*x^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))+1/3*x^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[x*sqrt(c-c/(a*x))/exp(3*arctanh(a*x)),x,7,47/4*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(a^(3/2)*sqrt(1-a*x))+8*x^2*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))-47/4*x*sqrt(c-c/(a*x))*sqrt(1+a*x)/(a*sqrt(1-a*x))+1/2*x^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[sqrt(c-c/(a*x))/exp(3*arctanh(a*x)),x,6,-7*arcsinh(sqrt(a)*sqrt(x))*sqrt(c-c/(a*x))*sqrt(x)/(sqrt(a)*sqrt(1-a*x))+8*x*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))+x*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[sqrt(c-c/(a*x))/(exp(3*arctanh(a*x))*x),x,6,2*arcsinh(sqrt(a)*sqrt(x))*sqrt(a)*sqrt(c-c/(a*x))*sqrt(x)/sqrt(1-a*x)-2*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))-10*a*x*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a*x))/(exp(3*arctanh(a*x))*x^2),x,5,20/3*a*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))-2/3*sqrt(c-c/(a*x))/(x*sqrt(1-a*x)*sqrt(1+a*x))+46/3*a^2*x*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a*x))/(exp(3*arctanh(a*x))*x^3),x,6,158/15*a^2*sqrt(c-c/(a*x))/(sqrt(1-a*x)*sqrt(1+a*x))-2/5*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x)*sqrt(1+a*x))+32/15*a*sqrt(c-c/(a*x))/(x*sqrt(1-a*x)*sqrt(1+a*x))-316/15*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)],
[sqrt(c-c/(a*x))/(exp(3*arctanh(a*x))*x^4),x,7,-2/7*sqrt(c-c/(a*x))/(x^3*sqrt(1-a*x)*sqrt(1+a*x))+44/35*a*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x)*sqrt(1+a*x))+334/35*a^2*sqrt(c-c/(a*x))/(x*sqrt(1-a*x)*sqrt(1+a*x))+2672/105*a^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)-1336/105*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x*sqrt(1-a*x))],
[sqrt(c-c/(a*x))/(exp(3*arctanh(a*x))*x^5),x,8,-2/9*sqrt(c-c/(a*x))/(x^4*sqrt(1-a*x)*sqrt(1+a*x))+8/9*a*sqrt(c-c/(a*x))/(x^3*sqrt(1-a*x)*sqrt(1+a*x))+82/9*a^2*sqrt(c-c/(a*x))/(x^2*sqrt(1-a*x)*sqrt(1+a*x))-1312/45*a^4*sqrt(c-c/(a*x))*sqrt(1+a*x)/sqrt(1-a*x)-164/15*a^2*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x^2*sqrt(1-a*x))+656/45*a^3*sqrt(c-c/(a*x))*sqrt(1+a*x)/(x*sqrt(1-a*x))],

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a x))^p with n symbolic
[exp(n*arctanh(a*x))*(c-c/(a*x))^p,x,3,(c-c/(a*x))^p*x*AppellF1(1-p,1/2*(n-2*p),-1/2*n,2-p,a*x,-a*x)/((1-p)*(1-a*x)^p)],
[(c-c/(a*x))^p/exp(2*p*arctanh(a*x)),x,3,(c-c/(a*x))^p*x*AppellF1(1-p,-2*p,p,2-p,a*x,-a*x)/((1-p)*(1-a*x)^p)],
[exp(2*p*arctanh(a*x))*(c-c/(a*x))^p,x,3,(c-c/(a*x))^p*x*hypergeom([1-p,-p],[2-p],-a*x)/((1-p)*(1-a*x)^p)],
[exp(n*arctanh(a*x))*(c-c/(a*x))^2,x,5,4*c^2*(1+a*x)^(1/2*n)*hypergeom([2,1/2*n],[1/2*(2+n)],(1+a*x)/(1-a*x))/(a*n*(1-a*x)^(1/2*n))+2^(1/2*n)*c^2*(1-a*x)^(2-1/2*n)*hypergeom([1-1/2*n,2-1/2*n],[3-1/2*n],1/2*(1-a*x))/(a*(4-n))],
[exp(n*arctanh(a*x))*(c-c/(a*x)),x,6,c*(1-a*x)^(2-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a*(2-n))-2*c*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))*hypergeom([1,1/2*(-2+n)],[1/2*n],(1+a*x)/(1-a*x))/(a*(2-n))+2^(1/2*n)*c*(1-n)*(1-a*x)^(2-1/2*n)*hypergeom([1/2*(2-n),2-1/2*n],[3-1/2*n],1/2*(1-a*x))/(a*(8-6*n+n^2))],
[exp(n*arctanh(a*x))/(c-c/(a*x)),x,4,-(1+a*x)^(1/2*(2+n))/(a*c*n*(1-a*x)^(1/2*n))-2^(1+1/2*n)*(1+n)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a*c*(2-n)*n)],
[exp(n*arctanh(a*x))/(c-c/(a*x))^2,x,5,(3+n)*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(2+n))/(a*c^2*(2+n))-x*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(2+n))/c^2-2^(1+1/2*n)*(2+n)*hypergeom([-1/2*n,-1/2*n],[1/2*(2-n)],1/2*(1-a*x))/(a*c^2*n*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*(c-c/(a*x))^(3/2),x,3,-2*(c-c/(a*x))^(3/2)*x*AppellF1(-1/2,1/2*(-3+n),-1/2*n,1/2,a*x,-a*x)/(1-a*x)^(3/2)],
[exp(n*arctanh(a*x))*sqrt(c-c/(a*x)),x,3,2*x*AppellF1(1/2,1/2*(-1+n),-1/2*n,3/2,a*x,-a*x)*sqrt(c-c/(a*x))/sqrt(1-a*x)],
[exp(n*arctanh(a*x))/sqrt(c-c/(a*x)),x,3,2/3*x*AppellF1(3/2,1/2*(1+n),-1/2*n,5/2,a*x,-a*x)*sqrt(1-a*x)/sqrt(c-c/(a*x))],
[exp(n*arctanh(a*x))/(c-c/(a*x))^(3/2),x,3,2/5*x*(1-a*x)^(3/2)*AppellF1(5/2,1/2*(3+n),-1/2*n,7/2,a*x,-a*x)/(c-c/(a*x))^(3/2)],

# Integrands of the form u E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p

# n>0
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^4,x,11,-1/48*c^4*(16+35*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+1/120*c^4*(24+35*a*x)*(1-a^2*x^2)^(5/2)/(a^6*x^5)-1/42*c^4*(6+7*a*x)*(1-a^2*x^2)^(7/2)/(a^8*x^7)+c^4*arcsin(a*x)/a+35/16*c^4*arctanh(sqrt(1-a^2*x^2))/a+1/16*c^4*(16-35*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^3,x,10,-1/24*c^3*(8+15*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+1/20*c^3*(4+5*a*x)*(1-a^2*x^2)^(5/2)/(a^6*x^5)+c^3*arcsin(a*x)/a+15/8*c^3*arctanh(sqrt(1-a^2*x^2))/a+1/8*c^3*(8-15*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^2,x,9,-1/6*c^2*(2+3*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+c^2*arcsin(a*x)/a+3/2*c^2*arctanh(sqrt(1-a^2*x^2))/a+1/2*c^2*(2-3*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2)),x,8,c*arcsin(a*x)/a+c*arctanh(sqrt(1-a^2*x^2))/a+c*(1-a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(arctanh(a*x))/(c-c/(a^2*x^2)),x,6,arcsin(a*x)/(a*c)+(-1-a*x)/(a*c*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c)],
[exp(arctanh(a*x))/(c-c/(a^2*x^2))^2,x,6,1/3*a^2*x^3*(1+a*x)/(c^2*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^2)-1/3*x*(3+4*a*x)/(c^2*sqrt(1-a^2*x^2))-8/3*sqrt(1-a^2*x^2)/(a*c^2)],
[exp(arctanh(a*x))/(c-c/(a^2*x^2))^3,x,7,-1/5*a^4*x^5*(1+a*x)/(c^3*(1-a^2*x^2)^(5/2))+1/15*a^2*x^3*(5+6*a*x)/(c^3*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^3)-1/5*x*(5+8*a*x)/(c^3*sqrt(1-a^2*x^2))-16/5*sqrt(1-a^2*x^2)/(a*c^3)],
[exp(arctanh(a*x))/(c-c/(a^2*x^2))^4,x,8,1/7*a^6*x^7*(1+a*x)/(c^4*(1-a^2*x^2)^(7/2))-1/35*a^4*x^5*(7+8*a*x)/(c^4*(1-a^2*x^2)^(5/2))+1/105*a^2*x^3*(35+48*a*x)/(c^4*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^4)-1/35*x*(35+64*a*x)/(c^4*sqrt(1-a^2*x^2))-128/35*sqrt(1-a^2*x^2)/(a*c^4)],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^5,x,4,1/9*c^5/(a^10*x^9)+1/4*c^5/(a^9*x^8)-3/7*c^5/(a^8*x^7)-4/3*c^5/(a^7*x^6)+2/5*c^5/(a^6*x^5)+3*c^5/(a^5*x^4)+2/3*c^5/(a^4*x^3)-4*c^5/(a^3*x^2)-3*c^5/(a^2*x)-c^5*x-2*c^5*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^4,x,4,-1/7*c^4/(a^8*x^7)-1/3*c^4/(a^7*x^6)+2/5*c^4/(a^6*x^5)+3/2*c^4/(a^5*x^4)-3*c^4/(a^3*x^2)-2*c^4/(a^2*x)-c^4*x-2*c^4*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^3,x,4,1/5*c^3/(a^6*x^5)+1/2*c^3/(a^5*x^4)-1/3*c^3/(a^4*x^3)-2*c^3/(a^3*x^2)-c^3/(a^2*x)-c^3*x-2*c^3*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^2,x,4,-1/3*c^2/(a^4*x^3)-c^2/(a^3*x^2)-c^2*x-2*c^2*log(x)/a],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2)),x,4,c/(a^2*x)-c*x-2*c*log(x)/a],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2)),x,4,-x/c+(-1)/(a*c*(1-a*x))-2*log(1-a*x)/(a*c)],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^2,x,4,-x/c^2+1/4/(a*c^2*(1-a*x)^2)+(-7/4)/(a*c^2*(1-a*x))-17/8*log(1-a*x)/(a*c^2)+1/8*log(1+a*x)/(a*c^2)],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^3,x,4,-x/c^3+(-1/12)/(a*c^3*(1-a*x)^3)+5/8/(a*c^3*(1-a*x)^2)+(-39/16)/(a*c^3*(1-a*x))+1/16/(a*c^3*(1+a*x))-9/4*log(1-a*x)/(a*c^3)+1/4*log(1+a*x)/(a*c^3)],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^4,x,4,-x/c^4+1/32/(a*c^4*(1-a*x)^4)+(-13/48)/(a*c^4*(1-a*x)^3)+35/32/(a*c^4*(1-a*x)^2)+(-99/32)/(a*c^4*(1-a*x))+(-1/64)/(a*c^4*(1+a*x)^2)+11/64/(a*c^4*(1+a*x))-303/128*log(1-a*x)/(a*c^4)+47/128*log(1+a*x)/(a*c^4)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^4,x,12,1/16*c^4*(16+5*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)-1/40*c^4*(24+5*a*x)*(1-a^2*x^2)^(5/2)/(a^6*x^5)-1/7*c^4*(1-a^2*x^2)^(7/2)/(a^8*x^7)-1/2*c^4*(1-a^2*x^2)^(7/2)/(a^7*x^6)-3*c^4*arcsin(a*x)/a-15/16*c^4*arctanh(sqrt(1-a^2*x^2))/a-3/16*c^4*(16-5*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^3,x,11,1/8*c^3*(8+a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+1/5*c^3*(1-a^2*x^2)^(5/2)/(a^6*x^5)+3/4*c^3*(1-a^2*x^2)^(5/2)/(a^5*x^4)-3*c^3*arcsin(a*x)/a-3/8*c^3*arctanh(sqrt(1-a^2*x^2))/a-3/8*c^3*(8-a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^2,x,10,-1/3*c^2*(1-a^2*x^2)^(3/2)/(a^4*x^3)-3/2*c^2*(1-a^2*x^2)^(3/2)/(a^3*x^2)-3*c^2*arcsin(a*x)/a+1/2*c^2*arctanh(sqrt(1-a^2*x^2))/a-1/2*c^2*(6+a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2)),x,9,-3*c*arcsin(a*x)/a+3*c*arctanh(sqrt(1-a^2*x^2))/a+c*sqrt(1-a^2*x^2)/a+c*sqrt(1-a^2*x^2)/(a^2*x)],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2)),x,7,-1/3*(1+a*x)^3/(a*c*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c)+2*(1+a*x)^2/(a*c*sqrt(1-a^2*x^2))+3*sqrt(1-a^2*x^2)/(a*c)],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2))^2,x,7,1/5*(1+a*x)^3/(a*c^2*(1-a^2*x^2)^(5/2))-6/5*(1+a*x)^2/(a*c^2*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^2)+24/5*(1+a*x)/(a*c^2*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a*c^2)],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2))^3,x,8,-1/7*(1+a*x)^3/(a*c^3*(1-a^2*x^2)^(7/2))+38/35*(1+a*x)^2/(a*c^3*(1-a^2*x^2)^(5/2))-137/35*(1+a*x)/(a*c^3*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^3)+1/35*(245+181*a*x)/(a*c^3*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a*c^3)],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2))^4,x,9,1/9*(1+a*x)^3/(a*c^4*(1-a^2*x^2)^(9/2))-22/21*(1+a*x)^2/(a*c^4*(1-a^2*x^2)^(7/2))+478/105*(1+a*x)/(a*c^4*(1-a^2*x^2)^(5/2))-2/315*(1155+829*a*x)/(a*c^4*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^4)+4/315*(630+431*a*x)/(a*c^4*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a*c^4)],
[exp(4*arctanh(a*x))*(c-c/(a^2*x^2))^5,x,4,1/9*c^5/(a^10*x^9)+1/2*c^5/(a^9*x^8)+3/7*c^5/(a^8*x^7)-4/3*c^5/(a^7*x^6)-14/5*c^5/(a^6*x^5)+14/3*c^5/(a^4*x^3)+4*c^5/(a^3*x^2)-3*c^5/(a^2*x)+c^5*x+4*c^5*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a^2*x^2))^4,x,4,-1/7*c^4/(a^8*x^7)-2/3*c^4/(a^7*x^6)-4/5*c^4/(a^6*x^5)+c^4/(a^5*x^4)+10/3*c^4/(a^4*x^3)+2*c^4/(a^3*x^2)-4*c^4/(a^2*x)+c^4*x+4*c^4*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a^2*x^2))^3,x,4,1/5*c^3/(a^6*x^5)+c^3/(a^5*x^4)+5/3*c^3/(a^4*x^3)-5*c^3/(a^2*x)+c^3*x+4*c^3*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a^2*x^2))^2,x,4,-1/3*c^2/(a^4*x^3)-2*c^2/(a^3*x^2)-6*c^2/(a^2*x)+c^2*x+4*c^2*log(x)/a],
[exp(4*arctanh(a*x))*(c-c/(a^2*x^2)),x,4,c/(a^2*x)+c*x-4*c*log(x)/a+8*c*log(1-a*x)/a],
[exp(4*arctanh(a*x))/(c-c/(a^2*x^2)),x,4,x/c+(-1)/(a*c*(1-a*x)^2)+5/(a*c*(1-a*x))+4*log(1-a*x)/(a*c)],
[exp(4*arctanh(a*x))/(c-c/(a^2*x^2))^2,x,4,x/c^2+1/3/(a*c^2*(1-a*x)^3)+(-2)/(a*c^2*(1-a*x)^2)+6/(a*c^2*(1-a*x))+4*log(1-a*x)/(a*c^2)],
[exp(4*arctanh(a*x))/(c-c/(a^2*x^2))^3,x,4,x/c^3+(-1/8)/(a*c^3*(1-a*x)^4)+11/12/(a*c^3*(1-a*x)^3)+(-49/16)/(a*c^3*(1-a*x)^2)+111/16/(a*c^3*(1-a*x))+129/32*log(1-a*x)/(a*c^3)-1/32*log(1+a*x)/(a*c^3)],
[exp(4*arctanh(a*x))/(c-c/(a^2*x^2))^4,x,4,x/c^4+1/20/(a*c^4*(1-a*x)^5)+(-7/16)/(a*c^4*(1-a*x)^4)+83/48/(a*c^4*(1-a*x)^3)+(-67/16)/(a*c^4*(1-a*x)^2)+501/64/(a*c^4*(1-a*x))+(-1/64)/(a*c^4*(1+a*x))+261/64*log(1-a*x)/(a*c^4)-5/64*log(1+a*x)/(a*c^4)],

# n<0
[(c-c/(a^2*x^2))^4/exp(arctanh(a*x)),x,11,-1/48*c^4*(16-35*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+1/120*c^4*(24-35*a*x)*(1-a^2*x^2)^(5/2)/(a^6*x^5)-1/42*c^4*(6-7*a*x)*(1-a^2*x^2)^(7/2)/(a^8*x^7)+c^4*arcsin(a*x)/a-35/16*c^4*arctanh(sqrt(1-a^2*x^2))/a+1/16*c^4*(16+35*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a^2*x^2))^3/exp(arctanh(a*x)),x,10,-1/24*c^3*(8-15*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+1/20*c^3*(4-5*a*x)*(1-a^2*x^2)^(5/2)/(a^6*x^5)+c^3*arcsin(a*x)/a-15/8*c^3*arctanh(sqrt(1-a^2*x^2))/a+1/8*c^3*(8+15*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a^2*x^2))^2/exp(arctanh(a*x)),x,9,-1/6*c^2*(2-3*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+c^2*arcsin(a*x)/a-3/2*c^2*arctanh(sqrt(1-a^2*x^2))/a+1/2*c^2*(2+3*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a^2*x^2))/exp(arctanh(a*x)),x,8,c*arcsin(a*x)/a-c*arctanh(sqrt(1-a^2*x^2))/a+c*(1+a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))),x,6,arcsin(a*x)/(a*c)+(1-a*x)/(a*c*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a*c)],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))^2),x,6,1/3*a^2*x^3*(1-a*x)/(c^2*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^2)-1/3*x*(3-4*a*x)/(c^2*sqrt(1-a^2*x^2))+8/3*sqrt(1-a^2*x^2)/(a*c^2)],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))^3),x,7,-1/5*a^4*x^5*(1-a*x)/(c^3*(1-a^2*x^2)^(5/2))+1/15*a^2*x^3*(5-6*a*x)/(c^3*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^3)-1/5*x*(5-8*a*x)/(c^3*sqrt(1-a^2*x^2))+16/5*sqrt(1-a^2*x^2)/(a*c^3)],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))^4),x,8,1/7*a^6*x^7*(1-a*x)/(c^4*(1-a^2*x^2)^(7/2))-1/35*a^4*x^5*(7-8*a*x)/(c^4*(1-a^2*x^2)^(5/2))+1/105*a^2*x^3*(35-48*a*x)/(c^4*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a*c^4)-1/35*x*(35-64*a*x)/(c^4*sqrt(1-a^2*x^2))+128/35*sqrt(1-a^2*x^2)/(a*c^4)],
[(c-c/(a^2*x^2))^4/exp(2*arctanh(a*x)),x,4,-1/7*c^4/(a^8*x^7)+1/3*c^4/(a^7*x^6)+2/5*c^4/(a^6*x^5)-3/2*c^4/(a^5*x^4)+3*c^4/(a^3*x^2)-2*c^4/(a^2*x)-c^4*x+2*c^4*log(x)/a],
[(c-c/(a^2*x^2))^3/exp(2*arctanh(a*x)),x,4,1/5*c^3/(a^6*x^5)-1/2*c^3/(a^5*x^4)-1/3*c^3/(a^4*x^3)+2*c^3/(a^3*x^2)-c^3/(a^2*x)-c^3*x+2*c^3*log(x)/a],
[(c-c/(a^2*x^2))^2/exp(2*arctanh(a*x)),x,4,-1/3*c^2/(a^4*x^3)+c^2/(a^3*x^2)-c^2*x+2*c^2*log(x)/a],
[(c-c/(a^2*x^2))/exp(2*arctanh(a*x)),x,4,c/(a^2*x)-c*x+2*c*log(x)/a],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))),x,4,-x/c+1/(a*c*(1+a*x))+2*log(1+a*x)/(a*c)],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^2),x,4,-x/c^2+(-1/4)/(a*c^2*(1+a*x)^2)+7/4/(a*c^2*(1+a*x))-1/8*log(1-a*x)/(a*c^2)+17/8*log(1+a*x)/(a*c^2)],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^3),x,4,-x/c^3+(-1/16)/(a*c^3*(1-a*x))+1/12/(a*c^3*(1+a*x)^3)+(-5/8)/(a*c^3*(1+a*x)^2)+39/16/(a*c^3*(1+a*x))-1/4*log(1-a*x)/(a*c^3)+9/4*log(1+a*x)/(a*c^3)],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^4),x,4,-x/c^4+1/64/(a*c^4*(1-a*x)^2)+(-11/64)/(a*c^4*(1-a*x))+(-1/32)/(a*c^4*(1+a*x)^4)+13/48/(a*c^4*(1+a*x)^3)+(-35/32)/(a*c^4*(1+a*x)^2)+99/32/(a*c^4*(1+a*x))-47/128*log(1-a*x)/(a*c^4)+303/128*log(1+a*x)/(a*c^4)],
[(c-c/(a^2*x^2))^4/exp(3*arctanh(a*x)),x,12,1/16*c^4*(16-5*a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)-1/40*c^4*(24-5*a*x)*(1-a^2*x^2)^(5/2)/(a^6*x^5)-1/7*c^4*(1-a^2*x^2)^(7/2)/(a^8*x^7)+1/2*c^4*(1-a^2*x^2)^(7/2)/(a^7*x^6)-3*c^4*arcsin(a*x)/a+15/16*c^4*arctanh(sqrt(1-a^2*x^2))/a-3/16*c^4*(16+5*a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a^2*x^2))^3/exp(3*arctanh(a*x)),x,11,1/8*c^3*(8-a*x)*(1-a^2*x^2)^(3/2)/(a^4*x^3)+1/5*c^3*(1-a^2*x^2)^(5/2)/(a^6*x^5)-3/4*c^3*(1-a^2*x^2)^(5/2)/(a^5*x^4)-3*c^3*arcsin(a*x)/a+3/8*c^3*arctanh(sqrt(1-a^2*x^2))/a-3/8*c^3*(8+a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a^2*x^2))^2/exp(3*arctanh(a*x)),x,10,-1/3*c^2*(1-a^2*x^2)^(3/2)/(a^4*x^3)+3/2*c^2*(1-a^2*x^2)^(3/2)/(a^3*x^2)-3*c^2*arcsin(a*x)/a-1/2*c^2*arctanh(sqrt(1-a^2*x^2))/a-1/2*c^2*(6-a*x)*sqrt(1-a^2*x^2)/(a^2*x)],
[(c-c/(a^2*x^2))/exp(3*arctanh(a*x)),x,9,-3*c*arcsin(a*x)/a-3*c*arctanh(sqrt(1-a^2*x^2))/a-c*sqrt(1-a^2*x^2)/a+c*sqrt(1-a^2*x^2)/(a^2*x)],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))),x,7,1/3*(1-a*x)^3/(a*c*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c)-2*(1-a*x)^2/(a*c*sqrt(1-a^2*x^2))-3*sqrt(1-a^2*x^2)/(a*c)],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^2),x,7,-1/5*(1-a*x)^3/(a*c^2*(1-a^2*x^2)^(5/2))+6/5*(1-a*x)^2/(a*c^2*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^2)-24/5*(1-a*x)/(a*c^2*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c^2)],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^3),x,8,1/7*(1-a*x)^3/(a*c^3*(1-a^2*x^2)^(7/2))-38/35*(1-a*x)^2/(a*c^3*(1-a^2*x^2)^(5/2))+137/35*(1-a*x)/(a*c^3*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^3)+1/35*(-245+181*a*x)/(a*c^3*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c^3)],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^4),x,9,-1/9*(1-a*x)^3/(a*c^4*(1-a^2*x^2)^(9/2))+22/21*(1-a*x)^2/(a*c^4*(1-a^2*x^2)^(7/2))-478/105*(1-a*x)/(a*c^4*(1-a^2*x^2)^(5/2))+2/315*(1155-829*a*x)/(a*c^4*(1-a^2*x^2)^(3/2))-3*arcsin(a*x)/(a*c^4)-4/315*(630-431*a*x)/(a*c^4*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*c^4)],

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^(p/2)

# n>0
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^(9/2),x,4,-1/8*(c-c/(a^2*x^2))^(9/2)*x/(1-a^2*x^2)^(9/2)-1/7*a*(c-c/(a^2*x^2))^(9/2)*x^2/(1-a^2*x^2)^(9/2)+2/3*a^2*(c-c/(a^2*x^2))^(9/2)*x^3/(1-a^2*x^2)^(9/2)+4/5*a^3*(c-c/(a^2*x^2))^(9/2)*x^4/(1-a^2*x^2)^(9/2)-3/2*a^4*(c-c/(a^2*x^2))^(9/2)*x^5/(1-a^2*x^2)^(9/2)-2*a^5*(c-c/(a^2*x^2))^(9/2)*x^6/(1-a^2*x^2)^(9/2)+2*a^6*(c-c/(a^2*x^2))^(9/2)*x^7/(1-a^2*x^2)^(9/2)+4*a^7*(c-c/(a^2*x^2))^(9/2)*x^8/(1-a^2*x^2)^(9/2)+a^9*(c-c/(a^2*x^2))^(9/2)*x^10/(1-a^2*x^2)^(9/2)+a^8*(c-c/(a^2*x^2))^(9/2)*x^9*log(x)/(1-a^2*x^2)^(9/2)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^(7/2),x,4,-1/6*(c-c/(a^2*x^2))^(7/2)*x/(1-a^2*x^2)^(7/2)-1/5*a*(c-c/(a^2*x^2))^(7/2)*x^2/(1-a^2*x^2)^(7/2)+3/4*a^2*(c-c/(a^2*x^2))^(7/2)*x^3/(1-a^2*x^2)^(7/2)+a^3*(c-c/(a^2*x^2))^(7/2)*x^4/(1-a^2*x^2)^(7/2)-3/2*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/(1-a^2*x^2)^(7/2)-3*a^5*(c-c/(a^2*x^2))^(7/2)*x^6/(1-a^2*x^2)^(7/2)-a^7*(c-c/(a^2*x^2))^(7/2)*x^8/(1-a^2*x^2)^(7/2)-a^6*(c-c/(a^2*x^2))^(7/2)*x^7*log(x)/(1-a^2*x^2)^(7/2)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^(5/2),x,4,-1/4*(c-c/(a^2*x^2))^(5/2)*x/(1-a^2*x^2)^(5/2)-1/3*a*(c-c/(a^2*x^2))^(5/2)*x^2/(1-a^2*x^2)^(5/2)+a^2*(c-c/(a^2*x^2))^(5/2)*x^3/(1-a^2*x^2)^(5/2)+2*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/(1-a^2*x^2)^(5/2)+a^5*(c-c/(a^2*x^2))^(5/2)*x^6/(1-a^2*x^2)^(5/2)+a^4*(c-c/(a^2*x^2))^(5/2)*x^5*log(x)/(1-a^2*x^2)^(5/2)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^(3/2),x,4,-1/2*(c-c/(a^2*x^2))^(3/2)*x/(1-a^2*x^2)^(3/2)-a*(c-c/(a^2*x^2))^(3/2)*x^2/(1-a^2*x^2)^(3/2)-a^3*(c-c/(a^2*x^2))^(3/2)*x^4/(1-a^2*x^2)^(3/2)-a^2*(c-c/(a^2*x^2))^(3/2)*x^3*log(x)/(1-a^2*x^2)^(3/2)],
[exp(arctanh(a*x))*sqrt(c-c/(a^2*x^2)),x,4,a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))/sqrt(c-c/(a^2*x^2)),x,4,-sqrt(1-a^2*x^2)/(a*sqrt(c-c/(a^2*x^2)))-log(1-a*x)*sqrt(1-a^2*x^2)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[exp(arctanh(a*x))/(c-c/(a^2*x^2))^(3/2),x,4,(1-a^2*x^2)^(3/2)/(a^3*(c-c/(a^2*x^2))^(3/2)*x^2)+1/2*(1-a^2*x^2)^(3/2)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3*(1-a*x))+5/4*(1-a^2*x^2)^(3/2)*log(1-a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)-1/4*(1-a^2*x^2)^(3/2)*log(1+a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[exp(arctanh(a*x))/(c-c/(a^2*x^2))^(5/2),x,4,-(1-a^2*x^2)^(5/2)/(a^5*(c-c/(a^2*x^2))^(5/2)*x^4)+1/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1-a*x)^2)-(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1-a*x))+1/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1+a*x))-23/16*(1-a^2*x^2)^(5/2)*log(1-a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)+7/16*(1-a^2*x^2)^(5/2)*log(1+a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[exp(arctanh(a*x))/(c-c/(a^2*x^2))^(7/2),x,4,(1-a^2*x^2)^(7/2)/(a^7*(c-c/(a^2*x^2))^(7/2)*x^6)+1/24*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x)^3)-11/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x)^2)+3/2*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x))+1/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x)^2)-5/16*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x))+51/32*(1-a^2*x^2)^(7/2)*log(1-a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)-19/32*(1-a^2*x^2)^(7/2)*log(1+a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(9/2),x,16,295/1344*a^4*(c-c/(a^2*x^2))^(9/2)*x^5/(1-a*x)^4-501/128*a^8*(c-c/(a^2*x^2))^(9/2)*x^9/((1-a*x)^4*(1+a*x)^4)+373/192*a^7*(c-c/(a^2*x^2))^(9/2)*x^8/((1-a*x)^4*(1+a*x)^3)+501/640*a^6*(c-c/(a^2*x^2))^(9/2)*x^7/((1-a*x)^4*(1+a*x)^2)+661/1680*a^5*(c-c/(a^2*x^2))^(9/2)*x^6/((1-a*x)^4*(1+a*x))-127/420*a^3*(c-c/(a^2*x^2))^(9/2)*x^4*(1+a*x)/(1-a*x)^4+71/336*a^2*(c-c/(a^2*x^2))^(9/2)*x^3*(1+a*x)/(1-a*x)^3-1/28*a*(c-c/(a^2*x^2))^(9/2)*x^2*(1+a*x)/(1-a*x)^2-1/8*(c-c/(a^2*x^2))^(9/2)*x*(1+a*x)/(1-a*x)+2*a^8*(c-c/(a^2*x^2))^(9/2)*x^9*arcsin(a*x)/((1-a*x)^(9/2)*(1+a*x)^(9/2))+245/128*a^8*(c-c/(a^2*x^2))^(9/2)*x^9*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(9/2)*(1+a*x)^(9/2))],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(7/2),x,14,-11/30*a^3*(c-c/(a^2*x^2))^(7/2)*x^4/(1-a*x)^3+57/16*a^6*(c-c/(a^2*x^2))^(7/2)*x^7/((1-a*x)^3*(1+a*x)^3)-41/24*a^5*(c-c/(a^2*x^2))^(7/2)*x^6/((1-a*x)^3*(1+a*x)^2)-57/80*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/((1-a*x)^3*(1+a*x))+13/40*a^2*(c-c/(a^2*x^2))^(7/2)*x^3*(1+a*x)/(1-a*x)^3-1/15*a*(c-c/(a^2*x^2))^(7/2)*x^2*(1+a*x)/(1-a*x)^2-1/6*(c-c/(a^2*x^2))^(7/2)*x*(1+a*x)/(1-a*x)-2*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*arcsin(a*x)/((1-a*x)^(7/2)*(1+a*x)^(7/2))-25/16*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(7/2)*(1+a*x)^(7/2))],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(5/2),x,12,5/8*a^2*(c-c/(a^2*x^2))^(5/2)*x^3/(1-a*x)^2-25/8*a^4*(c-c/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(1+a*x)^2)+17/12*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/((1-a*x)^2*(1+a*x))-1/6*a*(c-c/(a^2*x^2))^(5/2)*x^2*(1+a*x)/(1-a*x)^2-1/4*(c-c/(a^2*x^2))^(5/2)*x*(1+a*x)/(1-a*x)+2*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*arcsin(a*x)/((1-a*x)^(5/2)*(1+a*x)^(5/2))+9/8*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(5/2)*(1+a*x)^(5/2))],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(3/2),x,10,-a*(c-c/(a^2*x^2))^(3/2)*x^2/(1-a*x)+5/2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(1+a*x))-1/2*(c-c/(a^2*x^2))^(3/2)*x*(1+a*x)/(1-a*x)-2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*arcsin(a*x)/((1-a*x)^(3/2)*(1+a*x)^(3/2))-1/2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(3/2)*(1+a*x)^(3/2))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2)),x,8,-x*sqrt(c-c/(a^2*x^2))+2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))-x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))/sqrt(c-c/(a^2*x^2)),x,6,2*(1-a*x)*(1+a*x)/(a^2*x*sqrt(c-c/(a^2*x^2)))+(1+a*x)^2/(a^2*x*sqrt(c-c/(a^2*x^2)))-2*arcsin(a*x)*sqrt(1-a*x)*sqrt(1+a*x)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^(3/2),x,6,1/3*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(3/2)*x)-2/3*(5-2*a*x)*(1-a*x)*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)+2*(1-a*x)^(3/2)*(1+a*x)^(3/2)*arcsin(a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^(5/2),x,8,1/5*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(5/2)*x)-2/3*(1-a*x)*(1+a*x)^2/(a^3*(c-c/(a^2*x^2))^(5/2)*x^2)+58/15*(1-a*x)^2*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(5/2)*x^3)+2/15*(1-a*x)^3*(1+a*x)^2*(28+43*a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)-2*(1-a*x)^(5/2)*(1+a*x)^(5/2)*arcsin(a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^(7/2),x,10,1/7*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(7/2)*x)-2/5*(1-a*x)*(1+a*x)^2/(a^3*(c-c/(a^2*x^2))^(7/2)*x^2)+124/105*(1-a*x)^2*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(7/2)*x^3)-782/105*(1-a*x)^3*(1+a*x)^2/(a^5*(c-c/(a^2*x^2))^(7/2)*x^4)-142/35*(1-a*x)^4*(1+a*x)^2/(a^6*(c-c/(a^2*x^2))^(7/2)*x^5)-2/35*(1-a*x)^4*(1+a*x)^3*(72+107*a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)+2*(1-a*x)^(7/2)*(1+a*x)^(7/2)*arcsin(a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],
[exp(2*arctanh(a*x))/(c-c/(a^2*x^2))^(9/2),x,12,1/9*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(9/2)*x)-2/7*(1-a*x)*(1+a*x)^2/(a^3*(c-c/(a^2*x^2))^(9/2)*x^2)+214/315*(1-a*x)^2*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(9/2)*x^3)-646/315*(1-a*x)^3*(1+a*x)^2/(a^5*(c-c/(a^2*x^2))^(9/2)*x^4)+302/21*(1-a*x)^4*(1+a*x)^2/(a^6*(c-c/(a^2*x^2))^(9/2)*x^5)+2458/315*(1-a*x)^5*(1+a*x)^2/(a^7*(c-c/(a^2*x^2))^(9/2)*x^6)+1334/315*(1-a*x)^5*(1+a*x)^3/(a^8*(c-c/(a^2*x^2))^(9/2)*x^7)+2/315*(1-a*x)^5*(1+a*x)^4*(704+1019*a*x)/(a^10*(c-c/(a^2*x^2))^(9/2)*x^9)-2*(1-a*x)^(9/2)*(1+a*x)^(9/2)*arcsin(a*x)/(a^10*(c-c/(a^2*x^2))^(9/2)*x^9)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(9/2),x,4,-1/8*(c-c/(a^2*x^2))^(9/2)*x/(1-a^2*x^2)^(9/2)-3/7*a*(c-c/(a^2*x^2))^(9/2)*x^2/(1-a^2*x^2)^(9/2)+8/5*a^3*(c-c/(a^2*x^2))^(9/2)*x^4/(1-a^2*x^2)^(9/2)+3/2*a^4*(c-c/(a^2*x^2))^(9/2)*x^5/(1-a^2*x^2)^(9/2)-2*a^5*(c-c/(a^2*x^2))^(9/2)*x^6/(1-a^2*x^2)^(9/2)-4*a^6*(c-c/(a^2*x^2))^(9/2)*x^7/(1-a^2*x^2)^(9/2)-a^9*(c-c/(a^2*x^2))^(9/2)*x^10/(1-a^2*x^2)^(9/2)-3*a^8*(c-c/(a^2*x^2))^(9/2)*x^9*log(x)/(1-a^2*x^2)^(9/2)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(7/2),x,4,-1/6*(c-c/(a^2*x^2))^(7/2)*x/(1-a^2*x^2)^(7/2)-3/5*a*(c-c/(a^2*x^2))^(7/2)*x^2/(1-a^2*x^2)^(7/2)-1/4*a^2*(c-c/(a^2*x^2))^(7/2)*x^3/(1-a^2*x^2)^(7/2)+5/3*a^3*(c-c/(a^2*x^2))^(7/2)*x^4/(1-a^2*x^2)^(7/2)+5/2*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/(1-a^2*x^2)^(7/2)-a^5*(c-c/(a^2*x^2))^(7/2)*x^6/(1-a^2*x^2)^(7/2)+a^7*(c-c/(a^2*x^2))^(7/2)*x^8/(1-a^2*x^2)^(7/2)+3*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*log(x)/(1-a^2*x^2)^(7/2)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(5/2),x,4,-1/4*(c-c/(a^2*x^2))^(5/2)*x/(1-a^2*x^2)^(5/2)-a*(c-c/(a^2*x^2))^(5/2)*x^2/(1-a^2*x^2)^(5/2)-a^2*(c-c/(a^2*x^2))^(5/2)*x^3/(1-a^2*x^2)^(5/2)+2*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/(1-a^2*x^2)^(5/2)-a^5*(c-c/(a^2*x^2))^(5/2)*x^6/(1-a^2*x^2)^(5/2)-3*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*log(x)/(1-a^2*x^2)^(5/2)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(3/2),x,4,-1/2*(c-c/(a^2*x^2))^(3/2)*x/(1-a^2*x^2)^(3/2)-3*a*(c-c/(a^2*x^2))^(3/2)*x^2/(1-a^2*x^2)^(3/2)+a^3*(c-c/(a^2*x^2))^(3/2)*x^4/(1-a^2*x^2)^(3/2)+3*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*log(x)/(1-a^2*x^2)^(3/2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2)),x,4,-a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))/sqrt(c-c/(a^2*x^2)),x,4,sqrt(1-a^2*x^2)/(a*sqrt(c-c/(a^2*x^2)))+2*sqrt(1-a^2*x^2)/(a^2*x*(1-a*x)*sqrt(c-c/(a^2*x^2)))+3*log(1-a*x)*sqrt(1-a^2*x^2)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2))^(3/2),x,4,-(1-a^2*x^2)^(3/2)/(a^3*(c-c/(a^2*x^2))^(3/2)*x^2)+1/2*(1-a^2*x^2)^(3/2)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3*(1-a*x)^2)-3*(1-a^2*x^2)^(3/2)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3*(1-a*x))-3*(1-a^2*x^2)^(3/2)*log(1-a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2))^(5/2),x,4,(1-a^2*x^2)^(5/2)/(a^5*(c-c/(a^2*x^2))^(5/2)*x^4)+1/6*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1-a*x)^3)-9/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1-a*x)^2)+31/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1-a*x))+49/16*(1-a^2*x^2)^(5/2)*log(1-a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)-1/16*(1-a^2*x^2)^(5/2)*log(1+a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[exp(3*arctanh(a*x))/(c-c/(a^2*x^2))^(7/2),x,4,-(1-a^2*x^2)^(7/2)/(a^7*(c-c/(a^2*x^2))^(7/2)*x^6)+1/16*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x)^4)-1/2*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x)^3)+59/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x)^2)-75/16*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x))+1/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x))-201/64*(1-a^2*x^2)^(7/2)*log(1-a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)+9/64*(1-a^2*x^2)^(7/2)*log(1+a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],

# n<0
[(c-c/(a^2*x^2))^(9/2)/exp(arctanh(a*x)),x,4,-1/8*(c-c/(a^2*x^2))^(9/2)*x/(1-a^2*x^2)^(9/2)+1/7*a*(c-c/(a^2*x^2))^(9/2)*x^2/(1-a^2*x^2)^(9/2)+2/3*a^2*(c-c/(a^2*x^2))^(9/2)*x^3/(1-a^2*x^2)^(9/2)-4/5*a^3*(c-c/(a^2*x^2))^(9/2)*x^4/(1-a^2*x^2)^(9/2)-3/2*a^4*(c-c/(a^2*x^2))^(9/2)*x^5/(1-a^2*x^2)^(9/2)+2*a^5*(c-c/(a^2*x^2))^(9/2)*x^6/(1-a^2*x^2)^(9/2)+2*a^6*(c-c/(a^2*x^2))^(9/2)*x^7/(1-a^2*x^2)^(9/2)-4*a^7*(c-c/(a^2*x^2))^(9/2)*x^8/(1-a^2*x^2)^(9/2)-a^9*(c-c/(a^2*x^2))^(9/2)*x^10/(1-a^2*x^2)^(9/2)+a^8*(c-c/(a^2*x^2))^(9/2)*x^9*log(x)/(1-a^2*x^2)^(9/2)],
[(c-c/(a^2*x^2))^(7/2)/exp(arctanh(a*x)),x,4,-1/6*(c-c/(a^2*x^2))^(7/2)*x/(1-a^2*x^2)^(7/2)+1/5*a*(c-c/(a^2*x^2))^(7/2)*x^2/(1-a^2*x^2)^(7/2)+3/4*a^2*(c-c/(a^2*x^2))^(7/2)*x^3/(1-a^2*x^2)^(7/2)-a^3*(c-c/(a^2*x^2))^(7/2)*x^4/(1-a^2*x^2)^(7/2)-3/2*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/(1-a^2*x^2)^(7/2)+3*a^5*(c-c/(a^2*x^2))^(7/2)*x^6/(1-a^2*x^2)^(7/2)+a^7*(c-c/(a^2*x^2))^(7/2)*x^8/(1-a^2*x^2)^(7/2)-a^6*(c-c/(a^2*x^2))^(7/2)*x^7*log(x)/(1-a^2*x^2)^(7/2)],
[(c-c/(a^2*x^2))^(5/2)/exp(arctanh(a*x)),x,4,-1/4*(c-c/(a^2*x^2))^(5/2)*x/(1-a^2*x^2)^(5/2)+1/3*a*(c-c/(a^2*x^2))^(5/2)*x^2/(1-a^2*x^2)^(5/2)+a^2*(c-c/(a^2*x^2))^(5/2)*x^3/(1-a^2*x^2)^(5/2)-2*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/(1-a^2*x^2)^(5/2)-a^5*(c-c/(a^2*x^2))^(5/2)*x^6/(1-a^2*x^2)^(5/2)+a^4*(c-c/(a^2*x^2))^(5/2)*x^5*log(x)/(1-a^2*x^2)^(5/2)],
[(c-c/(a^2*x^2))^(3/2)/exp(arctanh(a*x)),x,4,-1/2*(c-c/(a^2*x^2))^(3/2)*x/(1-a^2*x^2)^(3/2)+a*(c-c/(a^2*x^2))^(3/2)*x^2/(1-a^2*x^2)^(3/2)+a^3*(c-c/(a^2*x^2))^(3/2)*x^4/(1-a^2*x^2)^(3/2)-a^2*(c-c/(a^2*x^2))^(3/2)*x^3*log(x)/(1-a^2*x^2)^(3/2)],
[sqrt(c-c/(a^2*x^2))/exp(arctanh(a*x)),x,4,-a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[1/(exp(arctanh(a*x))*sqrt(c-c/(a^2*x^2))),x,4,sqrt(1-a^2*x^2)/(a*sqrt(c-c/(a^2*x^2)))-log(1+a*x)*sqrt(1-a^2*x^2)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))^(3/2)),x,4,-(1-a^2*x^2)^(3/2)/(a^3*(c-c/(a^2*x^2))^(3/2)*x^2)+1/2*(1-a^2*x^2)^(3/2)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3*(1+a*x))-1/4*(1-a^2*x^2)^(3/2)*log(1-a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)+5/4*(1-a^2*x^2)^(3/2)*log(1+a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))^(5/2)),x,4,(1-a^2*x^2)^(5/2)/(a^5*(c-c/(a^2*x^2))^(5/2)*x^4)+1/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1-a*x))+1/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1+a*x)^2)-(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1+a*x))+7/16*(1-a^2*x^2)^(5/2)*log(1-a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)-23/16*(1-a^2*x^2)^(5/2)*log(1+a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[1/(exp(arctanh(a*x))*(c-c/(a^2*x^2))^(7/2)),x,4,-(1-a^2*x^2)^(7/2)/(a^7*(c-c/(a^2*x^2))^(7/2)*x^6)+1/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x)^2)-5/16*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x))+1/24*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x)^3)-11/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x)^2)+3/2*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x))-19/32*(1-a^2*x^2)^(7/2)*log(1-a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)+51/32*(1-a^2*x^2)^(7/2)*log(1+a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],
[(c-c/(a^2*x^2))^(9/2)/exp(2*arctanh(a*x)),x,16,11/128*a^8*(c-c/(a^2*x^2))^(9/2)*x^9/((1-a*x)^4*(1+a*x)^4)+39/64*a^7*(c-c/(a^2*x^2))^(9/2)*x^8/((1-a*x)^4*(1+a*x)^3)-11/640*a^6*(c-c/(a^2*x^2))^(9/2)*x^7/((1-a*x)^4*(1+a*x)^2)+1/28*a*(c-c/(a^2*x^2))^(9/2)*x^2/(1+a*x)-103/160*a^5*(c-c/(a^2*x^2))^(9/2)*x^6/((1-a*x)^4*(1+a*x))+629/960*a^4*(c-c/(a^2*x^2))^(9/2)*x^5/((1-a*x)^3*(1+a*x))-2/5*a^3*(c-c/(a^2*x^2))^(9/2)*x^4/((1-a*x)^2*(1+a*x))+47/336*a^2*(c-c/(a^2*x^2))^(9/2)*x^3/((1-a*x)*(1+a*x))-1/8*(c-c/(a^2*x^2))^(9/2)*x*(1-a*x)/(1+a*x)-2*a^8*(c-c/(a^2*x^2))^(9/2)*x^9*arcsin(a*x)/((1-a*x)^(9/2)*(1+a*x)^(9/2))+245/128*a^8*(c-c/(a^2*x^2))^(9/2)*x^9*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(9/2)*(1+a*x)^(9/2))],
[(c-c/(a^2*x^2))^(7/2)/exp(2*arctanh(a*x)),x,14,-7/16*a^6*(c-c/(a^2*x^2))^(7/2)*x^7/((1-a*x)^3*(1+a*x)^3)-3/8*a^5*(c-c/(a^2*x^2))^(7/2)*x^6/((1-a*x)^3*(1+a*x)^2)+1/15*a*(c-c/(a^2*x^2))^(7/2)*x^2/(1+a*x)+19/16*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/((1-a*x)^3*(1+a*x))-2/3*a^3*(c-c/(a^2*x^2))^(7/2)*x^4/((1-a*x)^2*(1+a*x))+23/120*a^2*(c-c/(a^2*x^2))^(7/2)*x^3/((1-a*x)*(1+a*x))-1/6*(c-c/(a^2*x^2))^(7/2)*x*(1-a*x)/(1+a*x)+2*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*arcsin(a*x)/((1-a*x)^(7/2)*(1+a*x)^(7/2))-25/16*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(7/2)*(1+a*x)^(7/2))],
[(c-c/(a^2*x^2))^(5/2)/exp(2*arctanh(a*x)),x,12,7/8*a^4*(c-c/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(1+a*x)^2)+1/6*a*(c-c/(a^2*x^2))^(5/2)*x^2/(1+a*x)-2*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/((1-a*x)^2*(1+a*x))+7/24*a^2*(c-c/(a^2*x^2))^(5/2)*x^3/((1-a*x)*(1+a*x))-1/4*(c-c/(a^2*x^2))^(5/2)*x*(1-a*x)/(1+a*x)-2*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*arcsin(a*x)/((1-a*x)^(5/2)*(1+a*x)^(5/2))+9/8*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(5/2)*(1+a*x)^(5/2))],
[(c-c/(a^2*x^2))^(3/2)/exp(2*arctanh(a*x)),x,10,a*(c-c/(a^2*x^2))^(3/2)*x^2/(1+a*x)+5/2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(1+a*x))-1/2*(c-c/(a^2*x^2))^(3/2)*x*(1-a*x)/(1+a*x)+2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*arcsin(a*x)/((1-a*x)^(3/2)*(1+a*x)^(3/2))-1/2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(3/2)*(1+a*x)^(3/2))],
[sqrt(c-c/(a^2*x^2))/exp(2*arctanh(a*x)),x,8,-x*sqrt(c-c/(a^2*x^2))-2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))-x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[1/(exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2))),x,6,(1-a*x)^2/(a^2*x*sqrt(c-c/(a^2*x^2)))+2*(1-a*x)*(1+a*x)/(a^2*x*sqrt(c-c/(a^2*x^2)))+2*arcsin(a*x)*sqrt(1-a*x)*sqrt(1+a*x)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(3/2)),x,6,1/3*(1-a*x)^2/(a^2*(c-c/(a^2*x^2))^(3/2)*x)-2/3*(1-a*x)^2*(1+a*x)*(5+2*a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)-2*(1-a*x)^(3/2)*(1+a*x)^(3/2)*arcsin(a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(5/2)),x,8,(1-a*x)^2/(a^2*(c-c/(a^2*x^2))^(5/2)*x)+2/5*(1-a*x)^3/(a^3*(c-c/(a^2*x^2))^(5/2)*x^2)-2/15*(1-a*x)^3*(1+a*x)/(a^4*(c-c/(a^2*x^2))^(5/2)*x^3)+2/15*(1-a*x)^3*(1+a*x)^2*(28+13*a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)+2*(1-a*x)^(5/2)*(1+a*x)^(5/2)*arcsin(a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[1/(exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^(7/2)),x,10,1/3*(1-a*x)^2/(a^2*(c-c/(a^2*x^2))^(7/2)*x)-10/3*(1-a*x)^3/(a^3*(c-c/(a^2*x^2))^(7/2)*x^2)-12/7*(1-a*x)^4/(a^4*(c-c/(a^2*x^2))^(7/2)*x^3)-82/105*(1-a*x)^4*(1+a*x)/(a^5*(c-c/(a^2*x^2))^(7/2)*x^4)-2/35*(1-a*x)^4*(1+a*x)^2/(a^6*(c-c/(a^2*x^2))^(7/2)*x^5)-2/35*(1-a*x)^4*(1+a*x)^3*(72+37*a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)-2*(1-a*x)^(7/2)*(1+a*x)^(7/2)*arcsin(a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],
[(c-c/(a^2*x^2))^(9/2)/exp(3*arctanh(a*x)),x,4,-1/8*(c-c/(a^2*x^2))^(9/2)*x/(1-a^2*x^2)^(9/2)+3/7*a*(c-c/(a^2*x^2))^(9/2)*x^2/(1-a^2*x^2)^(9/2)-8/5*a^3*(c-c/(a^2*x^2))^(9/2)*x^4/(1-a^2*x^2)^(9/2)+3/2*a^4*(c-c/(a^2*x^2))^(9/2)*x^5/(1-a^2*x^2)^(9/2)+2*a^5*(c-c/(a^2*x^2))^(9/2)*x^6/(1-a^2*x^2)^(9/2)-4*a^6*(c-c/(a^2*x^2))^(9/2)*x^7/(1-a^2*x^2)^(9/2)+a^9*(c-c/(a^2*x^2))^(9/2)*x^10/(1-a^2*x^2)^(9/2)-3*a^8*(c-c/(a^2*x^2))^(9/2)*x^9*log(x)/(1-a^2*x^2)^(9/2)],
[(c-c/(a^2*x^2))^(7/2)/exp(3*arctanh(a*x)),x,4,-1/6*(c-c/(a^2*x^2))^(7/2)*x/(1-a^2*x^2)^(7/2)+3/5*a*(c-c/(a^2*x^2))^(7/2)*x^2/(1-a^2*x^2)^(7/2)-1/4*a^2*(c-c/(a^2*x^2))^(7/2)*x^3/(1-a^2*x^2)^(7/2)-5/3*a^3*(c-c/(a^2*x^2))^(7/2)*x^4/(1-a^2*x^2)^(7/2)+5/2*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/(1-a^2*x^2)^(7/2)+a^5*(c-c/(a^2*x^2))^(7/2)*x^6/(1-a^2*x^2)^(7/2)-a^7*(c-c/(a^2*x^2))^(7/2)*x^8/(1-a^2*x^2)^(7/2)+3*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*log(x)/(1-a^2*x^2)^(7/2)],
[(c-c/(a^2*x^2))^(5/2)/exp(3*arctanh(a*x)),x,4,-1/4*(c-c/(a^2*x^2))^(5/2)*x/(1-a^2*x^2)^(5/2)+a*(c-c/(a^2*x^2))^(5/2)*x^2/(1-a^2*x^2)^(5/2)-a^2*(c-c/(a^2*x^2))^(5/2)*x^3/(1-a^2*x^2)^(5/2)-2*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/(1-a^2*x^2)^(5/2)+a^5*(c-c/(a^2*x^2))^(5/2)*x^6/(1-a^2*x^2)^(5/2)-3*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*log(x)/(1-a^2*x^2)^(5/2)],
[(c-c/(a^2*x^2))^(3/2)/exp(3*arctanh(a*x)),x,4,-1/2*(c-c/(a^2*x^2))^(3/2)*x/(1-a^2*x^2)^(3/2)+3*a*(c-c/(a^2*x^2))^(3/2)*x^2/(1-a^2*x^2)^(3/2)-a^3*(c-c/(a^2*x^2))^(3/2)*x^4/(1-a^2*x^2)^(3/2)+3*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*log(x)/(1-a^2*x^2)^(3/2)],
[sqrt(c-c/(a^2*x^2))/exp(3*arctanh(a*x)),x,4,a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[1/(exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2))),x,4,-sqrt(1-a^2*x^2)/(a*sqrt(c-c/(a^2*x^2)))+2*sqrt(1-a^2*x^2)/(a^2*x*(1+a*x)*sqrt(c-c/(a^2*x^2)))+3*log(1+a*x)*sqrt(1-a^2*x^2)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(3/2)),x,4,(1-a^2*x^2)^(3/2)/(a^3*(c-c/(a^2*x^2))^(3/2)*x^2)+1/2*(1-a^2*x^2)^(3/2)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3*(1+a*x)^2)-3*(1-a^2*x^2)^(3/2)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3*(1+a*x))-3*(1-a^2*x^2)^(3/2)*log(1+a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(5/2)),x,4,-(1-a^2*x^2)^(5/2)/(a^5*(c-c/(a^2*x^2))^(5/2)*x^4)+1/6*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1+a*x)^3)-9/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1+a*x)^2)+31/8*(1-a^2*x^2)^(5/2)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5*(1+a*x))-1/16*(1-a^2*x^2)^(5/2)*log(1-a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)+49/16*(1-a^2*x^2)^(5/2)*log(1+a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[1/(exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^(7/2)),x,4,(1-a^2*x^2)^(7/2)/(a^7*(c-c/(a^2*x^2))^(7/2)*x^6)+1/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1-a*x))+1/16*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x)^4)-1/2*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x)^3)+59/32*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x)^2)-75/16*(1-a^2*x^2)^(7/2)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7*(1+a*x))+9/64*(1-a^2*x^2)^(7/2)*log(1-a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)-201/64*(1-a^2*x^2)^(7/2)*log(1+a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^(p/2)

# n>0
[exp(arctanh(a*x))*x^m*sqrt(c-c/(a^2*x^2)),x,4,x^(1+m)*sqrt(c-c/(a^2*x^2))/(m*sqrt(1-a^2*x^2))+a*x^(2+m)*sqrt(c-c/(a^2*x^2))/((1+m)*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^2*sqrt(c-c/(a^2*x^2)),x,4,1/2*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+1/3*a*x^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*x*sqrt(c-c/(a^2*x^2)),x,3,x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+1/2*a*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*sqrt(c-c/(a^2*x^2)),x,4,a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x,x,4,-sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+a*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^2,x,3,-1/2*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))],
[exp(2*arctanh(a*x))*x^3*sqrt(c-c/(a^2*x^2)),x,8,-7/8*x*sqrt(c-c/(a^2*x^2))/a^3-7/24*x*(1+a*x)*sqrt(c-c/(a^2*x^2))/a^3-1/6*x*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/a^3-1/4*x^2*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/a^2+7/8*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*x^2*sqrt(c-c/(a^2*x^2)),x,7,-x*sqrt(c-c/(a^2*x^2))/a^2-1/3*x*(1+a*x)*sqrt(c-c/(a^2*x^2))/a^2-1/3*x*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/a^2+x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*x*sqrt(c-c/(a^2*x^2)),x,6,-3/2*x*sqrt(c-c/(a^2*x^2))/a-1/2*x*(1+a*x)*sqrt(c-c/(a^2*x^2))/a+3/2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2)),x,8,-x*sqrt(c-c/(a^2*x^2))+2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))-x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x,x,8,-sqrt(c-c/(a^2*x^2))+a*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))-2*a*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^2,x,6,-3/2*a*sqrt(c-c/(a^2*x^2))-1/2*(1+a*x)*sqrt(c-c/(a^2*x^2))/x-3/2*a^2*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^3,x,7,-a^2*sqrt(c-c/(a^2*x^2))-1/3*a*(1+a*x)*sqrt(c-c/(a^2*x^2))/x-1/3*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/x^2-a^3*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^4,x,9,-4/3*a^3*sqrt(c-c/(a^2*x^2))-1/4*sqrt(c-c/(a^2*x^2))/x^3-2/3*a*sqrt(c-c/(a^2*x^2))/x^2-7/8*a^2*sqrt(c-c/(a^2*x^2))/x-7/8*a^4*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^5,x,10,-6/5*a^4*sqrt(c-c/(a^2*x^2))-1/5*sqrt(c-c/(a^2*x^2))/x^4-1/2*a*sqrt(c-c/(a^2*x^2))/x^3-3/5*a^2*sqrt(c-c/(a^2*x^2))/x^2-3/4*a^3*sqrt(c-c/(a^2*x^2))/x-3/4*a^5*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(3*arctanh(a*x))*x^3*sqrt(c-c/(a^2*x^2)),x,4,-4*x^2*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a^2*x^2))-2*x^3*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a^2*x^2))-x^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/4*a*x^5*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*x^2*sqrt(c-c/(a^2*x^2)),x,4,-4*x^2*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a^2*x^2))-3/2*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/3*a*x^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*x*sqrt(c-c/(a^2*x^2)),x,4,-3*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/2*a*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2)),x,4,-a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x,x,4,-sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+3*a*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^2,x,4,-3*a*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))+4*a^2*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a^2*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^3,x,4,-4*a^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/3*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1-a^2*x^2))-3/2*a*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))+4*a^3*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a^3*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^4,x,4,-4*a^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/4*sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1-a^2*x^2))-a*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1-a^2*x^2))-2*a^2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))+4*a^4*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a^4*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*sqrt(c-c/(a^2*x^2))/x^5,x,4,-4*a^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/5*sqrt(c-c/(a^2*x^2))/(x^4*sqrt(1-a^2*x^2))-3/4*a*sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1-a^2*x^2))-4/3*a^2*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1-a^2*x^2))-2*a^3*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))+4*a^5*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a^5*x*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],

# n<0
[x^m*sqrt(c-c/(a^2*x^2))/exp(arctanh(a*x)),x,4,x^(1+m)*sqrt(c-c/(a^2*x^2))/(m*sqrt(1-a^2*x^2))-a*x^(2+m)*sqrt(c-c/(a^2*x^2))/((1+m)*sqrt(1-a^2*x^2))],
[x^2*sqrt(c-c/(a^2*x^2))/exp(arctanh(a*x)),x,4,1/2*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/3*a*x^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[x*sqrt(c-c/(a^2*x^2))/exp(arctanh(a*x)),x,3,x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/2*a*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/exp(arctanh(a*x)),x,4,-a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(arctanh(a*x))*x),x,4,-sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-a*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(arctanh(a*x))*x^2),x,3,-1/2*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))],
[x^3*sqrt(c-c/(a^2*x^2))/exp(2*arctanh(a*x)),x,8,7/8*x*sqrt(c-c/(a^2*x^2))/a^3+7/24*x*(1-a*x)*sqrt(c-c/(a^2*x^2))/a^3+1/6*x*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/a^3-1/4*x^2*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/a^2+7/8*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1-a*x)*sqrt(1+a*x))],
[x^2*sqrt(c-c/(a^2*x^2))/exp(2*arctanh(a*x)),x,7,-x*sqrt(c-c/(a^2*x^2))/a^2-1/3*x*(1-a*x)*sqrt(c-c/(a^2*x^2))/a^2-1/3*x*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/a^2-x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a*x)*sqrt(1+a*x))],
[x*sqrt(c-c/(a^2*x^2))/exp(2*arctanh(a*x)),x,6,3/2*x*sqrt(c-c/(a^2*x^2))/a+1/2*x*(1-a*x)*sqrt(c-c/(a^2*x^2))/a+3/2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/exp(2*arctanh(a*x)),x,8,-x*sqrt(c-c/(a^2*x^2))-2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))-x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arctanh(a*x))*x),x,8,-sqrt(c-c/(a^2*x^2))+a*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))+2*a*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arctanh(a*x))*x^2),x,6,3/2*a*sqrt(c-c/(a^2*x^2))-1/2*(1-a*x)*sqrt(c-c/(a^2*x^2))/x-3/2*a^2*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arctanh(a*x))*x^3),x,7,-a^2*sqrt(c-c/(a^2*x^2))+1/3*a*(1-a*x)*sqrt(c-c/(a^2*x^2))/x-1/3*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/x^2+a^3*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arctanh(a*x))*x^4),x,9,4/3*a^3*sqrt(c-c/(a^2*x^2))-1/4*sqrt(c-c/(a^2*x^2))/x^3+2/3*a*sqrt(c-c/(a^2*x^2))/x^2-7/8*a^2*sqrt(c-c/(a^2*x^2))/x-7/8*a^4*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arctanh(a*x))*x^5),x,10,-6/5*a^4*sqrt(c-c/(a^2*x^2))-1/5*sqrt(c-c/(a^2*x^2))/x^4+1/2*a*sqrt(c-c/(a^2*x^2))/x^3-3/5*a^2*sqrt(c-c/(a^2*x^2))/x^2+3/4*a^3*sqrt(c-c/(a^2*x^2))/x+3/4*a^5*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[x^3*sqrt(c-c/(a^2*x^2))/exp(3*arctanh(a*x)),x,4,-4*x^2*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a^2*x^2))+2*x^3*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a^2*x^2))-x^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+1/4*a*x^5*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+4*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1-a^2*x^2))],
[x^2*sqrt(c-c/(a^2*x^2))/exp(3*arctanh(a*x)),x,4,4*x^2*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a^2*x^2))-3/2*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+1/3*a*x^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a^2*x^2))],
[x*sqrt(c-c/(a^2*x^2))/exp(3*arctanh(a*x)),x,4,-3*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+1/2*a*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+4*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/exp(3*arctanh(a*x)),x,4,a*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(3*arctanh(a*x))*x),x,4,-sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-3*a*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+4*a*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(3*arctanh(a*x))*x^2),x,4,3*a*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))+4*a^2*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a^2*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(3*arctanh(a*x))*x^3),x,4,-4*a^2*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/3*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1-a^2*x^2))+3/2*a*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))-4*a^3*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+4*a^3*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(3*arctanh(a*x))*x^4),x,4,4*a^3*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/4*sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1-a^2*x^2))+a*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1-a^2*x^2))-2*a^2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))+4*a^4*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-4*a^4*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],
[sqrt(c-c/(a^2*x^2))/(exp(3*arctanh(a*x))*x^5),x,4,-4*a^4*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)-1/5*sqrt(c-c/(a^2*x^2))/(x^4*sqrt(1-a^2*x^2))+3/4*a*sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1-a^2*x^2))-4/3*a^2*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1-a^2*x^2))+2*a^3*sqrt(c-c/(a^2*x^2))/(x*sqrt(1-a^2*x^2))-4*a^5*x*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)+4*a^5*x*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1-a^2*x^2)],

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p with n symbolic
[(c-c/(a^2*x^2))^p/exp(2*p*arctanh(a*x)),x,3,(c-c/(a^2*x^2))^p*x*hypergeom([1-2*p,-2*p],[2-2*p],a*x)/((1-2*p)*(1-a^2*x^2)^p)],
[exp(2*p*arctanh(a*x))*(c-c/(a^2*x^2))^p,x,3,(c-c/(a^2*x^2))^p*x*hypergeom([1-2*p,-2*p],[2-2*p],-a*x)/((1-2*p)*(1-a^2*x^2)^p)],
[exp(n*arctanh(a*x))*(c-c/(a^2*x^2))^2,x,10,-4*c^2*(1-a*x)^(3-1/2*n)*(1+a*x)^(1/2*(-4+n))/(a*(4-n))-1/3*c^2*(1-a*x)^(3-1/2*n)*(1+a*x)^(1/2*(-4+n))/(a^4*x^3)-1/6*c^2*(10+n)*(1-a*x)^(3-1/2*n)*(1+a*x)^(1/2*(-4+n))/(a^3*x^2)-1/6*c^2*(14+5*n+n^2)*(1-a*x)^(3-1/2*n)*(1+a*x)^(1/2*(-4+n))/(a^2*x)-1/3*c^2*n*(10-n^2)*(1-a*x)^(2-1/2*n)*(1+a*x)^(1/2*(-4+n))*hypergeom([1,1/2*(-4+n)],[1/2*(-2+n)],(1+a*x)/(1-a*x))/(a*(4-n))+2^(-1+1/2*n)*c^2*n*(1-a*x)^(3-1/2*n)*hypergeom([1/2*(4-n),3-1/2*n],[4-1/2*n],1/2*(1-a*x))/(a*(24-10*n+n^2))],
[exp(n*arctanh(a*x))*(c-c/(a^2*x^2)),x,5,4*c*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(1-a*x)/(1+a*x))/(a*(2-n))-2^(1+1/2*n)*c*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a*(2-n))],
[exp(n*arctanh(a*x))/(c-c/(a^2*x^2)),x,5,-(1-n)*(1+a*x)^(1/2*n)/(a*c*n*(1-a*x)^(1/2*n))+x*(1+a*x)^(1/2*n)/(c*(1-a*x)^(1/2*n))-2^(1+1/2*n)*hypergeom([-1/2*n,-1/2*n],[1-1/2*n],1/2*(1-a*x))/(a*c*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(c-c/(a^2*x^2))^2,x,11,(1-n)*(3+n)*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a*c^2*(2-n))+(3+n)*x*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/c^2-a^2*x^3*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/c^2+(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a*c^2*(2-n))-(1+a*x)^(1/2*(-2+n))/(a*c^2*(1-a*x)^(1/2*n))-(3+n)*(2-n^2)*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*n)/(a*c^2*(4-n^2))-(3+n)*(2-n^2)*(1+a*x)^(1/2*n)/(a*c^2*n*(4-n^2)*(1-a*x)^(1/2*n))-2^(1/2*n)*n*(1-a*x)^(1-1/2*n)*hypergeom([1/2*(2-n),1-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a*c^2*(2-n))],
[exp(n*arctanh(a*x))*(c-c/(a^2*x^2))^(3/2),x,9,-1/2*(c-c/(a^2*x^2))^(3/2)*x*(1-a*x)^(1/2*(5-n))*(1+a*x)^(1/2*(-3+n))/(1-a^2*x^2)^(3/2)-1/2*a*(4+n)*(c-c/(a^2*x^2))^(3/2)*x^2*(1-a*x)^(1/2*(5-n))*(1+a*x)^(1/2*(-3+n))/(1-a^2*x^2)^(3/2)-3*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*(1-a*x)^(1/2*(5-n))*(1+a*x)^(1/2*(-3+n))/((3-n)*(1-a^2*x^2)^(3/2))-a^2*(3-n^2)*(c-c/(a^2*x^2))^(3/2)*x^3*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-3+n))*hypergeom([1,1/2*(-3+n)],[1/2*(-1+n)],(1+a*x)/(1-a*x))/((3-n)*(1-a^2*x^2)^(3/2))+2^(1/2*(-1+n))*a^2*n*(c-c/(a^2*x^2))^(3/2)*x^3*(1-a*x)^(1/2*(5-n))*hypergeom([1/2*(3-n),1/2*(5-n)],[1/2*(7-n)],1/2*(1-a*x))/((3-n)*(5-n)*(1-a^2*x^2)^(3/2))],
[exp(n*arctanh(a*x))*(c-c/(a^2*x^2))^(1/2),x,6,-x*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-1+n))*sqrt(c-c/(a^2*x^2))/((1-n)*sqrt(1-a^2*x^2))+2*x*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(c-c/(a^2*x^2))/((1-n)*sqrt(1-a^2*x^2))+2^(1/2*(1+n))*n*x*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(c-c/(a^2*x^2))/((3-4*n+n^2)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))/(c-c/(a^2*x^2))^(1/2),x,4,-(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(a^2*(1+n)*x*sqrt(c-c/(a^2*x^2)))-2^(1/2*(3+n))*n*(1-a*x)^(1/2*(1-n))*hypergeom([1/2*(-1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a^2*(1-n^2)*x*sqrt(c-c/(a^2*x^2)))],
[exp(n*arctanh(a*x))/(c-c/(a^2*x^2))^(3/2),x,5,-(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*(1-a^2*x^2)^(3/2)/(a^2*(c-c/(a^2*x^2))^(3/2)*x)+(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*(2+2*n+n^2-a*n*(3+2*n)*x)*(1-a^2*x^2)^(3/2)/(a^4*(1-n^2)*(c-c/(a^2*x^2))^(3/2)*x^3)-2^(1/2*(-1+n))*n*(1-a*x)^(1/2*(3-n))*(1-a^2*x^2)^(3/2)*hypergeom([1/2*(3-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))/(a^4*(3-n)*(c-c/(a^2*x^2))^(3/2)*x^3)],
[exp(n*arctanh(a*x))/(c-c/(a^2*x^2))^(5/2),x,18,(4+n)*(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^3*(3+n)*(c-c/(a^2*x^2))^(5/2)*x^2)-(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^2*(c-c/(a^2*x^2))^(5/2)*x)+n*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^6*(1+n)*(c-c/(a^2*x^2))^(5/2)*x^5)-3*(2-n)*(4+n)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^6*(9-n^2)*(c-c/(a^2*x^2))^(5/2)*x^5)-3*(4+n)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^5*(3+n)*(c-c/(a^2*x^2))^(5/2)*x^4)-2*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^6*(1-n^2)*(c-c/(a^2*x^2))^(5/2)*x^5)+2*n*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-3+n))*(1-a^2*x^2)^(5/2)/(a^6*(3-n-3*n^2+n^3)*(c-c/(a^2*x^2))^(5/2)*x^5)-3*n*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*(1-a^2*x^2)^(5/2)/(a^6*(1+n)*(c-c/(a^2*x^2))^(5/2)*x^5)+3*(4+n)*(1+2*n-n^2)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*(1-a^2*x^2)^(5/2)/(a^6*(9+9*n-n^2-n^3)*(c-c/(a^2*x^2))^(5/2)*x^5)+3*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*(1-a^2*x^2)^(5/2)/(a^6*(1-n^2)*(c-c/(a^2*x^2))^(5/2)*x^5)-3*(4+n)*(1+2*n-n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*(1-a^2*x^2)^(5/2)/(a^6*(9-10*n^2+n^4)*(c-c/(a^2*x^2))^(5/2)*x^5)+3*n*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(1+n))*(1-a^2*x^2)^(5/2)/(a^6*(1+n)*(c-c/(a^2*x^2))^(5/2)*x^5)-2^(1/2*(3+n))*n*(1-a*x)^(1/2*(-1-n))*(1-a^2*x^2)^(5/2)*hypergeom([1/2*(-1-n),1/2*(-1-n)],[1/2*(1-n)],1/2*(1-a*x))/(a^6*(1+n)*(c-c/(a^2*x^2))^(5/2)*x^5)],

# Integrands of the form E^(n ArcTanh[a x]) (c-c/(a^2 x^2))^p with p symbolic
[exp(n*arctanh(a*x))*(c-c/(a^2*x^2))^p,x,3,(c-c/(a^2*x^2))^p*x*AppellF1(1-2*p,1/2*(n-2*p),-1/2*n-p,2-2*p,a*x,-a*x)/((1-2*p)*(1-a^2*x^2)^p)],
[exp(4*arctanh(a*x))*(c-c/(a^2*x^2))^p,x,13,2*a*(c-c/(a^2*x^2))^p*x^2/((1-p)*(1-a*x)*(1+a*x))+(c-c/(a^2*x^2))^p*x*hypergeom([1/2*(1-2*p),2-p],[1/2*(3-2*p)],a^2*x^2)/((1-2*p)*(1-a*x)^p*(1+a*x)^p)+6*a^2*(c-c/(a^2*x^2))^p*x^3*hypergeom([1/2*(3-2*p),2-p],[1/2*(5-2*p)],a^2*x^2)/((3-2*p)*(1-a*x)^p*(1+a*x)^p)+a^4*(c-c/(a^2*x^2))^p*x^5*hypergeom([1/2*(5-2*p),2-p],[1/2*(7-2*p)],a^2*x^2)/((5-2*p)*(1-a*x)^p*(1+a*x)^p)+2*a^3*(c-c/(a^2*x^2))^p*x^4*hypergeom([2-p,2-p],[3-p],a^2*x^2)/((2-p)*(1-a*x)^p*(1+a*x)^p)],
[exp(3*arctanh(a*x))*(c-c/(a^2*x^2))^p,x,7,3*a^2*(c-c/(a^2*x^2))^p*x^3*hypergeom([1/2*(3-2*p),3/2-p],[1/2*(5-2*p)],a^2*x^2)/((3-2*p)*(1-a^2*x^2)^p)+1/2*a*(5-2*p)*(c-c/(a^2*x^2))^p*x^2*hypergeom([1-p,3/2-p],[2-p],a^2*x^2)/((1-p)*(1-a^2*x^2)^p)+(c-c/(a^2*x^2))^p*x/((1-2*p)*sqrt(1-a^2*x^2))-a*(c-c/(a^2*x^2))^p*x^2/sqrt(1-a^2*x^2)],
[exp(2*arctanh(a*x))*(c-c/(a^2*x^2))^p,x,10,(c-c/(a^2*x^2))^p*x*hypergeom([1/2*(1-2*p),1-p],[1/2*(3-2*p)],a^2*x^2)/((1-2*p)*(1-a*x)^p*(1+a*x)^p)+a^2*(c-c/(a^2*x^2))^p*x^3*hypergeom([1/2*(3-2*p),1-p],[1/2*(5-2*p)],a^2*x^2)/((3-2*p)*(1-a*x)^p*(1+a*x)^p)+a*(c-c/(a^2*x^2))^p*x^2*hypergeom([1-p,1-p],[2-p],a^2*x^2)/((1-p)*(1-a*x)^p*(1+a*x)^p)],
[exp(arctanh(a*x))*(c-c/(a^2*x^2))^p,x,5,(c-c/(a^2*x^2))^p*x*hypergeom([1/2*(1-2*p),1/2-p],[1/2*(3-2*p)],a^2*x^2)/((1-2*p)*(1-a^2*x^2)^p)+1/2*a*(c-c/(a^2*x^2))^p*x^2*hypergeom([1/2-p,1-p],[2-p],a^2*x^2)/((1-p)*(1-a^2*x^2)^p)],
[(c-c/(a^2*x^2))^p/exp(arctanh(a*x)),x,5,(c-c/(a^2*x^2))^p*x*hypergeom([1/2*(1-2*p),1/2-p],[1/2*(3-2*p)],a^2*x^2)/((1-2*p)*(1-a^2*x^2)^p)-1/2*a*(c-c/(a^2*x^2))^p*x^2*hypergeom([1/2-p,1-p],[2-p],a^2*x^2)/((1-p)*(1-a^2*x^2)^p)],
[(c-c/(a^2*x^2))^p/exp(2*arctanh(a*x)),x,10,(c-c/(a^2*x^2))^p*x*hypergeom([1/2*(1-2*p),1-p],[1/2*(3-2*p)],a^2*x^2)/((1-2*p)*(1-a*x)^p*(1+a*x)^p)+a^2*(c-c/(a^2*x^2))^p*x^3*hypergeom([1/2*(3-2*p),1-p],[1/2*(5-2*p)],a^2*x^2)/((3-2*p)*(1-a*x)^p*(1+a*x)^p)-a*(c-c/(a^2*x^2))^p*x^2*hypergeom([1-p,1-p],[2-p],a^2*x^2)/((1-p)*(1-a*x)^p*(1+a*x)^p)],
[(c-c/(a^2*x^2))^p/exp(3*arctanh(a*x)),x,7,3*a^2*(c-c/(a^2*x^2))^p*x^3*hypergeom([1/2*(3-2*p),3/2-p],[1/2*(5-2*p)],a^2*x^2)/((3-2*p)*(1-a^2*x^2)^p)-1/2*a*(5-2*p)*(c-c/(a^2*x^2))^p*x^2*hypergeom([1-p,3/2-p],[2-p],a^2*x^2)/((1-p)*(1-a^2*x^2)^p)+(c-c/(a^2*x^2))^p*x/((1-2*p)*sqrt(1-a^2*x^2))+a*(c-c/(a^2*x^2))^p*x^2/sqrt(1-a^2*x^2)],

# Integrands of the form u Sin[a x] E^(n ArcTanh[a x])

# Integrands of the form x^m Sin[a x] E^ArcTanh[a x] (1+a x)^n

# n/2>0
[exp(arctanh(x))*x*(1+x)^(1/2)*sin(x),x,16,-(1-x)^(3/2)*cos(x)-3*cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)-3/2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)+3/2*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)-3*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)+2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)-2*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)+3*cos(x)*sqrt(1-x)-3/2*sin(x)*sqrt(1-x)],
[exp(arctanh(x))*(1+x)^(1/2)*sin(x),x,11,-cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)-FresnelS(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)+2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)-2*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)+cos(x)*sqrt(1-x)],
[exp(arctanh(x))*(1-x)^(1/2)*x*sin(x),x,13,-(1+x)^(3/2)*cos(x)-cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sqrt(1/2*Pi)-3/2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1+x))*sqrt(1/2*Pi)+3/2*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(1/2*Pi)-FresnelS(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(1/2*Pi)+cos(x)*sqrt(1+x)+3/2*sin(x)*sqrt(1+x)],
[exp(arctanh(x))*(1-x)^(1/2)*sin(x),x,7,cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sqrt(1/2*Pi)+FresnelS(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(1/2*Pi)-cos(x)*sqrt(1+x)],
[exp(arctanh(x))*x*(1+x)^(3/2)*sin(x),x,22,-5*(1-x)^(3/2)*cos(x)+(1-x)^(5/2)*cos(x)+5/2*(1-x)^(3/2)*sin(x)+15/4*cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)-15/2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)+15/2*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)+15/4*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)-4*cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)+4*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)-4*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)-4*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)+17/4*cos(x)*sqrt(1-x)-15/2*sin(x)*sqrt(1-x)],
[exp(arctanh(x))*(1+x)^(3/2)*sin(x),x,16,-(1-x)^(3/2)*cos(x)-3/2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)+3/2*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)-2*cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)+4*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)-4*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)-2*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)+4*cos(x)*sqrt(1-x)-3/2*sin(x)*sqrt(1-x)],
[exp(arctanh(x))*(1-x)^(3/2)*x*sin(x),x,19,-3*(1+x)^(3/2)*cos(x)+(1+x)^(5/2)*cos(x)-5/2*(1+x)^(3/2)*sin(x)+7/4*cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sqrt(1/2*Pi)-9/2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1+x))*sqrt(1/2*Pi)+9/2*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(1/2*Pi)+7/4*FresnelS(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(1/2*Pi)-7/4*cos(x)*sqrt(1+x)+9/2*sin(x)*sqrt(1+x)],
[exp(arctanh(x))*(1-x)^(3/2)*sin(x),x,13,(1+x)^(3/2)*cos(x)+3/2*cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1+x))*sqrt(1/2*Pi)-3/2*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(1/2*Pi)+cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1+x))*sqrt(2*Pi)+FresnelS(sqrt(2/Pi)*sqrt(1+x))*sin(1)*sqrt(2*Pi)-2*cos(x)*sqrt(1+x)-3/2*sin(x)*sqrt(1+x)],

# n/2<0
[exp(arctanh(x))*x*sin(x)/(1+x)^(1/2),x,11,-cos(1)*FresnelC(sqrt(2/Pi)*sqrt(1-x))*sqrt(1/2*Pi)-FresnelS(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(1/2*Pi)+cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)-FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)+cos(x)*sqrt(1-x)],
[exp(arctanh(x))*sin(x)/(1+x)^(1/2),x,6,cos(1)*FresnelS(sqrt(2/Pi)*sqrt(1-x))*sqrt(2*Pi)-FresnelC(sqrt(2/Pi)*sqrt(1-x))*sin(1)*sqrt(2*Pi)],

#  {x*Sin[x]*E^ArcTanh[x]/(1 - x)^(1/2), x, 0, 0} 

#  {Sin[x]*E^ArcTanh[x]/(1 - x)^(1/2), x, 0, 0} 

#  {x*Sin[x]*E^ArcTanh[x]/(1 + x)^(3/2), x, 0, 0} 

#  {Sin[x]*E^ArcTanh[x]/(1 + x)^(3/2), x, 0, 0} 

#  {x*Sin[x]*E^ArcTanh[x]/(1 - x)^(3/2), x, 0, 0} 

#  {Sin[x]*E^ArcTanh[x]/(1 - x)^(3/2), x, 0, 0} 

# Integrands of the form u E^(n ArcTanh[a+b x])

# Integrands of the form u E^(n ArcTanh[a+b x])

# Integrands of the form x^m E^(n ArcTanh[a+b x])

# n>0 integer
[exp(arctanh(a+b*x))*x^3,x,7,1/8*(3-12*a+12*a^2-8*a^3)*arcsin(a+b*x)/b^4-1/4*x^2*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^2-1/24*(1+a+b*x)^(3/2)*(7-10*a+18*a^2+2*(1-6*a)*b*x)*sqrt(1-a-b*x)/b^4-1/8*(3-12*a+12*a^2-8*a^3)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^4],
[exp(arctanh(a+b*x))*x^2,x,7,1/2*(1-2*a+2*a^2)*arcsin(a+b*x)/b^3-1/6*(1-4*a)*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^3-1/3*x*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^2-1/2*(1-2*a+2*a^2)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^3],
[exp(arctanh(a+b*x))*x,x,6,1/2*(1-2*a)*arcsin(a+b*x)/b^2-1/2*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^2-1/2*(1-2*a)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^2],
[exp(arctanh(a+b*x)),x,5,arcsin(a+b*x)/b-sqrt(1-a-b*x)*sqrt(1+a+b*x)/b],
[exp(arctanh(a+b*x))/x,x,8,arcsin(a+b*x)-2*(1+a)*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/sqrt(1-a^2)],
[exp(arctanh(a+b*x))/x^2,x,4,-2*b*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)*sqrt(1-a^2))-sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)*x)],
[exp(arctanh(a+b*x))/x^3,x,5,-(1+2*a)*b^2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)^2*(1+a)*sqrt(1-a^2))-1/2*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/((1-a^2)*x^2)-1/2*(1+2*a)*b*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)^2*(1+a)*x)],
[exp(arctanh(a+b*x))/x^4,x,7,-(1+2*a+2*a^2)*b^3*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)*(1-a^2)^(5/2))-1/3*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)*x^3)-1/6*(3+2*a)*b*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)^2*(1+a)*x^2)-1/6*(4+a)*(1+2*a)*b^2*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)^3*(1+a)^2*x)],
[exp(2*arctanh(a+b*x))*x^4,x,3,-2*(1-a)^3*x/b^4-(1-a)^2*x^2/b^3-2/3*(1-a)*x^3/b^2-1/2*x^4/b-1/5*x^5-2*(1-a)^4*log(1-a-b*x)/b^5],
[exp(2*arctanh(a+b*x))*x^3,x,3,-2*(1-a)^2*x/b^3-(1-a)*x^2/b^2-2/3*x^3/b-1/4*x^4-2*(1-a)^3*log(1-a-b*x)/b^4],
[exp(2*arctanh(a+b*x))*x^2,x,3,-2*(1-a)*x/b^2-x^2/b-1/3*x^3-2*(1-a)^2*log(1-a-b*x)/b^3],
[exp(2*arctanh(a+b*x))*x,x,3,-2*x/b-1/2*x^2-2*(1-a)*log(1-a-b*x)/b^2],
[exp(2*arctanh(a+b*x)),x,3,-x-2*log(1-a-b*x)/b],
[exp(2*arctanh(a+b*x))/x,x,3,(1+a)*log(x)/(1-a)-2*log(1-a-b*x)/(1-a)],
[exp(2*arctanh(a+b*x))/x^2,x,3,(-1-a)/((1-a)*x)+2*b*log(x)/(1-a)^2-2*b*log(1-a-b*x)/(1-a)^2],
[exp(2*arctanh(a+b*x))/x^3,x,3,1/2*(-1-a)/((1-a)*x^2)-2*b/((1-a)^2*x)+2*b^2*log(x)/(1-a)^3-2*b^2*log(1-a-b*x)/(1-a)^3],
[exp(2*arctanh(a+b*x))/x^4,x,3,1/3*(-1-a)/((1-a)*x^3)-b/((1-a)^2*x^2)-2*b^2/((1-a)^3*x)+2*b^3*log(x)/(1-a)^4-2*b^3*log(1-a-b*x)/(1-a)^4],
[exp(3*arctanh(a+b*x))*x^3,x,8,-3/8*(17-44*a+36*a^2-8*a^3)*arcsin(a+b*x)/b^4+2*x^3*(1+a+b*x)^(3/2)/(b*sqrt(1-a-b*x))+9/4*x^2*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^2+1/8*(1+a+b*x)^(3/2)*(29-54*a+22*a^2+2*(11-10*a)*b*x)*sqrt(1-a-b*x)/b^4+3/8*(17-44*a+36*a^2-8*a^3)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^4],
[exp(3*arctanh(a+b*x))*x^2,x,8,-1/2*(11-18*a+6*a^2)*arcsin(a+b*x)/b^3+(1-a)^2*(1+a+b*x)^(5/2)/(b^3*sqrt(1-a-b*x))+1/6*(11-18*a+6*a^2)*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^3+1/3*(1+a+b*x)^(5/2)*sqrt(1-a-b*x)/b^3+1/2*(11-18*a+6*a^2)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^3],
[exp(3*arctanh(a+b*x))*x,x,7,-3/2*(3-2*a)*arcsin(a+b*x)/b^2+(1-a)*(1+a+b*x)^(5/2)/(b^2*sqrt(1-a-b*x))+1/2*(3-2*a)*(1+a+b*x)^(3/2)*sqrt(1-a-b*x)/b^2+3/2*(3-2*a)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^2],
[exp(3*arctanh(a+b*x)),x,6,-3*arcsin(a+b*x)/b+2*(1+a+b*x)^(3/2)/(b*sqrt(1-a-b*x))+3*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b],
[exp(3*arctanh(a+b*x))/x,x,8,-arcsin(a+b*x)-2*(1+a)^2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)*sqrt(1-a^2))+4*sqrt(1+a+b*x)/((1-a)*sqrt(1-a-b*x))],
[exp(3*arctanh(a+b*x))/x^2,x,5,-6*(1+a)*b*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)^2*sqrt(1-a^2))-(1+a+b*x)^(3/2)/((1-a)*x*sqrt(1-a-b*x))+6*b*sqrt(1+a+b*x)/((1-a)^2*sqrt(1-a-b*x))],
[exp(3*arctanh(a+b*x))/x^3,x,6,-3*(3+2*a)*b^2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)^3*sqrt(1-a^2))-1/2*(3+2*a)*b*(1+a+b*x)^(3/2)/((1-a)^2*(1+a)*x*sqrt(1-a-b*x))-1/2*(1+a+b*x)^(5/2)/((1-a^2)*x^2*sqrt(1-a-b*x))+3*(3+2*a)*b^2*sqrt(1+a+b*x)/((1-a)^3*(1+a)*sqrt(1-a-b*x))],
[exp(3*arctanh(a+b*x))/x^4,x,8,-(11+18*a+6*a^2)*b^3*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)^4*(1+a)*sqrt(1-a^2))+1/6*(52+51*a+2*a^2)*b^3*sqrt(1+a+b*x)/((1-a)^4*(1+a)*sqrt(1-a-b*x))-1/3*(1+a)*sqrt(1+a+b*x)/((1-a)*x^3*sqrt(1-a-b*x))-7/6*b*sqrt(1+a+b*x)/((1-a)^2*x^2*sqrt(1-a-b*x))-1/6*(19+16*a)*b^2*sqrt(1+a+b*x)/((1-a)^3*(1+a)*x*sqrt(1-a-b*x))],

# n<0 integer
[x^3/exp(arctanh(a+b*x)),x,7,-1/8*(3+12*a+12*a^2+8*a^3)*arcsin(a+b*x)/b^4-1/4*x^2*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^2-1/24*(1-a-b*x)^(3/2)*(7+10*a+18*a^2-2*(1+6*a)*b*x)*sqrt(1+a+b*x)/b^4-1/8*(3+12*a+12*a^2+8*a^3)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^4],
[x^2/exp(arctanh(a+b*x)),x,7,1/2*(1+2*a+2*a^2)*arcsin(a+b*x)/b^3+1/6*(1+4*a)*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^3-1/3*x*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^2+1/2*(1+2*a+2*a^2)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^3],
[x/exp(arctanh(a+b*x)),x,6,-1/2*(1+2*a)*arcsin(a+b*x)/b^2-1/2*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^2-1/2*(1+2*a)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^2],
[exp(-arctanh(a+b*x)),x,5,arcsin(a+b*x)/b+sqrt(1-a-b*x)*sqrt(1+a+b*x)/b],
[1/(exp(arctanh(a+b*x))*x),x,8,-arcsin(a+b*x)-2*(1-a)*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/sqrt(1-a^2)],
[1/(exp(arctanh(a+b*x))*x^2),x,4,2*b*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1+a)*sqrt(1-a^2))-sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1+a)*x)],
[1/(exp(arctanh(a+b*x))*x^3),x,5,-(1-2*a)*b^2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)*(1+a)^2*sqrt(1-a^2))-1/2*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/((1-a^2)*x^2)+1/2*(1-2*a)*b*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)*(1+a)^2*x)],
[1/(exp(arctanh(a+b*x))*x^4),x,7,(1-2*a+2*a^2)*b^3*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1+a)*(1-a^2)^(5/2))-1/3*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1+a)*x^3)+1/6*(3-2*a)*b*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)*(1+a)^2*x^2)-1/6*(1-2*a)*(4-a)*b^2*sqrt(1-a-b*x)*sqrt(1+a+b*x)/((1-a)^2*(1+a)^3*x)],
[x^4/exp(2*arctanh(a+b*x)),x,3,-2*(1+a)^3*x/b^4+(1+a)^2*x^2/b^3-2/3*(1+a)*x^3/b^2+1/2*x^4/b-1/5*x^5+2*(1+a)^4*log(1+a+b*x)/b^5],
[x^3/exp(2*arctanh(a+b*x)),x,3,2*(1+a)^2*x/b^3-(1+a)*x^2/b^2+2/3*x^3/b-1/4*x^4-2*(1+a)^3*log(1+a+b*x)/b^4],
[x^2/exp(2*arctanh(a+b*x)),x,3,-2*(1+a)*x/b^2+x^2/b-1/3*x^3+2*(1+a)^2*log(1+a+b*x)/b^3],
[x/exp(2*arctanh(a+b*x)),x,3,2*x/b-1/2*x^2-2*(1+a)*log(1+a+b*x)/b^2],
[exp(-2*arctanh(a+b*x)),x,3,-x+2*log(1+a+b*x)/b],
[1/(exp(2*arctanh(a+b*x))*x),x,3,(1-a)*log(x)/(1+a)-2*log(1+a+b*x)/(1+a)],
[1/(exp(2*arctanh(a+b*x))*x^2),x,3,(-1+a)/((1+a)*x)-2*b*log(x)/(1+a)^2+2*b*log(1+a+b*x)/(1+a)^2],
[1/(exp(2*arctanh(a+b*x))*x^3),x,3,1/2*(-1+a)/((1+a)*x^2)+2*b/((1+a)^2*x)+2*b^2*log(x)/(1+a)^3-2*b^2*log(1+a+b*x)/(1+a)^3],
[1/(exp(2*arctanh(a+b*x))*x^4),x,3,1/3*(-1+a)/((1+a)*x^3)+b/((1+a)^2*x^2)-2*b^2/((1+a)^3*x)-2*b^3*log(x)/(1+a)^4+2*b^3*log(1+a+b*x)/(1+a)^4],
[x^3/exp(3*arctanh(a+b*x)),x,8,3/8*(17+44*a+36*a^2+8*a^3)*arcsin(a+b*x)/b^4-2*x^3*(1-a-b*x)^(3/2)/(b*sqrt(1+a+b*x))+9/4*x^2*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^2+1/8*(1-a-b*x)^(3/2)*(29+54*a+22*a^2-2*(11+10*a)*b*x)*sqrt(1+a+b*x)/b^4+3/8*(17+44*a+36*a^2+8*a^3)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^4],
[x^2/exp(3*arctanh(a+b*x)),x,8,-1/2*(11+18*a+6*a^2)*arcsin(a+b*x)/b^3-(1+a)^2*(1-a-b*x)^(5/2)/(b^3*sqrt(1+a+b*x))-1/6*(11+18*a+6*a^2)*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^3-1/3*(1-a-b*x)^(5/2)*sqrt(1+a+b*x)/b^3-1/2*(11+18*a+6*a^2)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^3],
[x/exp(3*arctanh(a+b*x)),x,7,3/2*(3+2*a)*arcsin(a+b*x)/b^2+(1+a)*(1-a-b*x)^(5/2)/(b^2*sqrt(1+a+b*x))+1/2*(3+2*a)*(1-a-b*x)^(3/2)*sqrt(1+a+b*x)/b^2+3/2*(3+2*a)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^2],
[exp(-3*arctanh(a+b*x)),x,6,-3*arcsin(a+b*x)/b-2*(1-a-b*x)^(3/2)/(b*sqrt(1+a+b*x))-3*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b],
[1/(exp(3*arctanh(a+b*x))*x),x,8,arcsin(a+b*x)-2*(1-a)^2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1+a)*sqrt(1-a^2))+4*sqrt(1-a-b*x)/((1+a)*sqrt(1+a+b*x))],
[1/(exp(3*arctanh(a+b*x))*x^2),x,5,6*(1-a)*b*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1+a)^2*sqrt(1-a^2))-(1-a-b*x)^(3/2)/((1+a)*x*sqrt(1+a+b*x))-6*b*sqrt(1-a-b*x)/((1+a)^2*sqrt(1+a+b*x))],
[1/(exp(3*arctanh(a+b*x))*x^3),x,6,-3*(3-2*a)*b^2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1+a)^3*sqrt(1-a^2))+1/2*(3-2*a)*b*(1-a-b*x)^(3/2)/((1-a)*(1+a)^2*x*sqrt(1+a+b*x))-1/2*(1-a-b*x)^(5/2)/((1-a^2)*x^2*sqrt(1+a+b*x))+3*(3-2*a)*b^2*sqrt(1-a-b*x)/((1-a)*(1+a)^3*sqrt(1+a+b*x))],
[1/(exp(3*arctanh(a+b*x))*x^4),x,8,(11-18*a+6*a^2)*b^3*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)*(1+a)^4*sqrt(1-a^2))-1/6*(52-51*a+2*a^2)*b^3*sqrt(1-a-b*x)/((1-a)*(1+a)^4*sqrt(1+a+b*x))-1/3*(1-a)*sqrt(1-a-b*x)/((1+a)*x^3*sqrt(1+a+b*x))+7/6*b*sqrt(1-a-b*x)/((1+a)^2*x^2*sqrt(1+a+b*x))-1/6*(19-16*a)*b^2*sqrt(1-a-b*x)/((1-a)*(1+a)^3*x*sqrt(1+a+b*x))],

# Integrands of the form x^m E^(n ArcTanh[a+b x]) (c+d x)^p
[exp(arctanh(1+b*x))/(2+b*x),x,4,arcsin(1+b*x)/b],

# Integrands of the form x^m E^ArcTanh[a+b x] / (1-(a+b x)^2)
[exp(arctanh(a+b*x))*x^3/(1-a^2-2*a*b*x-b^2*x^2),x,6,-3/2*(1-2*a+2*a^2)*arcsin(a+b*x)/b^4+(1-a)*x^2*sqrt(1+a+b*x)/(b^2*sqrt(1-a-b*x))+1/2*((1-2*a)*(4-a)+(3-2*a)*b*x)*sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^4],
[exp(arctanh(a+b*x))*x^2/(1-a^2-2*a*b*x-b^2*x^2),x,6,-(1-2*a)*arcsin(a+b*x)/b^3+(1-a)^2*sqrt(1+a+b*x)/(b^3*sqrt(1-a-b*x))+sqrt(1-a-b*x)*sqrt(1+a+b*x)/b^3],
[exp(arctanh(a+b*x))*x/(1-a^2-2*a*b*x-b^2*x^2),x,5,-arcsin(a+b*x)/b^2+(1-a)*sqrt(1+a+b*x)/(b^2*sqrt(1-a-b*x))],
[exp(arctanh(a+b*x))/(1-a^2-2*a*b*x-b^2*x^2),x,2,sqrt(1+a+b*x)/(b*sqrt(1-a-b*x))],
[exp(arctanh(a+b*x))/(x*(1-a^2-2*a*b*x-b^2*x^2)),x,4,-2*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)*sqrt(1-a^2))+sqrt(1+a+b*x)/((1-a)*sqrt(1-a-b*x))],
[exp(arctanh(a+b*x))/(x^2*(1-a^2-2*a*b*x-b^2*x^2)),x,6,-2*(1+2*a)*b*arctanh(sqrt(1-a)*sqrt(1+a+b*x)/(sqrt(1+a)*sqrt(1-a-b*x)))/((1-a)^2*(1+a)*sqrt(1-a^2))+(2+a)*b*sqrt(1+a+b*x)/((1-a)^2*(1+a)*sqrt(1-a-b*x))-sqrt(1+a+b*x)/((1-a^2)*x*sqrt(1-a-b*x))],

# Integrands of the form x^m E^(n ArcTanh[a+b x])
[exp(n*arctanh(a+b*x))*x^m,x,4,x^(1+m)*(1+a+b*x)^(1/2*n)*(1-b*x/(1-a))^(1/2*n)*AppellF1(1+m,1/2*n,-1/2*n,2+m,b*x/(1-a),-b*x/(1+a))/((1+m)*(1-a-b*x)^(1/2*n)*(1+b*x/(1+a))^(1/2*n))],
[exp(n*arctanh(a+b*x))*x^3,x,4,-1/4*x^2*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(2+n))/b^2-1/24*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(2+n))*(6+18*a^2-10*a*n+n^2-2*b*(6*a-n)*x)/b^4+1/3*2^(-2+1/2*n)*(24*a^3-36*a^2*n+12*a*(2+n^2)-n*(8+n^2))*(1-a-b*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a-b*x))/(b^4*(2-n))],
[exp(n*arctanh(a+b*x))*x^2,x,4,1/6*(4*a-n)*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(2+n))/b^3-1/3*x*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(2+n))/b^2-1/3*2^(1/2*n)*(2+6*a^2-6*a*n+n^2)*(1-a-b*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a-b*x))/(b^3*(2-n))],
[exp(n*arctanh(a+b*x))*x,x,3,-1/2*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(2+n))/b^2+2^(1/2*n)*(2*a-n)*(1-a-b*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a-b*x))/(b^2*(2-n))],
[exp(n*arctanh(a+b*x)),x,2,-2^(1+1/2*n)*(1-a-b*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a-b*x))/(b*(2-n))],
[exp(n*arctanh(a+b*x))/x,x,5,2*(1+a+b*x)^(1/2*n)*hypergeom([1,-1/2*n],[1-1/2*n],(1+a)*(1-a-b*x)/((1-a)*(1+a+b*x)))/(n*(1-a-b*x)^(1/2*n))-2^(1+1/2*n)*hypergeom([-1/2*n,-1/2*n],[1-1/2*n],1/2*(1-a-b*x))/(n*(1-a-b*x)^(1/2*n))],
[exp(n*arctanh(a+b*x))/x^2,x,2,-4*b*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(1+a)*(1-a-b*x)/((1-a)*(1+a+b*x)))/((1-a)^2*(2-n))],
[exp(n*arctanh(a+b*x))/x^3,x,3,-1/2*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(2+n))/((1-a^2)*x^2)-2*b^2*(2*a+n)*(1-a-b*x)^(1-1/2*n)*(1+a+b*x)^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(1+a)*(1-a-b*x)/((1-a)*(1+a+b*x)))/((1-a)^3*(1+a)*(2-n))],

# Integrands of the form x^m (c-a^2 c x^2)^p E^(n ArcTanh[a x])

# Integrands of the form x^m (c-a^2 c x^2)^p E^(1 ArcTanh[a x])

# Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^p

# p>0
[exp(arctanh(a*x))*(c-a^2*c*x^2)^4,x,7,35/192*c^4*x*(1-a^2*x^2)^(3/2)+7/48*c^4*x*(1-a^2*x^2)^(5/2)+1/8*c^4*x*(1-a^2*x^2)^(7/2)-1/9*c^4*(1-a^2*x^2)^(9/2)/a+35/128*c^4*arcsin(a*x)/a+35/128*c^4*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^3,x,6,5/24*c^3*x*(1-a^2*x^2)^(3/2)+1/6*c^3*x*(1-a^2*x^2)^(5/2)-1/7*c^3*(1-a^2*x^2)^(7/2)/a+5/16*c^3*arcsin(a*x)/a+5/16*c^3*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^2,x,5,1/4*c^2*x*(1-a^2*x^2)^(3/2)-1/5*c^2*(1-a^2*x^2)^(5/2)/a+3/8*c^2*arcsin(a*x)/a+3/8*c^2*x*sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2),x,4,-1/3*c*(1-a^2*x^2)^(3/2)/a+1/2*c*arcsin(a*x)/a+1/2*c*x*sqrt(1-a^2*x^2)],

# p<0
[exp(arctanh(a*x))*x^4/(c-a^2*c*x^2),x,5,-3/2*arcsin(a*x)/(a^5*c)+x^3*(1+a*x)/(a^2*c*sqrt(1-a^2*x^2))+4/3*x^2*sqrt(1-a^2*x^2)/(a^3*c)+1/6*(16+9*a*x)*sqrt(1-a^2*x^2)/(a^5*c)],
[exp(arctanh(a*x))*x^3/(c-a^2*c*x^2),x,4,-3/2*arcsin(a*x)/(a^4*c)+x^2*(1+a*x)/(a^2*c*sqrt(1-a^2*x^2))+1/2*(4+3*a*x)*sqrt(1-a^2*x^2)/(a^4*c)],
[exp(arctanh(a*x))*x^2/(c-a^2*c*x^2),x,5,-arcsin(a*x)/(a^3*c)+(1+a*x)/(a^3*c*sqrt(1-a^2*x^2))+sqrt(1-a^2*x^2)/(a^3*c)],
[exp(arctanh(a*x))*x/(c-a^2*c*x^2),x,3,-arcsin(a*x)/(a^2*c)+(1+a*x)/(a^2*c*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2),x,1,exp(arctanh(a*x))/(a*c)],
[exp(arctanh(a*x))/(x*(c-a^2*c*x^2)),x,6,-arctanh(sqrt(1-a^2*x^2))/c+(1+a*x)/(c*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a^2*c*x^2)),x,6,-a*arctanh(sqrt(1-a^2*x^2))/c+(1+a*x)/(c*x*sqrt(1-a^2*x^2))-2*sqrt(1-a^2*x^2)/(c*x)],
[exp(arctanh(a*x))/(x^3*(c-a^2*c*x^2)),x,7,-3/2*a^2*arctanh(sqrt(1-a^2*x^2))/c+(1+a*x)/(c*x^2*sqrt(1-a^2*x^2))-3/2*sqrt(1-a^2*x^2)/(c*x^2)-2*a*sqrt(1-a^2*x^2)/(c*x)],
[exp(arctanh(a*x))/(x^4*(c-a^2*c*x^2)),x,8,-3/2*a^3*arctanh(sqrt(1-a^2*x^2))/c+(1+a*x)/(c*x^3*sqrt(1-a^2*x^2))-4/3*sqrt(1-a^2*x^2)/(c*x^3)-3/2*a*sqrt(1-a^2*x^2)/(c*x^2)-8/3*a^2*sqrt(1-a^2*x^2)/(c*x)],
[exp(arctanh(a*x))*x^6/(c-a^2*c*x^2)^2,x,6,1/3*x^5*(1+a*x)/(a^2*c^2*(1-a^2*x^2)^(3/2))+5/2*arcsin(a*x)/(a^7*c^2)-1/3*x^3*(5+6*a*x)/(a^4*c^2*sqrt(1-a^2*x^2))-8/3*x^2*sqrt(1-a^2*x^2)/(a^5*c^2)-1/6*(32+15*a*x)*sqrt(1-a^2*x^2)/(a^7*c^2)],
[exp(arctanh(a*x))*x^5/(c-a^2*c*x^2)^2,x,5,1/3*x^4*(1+a*x)/(a^2*c^2*(1-a^2*x^2)^(3/2))+5/2*arcsin(a*x)/(a^6*c^2)-1/3*x^2*(4+5*a*x)/(a^4*c^2*sqrt(1-a^2*x^2))-1/6*(16+15*a*x)*sqrt(1-a^2*x^2)/(a^6*c^2)],
[exp(arctanh(a*x))*x^4/(c-a^2*c*x^2)^2,x,5,1/3*x^3*(1+a*x)/(a^2*c^2*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a^5*c^2)-1/3*x*(3+4*a*x)/(a^4*c^2*sqrt(1-a^2*x^2))-8/3*sqrt(1-a^2*x^2)/(a^5*c^2)],
[exp(arctanh(a*x))*x^3/(c-a^2*c*x^2)^2,x,4,1/3*x^2*(1+a*x)/(a^2*c^2*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a^4*c^2)+1/3*(-2-3*a*x)/(a^4*c^2*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^2/(c-a^2*c*x^2)^2,x,4,1/3*x^2*(1+a*x)/(a*c^2*(1-a^2*x^2)^(3/2))+(-2/3)/(a^3*c^2*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x/(c-a^2*c*x^2)^2,x,3,1/3*(1+a*x)/(a^2*c^2*(1-a^2*x^2)^(3/2))-1/3*x/(a*c^2*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^2,x,3,1/3*(1+a*x)/(a*c^2*(1-a^2*x^2)^(3/2))+2/3*x/(c^2*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x*(c-a^2*c*x^2)^2),x,7,1/3*(1+a*x)/(c^2*(1-a^2*x^2)^(3/2))-arctanh(sqrt(1-a^2*x^2))/c^2+1/3*(3+2*a*x)/(c^2*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a^2*c*x^2)^2),x,7,1/3*(1+a*x)/(c^2*x*(1-a^2*x^2)^(3/2))-a*arctanh(sqrt(1-a^2*x^2))/c^2+1/3*(4+3*a*x)/(c^2*x*sqrt(1-a^2*x^2))-8/3*sqrt(1-a^2*x^2)/(c^2*x)],
[exp(arctanh(a*x))/(x^3*(c-a^2*c*x^2)^2),x,8,1/3*(1+a*x)/(c^2*x^2*(1-a^2*x^2)^(3/2))-5/2*a^2*arctanh(sqrt(1-a^2*x^2))/c^2+1/3*(5+4*a*x)/(c^2*x^2*sqrt(1-a^2*x^2))-5/2*sqrt(1-a^2*x^2)/(c^2*x^2)-8/3*a*sqrt(1-a^2*x^2)/(c^2*x)],
[exp(arctanh(a*x))/(x^4*(c-a^2*c*x^2)^2),x,9,1/3*(1+a*x)/(c^2*x^3*(1-a^2*x^2)^(3/2))-5/2*a^3*arctanh(sqrt(1-a^2*x^2))/c^2+1/3*(6+5*a*x)/(c^2*x^3*sqrt(1-a^2*x^2))-8/3*sqrt(1-a^2*x^2)/(c^2*x^3)-5/2*a*sqrt(1-a^2*x^2)/(c^2*x^2)-16/3*a^2*sqrt(1-a^2*x^2)/(c^2*x)],
[exp(arctanh(a*x))*x^7/(c-a^2*c*x^2)^3,x,6,1/5*x^6*(1+a*x)/(a^2*c^3*(1-a^2*x^2)^(5/2))-1/15*x^4*(6+7*a*x)/(a^4*c^3*(1-a^2*x^2)^(3/2))-7/2*arcsin(a*x)/(a^8*c^3)+1/15*x^2*(24+35*a*x)/(a^6*c^3*sqrt(1-a^2*x^2))+1/10*(32+35*a*x)*sqrt(1-a^2*x^2)/(a^8*c^3)],
[exp(arctanh(a*x))*x^6/(c-a^2*c*x^2)^3,x,6,1/5*x^5*(1+a*x)/(a^2*c^3*(1-a^2*x^2)^(5/2))-1/15*x^3*(5+6*a*x)/(a^4*c^3*(1-a^2*x^2)^(3/2))-arcsin(a*x)/(a^7*c^3)+1/5*x*(5+8*a*x)/(a^6*c^3*sqrt(1-a^2*x^2))+16/5*sqrt(1-a^2*x^2)/(a^7*c^3)],
[exp(arctanh(a*x))*x^5/(c-a^2*c*x^2)^3,x,5,1/5*x^4*(1+a*x)/(a^2*c^3*(1-a^2*x^2)^(5/2))-1/15*x^2*(4+5*a*x)/(a^4*c^3*(1-a^2*x^2)^(3/2))-arcsin(a*x)/(a^6*c^3)+1/15*(8+15*a*x)/(a^6*c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^4/(c-a^2*c*x^2)^3,x,5,1/5*x^4*(1+a*x)/(a*c^3*(1-a^2*x^2)^(5/2))+(-4/15)/(a^5*c^3*(1-a^2*x^2)^(3/2))+4/5/(a^5*c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^3/(c-a^2*c*x^2)^3,x,4,1/5/(a^4*c^3*(1-a^2*x^2)^(5/2))+1/5*a*x^5/(c^3*(1-a^2*x^2)^(5/2))+(-1/3)/(a^4*c^3*(1-a^2*x^2)^(3/2)),1/5*x^2*(1+a*x)/(a^2*c^3*(1-a^2*x^2)^(5/2))+1/15*(-2-3*a*x)/(a^4*c^3*(1-a^2*x^2)^(3/2))+1/5*x/(a^3*c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^2/(c-a^2*c*x^2)^3,x,4,1/5*x^2*(1+a*x)/(a*c^3*(1-a^2*x^2)^(5/2))-2/15*(1-a*x)/(a^3*c^3*(1-a^2*x^2)^(3/2))-2/15*x/(a^2*c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x/(c-a^2*c*x^2)^3,x,4,1/5*(1+a*x)/(a^2*c^3*(1-a^2*x^2)^(5/2))-1/15*x/(a*c^3*(1-a^2*x^2)^(3/2))-2/15*x/(a*c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^3,x,4,1/5*(1+a*x)/(a*c^3*(1-a^2*x^2)^(5/2))+4/15*x/(c^3*(1-a^2*x^2)^(3/2))+8/15*x/(c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x*(c-a^2*c*x^2)^3),x,8,1/5*(1+a*x)/(c^3*(1-a^2*x^2)^(5/2))+1/15*(5+4*a*x)/(c^3*(1-a^2*x^2)^(3/2))-arctanh(sqrt(1-a^2*x^2))/c^3+1/15*(15+8*a*x)/(c^3*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a^2*c*x^2)^3),x,8,1/5*(1+a*x)/(c^3*x*(1-a^2*x^2)^(5/2))+1/15*(6+5*a*x)/(c^3*x*(1-a^2*x^2)^(3/2))-a*arctanh(sqrt(1-a^2*x^2))/c^3+1/5*(8+5*a*x)/(c^3*x*sqrt(1-a^2*x^2))-16/5*sqrt(1-a^2*x^2)/(c^3*x)],
[exp(arctanh(a*x))/(x^3*(c-a^2*c*x^2)^3),x,9,1/5*(1+a*x)/(c^3*x^2*(1-a^2*x^2)^(5/2))+1/15*(7+6*a*x)/(c^3*x^2*(1-a^2*x^2)^(3/2))-7/2*a^2*arctanh(sqrt(1-a^2*x^2))/c^3+1/15*(35+24*a*x)/(c^3*x^2*sqrt(1-a^2*x^2))-7/2*sqrt(1-a^2*x^2)/(c^3*x^2)-16/5*a*sqrt(1-a^2*x^2)/(c^3*x)],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^4,x,5,1/7*(1+a*x)/(a*c^4*(1-a^2*x^2)^(7/2))+6/35*x/(c^4*(1-a^2*x^2)^(5/2))+8/35*x/(c^4*(1-a^2*x^2)^(3/2))+16/35*x/(c^4*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^5,x,6,1/9*(1+a*x)/(a*c^5*(1-a^2*x^2)^(9/2))+8/63*x/(c^5*(1-a^2*x^2)^(7/2))+16/105*x/(c^5*(1-a^2*x^2)^(5/2))+64/315*x/(c^5*(1-a^2*x^2)^(3/2))+128/315*x/(c^5*sqrt(1-a^2*x^2))],

# Integrands of the form x^m E^ArcTanh[a x] (1-a^2 x^2)^(p/2)

# p>0

# p<0
[exp(arctanh(a*x))*x^4/sqrt(1-a^2*x^2),x,3,-x/a^4-1/2*x^2/a^3-1/3*x^3/a^2-1/4*x^4/a-log(1-a*x)/a^5],
[exp(arctanh(a*x))*x^3/sqrt(1-a^2*x^2),x,3,-x/a^3-1/2*x^2/a^2-1/3*x^3/a-log(1-a*x)/a^4],
[exp(arctanh(a*x))*x^2/sqrt(1-a^2*x^2),x,3,-x/a^2-1/2*x^2/a-log(1-a*x)/a^3],
[exp(arctanh(a*x))*x/sqrt(1-a^2*x^2),x,3,-x/a-log(1-a*x)/a^2],
[exp(arctanh(a*x))/sqrt(1-a^2*x^2),x,2,-log(1-a*x)/a],
[exp(arctanh(a*x))/(x*sqrt(1-a^2*x^2)),x,4,log(x)-log(1-a*x)],
[exp(arctanh(a*x))/(x^2*sqrt(1-a^2*x^2)),x,3,(-1)/x+a*log(x)-a*log(1-a*x)],
[exp(arctanh(a*x))/(x^3*sqrt(1-a^2*x^2)),x,3,(-1/2)/x^2-a/x+a^2*log(x)-a^2*log(1-a*x)],
[exp(arctanh(a*x))/(x^4*sqrt(1-a^2*x^2)),x,3,(-1/3)/x^3-1/2*a/x^2-a^2/x+a^3*log(x)-a^3*log(1-a*x)],
[exp(arctanh(a*x))*x^4/(1-a^2*x^2)^(3/2),x,3,x/a^4+1/2*x^2/a^3+1/2/(a^5*(1-a*x))+7/4*log(1-a*x)/a^5+1/4*log(1+a*x)/a^5],
[exp(arctanh(a*x))*x^3/(1-a^2*x^2)^(3/2),x,3,x/a^3+1/2/(a^4*(1-a*x))+5/4*log(1-a*x)/a^4-1/4*log(1+a*x)/a^4],
[exp(arctanh(a*x))*x^2/(1-a^2*x^2)^(3/2),x,3,1/2/(a^3*(1-a*x))+3/4*log(1-a*x)/a^3+1/4*log(1+a*x)/a^3],
[exp(arctanh(a*x))*x/(1-a^2*x^2)^(3/2),x,4,1/2/(a^2*(1-a*x))-1/2*arctanh(a*x)/a^2],
[exp(arctanh(a*x))/(1-a^2*x^2)^(3/2),x,4,1/2/(a*(1-a*x))+1/2*arctanh(a*x)/a],
[exp(arctanh(a*x))/(x*(1-a^2*x^2)^(3/2)),x,3,1/2/(1-a*x)+log(x)-3/4*log(1-a*x)-1/4*log(1+a*x)],
[exp(arctanh(a*x))/(x^2*(1-a^2*x^2)^(3/2)),x,3,(-1)/x+1/2*a/(1-a*x)+a*log(x)-5/4*a*log(1-a*x)+1/4*a*log(1+a*x)],
[exp(arctanh(a*x))/(x^3*(1-a^2*x^2)^(3/2)),x,3,(-1/2)/x^2-a/x+1/2*a^2/(1-a*x)+2*a^2*log(x)-7/4*a^2*log(1-a*x)-1/4*a^2*log(1+a*x)],
[exp(arctanh(a*x))/(x^4*(1-a^2*x^2)^(3/2)),x,3,(-1/3)/x^3-1/2*a/x^2-2*a^2/x+1/2*a^3/(1-a*x)+2*a^3*log(x)-9/4*a^3*log(1-a*x)+1/4*a^3*log(1+a*x)],
[exp(arctanh(a*x))*x^6/(1-a^2*x^2)^(5/2),x,3,-x/a^6-1/2*x^2/a^5+1/8/(a^7*(1-a*x)^2)+(-5/4)/(a^7*(1-a*x))+(-1/8)/(a^7*(1+a*x))-39/16*log(1-a*x)/a^7-9/16*log(1+a*x)/a^7],
[exp(arctanh(a*x))*x^5/(1-a^2*x^2)^(5/2),x,3,-x/a^5+1/8/(a^6*(1-a*x)^2)+(-1)/(a^6*(1-a*x))+1/8/(a^6*(1+a*x))-23/16*log(1-a*x)/a^6+7/16*log(1+a*x)/a^6],
[exp(arctanh(a*x))*x^4/(1-a^2*x^2)^(5/2),x,3,1/8/(a^5*(1-a*x)^2)+(-3/4)/(a^5*(1-a*x))+(-1/8)/(a^5*(1+a*x))-11/16*log(1-a*x)/a^5-5/16*log(1+a*x)/a^5],
[exp(arctanh(a*x))*x^3/(1-a^2*x^2)^(5/2),x,4,1/8/(a^4*(1-a*x)^2)+(-1/2)/(a^4*(1-a*x))+1/8/(a^4*(1+a*x))+3/8*arctanh(a*x)/a^4],
[exp(arctanh(a*x))*x^2/(1-a^2*x^2)^(5/2),x,4,1/8/(a^3*(1-a*x)^2)+(-1/4)/(a^3*(1-a*x))+(-1/8)/(a^3*(1+a*x))-1/8*arctanh(a*x)/a^3],
[exp(arctanh(a*x))*x/(1-a^2*x^2)^(5/2),x,4,1/8/(a^2*(1-a*x)^2)+1/8/(a^2*(1+a*x))-1/8*arctanh(a*x)/a^2],
[exp(arctanh(a*x))/(1-a^2*x^2)^(5/2),x,4,1/8/(a*(1-a*x)^2)+1/4/(a*(1-a*x))+(-1/8)/(a*(1+a*x))+3/8*arctanh(a*x)/a],
[exp(arctanh(a*x))/(x*(1-a^2*x^2)^(5/2)),x,3,1/8/(1-a*x)^2+1/2/(1-a*x)+1/8/(1+a*x)+log(x)-11/16*log(1-a*x)-5/16*log(1+a*x)],
[exp(arctanh(a*x))/(x^2*(1-a^2*x^2)^(5/2)),x,3,(-1)/x+1/8*a/(1-a*x)^2+3/4*a/(1-a*x)-1/8*a/(1+a*x)+a*log(x)-23/16*a*log(1-a*x)+7/16*a*log(1+a*x)],
[exp(arctanh(a*x))/(x^3*(1-a^2*x^2)^(5/2)),x,3,(-1/2)/x^2-a/x+1/8*a^2/(1-a*x)^2+a^2/(1-a*x)+1/8*a^2/(1+a*x)+3*a^2*log(x)-39/16*a^2*log(1-a*x)-9/16*a^2*log(1+a*x)],
[exp(arctanh(a*x))/(x^4*(1-a^2*x^2)^(5/2)),x,3,(-1/3)/x^3-1/2*a/x^2-3*a^2/x+1/8*a^3/(1-a*x)^2+5/4*a^3/(1-a*x)-1/8*a^3/(1+a*x)+3*a^3*log(x)-59/16*a^3*log(1-a*x)+11/16*a^3*log(1+a*x)],

# Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^(p/2)

# p>0
[exp(arctanh(a*x))*x^2*(c-a^2*c*x^2)^(1/2),x,4,1/3*x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/4*a*x^4*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*x*(c-a^2*c*x^2)^(1/2),x,4,1/2*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/3*a*x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^(1/2),x,3,x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/2*a*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x,x,4,a*x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^2,x,4,-sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+a*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^(3/2),x,4,2/3*c*(1+a*x)^3*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/4*c*(1+a*x)^4*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^(5/2),x,4,c^2*(1+a*x)^4*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-4/5*c^2*(1+a*x)^5*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+1/6*c^2*(1+a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^(7/2),x,4,8/5*c^3*(1+a*x)^5*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-2*c^3*(1+a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+6/7*c^3*(1+a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/8*c^3*(1+a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],

# p<0
[exp(arctanh(a*x))*x^4/sqrt(c-a^2*c*x^2),x,4,-x*sqrt(1-a^2*x^2)/(a^4*sqrt(c-a^2*c*x^2))-1/2*x^2*sqrt(1-a^2*x^2)/(a^3*sqrt(c-a^2*c*x^2))-1/3*x^3*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a^2*c*x^2))-1/4*x^4*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-log(1-a*x)*sqrt(1-a^2*x^2)/(a^5*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^3/sqrt(c-a^2*c*x^2),x,4,-x*sqrt(1-a^2*x^2)/(a^3*sqrt(c-a^2*c*x^2))-1/2*x^2*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a^2*c*x^2))-1/3*x^3*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-log(1-a*x)*sqrt(1-a^2*x^2)/(a^4*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^2/sqrt(c-a^2*c*x^2),x,4,-x*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a^2*c*x^2))-1/2*x^2*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-log(1-a*x)*sqrt(1-a^2*x^2)/(a^3*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x/sqrt(c-a^2*c*x^2),x,4,-x*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-log(1-a*x)*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/sqrt(c-a^2*c*x^2),x,3,-log(1-a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x*sqrt(c-a^2*c*x^2)),x,5,log(x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)-log(1-a*x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)],
[exp(arctanh(a*x))/(x^2*sqrt(c-a^2*c*x^2)),x,4,-sqrt(1-a^2*x^2)/(x*sqrt(c-a^2*c*x^2))+a*log(x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)-a*log(1-a*x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)],
[exp(arctanh(a*x))/(x^3*sqrt(c-a^2*c*x^2)),x,4,-1/2*sqrt(1-a^2*x^2)/(x^2*sqrt(c-a^2*c*x^2))-a*sqrt(1-a^2*x^2)/(x*sqrt(c-a^2*c*x^2))+a^2*log(x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)-a^2*log(1-a*x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)],
[exp(arctanh(a*x))/(x^4*sqrt(c-a^2*c*x^2)),x,4,-1/3*sqrt(1-a^2*x^2)/(x^3*sqrt(c-a^2*c*x^2))-1/2*a*sqrt(1-a^2*x^2)/(x^2*sqrt(c-a^2*c*x^2))-a^2*sqrt(1-a^2*x^2)/(x*sqrt(c-a^2*c*x^2))+a^3*log(x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)-a^3*log(1-a*x)*sqrt(1-a^2*x^2)/sqrt(c-a^2*c*x^2)],
[exp(arctanh(a*x))*x^5/(c-a^2*c*x^2)^(3/2),x,4,2*x*sqrt(1-a^2*x^2)/(a^5*c*sqrt(c-a^2*c*x^2))+1/2*x^2*sqrt(1-a^2*x^2)/(a^4*c*sqrt(c-a^2*c*x^2))+1/3*x^3*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))+1/2*sqrt(1-a^2*x^2)/(a^6*c*(1-a*x)*sqrt(c-a^2*c*x^2))+9/4*log(1-a*x)*sqrt(1-a^2*x^2)/(a^6*c*sqrt(c-a^2*c*x^2))-1/4*log(1+a*x)*sqrt(1-a^2*x^2)/(a^6*c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^4/(c-a^2*c*x^2)^(3/2),x,4,x*sqrt(1-a^2*x^2)/(a^4*c*sqrt(c-a^2*c*x^2))+1/2*x^2*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))+1/2*sqrt(1-a^2*x^2)/(a^5*c*(1-a*x)*sqrt(c-a^2*c*x^2))+7/4*log(1-a*x)*sqrt(1-a^2*x^2)/(a^5*c*sqrt(c-a^2*c*x^2))+1/4*log(1+a*x)*sqrt(1-a^2*x^2)/(a^5*c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^3/(c-a^2*c*x^2)^(3/2),x,4,x*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))+1/2*sqrt(1-a^2*x^2)/(a^4*c*(1-a*x)*sqrt(c-a^2*c*x^2))+5/4*log(1-a*x)*sqrt(1-a^2*x^2)/(a^4*c*sqrt(c-a^2*c*x^2))-1/4*log(1+a*x)*sqrt(1-a^2*x^2)/(a^4*c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^2/(c-a^2*c*x^2)^(3/2),x,4,1/2*sqrt(1-a^2*x^2)/(a^3*c*(1-a*x)*sqrt(c-a^2*c*x^2))+3/4*log(1-a*x)*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))+1/4*log(1+a*x)*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x/(c-a^2*c*x^2)^(3/2),x,5,1/2*sqrt(1-a^2*x^2)/(a^2*c*(1-a*x)*sqrt(c-a^2*c*x^2))-1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/(a^2*c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^(3/2),x,5,1/2*sqrt(1-a^2*x^2)/(a*c*(1-a*x)*sqrt(c-a^2*c*x^2))+1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x*(c-a^2*c*x^2)^(3/2)),x,4,1/2*sqrt(1-a^2*x^2)/(c*(1-a*x)*sqrt(c-a^2*c*x^2))+log(x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))-3/4*log(1-a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))-1/4*log(1+a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(3/2)),x,4,-sqrt(1-a^2*x^2)/(c*x*sqrt(c-a^2*c*x^2))+1/2*a*sqrt(1-a^2*x^2)/(c*(1-a*x)*sqrt(c-a^2*c*x^2))+a*log(x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))-5/4*a*log(1-a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))+1/4*a*log(1+a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x^3*(c-a^2*c*x^2)^(3/2)),x,4,-1/2*sqrt(1-a^2*x^2)/(c*x^2*sqrt(c-a^2*c*x^2))-a*sqrt(1-a^2*x^2)/(c*x*sqrt(c-a^2*c*x^2))+1/2*a^2*sqrt(1-a^2*x^2)/(c*(1-a*x)*sqrt(c-a^2*c*x^2))+2*a^2*log(x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))-7/4*a^2*log(1-a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))-1/4*a^2*log(1+a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x^4*(c-a^2*c*x^2)^(3/2)),x,4,-1/3*sqrt(1-a^2*x^2)/(c*x^3*sqrt(c-a^2*c*x^2))-1/2*a*sqrt(1-a^2*x^2)/(c*x^2*sqrt(c-a^2*c*x^2))-2*a^2*sqrt(1-a^2*x^2)/(c*x*sqrt(c-a^2*c*x^2))+1/2*a^3*sqrt(1-a^2*x^2)/(c*(1-a*x)*sqrt(c-a^2*c*x^2))+2*a^3*log(x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))-9/4*a^3*log(1-a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))+1/4*a^3*log(1+a*x)*sqrt(1-a^2*x^2)/(c*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^6/(c-a^2*c*x^2)^(5/2),x,4,-x*sqrt(1-a^2*x^2)/(a^6*c^2*sqrt(c-a^2*c*x^2))-1/2*x^2*sqrt(1-a^2*x^2)/(a^5*c^2*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a^7*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))-5/4*sqrt(1-a^2*x^2)/(a^7*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a^7*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))-39/16*log(1-a*x)*sqrt(1-a^2*x^2)/(a^7*c^2*sqrt(c-a^2*c*x^2))-9/16*log(1+a*x)*sqrt(1-a^2*x^2)/(a^7*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^5/(c-a^2*c*x^2)^(5/2),x,4,-x*sqrt(1-a^2*x^2)/(a^5*c^2*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a^6*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))-sqrt(1-a^2*x^2)/(a^6*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a^6*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))-23/16*log(1-a*x)*sqrt(1-a^2*x^2)/(a^6*c^2*sqrt(c-a^2*c*x^2))+7/16*log(1+a*x)*sqrt(1-a^2*x^2)/(a^6*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^4/(c-a^2*c*x^2)^(5/2),x,4,1/8*sqrt(1-a^2*x^2)/(a^5*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))-3/4*sqrt(1-a^2*x^2)/(a^5*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a^5*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))-11/16*log(1-a*x)*sqrt(1-a^2*x^2)/(a^5*c^2*sqrt(c-a^2*c*x^2))-5/16*log(1+a*x)*sqrt(1-a^2*x^2)/(a^5*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^3/(c-a^2*c*x^2)^(5/2),x,5,1/8*sqrt(1-a^2*x^2)/(a^4*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))-1/2*sqrt(1-a^2*x^2)/(a^4*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a^4*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+3/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a^4*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^2/(c-a^2*c*x^2)^(5/2),x,5,1/8*sqrt(1-a^2*x^2)/(a^3*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))-1/4*sqrt(1-a^2*x^2)/(a^3*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a^3*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))-1/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a^3*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x/(c-a^2*c*x^2)^(5/2),x,5,1/8*sqrt(1-a^2*x^2)/(a^2*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a^2*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))-1/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a^2*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^(5/2),x,5,1/8*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))+1/4*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+3/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x*(c-a^2*c*x^2)^(5/2)),x,4,1/8*sqrt(1-a^2*x^2)/(c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))+1/2*sqrt(1-a^2*x^2)/(c^2*(1-a*x)*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+log(x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))-11/16*log(1-a*x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))-5/16*log(1+a*x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(5/2)),x,4,-sqrt(1-a^2*x^2)/(c^2*x*sqrt(c-a^2*c*x^2))+1/8*a*sqrt(1-a^2*x^2)/(c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))+3/4*a*sqrt(1-a^2*x^2)/(c^2*(1-a*x)*sqrt(c-a^2*c*x^2))-1/8*a*sqrt(1-a^2*x^2)/(c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+a*log(x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))-23/16*a*log(1-a*x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))+7/16*a*log(1+a*x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(x^3*(c-a^2*c*x^2)^(5/2)),x,4,-1/2*sqrt(1-a^2*x^2)/(c^2*x^2*sqrt(c-a^2*c*x^2))-a*sqrt(1-a^2*x^2)/(c^2*x*sqrt(c-a^2*c*x^2))+1/8*a^2*sqrt(1-a^2*x^2)/(c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))+a^2*sqrt(1-a^2*x^2)/(c^2*(1-a*x)*sqrt(c-a^2*c*x^2))+1/8*a^2*sqrt(1-a^2*x^2)/(c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+3*a^2*log(x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))-39/16*a^2*log(1-a*x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))-9/16*a^2*log(1+a*x)*sqrt(1-a^2*x^2)/(c^2*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))/(c-a^2*c*x^2)^(7/2),x,5,1/24*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^3*sqrt(c-a^2*c*x^2))+3/32*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^2*sqrt(c-a^2*c*x^2))+3/16*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)*sqrt(c-a^2*c*x^2))-1/32*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)^2*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)*sqrt(c-a^2*c*x^2))+5/16*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^3*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^p with m symbolic
[exp(arctanh(a*x))*x^m*(c-a^2*c*x^2)^2,x,4,c^2*x^(1+m)*hypergeom([-3/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+a*c^2*x^(2+m)*hypergeom([-3/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[exp(arctanh(a*x))*x^m*(c-a^2*c*x^2),x,4,c*x^(1+m)*hypergeom([-1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+a*c*x^(2+m)*hypergeom([-1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[exp(arctanh(a*x))*x^m/(c-a^2*c*x^2),x,4,x^(1+m)*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(c*(1+m))+a*x^(2+m)*hypergeom([3/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(c*(2+m))],
[exp(arctanh(a*x))*x^m/(c-a^2*c*x^2)^2,x,4,x^(1+m)*hypergeom([5/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(c^2*(1+m))+a*x^(2+m)*hypergeom([5/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(c^2*(2+m))],
[exp(arctanh(a*x))*x^m/(c-a^2*c*x^2)^3,x,4,x^(1+m)*hypergeom([7/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(c^3*(1+m))+a*x^(2+m)*hypergeom([7/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(c^3*(2+m))],
[exp(arctanh(a*x))*x^m*(1-a^2*x^2)^(5/2),x,3,x^(1+m)/(1+m)+a*x^(2+m)/(2+m)-2*a^2*x^(3+m)/(3+m)-2*a^3*x^(4+m)/(4+m)+a^4*x^(5+m)/(5+m)+a^5*x^(6+m)/(6+m)],
[exp(arctanh(a*x))*x^m*(1-a^2*x^2)^(3/2),x,3,x^(1+m)/(1+m)+a*x^(2+m)/(2+m)-a^2*x^(3+m)/(3+m)-a^3*x^(4+m)/(4+m)],
[exp(arctanh(a*x))*x^m*(1-a^2*x^2)^(1/2),x,3,x^(1+m)/(1+m)+a*x^(2+m)/(2+m)],
[exp(arctanh(a*x))*x^m/(1-a^2*x^2)^(1/2),x,2,x^(1+m)*hypergeom([1,1+m],[2+m],a*x)/(1+m)],
[exp(arctanh(a*x))*x^m/(1-a^2*x^2)^(3/2),x,6,x^(1+m)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+a*x^(2+m)*hypergeom([2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[exp(arctanh(a*x))*x^m/(1-a^2*x^2)^(5/2),x,6,x^(1+m)*hypergeom([3,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)/(1+m)+a*x^(2+m)*hypergeom([3,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)/(2+m)],
[exp(arctanh(a*x))*x^m*(c-a^2*c*x^2)^(5/2),x,4,c^2*x^(1+m)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))+a*c^2*x^(2+m)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))-2*a^2*c^2*x^(3+m)*sqrt(c-a^2*c*x^2)/((3+m)*sqrt(1-a^2*x^2))-2*a^3*c^2*x^(4+m)*sqrt(c-a^2*c*x^2)/((4+m)*sqrt(1-a^2*x^2))+a^4*c^2*x^(5+m)*sqrt(c-a^2*c*x^2)/((5+m)*sqrt(1-a^2*x^2))+a^5*c^2*x^(6+m)*sqrt(c-a^2*c*x^2)/((6+m)*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^m*(c-a^2*c*x^2)^(3/2),x,4,c*x^(1+m)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))+a*c*x^(2+m)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))-a^2*c*x^(3+m)*sqrt(c-a^2*c*x^2)/((3+m)*sqrt(1-a^2*x^2))-a^3*c*x^(4+m)*sqrt(c-a^2*c*x^2)/((4+m)*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^m*(c-a^2*c*x^2)^(1/2),x,4,x^(1+m)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))+a*x^(2+m)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))],
[exp(arctanh(a*x))*x^m/(c-a^2*c*x^2)^(1/2),x,3,x^(1+m)*hypergeom([1,1+m],[2+m],a*x)*sqrt(1-a^2*x^2)/((1+m)*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^m/(c-a^2*c*x^2)^(3/2),x,7,x^(1+m)*hypergeom([2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(1-a^2*x^2)/(c*(1+m)*sqrt(c-a^2*c*x^2))+a*x^(2+m)*hypergeom([2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(1-a^2*x^2)/(c*(2+m)*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^m/(c-a^2*c*x^2)^(5/2),x,7,x^(1+m)*hypergeom([3,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(1-a^2*x^2)/(c^2*(1+m)*sqrt(c-a^2*c*x^2))+a*x^(2+m)*hypergeom([3,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(1-a^2*x^2)/(c^2*(2+m)*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^ArcTanh[a x] (c-a^2 c x^2)^p with p symbolic
[exp(arctanh(a*x))*x^m*(c-a^2*c*x^2)^p,x,5,x^(1+m)*(c-a^2*c*x^2)^p*hypergeom([1/2*(1+m),1/2-p],[1/2*(3+m)],a^2*x^2)/((1+m)*(1-a^2*x^2)^p)+a*x^(2+m)*(c-a^2*c*x^2)^p*hypergeom([1/2*(2+m),1/2-p],[1/2*(4+m)],a^2*x^2)/((2+m)*(1-a^2*x^2)^p)],
[exp(arctanh(a*x))*x^3*(1-a^2*x^2)^p,x,6,-(1-a^2*x^2)^(1/2+p)/(a^4*(1+2*p))+(1-a^2*x^2)^(3/2+p)/(a^4*(3+2*p))+1/5*a*x^5*hypergeom([5/2,1/2-p],[7/2],a^2*x^2)],
[exp(arctanh(a*x))*x^2*(1-a^2*x^2)^p,x,6,-(1-a^2*x^2)^(1/2+p)/(a^3*(1+2*p))+(1-a^2*x^2)^(3/2+p)/(a^3*(3+2*p))+1/3*x^3*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)],
[exp(arctanh(a*x))*x*(1-a^2*x^2)^p,x,4,-(1-a^2*x^2)^(1/2+p)/(a^2*(1+2*p))+1/3*a*x^3*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)],
[exp(arctanh(a*x))*(1-a^2*x^2)^p,x,2,-2^(3/2+p)*(1-a*x)^(1/2+p)*hypergeom([-1/2-p,1/2+p],[3/2+p],1/2*(1-a*x))/(a*(1+2*p))],
[exp(arctanh(a*x))*(1-a^2*x^2)^p/x,x,5,a*x*hypergeom([1/2,1/2-p],[3/2],a^2*x^2)-(1-a^2*x^2)^(1/2+p)*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)/(1+2*p)],
[exp(arctanh(a*x))*(1-a^2*x^2)^p/x^2,x,5,-hypergeom([-1/2,1/2-p],[1/2],a^2*x^2)/x-a*(1-a^2*x^2)^(1/2+p)*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)/(1+2*p)],
[exp(arctanh(a*x))*(1-a^2*x^2)^p/x^3,x,5,-a*hypergeom([-1/2,1/2-p],[1/2],a^2*x^2)/x-a^2*(1-a^2*x^2)^(1/2+p)*hypergeom([2,1/2+p],[3/2+p],1-a^2*x^2)/(1+2*p)],
[exp(arctanh(a*x))*x^3*(c-a^2*c*x^2)^p,x,7,(1-a^2*x^2)^(3/2)*(c-a^2*c*x^2)^p/(a^4*(3+2*p))+1/5*a*x^5*(c-a^2*c*x^2)^p*hypergeom([5/2,1/2-p],[7/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^4*(1+2*p))],
[exp(arctanh(a*x))*x^2*(c-a^2*c*x^2)^p,x,7,(1-a^2*x^2)^(3/2)*(c-a^2*c*x^2)^p/(a^3*(3+2*p))+1/3*x^3*(c-a^2*c*x^2)^p*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^3*(1+2*p))],
[exp(arctanh(a*x))*x*(c-a^2*c*x^2)^p,x,5,1/3*a*x^3*(c-a^2*c*x^2)^p*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^2*(1+2*p))],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^p,x,3,-2^(3/2+p)*(1-a*x)^(1/2+p)*(c-a^2*c*x^2)^p*hypergeom([-1/2-p,1/2+p],[3/2+p],1/2*(1-a*x))/(a*(1+2*p)*(1-a^2*x^2)^p)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^p/x,x,6,a*x*(c-a^2*c*x^2)^p*hypergeom([1/2,1/2-p],[3/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^p/x^2,x,6,-(c-a^2*c*x^2)^p*hypergeom([-1/2,1/2-p],[1/2],a^2*x^2)/(x*(1-a^2*x^2)^p)-a*(c-a^2*c*x^2)^p*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],
[exp(arctanh(a*x))*(c-a^2*c*x^2)^p/x^3,x,6,-a*(c-a^2*c*x^2)^p*hypergeom([-1/2,1/2-p],[1/2],a^2*x^2)/(x*(1-a^2*x^2)^p)-a^2*(c-a^2*c*x^2)^p*hypergeom([2,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],

# Integrands of the form x^m (c-a^2 c x^2)^p E^(2 ArcTanh[a x])

# Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[exp(2*arctanh(a*x))*x^4*(c-a^2*c*x^2),x,3,1/5*c*x^5+1/3*a*c*x^6+1/7*a^2*c*x^7],
[exp(2*arctanh(a*x))*x^3*(c-a^2*c*x^2),x,3,1/4*c*x^4+2/5*a*c*x^5+1/6*a^2*c*x^6],
[exp(2*arctanh(a*x))*x^2*(c-a^2*c*x^2),x,3,1/3*c*x^3+1/2*a*c*x^4+1/5*a^2*c*x^5],
[exp(2*arctanh(a*x))*x*(c-a^2*c*x^2),x,3,1/2*c*x^2+2/3*a*c*x^3+1/4*a^2*c*x^4],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2),x,2,1/3*c*(1+a*x)^3/a],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)/x,x,3,2*a*c*x+1/2*a^2*c*x^2+c*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)/x^2,x,3,-c/x+a^2*c*x+2*a*c*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)/x^3,x,3,-1/2*c/x^2-2*a*c/x+a^2*c*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)/x^4,x,2,-1/3*c*(1+a*x)^3/x^3],
[exp(2*arctanh(a*x))*x^4*(c-a^2*c*x^2)^2,x,3,1/5*c^2*x^5+1/3*a*c^2*x^6-1/4*a^3*c^2*x^8-1/9*a^4*c^2*x^9],
[exp(2*arctanh(a*x))*x^3*(c-a^2*c*x^2)^2,x,3,1/4*c^2*x^4+2/5*a*c^2*x^5-2/7*a^3*c^2*x^7-1/8*a^4*c^2*x^8],
[exp(2*arctanh(a*x))*x^2*(c-a^2*c*x^2)^2,x,3,1/3*c^2*x^3+1/2*a*c^2*x^4-1/3*a^3*c^2*x^6-1/7*a^4*c^2*x^7],
[exp(2*arctanh(a*x))*x*(c-a^2*c*x^2)^2,x,3,1/2*c^2*x^2+2/3*a*c^2*x^3-2/5*a^3*c^2*x^5-1/6*a^4*c^2*x^6],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2,x,3,1/2*c^2*(1+a*x)^4/a-1/5*c^2*(1+a*x)^5/a],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2/x,x,3,2*a*c^2*x-2/3*a^3*c^2*x^3-1/4*a^4*c^2*x^4+c^2*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2/x^2,x,3,-c^2/x-a^3*c^2*x^2-1/3*a^4*c^2*x^3+2*a*c^2*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2/x^3,x,2,-1/2*c^2*(1+a*x)^4/x^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2/x^4,x,3,-1/3*c^2/x^3-a*c^2/x^2-a^4*c^2*x-2*a^3*c^2*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2/x^5,x,3,-1/4*c^2/x^4-2/3*a*c^2/x^3+2*a^3*c^2/x-a^4*c^2*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2/x^6,x,3,-1/5*c^2/x^5-1/2*a*c^2/x^4+a^3*c^2/x^2+a^4*c^2/x],
[exp(2*arctanh(a*x))*x^4*(c-a^2*c*x^2)^3,x,3,1/5*c^3*x^5+1/3*a*c^3*x^6-1/7*a^2*c^3*x^7-1/2*a^3*c^3*x^8-1/9*a^4*c^3*x^9+1/5*a^5*c^3*x^10+1/11*a^6*c^3*x^11],
[exp(2*arctanh(a*x))*x^3*(c-a^2*c*x^2)^3,x,3,1/4*c^3*x^4+2/5*a*c^3*x^5-1/6*a^2*c^3*x^6-4/7*a^3*c^3*x^7-1/8*a^4*c^3*x^8+2/9*a^5*c^3*x^9+1/10*a^6*c^3*x^10],
[exp(2*arctanh(a*x))*x^2*(c-a^2*c*x^2)^3,x,3,4/5*c^3*(1+a*x)^5/a^3-2*c^3*(1+a*x)^6/a^3+13/7*c^3*(1+a*x)^7/a^3-3/4*c^3*(1+a*x)^8/a^3+1/9*c^3*(1+a*x)^9/a^3],
[exp(2*arctanh(a*x))*x*(c-a^2*c*x^2)^3,x,3,-4/5*c^3*(1+a*x)^5/a^2+4/3*c^3*(1+a*x)^6/a^2-5/7*c^3*(1+a*x)^7/a^2+1/8*c^3*(1+a*x)^8/a^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3,x,3,4/5*c^3*(1+a*x)^5/a-2/3*c^3*(1+a*x)^6/a+1/7*c^3*(1+a*x)^7/a],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3/x,x,3,2*a*c^3*x-1/2*a^2*c^3*x^2-4/3*a^3*c^3*x^3-1/4*a^4*c^3*x^4+2/5*a^5*c^3*x^5+1/6*a^6*c^3*x^6+c^3*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3/x^2,x,3,-c^3/x-a^2*c^3*x-2*a^3*c^3*x^2-1/3*a^4*c^3*x^3+1/2*a^5*c^3*x^4+1/5*a^6*c^3*x^5+2*a*c^3*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3/x^3,x,3,-1/2*c^3/x^2-2*a*c^3/x-4*a^3*c^3*x-1/2*a^4*c^3*x^2+2/3*a^5*c^3*x^3+1/4*a^6*c^3*x^4-a^2*c^3*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3/x^4,x,3,-1/3*c^3/x^3-a*c^3/x^2+a^2*c^3/x-a^4*c^3*x+a^5*c^3*x^2+1/3*a^6*c^3*x^3-4*a^3*c^3*log(x)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^4,x,3,4/3*c^4*(1+a*x)^6/a-12/7*c^4*(1+a*x)^7/a+3/4*c^4*(1+a*x)^8/a-1/9*c^4*(1+a*x)^9/a],

# p<0
[exp(2*arctanh(a*x))*x^4/(c-a^2*c*x^2),x,3,3*x/(a^4*c)+x^2/(a^3*c)+1/3*x^3/(a^2*c)+1/(a^5*c*(1-a*x))+4*log(1-a*x)/(a^5*c)],
[exp(2*arctanh(a*x))*x^3/(c-a^2*c*x^2),x,3,2*x/(a^3*c)+1/2*x^2/(a^2*c)+1/(a^4*c*(1-a*x))+3*log(1-a*x)/(a^4*c)],
[exp(2*arctanh(a*x))*x^2/(c-a^2*c*x^2),x,3,x/(a^2*c)+1/(a^3*c*(1-a*x))+2*log(1-a*x)/(a^3*c)],
[exp(2*arctanh(a*x))*x/(c-a^2*c*x^2),x,3,1/(a^2*c*(1-a*x))+log(1-a*x)/(a^2*c)],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2),x,2,1/(a*c*(1-a*x))],
[exp(2*arctanh(a*x))/(x*(c-a^2*c*x^2)),x,3,1/(c*(1-a*x))+log(x)/c-log(1-a*x)/c],
[exp(2*arctanh(a*x))/(x^2*(c-a^2*c*x^2)),x,3,(-1)/(c*x)+a/(c*(1-a*x))+2*a*log(x)/c-2*a*log(1-a*x)/c],
[exp(2*arctanh(a*x))/(x^3*(c-a^2*c*x^2)),x,3,(-1/2)/(c*x^2)-2*a/(c*x)+a^2/(c*(1-a*x))+3*a^2*log(x)/c-3*a^2*log(1-a*x)/c],
[exp(2*arctanh(a*x))/(x^4*(c-a^2*c*x^2)),x,3,(-1/3)/(c*x^3)-a/(c*x^2)-3*a^2/(c*x)+a^3/(c*(1-a*x))+4*a^3*log(x)/c-4*a^3*log(1-a*x)/c],
[exp(2*arctanh(a*x))*x^4/(c-a^2*c*x^2)^2,x,3,-x/(a^4*c^2)+1/4/(a^5*c^2*(1-a*x)^2)+(-7/4)/(a^5*c^2*(1-a*x))-17/8*log(1-a*x)/(a^5*c^2)+1/8*log(1+a*x)/(a^5*c^2)],
[exp(2*arctanh(a*x))*x^3/(c-a^2*c*x^2)^2,x,3,1/4/(a^4*c^2*(1-a*x)^2)+(-5/4)/(a^4*c^2*(1-a*x))-7/8*log(1-a*x)/(a^4*c^2)-1/8*log(1+a*x)/(a^4*c^2)],
[exp(2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^2,x,4,1/4/(a^3*c^2*(1-a*x)^2)+(-3/4)/(a^3*c^2*(1-a*x))+1/4*arctanh(a*x)/(a^3*c^2)],
[exp(2*arctanh(a*x))*x/(c-a^2*c*x^2)^2,x,4,1/4/(a^2*c^2*(1-a*x)^2)+(-1/4)/(a^2*c^2*(1-a*x))-1/4*arctanh(a*x)/(a^2*c^2)],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^2,x,4,1/4/(a*c^2*(1-a*x)^2)+1/4/(a*c^2*(1-a*x))+1/4*arctanh(a*x)/(a*c^2)],
[exp(2*arctanh(a*x))/(x*(c-a^2*c*x^2)^2),x,3,1/4/(c^2*(1-a*x)^2)+3/4/(c^2*(1-a*x))+log(x)/c^2-7/8*log(1-a*x)/c^2-1/8*log(1+a*x)/c^2],
[exp(2*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^2),x,3,(-1)/(c^2*x)+1/4*a/(c^2*(1-a*x)^2)+5/4*a/(c^2*(1-a*x))+2*a*log(x)/c^2-17/8*a*log(1-a*x)/c^2+1/8*a*log(1+a*x)/c^2],
[exp(2*arctanh(a*x))/(x^3*(c-a^2*c*x^2)^2),x,3,(-1/2)/(c^2*x^2)-2*a/(c^2*x)+1/4*a^2/(c^2*(1-a*x)^2)+7/4*a^2/(c^2*(1-a*x))+4*a^2*log(x)/c^2-31/8*a^2*log(1-a*x)/c^2-1/8*a^2*log(1+a*x)/c^2],
[exp(2*arctanh(a*x))/(x^4*(c-a^2*c*x^2)^2),x,3,(-1/3)/(c^2*x^3)-a/(c^2*x^2)-4*a^2/(c^2*x)+1/4*a^3/(c^2*(1-a*x)^2)+9/4*a^3/(c^2*(1-a*x))+6*a^3*log(x)/c^2-49/8*a^3*log(1-a*x)/c^2+1/8*a^3*log(1+a*x)/c^2],
[exp(2*arctanh(a*x))*x^5/(c-a^2*c*x^2)^3,x,3,1/12/(a^6*c^3*(1-a*x)^3)+(-1/2)/(a^6*c^3*(1-a*x)^2)+23/16/(a^6*c^3*(1-a*x))+1/16/(a^6*c^3*(1+a*x))+13/16*log(1-a*x)/(a^6*c^3)+3/16*log(1+a*x)/(a^6*c^3)],
[exp(2*arctanh(a*x))*x^4/(c-a^2*c*x^2)^3,x,4,1/12/(a^5*c^3*(1-a*x)^3)+(-3/8)/(a^5*c^3*(1-a*x)^2)+11/16/(a^5*c^3*(1-a*x))+(-1/16)/(a^5*c^3*(1+a*x))-1/4*arctanh(a*x)/(a^5*c^3)],
[exp(2*arctanh(a*x))*x^3/(c-a^2*c*x^2)^3,x,4,1/12/(a^4*c^3*(1-a*x)^3)+(-1/4)/(a^4*c^3*(1-a*x)^2)+3/16/(a^4*c^3*(1-a*x))+1/16/(a^4*c^3*(1+a*x))+1/8*arctanh(a*x)/(a^4*c^3)],
[exp(2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^3,x,2,1/6*(-1+2*a*x)/(a^3*c^3*(1-a*x)^3*(1+a*x))],
[exp(2*arctanh(a*x))*x/(c-a^2*c*x^2)^3,x,4,1/12/(a^2*c^3*(1-a*x)^3)+(-1/16)/(a^2*c^3*(1-a*x))+1/16/(a^2*c^3*(1+a*x))-1/8*arctanh(a*x)/(a^2*c^3)],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^3,x,4,1/12/(a*c^3*(1-a*x)^3)+1/8/(a*c^3*(1-a*x)^2)+3/16/(a*c^3*(1-a*x))+(-1/16)/(a*c^3*(1+a*x))+1/4*arctanh(a*x)/(a*c^3)],
[exp(2*arctanh(a*x))/(x*(c-a^2*c*x^2)^3),x,3,1/12/(c^3*(1-a*x)^3)+1/4/(c^3*(1-a*x)^2)+11/16/(c^3*(1-a*x))+1/16/(c^3*(1+a*x))+log(x)/c^3-13/16*log(1-a*x)/c^3-3/16*log(1+a*x)/c^3],
[exp(2*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^3),x,3,(-1)/(c^3*x)+1/12*a/(c^3*(1-a*x)^3)+3/8*a/(c^3*(1-a*x)^2)+23/16*a/(c^3*(1-a*x))-1/16*a/(c^3*(1+a*x))+2*a*log(x)/c^3-9/4*a*log(1-a*x)/c^3+1/4*a*log(1+a*x)/c^3],
[exp(2*arctanh(a*x))/(x^3*(c-a^2*c*x^2)^3),x,3,(-1/2)/(c^3*x^2)-2*a/(c^3*x)+1/12*a^2/(c^3*(1-a*x)^3)+1/2*a^2/(c^3*(1-a*x)^2)+39/16*a^2/(c^3*(1-a*x))+1/16*a^2/(c^3*(1+a*x))+5*a^2*log(x)/c^3-75/16*a^2*log(1-a*x)/c^3-5/16*a^2*log(1+a*x)/c^3],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^4,x,4,1/32/(a*c^4*(1-a*x)^4)+1/16/(a*c^4*(1-a*x)^3)+3/32/(a*c^4*(1-a*x)^2)+5/32/(a*c^4*(1-a*x))+(-1/64)/(a*c^4*(1+a*x)^2)+(-5/64)/(a*c^4*(1+a*x))+15/64*arctanh(a*x)/(a*c^4)],

# Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0
[exp(2*arctanh(a*x))*x^3*(c-a^2*c*x^2)^(1/2),x,7,3/4*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^4-3/5*x^2*sqrt(c-a^2*c*x^2)/a^2-1/2*x^3*sqrt(c-a^2*c*x^2)/a-1/5*x^4*sqrt(c-a^2*c*x^2)-3/20*(8+5*a*x)*sqrt(c-a^2*c*x^2)/a^4],
[exp(2*arctanh(a*x))*x^2*(c-a^2*c*x^2)^(1/2),x,6,7/8*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^3-2/3*x^2*sqrt(c-a^2*c*x^2)/a-1/4*x^3*sqrt(c-a^2*c*x^2)-1/24*(32+21*a*x)*sqrt(c-a^2*c*x^2)/a^3],
[exp(2*arctanh(a*x))*x*(c-a^2*c*x^2)^(1/2),x,5,arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^2-1/3*x^2*sqrt(c-a^2*c*x^2)-1/3*(5+3*a*x)*sqrt(c-a^2*c*x^2)/a^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2),x,5,3/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a-3/2*sqrt(c-a^2*c*x^2)/a-1/2*(1+a*x)*sqrt(c-a^2*c*x^2)/a],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x,x,8,2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)-arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^2,x,8,a*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)-2*a*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^3,x,6,-3/2*a^2*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-1/2*sqrt(c-a^2*c*x^2)/x^2-2*a*sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^4,x,7,-a^3*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-1/3*sqrt(c-a^2*c*x^2)/x^3-a*sqrt(c-a^2*c*x^2)/x^2-5/3*a^2*sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^5,x,8,-7/8*a^4*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-1/4*sqrt(c-a^2*c*x^2)/x^4-2/3*a*sqrt(c-a^2*c*x^2)/x^3-7/8*a^2*sqrt(c-a^2*c*x^2)/x^2-4/3*a^3*sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*x^3*(c-a^2*c*x^2)^(3/2),x,8,-11/35*x^2*(c-a^2*c*x^2)^(3/2)/a^2-1/3*x^3*(c-a^2*c*x^2)^(3/2)/a-1/7*x^4*(c-a^2*c*x^2)^(3/2)-1/420*(88+105*a*x)*(c-a^2*c*x^2)^(3/2)/a^4+1/8*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a^4+1/8*c*x*sqrt(c-a^2*c*x^2)/a^3],
[exp(2*arctanh(a*x))*x^2*(c-a^2*c*x^2)^(3/2),x,7,-2/5*x^2*(c-a^2*c*x^2)^(3/2)/a-1/6*x^3*(c-a^2*c*x^2)^(3/2)-1/120*(32+45*a*x)*(c-a^2*c*x^2)^(3/2)/a^3+3/16*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a^3+3/16*c*x*sqrt(c-a^2*c*x^2)/a^2],
[exp(2*arctanh(a*x))*x*(c-a^2*c*x^2)^(3/2),x,6,-1/5*x^2*(c-a^2*c*x^2)^(3/2)-1/30*(14+15*a*x)*(c-a^2*c*x^2)^(3/2)/a^2+1/4*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a^2+1/4*c*x*sqrt(c-a^2*c*x^2)/a],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2),x,6,-5/12*(c-a^2*c*x^2)^(3/2)/a-1/4*(1+a*x)*(c-a^2*c*x^2)^(3/2)/a+5/8*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a+5/8*c*x*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x,x,9,-1/3*(c-a^2*c*x^2)^(3/2)+c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))-c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))+c*(1+a*x)*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^2,x,9,-(c-a^2*c*x^2)^(3/2)/x-1/2*a*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))-2*a*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))+1/2*a*c*(4-a*x)*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^3,x,9,-1/2*(c-a^2*c*x^2)^(3/2)/x^2-2*a^2*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))-1/2*a^2*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-1/2*a*c*(4-a*x)*sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^4,x,9,-1/3*(c-a^2*c*x^2)^(3/2)/x^3-a^3*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))+a^3*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-a*c*(1+a*x)*sqrt(c-a^2*c*x^2)/x^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^5,x,7,-1/4*(c-a^2*c*x^2)^(3/2)/x^4-2/3*a*(c-a^2*c*x^2)^(3/2)/x^3+5/8*a^4*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-5/8*a^2*c*sqrt(c-a^2*c*x^2)/x^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^6,x,8,-1/5*(c-a^2*c*x^2)^(3/2)/x^5-1/2*a*(c-a^2*c*x^2)^(3/2)/x^4-7/15*a^2*(c-a^2*c*x^2)^(3/2)/x^3+1/4*a^5*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-1/4*a^3*c*sqrt(c-a^2*c*x^2)/x^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^7,x,9,-1/6*(c-a^2*c*x^2)^(3/2)/x^6-2/5*a*(c-a^2*c*x^2)^(3/2)/x^5-3/8*a^2*(c-a^2*c*x^2)^(3/2)/x^4-4/15*a^3*(c-a^2*c*x^2)^(3/2)/x^3+3/16*a^6*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-3/16*a^4*c*sqrt(c-a^2*c*x^2)/x^2],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)/x^8,x,10,-1/7*(c-a^2*c*x^2)^(3/2)/x^7-1/3*a*(c-a^2*c*x^2)^(3/2)/x^6-11/35*a^2*(c-a^2*c*x^2)^(3/2)/x^5-1/4*a^3*(c-a^2*c*x^2)^(3/2)/x^4-22/105*a^4*(c-a^2*c*x^2)^(3/2)/x^3+1/8*a^7*c^(3/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-1/8*a^5*c*sqrt(c-a^2*c*x^2)/x^2],
[exp(2*arctanh(a*x))*x^3*(c-a^2*c*x^2)^(5/2),x,9,1/32*c*x*(c-a^2*c*x^2)^(3/2)/a^3-13/63*x^2*(c-a^2*c*x^2)^(5/2)/a^2-1/4*x^3*(c-a^2*c*x^2)^(5/2)/a-1/9*x^4*(c-a^2*c*x^2)^(5/2)-1/2520*(208+315*a*x)*(c-a^2*c*x^2)^(5/2)/a^4+3/64*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a^4+3/64*c^2*x*sqrt(c-a^2*c*x^2)/a^3],
[exp(2*arctanh(a*x))*x^2*(c-a^2*c*x^2)^(5/2),x,8,11/192*c*x*(c-a^2*c*x^2)^(3/2)/a^2-2/7*x^2*(c-a^2*c*x^2)^(5/2)/a-1/8*x^3*(c-a^2*c*x^2)^(5/2)-1/1680*(192+385*a*x)*(c-a^2*c*x^2)^(5/2)/a^3+11/128*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a^3+11/128*c^2*x*sqrt(c-a^2*c*x^2)/a^2],
[exp(2*arctanh(a*x))*x*(c-a^2*c*x^2)^(5/2),x,7,1/12*c*x*(c-a^2*c*x^2)^(3/2)/a-1/7*x^2*(c-a^2*c*x^2)^(5/2)-1/105*(27+35*a*x)*(c-a^2*c*x^2)^(5/2)/a^2+1/8*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a^2+1/8*c^2*x*sqrt(c-a^2*c*x^2)/a],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2),x,7,7/24*c*x*(c-a^2*c*x^2)^(3/2)-7/30*(c-a^2*c*x^2)^(5/2)/a-1/6*(1+a*x)*(c-a^2*c*x^2)^(5/2)/a+7/16*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a+7/16*c^2*x*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)/x,x,10,1/6*c*(2+3*a*x)*(c-a^2*c*x^2)^(3/2)-1/5*(c-a^2*c*x^2)^(5/2)+3/4*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))-c^(5/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))+1/4*c^2*(4+3*a*x)*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)/x^2,x,10,1/12*a*c*(8-9*a*x)*(c-a^2*c*x^2)^(3/2)-(c-a^2*c*x^2)^(5/2)/x-9/8*a*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))-2*a*c^(5/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))+1/8*a*c^2*(16-9*a*x)*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)/x^3,x,10,-1/6*a*c*(12+a*x)*(c-a^2*c*x^2)^(3/2)/x-1/2*(c-a^2*c*x^2)^(5/2)/x^2-3*a^2*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))+1/2*a^2*c^(5/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-1/2*a^2*c^2*(1+6*a*x)*sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)/x^4,x,10,-1/6*a*c*(6-a*x)*(c-a^2*c*x^2)^(3/2)/x^2-1/3*(c-a^2*c*x^2)^(5/2)/x^3-1/2*a^3*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))+3*a^3*c^(5/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))-1/2*a^2*c^2*(1+6*a*x)*sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)/x^5,x,10,-1/24*a*c*(16+9*a*x)*(c-a^2*c*x^2)^(3/2)/x^3-1/4*(c-a^2*c*x^2)^(5/2)/x^4+2*a^4*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))+9/8*a^4*c^(5/2)*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))+1/8*a^3*c^2*(16-9*a*x)*sqrt(c-a^2*c*x^2)/x],
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(7/2),x,8,15/64*c^2*x*(c-a^2*c*x^2)^(3/2)+3/16*c*x*(c-a^2*c*x^2)^(5/2)-9/56*(c-a^2*c*x^2)^(7/2)/a-1/8*(1+a*x)*(c-a^2*c*x^2)^(7/2)/a+45/128*c^(7/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a+45/128*c^3*x*sqrt(c-a^2*c*x^2)],

# p<0
[exp(2*arctanh(a*x))*x^3/(c-a^2*c*x^2)^(1/2),x,7,-3*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a^4*sqrt(c))+(1+a*x)^2/(a^4*sqrt(c-a^2*c*x^2))+11/3*sqrt(c-a^2*c*x^2)/(a^4*c)+x*sqrt(c-a^2*c*x^2)/(a^3*c)+1/3*x^2*sqrt(c-a^2*c*x^2)/(a^2*c)],
[exp(2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(1/2),x,5,-5/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a^3*sqrt(c))+(1+a*x)^2/(a^3*sqrt(c-a^2*c*x^2))+1/2*(6+a*x)*sqrt(c-a^2*c*x^2)/(a^3*c)],
[exp(2*arctanh(a*x))*x/(c-a^2*c*x^2)^(1/2),x,5,-2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a^2*sqrt(c))+(1+a*x)^2/(a^2*sqrt(c-a^2*c*x^2))+2*sqrt(c-a^2*c*x^2)/(a^2*c)],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^(1/2),x,4,-arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a*sqrt(c))+2*(1+a*x)/(a*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))/(x*(c-a^2*c*x^2)^(1/2)),x,5,-arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/sqrt(c)+2*(1+a*x)/sqrt(c-a^2*c*x^2)],
[exp(2*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(1/2)),x,6,-2*a*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/sqrt(c)+2*a*(1+a*x)/sqrt(c-a^2*c*x^2)-sqrt(c-a^2*c*x^2)/(c*x)],
[exp(2*arctanh(a*x))/(x^3*(c-a^2*c*x^2)^(1/2)),x,7,-5/2*a^2*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/sqrt(c)+2*a^2*(1+a*x)/sqrt(c-a^2*c*x^2)-1/2*sqrt(c-a^2*c*x^2)/(c*x^2)-2*a*sqrt(c-a^2*c*x^2)/(c*x)],
[exp(2*arctanh(a*x))/(x^4*(c-a^2*c*x^2)^(1/2)),x,8,-3*a^3*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/sqrt(c)+2*a^3*(1+a*x)/sqrt(c-a^2*c*x^2)-1/3*sqrt(c-a^2*c*x^2)/(c*x^3)-a*sqrt(c-a^2*c*x^2)/(c*x^2)-8/3*a^2*sqrt(c-a^2*c*x^2)/(c*x)],
[exp(2*arctanh(a*x))*x^3/(c-a^2*c*x^2)^(3/2),x,6,1/3*(1+a*x)^2/(a^4*(c-a^2*c*x^2)^(3/2))+2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a^4*c^(3/2))-8/3*(1+a*x)/(a^4*c*sqrt(c-a^2*c*x^2))-sqrt(c-a^2*c*x^2)/(a^4*c^2)],
[exp(2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(3/2),x,5,1/3*(1+a*x)^2/(a^3*(c-a^2*c*x^2)^(3/2))+arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a^3*c^(3/2))-5/3*(1+a*x)/(a^3*c*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))*x/(c-a^2*c*x^2)^(3/2),x,3,1/3*(1+a*x)^2/(a^2*(c-a^2*c*x^2)^(3/2))-2/3*(1+a*x)/(a^2*c*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^(3/2),x,3,2/3*(1+a*x)/(a*(c-a^2*c*x^2)^(3/2))+1/3*x/(c*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))/(x*(c-a^2*c*x^2)^(3/2)),x,7,2/3*(1+a*x)/(c-a^2*c*x^2)^(3/2)-arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/c^(3/2)+1/3*(3+4*a*x)/(c*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(3/2)),x,7,2/3*a*(1+a*x)/(c-a^2*c*x^2)^(3/2)-2*a*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/c^(3/2)+1/3*a*(6+7*a*x)/(c*sqrt(c-a^2*c*x^2))-sqrt(c-a^2*c*x^2)/(c^2*x)],
[exp(2*arctanh(a*x))/(x^3*(c-a^2*c*x^2)^(3/2)),x,8,2/3*a^2*(1+a*x)/(c-a^2*c*x^2)^(3/2)-7/2*a^2*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))/c^(3/2)+1/3*a^2*(9+10*a*x)/(c*sqrt(c-a^2*c*x^2))-1/2*sqrt(c-a^2*c*x^2)/(c^2*x^2)-2*a*sqrt(c-a^2*c*x^2)/(c^2*x)],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^(5/2),x,4,2/5*(1+a*x)/(a*(c-a^2*c*x^2)^(5/2))+1/5*x/(c*(c-a^2*c*x^2)^(3/2))+2/5*x/(c^2*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))/(c-a^2*c*x^2)^(7/2),x,5,2/7*(1+a*x)/(a*(c-a^2*c*x^2)^(7/2))+1/7*x/(c*(c-a^2*c*x^2)^(5/2))+4/21*x/(c^2*(c-a^2*c*x^2)^(3/2))+8/21*x/(c^3*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic
[exp(2*arctanh(a*x))*x^m*(c-a^2*c*x^2)^3,x,3,c^3*x^(1+m)/(1+m)+2*a*c^3*x^(2+m)/(2+m)-a^2*c^3*x^(3+m)/(3+m)-4*a^3*c^3*x^(4+m)/(4+m)-a^4*c^3*x^(5+m)/(5+m)+2*a^5*c^3*x^(6+m)/(6+m)+a^6*c^3*x^(7+m)/(7+m)],
[exp(2*arctanh(a*x))*x^m*(c-a^2*c*x^2)^2,x,3,c^2*x^(1+m)/(1+m)+2*a*c^2*x^(2+m)/(2+m)-2*a^3*c^2*x^(4+m)/(4+m)-a^4*c^2*x^(5+m)/(5+m)],
[exp(2*arctanh(a*x))*x^m*(c-a^2*c*x^2),x,3,c*x^(1+m)/(1+m)+2*a*c*x^(2+m)/(2+m)+a^2*c*x^(3+m)/(3+m)],
[exp(2*arctanh(a*x))*x^m/(c-a^2*c*x^2),x,2,x^(1+m)*hypergeom([2,1+m],[2+m],a*x)/(c*(1+m))],
[exp(2*arctanh(a*x))*x^m/(c-a^2*c*x^2)^2,x,6,1/4*x^(1+m)/(c^2*(1-a*x)^2)+1/4*(2-m)*x^(1+m)/(c^2*(1-a*x))+1/8*x^(1+m)*hypergeom([1,1+m],[2+m],-a*x)/(c^2*(1+m))+1/8*(1-4*m+2*m^2)*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)/(c^2*(1+m))],
[exp(2*arctanh(a*x))*x^m/(c-a^2*c*x^2)^3,x,8,-1/24*(2-m)*(4-m)*x^(1+m)/(c^3*(1+a*x))+1/6*x^(1+m)/(c^3*(1-a*x)^3*(1+a*x))+1/12*(4-m)*x^(1+m)/(c^3*(1-a*x)^2*(1+a*x))+1/24*(7-2*m)*(2-m)*x^(1+m)/(c^3*(1-a*x)*(1+a*x))+1/16*(2-m)*x^(1+m)*hypergeom([1,1+m],[2+m],-a*x)/(c^3*(1+m))+1/48*(2-m)*(3-8*m+2*m^2)*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)/(c^3*(1+m))],
[exp(2*arctanh(a*x))*x^m*(c-a^2*c*x^2)^(5/2),x,7,-x^(1+m)*(c-a^2*c*x^2)^(5/2)/(6+m)+c^2*(7+2*m)*x^(1+m)*hypergeom([-3/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(c-a^2*c*x^2)/((6+7*m+m^2)*sqrt(1-a^2*x^2))+2*a*c^2*x^(2+m)*hypergeom([-3/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))],
[exp(2*arctanh(a*x))*x^m*(c-a^2*c*x^2)^(3/2),x,7,-x^(1+m)*(c-a^2*c*x^2)^(3/2)/(4+m)+c*(5+2*m)*x^(1+m)*hypergeom([-1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(c-a^2*c*x^2)/((4+5*m+m^2)*sqrt(1-a^2*x^2))+2*a*c*x^(2+m)*hypergeom([-1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))],
[exp(2*arctanh(a*x))*x^m*(c-a^2*c*x^2)^(1/2),x,7,c*(3+2*m)*x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(1-a^2*x^2)/((2+3*m+m^2)*sqrt(c-a^2*c*x^2))+2*a*c*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(1-a^2*x^2)/((2+m)*sqrt(c-a^2*c*x^2))-x^(1+m)*sqrt(c-a^2*c*x^2)/(2+m)],
[exp(2*arctanh(a*x))*x^m/(c-a^2*c*x^2)^(1/2),x,7,2*x^(1+m)*(1+a*x)/sqrt(c-a^2*c*x^2)-(1+2*m)*x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(1-a^2*x^2)/((1+m)*sqrt(c-a^2*c*x^2))-2*a*(1+m)*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(1-a^2*x^2)/((2+m)*sqrt(c-a^2*c*x^2))],
[exp(2*arctanh(a*x))*x^m/(c-a^2*c*x^2)^(3/2),x,7,2/3*x^(1+m)*(1+a*x)/(c-a^2*c*x^2)^(3/2)+1/3*(1-2*m)*x^(1+m)*hypergeom([3/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(1-a^2*x^2)/(c*(1+m)*sqrt(c-a^2*c*x^2))+2/3*a*(1-m)*x^(2+m)*hypergeom([3/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(1-a^2*x^2)/(c*(2+m)*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(2 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[exp(2*arctanh(a*x))*(c-a^2*c*x^2)^p,x,3,-2^(1+p)*(c-a^2*c*x^2)^p*hypergeom([-1-p,p],[1+p],1/2*(1-a*x))/(a*p*(1+a*x)^p)],

# Integrands of the form x^m (c-a^2 c x^2)^p E^(3 ArcTanh[a x])

# Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[exp(3*arctanh(a*x))*x^3*(c-a^2*c*x^2),x,7,23/16*c*arcsin(a*x)/a^4-17/15*c*x^2*sqrt(1-a^2*x^2)/a^2-23/24*c*x^3*sqrt(1-a^2*x^2)/a-3/5*c*x^4*sqrt(1-a^2*x^2)-1/6*a*c*x^5*sqrt(1-a^2*x^2)-1/240*c*(544+345*a*x)*sqrt(1-a^2*x^2)/a^4],
[exp(3*arctanh(a*x))*x^2*(c-a^2*c*x^2),x,6,13/8*c*arcsin(a*x)/a^3-19/15*c*x^2*sqrt(1-a^2*x^2)/a-3/4*c*x^3*sqrt(1-a^2*x^2)-1/5*a*c*x^4*sqrt(1-a^2*x^2)-1/120*c*(304+195*a*x)*sqrt(1-a^2*x^2)/a^3],
[exp(3*arctanh(a*x))*x*(c-a^2*c*x^2),x,6,15/8*c*arcsin(a*x)/a^2-15/8*c*sqrt(1-a^2*x^2)/a^2-5/8*c*(1+a*x)*sqrt(1-a^2*x^2)/a^2-1/4*c*(1+a*x)^2*sqrt(1-a^2*x^2)/a^2-1/4*c*(1+a*x)^3*sqrt(1-a^2*x^2)/a^2],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2),x,5,5/2*c*arcsin(a*x)/a-5/2*c*sqrt(1-a^2*x^2)/a-5/6*c*(1+a*x)*sqrt(1-a^2*x^2)/a-1/3*c*(1+a*x)^2*sqrt(1-a^2*x^2)/a],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)/x,x,8,7/2*c*arcsin(a*x)-c*arctanh(sqrt(1-a^2*x^2))-3*c*sqrt(1-a^2*x^2)-1/2*a*c*x*sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)/x^2,x,8,3*a*c*arcsin(a*x)-3*a*c*arctanh(sqrt(1-a^2*x^2))-a*c*sqrt(1-a^2*x^2)-c*sqrt(1-a^2*x^2)/x],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)/x^3,x,8,a^2*c*arcsin(a*x)-7/2*a^2*c*arctanh(sqrt(1-a^2*x^2))-1/2*c*sqrt(1-a^2*x^2)/x^2-3*a*c*sqrt(1-a^2*x^2)/x],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)/x^4,x,7,-5/2*a^3*c*arctanh(sqrt(1-a^2*x^2))-1/3*c*sqrt(1-a^2*x^2)/x^3-3/2*a*c*sqrt(1-a^2*x^2)/x^2-11/3*a^2*c*sqrt(1-a^2*x^2)/x],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)/x^5,x,8,-15/8*a^4*c*arctanh(sqrt(1-a^2*x^2))-1/4*c*sqrt(1-a^2*x^2)/x^4-a*c*sqrt(1-a^2*x^2)/x^3-15/8*a^2*c*sqrt(1-a^2*x^2)/x^2-3*a^3*c*sqrt(1-a^2*x^2)/x],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)/x^6,x,9,-13/8*a^5*c*arctanh(sqrt(1-a^2*x^2))-1/5*c*sqrt(1-a^2*x^2)/x^5-3/4*a*c*sqrt(1-a^2*x^2)/x^4-19/15*a^2*c*sqrt(1-a^2*x^2)/x^3-13/8*a^3*c*sqrt(1-a^2*x^2)/x^2-38/15*a^4*c*sqrt(1-a^2*x^2)/x],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^2,x,6,-7/12*c^2*(1-a^2*x^2)^(3/2)/a-7/20*c^2*(1+a*x)*(1-a^2*x^2)^(3/2)/a-1/5*c^2*(1+a*x)^2*(1-a^2*x^2)^(3/2)/a+7/8*c^2*arcsin(a*x)/a+7/8*c^2*x*sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^3,x,7,3/8*c^3*x*(1-a^2*x^2)^(3/2)-3/10*c^3*(1-a^2*x^2)^(5/2)/a-3/14*c^3*(1+a*x)*(1-a^2*x^2)^(5/2)/a-1/7*c^3*(1+a*x)^2*(1-a^2*x^2)^(5/2)/a+9/16*c^3*arcsin(a*x)/a+9/16*c^3*x*sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^4,x,8,55/192*c^4*x*(1-a^2*x^2)^(3/2)+11/48*c^4*x*(1-a^2*x^2)^(5/2)-11/56*c^4*(1-a^2*x^2)^(7/2)/a-11/72*c^4*(1+a*x)*(1-a^2*x^2)^(7/2)/a-1/9*c^4*(1+a*x)^2*(1-a^2*x^2)^(7/2)/a+55/128*c^4*arcsin(a*x)/a+55/128*c^4*x*sqrt(1-a^2*x^2)],

# p<0
[exp(3*arctanh(a*x))*x^2/(c-a^2*c*x^2),x,6,1/3*(1+a*x)^3/(a^3*c*(1-a^2*x^2)^(3/2))+3*arcsin(a*x)/(a^3*c)-2*(1+a*x)^2/(a^3*c*sqrt(1-a^2*x^2))-3*sqrt(1-a^2*x^2)/(a^3*c)],
[exp(3*arctanh(a*x))*x/(c-a^2*c*x^2),x,4,1/3*(1+a*x)^3/(a^2*c*(1-a^2*x^2)^(3/2))+arcsin(a*x)/(a^2*c)-2*(1+a*x)/(a^2*c*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2),x,1,1/3*exp(3*arctanh(a*x))/(a*c)],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^2,x,5,1/5*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)^3)+2/15*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)^2)+2/15*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x))],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^3,x,6,8/35*x/(c^3*sqrt(1-a^2*x^2))+1/7/(a*c^3*(1-a*x)^3*sqrt(1-a^2*x^2))+4/35/(a*c^3*(1-a*x)^2*sqrt(1-a^2*x^2))+4/35/(a*c^3*(1-a*x)*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^4,x,7,8/63*x/(c^4*(1-a^2*x^2)^(3/2))+1/9/(a*c^4*(1-a*x)^3*(1-a^2*x^2)^(3/2))+2/21/(a*c^4*(1-a*x)^2*(1-a^2*x^2)^(3/2))+2/21/(a*c^4*(1-a*x)*(1-a^2*x^2)^(3/2))+16/63*x/(c^4*sqrt(1-a^2*x^2))],

# Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0
[exp(3*arctanh(a*x))*x^3*(c-a^2*c*x^2)^(1/2),x,4,-4*x*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1-a^2*x^2))-2*x^2*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))-4/3*x^3*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-3/4*x^4*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/5*a*x^5*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^4*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*x^2*(c-a^2*c*x^2)^(1/2),x,4,-4*x*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))-2*x^2*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/4*a*x^4*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*x*(c-a^2*c*x^2)^(1/2),x,4,-4*x*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-3/2*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/3*a*x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2),x,4,-3*x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/2*a*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x,x,4,-a*x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1-a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^2,x,4,-sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+3*a*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*a*log(1-a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^3,x,4,-1/2*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1-a^2*x^2))-3*a*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+4*a^2*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*a^2*log(1-a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^4,x,4,-1/3*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1-a^2*x^2))-3/2*a*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1-a^2*x^2))-4*a^2*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+4*a^3*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*a^3*log(1-a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)/x^5,x,4,-1/4*sqrt(c-a^2*c*x^2)/(x^4*sqrt(1-a^2*x^2))-a*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1-a^2*x^2))-2*a^2*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1-a^2*x^2))-4*a^3*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+4*a^4*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*a^4*log(1-a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(3/2),x,3,1/4*c*(1+a*x)^4*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(5/2),x,4,2/5*c^2*(1+a*x)^5*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/6*c^2*(1+a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(7/2),x,4,2/3*c^3*(1+a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-4/7*c^3*(1+a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+1/8*c^3*(1+a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(9/2),x,4,8/7*c^4*(1+a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-3/2*c^4*(1+a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+2/3*c^4*(1+a*x)^9*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/10*c^4*(1+a*x)^10*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],

# p<0
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^(1/2),x,4,2*sqrt(1-a^2*x^2)/(a*(1-a*x)*sqrt(c-a^2*c*x^2))+log(1-a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^(3/2),x,3,1/2*sqrt(1-a^2*x^2)/(a*c*(1-a*x)^2*sqrt(c-a^2*c*x^2))],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^(5/2),x,5,1/6*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)^3*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)^2*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))+1/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^2*sqrt(c-a^2*c*x^2))],
[exp(3*arctanh(a*x))/(c-a^2*c*x^2)^(7/2),x,5,1/16*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^4*sqrt(c-a^2*c*x^2))+1/12*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^3*sqrt(c-a^2*c*x^2))+3/32*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^2*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)*sqrt(c-a^2*c*x^2))-1/32*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)*sqrt(c-a^2*c*x^2))+5/32*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^3*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic
[exp(3*arctanh(a*x))*x^m*(c-a^2*c*x^2)^(1/2),x,5,-3*x^(1+m)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))-a*x^(2+m)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))+4*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))],

# Integrands of the form x^m E^(3 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[exp(3*arctanh(a*x))*x^m*(c-a^2*c*x^2)^p,x,7,(3+4*m+2*p)*x^(1+m)*(c-a^2*c*x^2)^p*hypergeom([1/2*(1+m),3/2-p],[1/2*(3+m)],a^2*x^2)/((1+m)*(m+2*p)*(1-a^2*x^2)^p)+a*(5+4*m+6*p)*x^(2+m)*(c-a^2*c*x^2)^p*hypergeom([1/2*(2+m),3/2-p],[1/2*(4+m)],a^2*x^2)/((2+m)*(1+m+2*p)*(1-a^2*x^2)^p)-3*x^(1+m)*(c-a^2*c*x^2)^p/((m+2*p)*sqrt(1-a^2*x^2))-a*x^(2+m)*(c-a^2*c*x^2)^p/((1+m+2*p)*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*x^3*(c-a^2*c*x^2)^p,x,8,-3*(1-a^2*x^2)^(3/2)*(c-a^2*c*x^2)^p/(a^4*(3+2*p))+1/10*a*(17+6*p)*x^5*(c-a^2*c*x^2)^p*hypergeom([5/2,3/2-p],[7/2],a^2*x^2)/((2+p)*(1-a^2*x^2)^p)+4*(c-a^2*c*x^2)^p/(a^4*(1-2*p)*sqrt(1-a^2*x^2))-1/2*a*x^5*(c-a^2*c*x^2)^p/((2+p)*sqrt(1-a^2*x^2))+7*(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^4*(1+2*p))],
[exp(3*arctanh(a*x))*x^2*(c-a^2*c*x^2)^p,x,8,-(1-a^2*x^2)^(3/2)*(c-a^2*c*x^2)^p/(a^3*(3+2*p))+1/6*(11+2*p)*x^3*(c-a^2*c*x^2)^p*hypergeom([3/2,3/2-p],[5/2],a^2*x^2)/((1+p)*(1-a^2*x^2)^p)+4*(c-a^2*c*x^2)^p/(a^3*(1-2*p)*sqrt(1-a^2*x^2))-3/2*x^3*(c-a^2*c*x^2)^p/((1+p)*sqrt(1-a^2*x^2))+5*(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^3*(1+2*p))],
[exp(3*arctanh(a*x))*x*(c-a^2*c*x^2)^p,x,5,3*2^(3/2+p)*(1-a*x)^(-1/2+p)*(c-a^2*c*x^2)^p*hypergeom([-3/2-p,-1/2+p],[1/2+p],1/2*(1-a*x))/(a^2*(1-p-2*p^2)*(1-a^2*x^2)^p)-1/2*(1+a*x)^3*(c-a^2*c*x^2)^p/(a^2*(1+p)*sqrt(1-a^2*x^2))],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^p,x,3,2^(5/2+p)*(1-a*x)^(-1/2+p)*(c-a^2*c*x^2)^p*hypergeom([-3/2-p,-1/2+p],[1/2+p],1/2*(1-a*x))/(a*(1-2*p)*(1-a^2*x^2)^p)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^p/x,x,8,1/2*a*(1+6*p)*x*(c-a^2*c*x^2)^p*hypergeom([1/2,3/2-p],[3/2],a^2*x^2)/(p*(1-a^2*x^2)^p)+4*(c-a^2*c*x^2)^p/((1-2*p)*sqrt(1-a^2*x^2))-1/2*a*x*(c-a^2*c*x^2)^p/(p*sqrt(1-a^2*x^2))-(c-a^2*c*x^2)^p*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^p/x^2,x,9,a^2*(5-2*p)*x*(c-a^2*c*x^2)^p*hypergeom([1/2,3/2-p],[3/2],a^2*x^2)/(1-a^2*x^2)^p+4*a*(c-a^2*c*x^2)^p/((1-2*p)*sqrt(1-a^2*x^2))-(c-a^2*c*x^2)^p/(x*sqrt(1-a^2*x^2))-3*a*(c-a^2*c*x^2)^p*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],
[exp(3*arctanh(a*x))*(c-a^2*c*x^2)^p/x^3,x,8,a^3*(7-6*p)*x*(c-a^2*c*x^2)^p*hypergeom([1/2,3/2-p],[3/2],a^2*x^2)/(1-a^2*x^2)^p-1/2*(c-a^2*c*x^2)^p/(x^2*sqrt(1-a^2*x^2))-3*a*(c-a^2*c*x^2)^p/(x*sqrt(1-a^2*x^2))+1/2*a^2*(9-2*p)*(c-a^2*c*x^2)^p*hypergeom([1,-1/2+p],[1/2+p],1-a^2*x^2)/((1-2*p)*sqrt(1-a^2*x^2))],

# Integrands of the form x^m (c-a^2 c x^2)^p E^(4 ArcTanh[a x])

# Integrands of the form x^m E^(4 ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[exp(4*arctanh(a*x))*(c-a^2*c*x^2)^5,x,3,c^5*(1+a*x)^8/a-4/3*c^5*(1+a*x)^9/a+3/5*c^5*(1+a*x)^10/a-1/11*c^5*(1+a*x)^11/a],
[exp(4*arctanh(a*x))*(c-a^2*c*x^2)^4,x,3,4/7*c^4*(1+a*x)^7/a-1/2*c^4*(1+a*x)^8/a+1/9*c^4*(1+a*x)^9/a],
[exp(4*arctanh(a*x))*(c-a^2*c*x^2)^3,x,3,1/3*c^3*(1+a*x)^6/a-1/7*c^3*(1+a*x)^7/a],
[exp(4*arctanh(a*x))*(c-a^2*c*x^2)^2,x,2,1/5*c^2*(1+a*x)^5/a],
[exp(4*arctanh(a*x))*(c-a^2*c*x^2),x,3,-7*c*x-2*a*c*x^2-1/3*a^2*c*x^3-8*c*log(1-a*x)/a,-4*c*x-c*(1+a*x)^2/a-1/3*c*(1+a*x)^3/a-8*c*log(1-a*x)/a],

# p<0
[exp(4*arctanh(a*x))/(c-a^2*c*x^2),x,2,x/(c*(1-a*x)^2)],
[exp(4*arctanh(a*x))/(c-a^2*c*x^2)^2,x,2,1/3/(a*c^2*(1-a*x)^3)],
[exp(4*arctanh(a*x))/(c-a^2*c*x^2)^3,x,4,1/8/(a*c^3*(1-a*x)^4)+1/12/(a*c^3*(1-a*x)^3)+1/16/(a*c^3*(1-a*x)^2)+1/16/(a*c^3*(1-a*x))+1/16*arctanh(a*x)/(a*c^3)],
[exp(4*arctanh(a*x))/(c-a^2*c*x^2)^4,x,4,1/20/(a*c^4*(1-a*x)^5)+1/16/(a*c^4*(1-a*x)^4)+1/16/(a*c^4*(1-a*x)^3)+1/16/(a*c^4*(1-a*x)^2)+5/64/(a*c^4*(1-a*x))+(-1/64)/(a*c^4*(1+a*x))+3/32*arctanh(a*x)/(a*c^4)],

# Integrands of the form x^m E^(4 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0

# p<0

# Integrands of the form x^m E^(4 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[exp(4*arctanh(a*x))*(c-a^2*c*x^2)^p,x,3,2^(2+p)*c*(1+a*x)^(1-p)*(c-a^2*c*x^2)^(-1+p)*hypergeom([-2-p,-1+p],[p],1/2*(1-a*x))/(a*(1-p))],

# Integrands of the form x^m (c-a^2 c x^2)^p / E^(1 ArcTanh[a x])

# Integrands of the form x^m E^(-ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[(c-a^2*c*x^2)^4/exp(arctanh(a*x)),x,7,35/192*c^4*x*(1-a^2*x^2)^(3/2)+7/48*c^4*x*(1-a^2*x^2)^(5/2)+1/8*c^4*x*(1-a^2*x^2)^(7/2)+1/9*c^4*(1-a^2*x^2)^(9/2)/a+35/128*c^4*arcsin(a*x)/a+35/128*c^4*x*sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^3/exp(arctanh(a*x)),x,6,5/24*c^3*x*(1-a^2*x^2)^(3/2)+1/6*c^3*x*(1-a^2*x^2)^(5/2)+1/7*c^3*(1-a^2*x^2)^(7/2)/a+5/16*c^3*arcsin(a*x)/a+5/16*c^3*x*sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^2/exp(arctanh(a*x)),x,5,1/4*c^2*x*(1-a^2*x^2)^(3/2)+1/5*c^2*(1-a^2*x^2)^(5/2)/a+3/8*c^2*arcsin(a*x)/a+3/8*c^2*x*sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)/exp(arctanh(a*x)),x,4,1/3*c*(1-a^2*x^2)^(3/2)/a+1/2*c*arcsin(a*x)/a+1/2*c*x*sqrt(1-a^2*x^2)],

# p<0
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)),x,1,(-1)/(a*c*exp(arctanh(a*x)))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^2),x,3,1/3*(-1+a*x)/(a*c^2*(1-a^2*x^2)^(3/2))+2/3*x/(c^2*sqrt(1-a^2*x^2))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^3),x,4,1/5*(-1+a*x)/(a*c^3*(1-a^2*x^2)^(5/2))+4/15*x/(c^3*(1-a^2*x^2)^(3/2))+8/15*x/(c^3*sqrt(1-a^2*x^2))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^4),x,5,1/7*(-1+a*x)/(a*c^4*(1-a^2*x^2)^(7/2))+6/35*x/(c^4*(1-a^2*x^2)^(5/2))+8/35*x/(c^4*(1-a^2*x^2)^(3/2))+16/35*x/(c^4*sqrt(1-a^2*x^2))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^5),x,6,1/9*(-1+a*x)/(a*c^5*(1-a^2*x^2)^(9/2))+8/63*x/(c^5*(1-a^2*x^2)^(7/2))+16/105*x/(c^5*(1-a^2*x^2)^(5/2))+64/315*x/(c^5*(1-a^2*x^2)^(3/2))+128/315*x/(c^5*sqrt(1-a^2*x^2))],

# Integrands of the form x^m E^(-ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0
[x^m*(c-a^2*c*x^2)^(1/2)/exp(arctanh(a*x)),x,4,x^(1+m)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))-a*x^(2+m)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))],
[x^2*(c-a^2*c*x^2)^(1/2)/exp(arctanh(a*x)),x,4,1/3*x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/4*a*x^4*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[x*(c-a^2*c*x^2)^(1/2)/exp(arctanh(a*x)),x,4,1/2*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/3*a*x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/exp(arctanh(a*x)),x,3,x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-1/2*a*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(arctanh(a*x))*x),x,4,-a*x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(arctanh(a*x))*x^2),x,4,-sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))-a*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(3/2)/exp(arctanh(a*x)),x,4,-2/3*c*(1-a*x)^3*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+1/4*c*(1-a*x)^4*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(5/2)/exp(arctanh(a*x)),x,4,-c^2*(1-a*x)^4*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+4/5*c^2*(1-a*x)^5*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/6*c^2*(1-a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(7/2)/exp(arctanh(a*x)),x,4,-8/5*c^3*(1-a*x)^5*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+2*c^3*(1-a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-6/7*c^3*(1-a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+1/8*c^3*(1-a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(9/2)/exp(arctanh(a*x)),x,4,-8/3*c^4*(1-a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+32/7*c^4*(1-a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-3*c^4*(1-a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+8/9*c^4*(1-a*x)^9*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/10*c^4*(1-a*x)^10*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],

# p<0
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^(1/2)),x,3,log(1+a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^(3/2)),x,5,-1/2*sqrt(1-a^2*x^2)/(a*c*(1+a*x)*sqrt(c-a^2*c*x^2))+1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c*sqrt(c-a^2*c*x^2))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^(5/2)),x,5,1/8*sqrt(1-a^2*x^2)/(a*c^2*(1-a*x)*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)^2*sqrt(c-a^2*c*x^2))-1/4*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+3/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^2*sqrt(c-a^2*c*x^2))],
[1/(exp(arctanh(a*x))*(c-a^2*c*x^2)^(7/2)),x,5,1/32*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)^2*sqrt(c-a^2*c*x^2))+1/8*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)*sqrt(c-a^2*c*x^2))-1/24*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)^3*sqrt(c-a^2*c*x^2))-3/32*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)^2*sqrt(c-a^2*c*x^2))-3/16*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)*sqrt(c-a^2*c*x^2))+5/16*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^3*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(-ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[x^m*(c-a^2*c*x^2)^p/exp(arctanh(a*x)),x,5,x^(1+m)*(c-a^2*c*x^2)^p*hypergeom([1/2*(1+m),1/2-p],[1/2*(3+m)],a^2*x^2)/((1+m)*(1-a^2*x^2)^p)-a*x^(2+m)*(c-a^2*c*x^2)^p*hypergeom([1/2*(2+m),1/2-p],[1/2*(4+m)],a^2*x^2)/((2+m)*(1-a^2*x^2)^p)],
[x^3*(1-a^2*x^2)^p/exp(arctanh(a*x)),x,6,-(1-a^2*x^2)^(1/2+p)/(a^4*(1+2*p))+(1-a^2*x^2)^(3/2+p)/(a^4*(3+2*p))-1/5*a*x^5*hypergeom([5/2,1/2-p],[7/2],a^2*x^2)],
[x^2*(1-a^2*x^2)^p/exp(arctanh(a*x)),x,6,(1-a^2*x^2)^(1/2+p)/(a^3*(1+2*p))-(1-a^2*x^2)^(3/2+p)/(a^3*(3+2*p))+1/3*x^3*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)],
[x*(1-a^2*x^2)^p/exp(arctanh(a*x)),x,4,-(1-a^2*x^2)^(1/2+p)/(a^2*(1+2*p))-1/3*a*x^3*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)],
[(1-a^2*x^2)^p/exp(arctanh(a*x)),x,2,-2^(1/2+p)*(1-a*x)^(3/2+p)*hypergeom([1/2-p,3/2+p],[5/2+p],1/2*(1-a*x))/(a*(3+2*p))],
[(1-a^2*x^2)^p/(exp(arctanh(a*x))*x),x,5,-a*x*hypergeom([1/2,1/2-p],[3/2],a^2*x^2)-(1-a^2*x^2)^(1/2+p)*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)/(1+2*p)],
[(1-a^2*x^2)^p/(exp(arctanh(a*x))*x^2),x,5,-hypergeom([-1/2,1/2-p],[1/2],a^2*x^2)/x+a*(1-a^2*x^2)^(1/2+p)*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)/(1+2*p)],
[x^3*(c-a^2*c*x^2)^p/exp(arctanh(a*x)),x,7,(1-a^2*x^2)^(3/2)*(c-a^2*c*x^2)^p/(a^4*(3+2*p))-1/5*a*x^5*(c-a^2*c*x^2)^p*hypergeom([5/2,1/2-p],[7/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^4*(1+2*p))],
[x^2*(c-a^2*c*x^2)^p/exp(arctanh(a*x)),x,7,-(1-a^2*x^2)^(3/2)*(c-a^2*c*x^2)^p/(a^3*(3+2*p))+1/3*x^3*(c-a^2*c*x^2)^p*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)/(1-a^2*x^2)^p+(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^3*(1+2*p))],
[x*(c-a^2*c*x^2)^p/exp(arctanh(a*x)),x,5,-1/3*a*x^3*(c-a^2*c*x^2)^p*hypergeom([3/2,1/2-p],[5/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*sqrt(1-a^2*x^2)/(a^2*(1+2*p))],
[(c-a^2*c*x^2)^p/exp(arctanh(a*x)),x,3,-2^(1/2+p)*(1-a*x)^(3/2+p)*(c-a^2*c*x^2)^p*hypergeom([1/2-p,3/2+p],[5/2+p],1/2*(1-a*x))/(a*(3+2*p)*(1-a^2*x^2)^p)],
[(c-a^2*c*x^2)^p/(exp(arctanh(a*x))*x),x,6,-a*x*(c-a^2*c*x^2)^p*hypergeom([1/2,1/2-p],[3/2],a^2*x^2)/(1-a^2*x^2)^p-(c-a^2*c*x^2)^p*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],
[(c-a^2*c*x^2)^p/(exp(arctanh(a*x))*x^2),x,6,-(c-a^2*c*x^2)^p*hypergeom([-1/2,1/2-p],[1/2],a^2*x^2)/(x*(1-a^2*x^2)^p)+a*(c-a^2*c*x^2)^p*hypergeom([1,1/2+p],[3/2+p],1-a^2*x^2)*sqrt(1-a^2*x^2)/(1+2*p)],

# Integrands of the form x^m (c-a^2 c x^2)^p / E^(2 ArcTanh[a x])

# Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[(c-a^2*c*x^2)^4/exp(2*arctanh(a*x)),x,3,-4/3*c^4*(1-a*x)^6/a+12/7*c^4*(1-a*x)^7/a-3/4*c^4*(1-a*x)^8/a+1/9*c^4*(1-a*x)^9/a],
[(c-a^2*c*x^2)^3/exp(2*arctanh(a*x)),x,3,-4/5*c^3*(1-a*x)^5/a+2/3*c^3*(1-a*x)^6/a-1/7*c^3*(1-a*x)^7/a],
[(c-a^2*c*x^2)^2/exp(2*arctanh(a*x)),x,3,-1/2*c^2*(1-a*x)^4/a+1/5*c^2*(1-a*x)^5/a],
[(c-a^2*c*x^2)/exp(2*arctanh(a*x)),x,2,-1/3*c*(1-a*x)^3/a],

# p<0
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)),x,2,(-1)/(a*c*(1+a*x))],
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^2),x,4,(-1/4)/(a*c^2*(1+a*x)^2)+(-1/4)/(a*c^2*(1+a*x))+1/4*arctanh(a*x)/(a*c^2)],
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3),x,4,1/16/(a*c^3*(1-a*x))+(-1/12)/(a*c^3*(1+a*x)^3)+(-1/8)/(a*c^3*(1+a*x)^2)+(-3/16)/(a*c^3*(1+a*x))+1/4*arctanh(a*x)/(a*c^3)],
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^4),x,4,1/64/(a*c^4*(1-a*x)^2)+5/64/(a*c^4*(1-a*x))+(-1/32)/(a*c^4*(1+a*x)^4)+(-1/16)/(a*c^4*(1+a*x)^3)+(-3/32)/(a*c^4*(1+a*x)^2)+(-5/32)/(a*c^4*(1+a*x))+15/64*arctanh(a*x)/(a*c^4)],

# Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0
[x^3*(c-a^2*c*x^2)^(1/2)/exp(2*arctanh(a*x)),x,7,-3/4*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^4-3/5*x^2*sqrt(c-a^2*c*x^2)/a^2+1/2*x^3*sqrt(c-a^2*c*x^2)/a-1/5*x^4*sqrt(c-a^2*c*x^2)-3/20*(8-5*a*x)*sqrt(c-a^2*c*x^2)/a^4],
[x^2*(c-a^2*c*x^2)^(1/2)/exp(2*arctanh(a*x)),x,6,7/8*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^3+2/3*x^2*sqrt(c-a^2*c*x^2)/a-1/4*x^3*sqrt(c-a^2*c*x^2)+1/24*(32-21*a*x)*sqrt(c-a^2*c*x^2)/a^3],
[x*(c-a^2*c*x^2)^(1/2)/exp(2*arctanh(a*x)),x,5,-arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^2-1/3*x^2*sqrt(c-a^2*c*x^2)-1/3*(5-3*a*x)*sqrt(c-a^2*c*x^2)/a^2],
[(c-a^2*c*x^2)^(1/2)/exp(2*arctanh(a*x)),x,5,3/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a+3/2*sqrt(c-a^2*c*x^2)/a+1/2*(1-a*x)*sqrt(c-a^2*c*x^2)/a],
[(c-a^2*c*x^2)^(1/2)/(exp(2*arctanh(a*x))*x),x,8,-2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)-arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-sqrt(c-a^2*c*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(2*arctanh(a*x))*x^2),x,8,a*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)+2*a*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-sqrt(c-a^2*c*x^2)/x],
[(c-a^2*c*x^2)^(1/2)/(exp(2*arctanh(a*x))*x^3),x,6,-3/2*a^2*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-1/2*sqrt(c-a^2*c*x^2)/x^2+2*a*sqrt(c-a^2*c*x^2)/x],
[(c-a^2*c*x^2)^(1/2)/(exp(2*arctanh(a*x))*x^4),x,7,a^3*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-1/3*sqrt(c-a^2*c*x^2)/x^3+a*sqrt(c-a^2*c*x^2)/x^2-5/3*a^2*sqrt(c-a^2*c*x^2)/x],
[(c-a^2*c*x^2)^(1/2)/(exp(2*arctanh(a*x))*x^5),x,8,-7/8*a^4*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)-1/4*sqrt(c-a^2*c*x^2)/x^4+2/3*a*sqrt(c-a^2*c*x^2)/x^3-7/8*a^2*sqrt(c-a^2*c*x^2)/x^2+4/3*a^3*sqrt(c-a^2*c*x^2)/x],
[(c-a^2*c*x^2)^(3/2)/exp(2*arctanh(a*x)),x,6,5/12*(c-a^2*c*x^2)^(3/2)/a+1/4*(1-a*x)*(c-a^2*c*x^2)^(3/2)/a+5/8*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a+5/8*c*x*sqrt(c-a^2*c*x^2)],
[(c-a^2*c*x^2)^(5/2)/exp(2*arctanh(a*x)),x,7,7/24*c*x*(c-a^2*c*x^2)^(3/2)+7/30*(c-a^2*c*x^2)^(5/2)/a+1/6*(1-a*x)*(c-a^2*c*x^2)^(5/2)/a+7/16*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a+7/16*c^2*x*sqrt(c-a^2*c*x^2)],
[(c-a^2*c*x^2)^(7/2)/exp(2*arctanh(a*x)),x,8,15/64*c^2*x*(c-a^2*c*x^2)^(3/2)+3/16*c*x*(c-a^2*c*x^2)^(5/2)+9/56*(c-a^2*c*x^2)^(7/2)/a+1/8*(1-a*x)*(c-a^2*c*x^2)^(7/2)/a+45/128*c^(7/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a+45/128*c^3*x*sqrt(c-a^2*c*x^2)],

# p<0
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)),x,4,-arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a*sqrt(c))-2*(1-a*x)/(a*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)),x,3,-2/3*(1-a*x)/(a*(c-a^2*c*x^2)^(3/2))+1/3*x/(c*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)),x,4,-2/5*(1-a*x)/(a*(c-a^2*c*x^2)^(5/2))+1/5*x/(c*(c-a^2*c*x^2)^(3/2))+2/5*x/(c^2*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^(7/2)),x,5,-2/7*(1-a*x)/(a*(c-a^2*c*x^2)^(7/2))+1/7*x/(c*(c-a^2*c*x^2)^(5/2))+4/21*x/(c^2*(c-a^2*c*x^2)^(3/2))+8/21*x/(c^3*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic
[x^m*sqrt(c-a^2*c*x^2)/exp(2*arctanh(a*x)),x,7,c*(3+2*m)*x^(1+m)*hypergeom([1/2,1/2*(1+m)],[1/2*(3+m)],a^2*x^2)*sqrt(1-a^2*x^2)/((2+3*m+m^2)*sqrt(c-a^2*c*x^2))-2*a*c*x^(2+m)*hypergeom([1/2,1/2*(2+m)],[1/2*(4+m)],a^2*x^2)*sqrt(1-a^2*x^2)/((2+m)*sqrt(c-a^2*c*x^2))-x^(1+m)*sqrt(c-a^2*c*x^2)/(2+m)],

# Integrands of the form x^m E^(-2 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[(c-a^2*c*x^2)^p/exp(2*arctanh(a*x)),x,3,2^(1+p)*(c-a^2*c*x^2)^p*hypergeom([-1-p,p],[1+p],1/2*(1+a*x))/(a*p*(1-a*x)^p)],

# Integrands of the form x^m (c-a^2 c x^2)^p / E^(3 ArcTanh[a x])

# Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[(c-a^2*c*x^2)^4/exp(3*arctanh(a*x)),x,8,55/192*c^4*x*(1-a^2*x^2)^(3/2)+11/48*c^4*x*(1-a^2*x^2)^(5/2)+11/56*c^4*(1-a^2*x^2)^(7/2)/a+11/72*c^4*(1-a*x)*(1-a^2*x^2)^(7/2)/a+1/9*c^4*(1-a*x)^2*(1-a^2*x^2)^(7/2)/a+55/128*c^4*arcsin(a*x)/a+55/128*c^4*x*sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^3/exp(3*arctanh(a*x)),x,7,3/8*c^3*x*(1-a^2*x^2)^(3/2)+3/10*c^3*(1-a^2*x^2)^(5/2)/a+3/14*c^3*(1-a*x)*(1-a^2*x^2)^(5/2)/a+1/7*c^3*(1-a*x)^2*(1-a^2*x^2)^(5/2)/a+9/16*c^3*arcsin(a*x)/a+9/16*c^3*x*sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^2/exp(3*arctanh(a*x)),x,6,7/12*c^2*(1-a^2*x^2)^(3/2)/a+7/20*c^2*(1-a*x)*(1-a^2*x^2)^(3/2)/a+1/5*c^2*(1-a*x)^2*(1-a^2*x^2)^(3/2)/a+7/8*c^2*arcsin(a*x)/a+7/8*c^2*x*sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)/exp(3*arctanh(a*x)),x,5,5/2*c*arcsin(a*x)/a+5/2*c*sqrt(1-a^2*x^2)/a+5/6*c*(1-a*x)*sqrt(1-a^2*x^2)/a+1/3*c*(1-a*x)^2*sqrt(1-a^2*x^2)/a],

# p<0
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)),x,1,(-1/3)/(a*c*exp(3*arctanh(a*x)))],
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^2),x,5,-1/5*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)^3)-2/15*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)^2)-2/15*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x))],
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^3),x,6,8/35*x/(c^3*sqrt(1-a^2*x^2))+(-1/7)/(a*c^3*(1+a*x)^3*sqrt(1-a^2*x^2))+(-4/35)/(a*c^3*(1+a*x)^2*sqrt(1-a^2*x^2))+(-4/35)/(a*c^3*(1+a*x)*sqrt(1-a^2*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^4),x,7,8/63*x/(c^4*(1-a^2*x^2)^(3/2))+(-1/9)/(a*c^4*(1+a*x)^3*(1-a^2*x^2)^(3/2))+(-2/21)/(a*c^4*(1+a*x)^2*(1-a^2*x^2)^(3/2))+(-2/21)/(a*c^4*(1+a*x)*(1-a^2*x^2)^(3/2))+16/63*x/(c^4*sqrt(1-a^2*x^2))],

# Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0
[x^3*(c-a^2*c*x^2)^(1/2)/exp(3*arctanh(a*x)),x,4,4*x*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1-a^2*x^2))-2*x^2*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))+4/3*x^3*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-3/4*x^4*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/5*a*x^5*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^4*sqrt(1-a^2*x^2))],
[x^2*(c-a^2*c*x^2)^(1/2)/exp(3*arctanh(a*x)),x,4,-4*x*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))+2*x^2*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/4*a*x^4*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1-a^2*x^2))],
[x*(c-a^2*c*x^2)^(1/2)/exp(3*arctanh(a*x)),x,4,4*x*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-3/2*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/3*a*x^3*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(1/2)/exp(3*arctanh(a*x)),x,4,-3*x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+1/2*a*x^2*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(1/2)/(exp(3*arctanh(a*x))*x),x,4,a*x*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*log(1+a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(3*arctanh(a*x))*x^2),x,4,-sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))-3*a*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+4*a*log(1+a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(3*arctanh(a*x))*x^3),x,4,-1/2*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1-a^2*x^2))+3*a*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+4*a^2*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*a^2*log(1+a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(3*arctanh(a*x))*x^4),x,4,-1/3*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1-a^2*x^2))+3/2*a*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1-a^2*x^2))-4*a^2*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))-4*a^3*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)+4*a^3*log(1+a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(1/2)/(exp(3*arctanh(a*x))*x^5),x,4,-1/4*sqrt(c-a^2*c*x^2)/(x^4*sqrt(1-a^2*x^2))+a*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1-a^2*x^2))-2*a^2*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1-a^2*x^2))+4*a^3*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))+4*a^4*log(x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)-4*a^4*log(1+a*x)*sqrt(c-a^2*c*x^2)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(9/2)/exp(3*arctanh(a*x)),x,4,-8/7*c^4*(1-a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+3/2*c^4*(1-a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-2/3*c^4*(1-a*x)^9*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+1/10*c^4*(1-a*x)^10*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(7/2)/exp(3*arctanh(a*x)),x,4,-2/3*c^3*(1-a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+4/7*c^3*(1-a*x)^7*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/8*c^3*(1-a*x)^8*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(5/2)/exp(3*arctanh(a*x)),x,4,-2/5*c^2*(1-a*x)^5*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+1/6*c^2*(1-a*x)^6*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],
[(c-a^2*c*x^2)^(3/2)/exp(3*arctanh(a*x)),x,3,-1/4*c*(1-a*x)^4*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))],

# p<0
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(1/2)),x,4,-2*sqrt(1-a^2*x^2)/(a*(1+a*x)*sqrt(c-a^2*c*x^2))-log(1+a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(3/2)),x,3,-1/2*sqrt(1-a^2*x^2)/(a*c*(1+a*x)^2*sqrt(c-a^2*c*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(5/2)),x,5,-1/6*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)^3*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)^2*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a*c^2*(1+a*x)*sqrt(c-a^2*c*x^2))+1/8*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^2*sqrt(c-a^2*c*x^2))],
[1/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(7/2)),x,5,1/32*sqrt(1-a^2*x^2)/(a*c^3*(1-a*x)*sqrt(c-a^2*c*x^2))-1/16*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)^4*sqrt(c-a^2*c*x^2))-1/12*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)^3*sqrt(c-a^2*c*x^2))-3/32*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)^2*sqrt(c-a^2*c*x^2))-1/8*sqrt(1-a^2*x^2)/(a*c^3*(1+a*x)*sqrt(c-a^2*c*x^2))+5/32*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*c^3*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic
[x^m*(c-a^2*c*x^2)^(1/2)/exp(3*arctanh(a*x)),x,5,-3*x^(1+m)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))+a*x^(2+m)*sqrt(c-a^2*c*x^2)/((2+m)*sqrt(1-a^2*x^2))+4*x^(1+m)*hypergeom([1,1+m],[2+m],-a*x)*sqrt(c-a^2*c*x^2)/((1+m)*sqrt(1-a^2*x^2))],

# Integrands of the form x^m E^(-3 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[(c-a^2*c*x^2)^p/exp(3*arctanh(a*x)),x,3,-2^(-1/2+p)*(1-a*x)^(5/2+p)*(c-a^2*c*x^2)^p*hypergeom([3/2-p,5/2+p],[7/2+p],1/2*(1-a*x))/(a*(5+2*p)*(1-a^2*x^2)^p)],

# Integrands of the form x^m (c-a^2 c x^2)^p E^(1/2 ArcTanh[a x])

# Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^p

# Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^(p/2)
[exp(1/2*arctanh(a*x))*(1-a^2*x^2)^(5/2),x,18,231/512*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a+231/1280*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a+77/960*(1-a*x)^(9/4)*(1+a*x)^(3/4)/a-77/480*(1-a*x)^(13/4)*(1+a*x)^(3/4)/a-11/60*(1-a*x)^(13/4)*(1+a*x)^(7/4)/a-1/6*(1-a*x)^(13/4)*(1+a*x)^(11/4)/a+231/512*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-231/512*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+231/1024*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-231/1024*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[exp(1/2*arctanh(a*x))*(1-a^2*x^2)^(3/2),x,16,35/64*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a+7/32*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a-7/24*(1-a*x)^(9/4)*(1+a*x)^(3/4)/a-1/4*(1-a*x)^(9/4)*(1+a*x)^(7/4)/a+35/64*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-35/64*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+35/128*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-35/128*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[exp(1/2*arctanh(a*x))*(1-a^2*x^2)^(1/2),x,14,3/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)/a-1/2*(1-a*x)^(5/4)*(1+a*x)^(3/4)/a+3/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))-3/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))/(a*sqrt(2))+3/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-3/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))],
[exp(1/2*arctanh(a*x))/(1-a^2*x^2)^(1/2),x,12,log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))-log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))/(a*sqrt(2))+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)/a-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)/a],
[exp(1/2*arctanh(a*x))/(1-a^2*x^2)^(3/2),x,1,-2/3*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))/(1-a^2*x^2)^(5/2),x,2,-2/35*exp(1/2*arctanh(a*x))*(1-6*a*x)/(a*(1-a^2*x^2)^(3/2))-16/35*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))/(1-a^2*x^2)^(7/2),x,3,-2/99*exp(1/2*arctanh(a*x))*(1-10*a*x)/(a*(1-a^2*x^2)^(5/2))-32/693*exp(1/2*arctanh(a*x))*(1-6*a*x)/(a*(1-a^2*x^2)^(3/2))-256/693*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))/(1-a^2*x^2)^(9/2),x,4,-2/195*exp(1/2*arctanh(a*x))*(1-14*a*x)/(a*(1-a^2*x^2)^(7/2))-112/6435*exp(1/2*arctanh(a*x))*(1-10*a*x)/(a*(1-a^2*x^2)^(5/2))-256/6435*exp(1/2*arctanh(a*x))*(1-6*a*x)/(a*(1-a^2*x^2)^(3/2))-2048/6435*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))*(c-a^2*c*x^2)^(5/2),x,19,231/512*c^2*(1-a*x)^(1/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+231/1280*c^2*(1-a*x)^(5/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+77/960*c^2*(1-a*x)^(9/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-77/480*c^2*(1-a*x)^(13/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-11/60*c^2*(1-a*x)^(13/4)*(1+a*x)^(7/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/6*c^2*(1-a*x)^(13/4)*(1+a*x)^(11/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+231/512*c^2*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))-231/512*c^2*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))+231/1024*c^2*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))-231/1024*c^2*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))*(c-a^2*c*x^2)^(3/2),x,17,35/64*c*(1-a*x)^(1/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+7/32*c*(1-a*x)^(5/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-7/24*c*(1-a*x)^(9/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/4*c*(1-a*x)^(9/4)*(1+a*x)^(7/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+35/64*c*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))-35/64*c*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))+35/128*c*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))-35/128*c*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))*(c-a^2*c*x^2)^(1/2),x,15,3/4*(1-a*x)^(1/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))-1/2*(1-a*x)^(5/4)*(1+a*x)^(3/4)*sqrt(c-a^2*c*x^2)/(a*sqrt(1-a^2*x^2))+3/4*arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))-3/4*arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))+3/8*log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))-3/8*log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(c-a^2*c*x^2)/(a*sqrt(2)*sqrt(1-a^2*x^2))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(1/2),x,13,log(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(2)*sqrt(c-a^2*c*x^2))-log(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4)+sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(2)*sqrt(c-a^2*c*x^2))+arctan(1-(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-arctan(1+(1-a*x)^(1/4)*sqrt(2)/(1+a*x)^(1/4))*sqrt(2)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(3/2),x,1,-2/3*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*c*sqrt(c-a^2*c*x^2))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(5/2),x,2,-2/35*exp(1/2*arctanh(a*x))*(1-6*a*x)/(a*c*(c-a^2*c*x^2)^(3/2))-16/35*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*c^2*sqrt(c-a^2*c*x^2))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(7/2),x,3,-2/99*exp(1/2*arctanh(a*x))*(1-10*a*x)/(a*c*(c-a^2*c*x^2)^(5/2))-32/693*exp(1/2*arctanh(a*x))*(1-6*a*x)/(a*c^2*(c-a^2*c*x^2)^(3/2))-256/693*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*c^3*sqrt(c-a^2*c*x^2))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(9/2),x,4,-2/195*exp(1/2*arctanh(a*x))*(1-14*a*x)/(a*c*(c-a^2*c*x^2)^(7/2))-112/6435*exp(1/2*arctanh(a*x))*(1-10*a*x)/(a*c^2*(c-a^2*c*x^2)^(5/2))-256/6435*exp(1/2*arctanh(a*x))*(1-6*a*x)/(a*c^3*(c-a^2*c*x^2)^(3/2))-2048/6435*exp(1/2*arctanh(a*x))*(1-2*a*x)/(a*c^4*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^(p/4)
[exp(1/2*arctanh(a*x))*x^3/(c-a^2*c*x^2)^(5/4),x,10,-2/3*(1-a*x)^(3/2)*(1-a^2*x^2)^(1/4)/(a^4*c*(c-a^2*c*x^2)^(1/4))+(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x)/sqrt(2))/(a^4*c*(c-a^2*c*x^2)^(1/4)*sqrt(2))+(1-a^2*x^2)^(1/4)/(a^4*c*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))+2*(1-a^2*x^2)^(1/4)*sqrt(1-a*x)/(a^4*c*(c-a^2*c*x^2)^(1/4))],
[exp(1/2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(5/4),x,8,-(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x)/sqrt(2))/(a^3*c*(c-a^2*c*x^2)^(1/4)*sqrt(2))+(1-a^2*x^2)^(1/4)/(a^3*c*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))+2*(1-a^2*x^2)^(1/4)*sqrt(1-a*x)/(a^3*c*(c-a^2*c*x^2)^(1/4))],
[exp(1/2*arctanh(a*x))*x/(c-a^2*c*x^2)^(5/4),x,5,(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x)/sqrt(2))/(a^2*c*(c-a^2*c*x^2)^(1/4)*sqrt(2))+(1-a^2*x^2)^(1/4)/(a^2*c*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(5/4),x,5,-(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x)/sqrt(2))/(a*c*(c-a^2*c*x^2)^(1/4)*sqrt(2))+(1-a^2*x^2)^(1/4)/(a*c*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))],
[exp(1/2*arctanh(a*x))/(x*(c-a^2*c*x^2)^(5/4)),x,8,-2*(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x))/(c*(c-a^2*c*x^2)^(1/4))+(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x)/sqrt(2))/(c*(c-a^2*c*x^2)^(1/4)*sqrt(2))+(1-a^2*x^2)^(1/4)/(c*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))],
[exp(1/2*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(5/4)),x,9,-a*(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x))/(c*(c-a^2*c*x^2)^(1/4))-a*(1-a^2*x^2)^(1/4)*arctanh(sqrt(1-a*x)/sqrt(2))/(c*(c-a^2*c*x^2)^(1/4)*sqrt(2))+2*a*(1-a^2*x^2)^(1/4)/(c*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))-(1-a^2*x^2)^(1/4)/(c*x*(c-a^2*c*x^2)^(1/4)*sqrt(1-a*x))],

# Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^(p/8)
[exp(1/2*arctanh(a*x))*x^3/(c-a^2*c*x^2)^(9/8),x,5,-4/7*x^2*(1+a*x)^(1/8)*(1-a^2*x^2)^(1/8)/(a^2*c*(1-a*x)^(3/8)*(c-a^2*c*x^2)^(1/8))+8/21*(6-a*x)*(1+a*x)^(1/8)*(1-a^2*x^2)^(1/8)/(a^4*c*(1-a*x)^(3/8)*(c-a^2*c*x^2)^(1/8))+64/105*2^(1/8)*(1-a*x)^(5/8)*(1-a^2*x^2)^(1/8)*hypergeom([5/8,7/8],[13/8],1/2*(1-a*x))/(a^4*c*(c-a^2*c*x^2)^(1/8))],
[exp(1/2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(9/8),x,1,4/3*exp(1/2*arctanh(a*x))*(2-a*x)/(a^3*c*(c-a^2*c*x^2)^(1/8))],
[exp(1/2*arctanh(a*x))*x/(c-a^2*c*x^2)^(9/8),x,4,4/3*(1+a*x)^(1/8)*(1-a^2*x^2)^(1/8)/(a^2*c*(1-a*x)^(3/8)*(c-a^2*c*x^2)^(1/8))+8/15*2^(1/8)*(1-a*x)^(5/8)*(1-a^2*x^2)^(1/8)*hypergeom([5/8,7/8],[13/8],1/2*(1-a*x))/(a^2*c*(c-a^2*c*x^2)^(1/8))],
[exp(1/2*arctanh(a*x))/(c-a^2*c*x^2)^(9/8),x,3,4/3*2^(1/8)*(1-a^2*x^2)^(1/8)*hypergeom([-3/8,7/8],[5/8],1/2*(1-a*x))/(a*c*(1-a*x)^(3/8)*(c-a^2*c*x^2)^(1/8))],
[exp(1/2*arctanh(a*x))/(x*(c-a^2*c*x^2)^(9/8)),x,3,-2*2^(5/8)*(1+a*x)^(1/8)*(1-a^2*x^2)^(1/8)*AppellF1(1/8,11/8,1,9/8,1/2*(1+a*x),1+a*x)/(c*(c-a^2*c*x^2)^(1/8))],

# Integrands of the form x^m E^(1/2 ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic

# Integrands of the form x^m (c-a^2 c x^2)^p E^(n ArcTanh[a x])

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^p

# p>0
[exp(n*arctanh(a*x))*(c-a^2*c*x^2),x,2,-2^(2+1/2*n)*c*(1-a*x)^(2-1/2*n)*hypergeom([-1-1/2*n,2-1/2*n],[3-1/2*n],1/2*(1-a*x))/(a*(4-n))],
[exp(n*arctanh(a*x))*(c-a^2*c*x^2)^2,x,2,-2^(3+1/2*n)*c^2*(1-a*x)^(3-1/2*n)*hypergeom([-2-1/2*n,3-1/2*n],[4-1/2*n],1/2*(1-a*x))/(a*(6-n))],
[exp(n*arctanh(a*x))*(c-a^2*c*x^2)^3,x,2,-2^(4+1/2*n)*c^3*(1-a*x)^(4-1/2*n)*hypergeom([-3-1/2*n,4-1/2*n],[5-1/2*n],1/2*(1-a*x))/(a*(8-n))],

# p<0
[exp(n*arctanh(a*x))*x^4/(c-a^2*c*x^2),x,5,-1/6*n*x^2*(1+a*x)^(1/2*n)/(a^3*c*(1-a*x)^(1/2*n))-1/3*x^3*(1+a*x)^(1/2*n)/(a^2*c*(1-a*x)^(1/2*n))+1/6*(1+a*x)^(1/2*n)*(6+8*n+n^2+n^3-a*n*(6+n^2)*x)/(a^5*c*n*(1-a*x)^(1/2*n))+1/3*2^(-1+1/2*n)*n*(8+n^2)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,1-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^5*c*(2-n))],
[exp(n*arctanh(a*x))*x^3/(c-a^2*c*x^2),x,4,-1/2*x^2*(1+a*x)^(1/2*n)/(a^2*c*(1-a*x)^(1/2*n))+1/2*(1+a*x)^(1/2*n)*(2+n+n^2-a*n^2*x)/(a^4*c*n*(1-a*x)^(1/2*n))+2^(-1+1/2*n)*(2+n^2)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,1-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^4*c*(2-n))],
[exp(n*arctanh(a*x))*x^2/(c-a^2*c*x^2),x,4,exp(n*arctanh(a*x))/(a^3*c*n)+2^(1+1/2*n)*(1-a*x)^(1-1/2*n)*hypergeom([1-1/2*n,-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^3*c*(2-n)),(1-n)*(1+a*x)^(1/2*n)/(a^3*c*n*(1-a*x)^(1/2*n))-x*(1+a*x)^(1/2*n)/(a^2*c*(1-a*x)^(1/2*n))+2^(1+1/2*n)*hypergeom([-1/2*n,-1/2*n],[1-1/2*n],1/2*(1-a*x))/(a^3*c*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*x/(c-a^2*c*x^2),x,3,-(1+a*x)^(1/2*n)/(a^2*c*n*(1-a*x)^(1/2*n))+2^(1+1/2*n)*hypergeom([-1/2*n,-1/2*n],[1-1/2*n],1/2*(1-a*x))/(a^2*c*n*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2),x,1,exp(n*arctanh(a*x))/(a*c*n)],
[exp(n*arctanh(a*x))/(x*(c-a^2*c*x^2)),x,3,(1+a*x)^(1/2*n)/(c*n*(1-a*x)^(1/2*n))-2*(1+a*x)^(1/2*n)*hypergeom([1,1/2*n],[1/2*(2+n)],(1+a*x)/(1-a*x))/(c*n*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(x^2*(c-a^2*c*x^2)),x,5,a*(1+n)*(1+a*x)^(1/2*n)/(c*n*(1-a*x)^(1/2*n))-(1+a*x)^(1/2*n)/(c*x*(1-a*x)^(1/2*n))-2*a*(1+a*x)^(1/2*n)*hypergeom([1,1/2*n],[1/2*(2+n)],(1+a*x)/(1-a*x))/(c*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))*x^4/(c-a^2*c*x^2)^2,x,10,(1-n)*(3+n)*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a^5*c^2*(2-n))+(3+n)*x*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a^4*c^2)-x^3*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a^2*c^2)+(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a^5*c^2*(2-n))-(1+a*x)^(1/2*(-2+n))/(a^5*c^2*(1-a*x)^(1/2*n))-(3+n)*(2-n^2)*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*n)/(a^5*c^2*(4-n^2))-(3+n)*(2-n^2)*(1+a*x)^(1/2*n)/(a^5*c^2*n*(4-n^2)*(1-a*x)^(1/2*n))-2^(1/2*n)*n*(1-a*x)^(1-1/2*n)*hypergeom([1/2*(2-n),1-1/2*n],[2-1/2*n],1/2*(1-a*x))/(a^5*c^2*(2-n))],
[exp(n*arctanh(a*x))*x^3/(c-a^2*c*x^2)^2,x,10,-(1-a*x)^(1/2*(-2-n))*(1+a*x)^(1/2*(-2+n))/(a^4*c^2*(2+n))+2*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(a^4*c^2*n*(4-n^2))-2*(1+a*x)^(1/2*(-2+n))/(a^4*c^2*n*(2+n)*(1-a*x)^(1/2*n))+3*(1-a*x)^(1/2*(-2-n))*(1+a*x)^(1/2*n)/(a^4*c^2*(2+n))+3*(1+a*x)^(1/2*n)/(a^4*c^2*n*(2+n)*(1-a*x)^(1/2*n))-3*(1-a*x)^(1/2*(-2-n))*(1+a*x)^(1/2*(2+n))/(a^4*c^2*(2+n))+2^(2+1/2*n)*(1-a*x)^(1/2*(-2-n))*hypergeom([1/2*(-2-n),-1-1/2*n],[-1/2*n],1/2*(1-a*x))/(a^4*c^2*(2+n))],
[exp(n*arctanh(a*x))*x^2/(c-a^2*c*x^2)^2,x,2,-exp(n*arctanh(a*x))*(2-n^2)/(a^3*c^2*n*(4-n^2))-exp(n*arctanh(a*x))*(n-2*a*x)/(a^3*c^2*(4-n^2)*(1-a^2*x^2))],
[exp(n*arctanh(a*x))*x/(c-a^2*c*x^2)^2,x,3,-exp(n*arctanh(a*x))/(a^2*c^2*(4-n^2))+exp(n*arctanh(a*x))*(2-a*n*x)/(a^2*c^2*(4-n^2)*(1-a^2*x^2)),-exp(n*arctanh(a*x))/(a^2*c^2*(4-n^2))+1/2*exp(n*arctanh(a*x))/(a^2*c^2*(1-a^2*x^2))+1/2*exp(n*arctanh(a*x))*n*(n-2*a*x)/(a^2*c^2*(4-n^2)*(1-a^2*x^2))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2)^2,x,2,2*exp(n*arctanh(a*x))/(a*c^2*n*(4-n^2))-exp(n*arctanh(a*x))*(n-2*a*x)/(a*c^2*(4-n^2)*(1-a^2*x^2))],
[exp(n*arctanh(a*x))/(x*(c-a^2*c*x^2)^2),x,6,(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(c^2*(2+n))-(4-n-n^2)*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(c^2*n*(4-n^2))+(4+n)*(1+a*x)^(1/2*(-2+n))/(c^2*n*(2+n)*(1-a*x)^(1/2*n))-2*(1+a*x)^(1/2*n)*hypergeom([1,1/2*n],[1/2*(2+n)],(1+a*x)/(1-a*x))/(c^2*n*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^2),x,7,a*(3+n)*(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(c^2*(2+n))-(1-a*x)^(-1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(c^2*x)-a*(6+4*n-n^2-n^3)*(1-a*x)^(1-1/2*n)*(1+a*x)^(1/2*(-2+n))/(c^2*n*(4-n^2))+a*(6+4*n+n^2)*(1+a*x)^(1/2*(-2+n))/(c^2*n*(2+n)*(1-a*x)^(1/2*n))-2*a*(1+a*x)^(1/2*n)*hypergeom([1,1/2*n],[1/2*(2+n)],(1+a*x)/(1-a*x))/(c^2*(1-a*x)^(1/2*n))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2)^3,x,3,24*exp(n*arctanh(a*x))/(a*c^3*n*(64-20*n^2+n^4))-exp(n*arctanh(a*x))*(n-4*a*x)/(a*c^3*(16-n^2)*(1-a^2*x^2)^2)-12*exp(n*arctanh(a*x))*(n-2*a*x)/(a*c^3*(64-20*n^2+n^4)*(1-a^2*x^2))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2)^4,x,4,720*exp(n*arctanh(a*x))/(a*c^4*n*(36-n^2)*(64-20*n^2+n^4))-exp(n*arctanh(a*x))*(n-6*a*x)/(a*c^4*(36-n^2)*(1-a^2*x^2)^3)-30*exp(n*arctanh(a*x))*(n-4*a*x)/(a*c^4*(576-52*n^2+n^4)*(1-a^2*x^2)^2)-360*exp(n*arctanh(a*x))*(n-2*a*x)/(a*c^4*(36-n^2)*(64-20*n^2+n^4)*(1-a^2*x^2))],

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^(p/2)

# p>0
[exp(n*arctanh(a*x))*x^3*sqrt(c-a^2*c*x^2),x,5,-1/5*x^2*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(3+n))*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))-1/60*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(3+n))*(8+n^2+3*a*n*x)*sqrt(c-a^2*c*x^2)/(a^4*sqrt(1-a^2*x^2))-1/15*2^(1/2*(-1+n))*n*(11+n^2)*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(-1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/(a^4*(3-n)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))*x^2*sqrt(c-a^2*c*x^2),x,5,-1/12*n*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(3+n))*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1-a^2*x^2))-1/4*x*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(3+n))*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))-1/3*2^(1/2*(-1+n))*(3+n^2)*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(-1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/(a^3*(3-n)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))*x*sqrt(c-a^2*c*x^2),x,4,-1/3*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(3+n))*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1-a^2*x^2))-1/3*2^(1/2*(3+n))*n*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(-1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/(a^2*(3-n)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))*sqrt(c-a^2*c*x^2),x,3,-2^(1/2*(3+n))*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(-1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/(a*(3-n)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))*sqrt(c-a^2*c*x^2)/x,x,6,-(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-1+n))*sqrt(c-a^2*c*x^2)/((1-n)*sqrt(1-a^2*x^2))+2*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(c-a^2*c*x^2)/((1-n)*sqrt(1-a^2*x^2))+2^(1/2*(1+n))*n*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/((3-4*n+n^2)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))*sqrt(c-a^2*c*x^2)/x^2,x,6,-(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(c-a^2*c*x^2)/(x*sqrt(1-a^2*x^2))-2*a*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(1-n)],[1/2*(3-n)],(1-a*x)/(1+a*x))*sqrt(c-a^2*c*x^2)/((1-n)*sqrt(1-a^2*x^2))+2^(1/2*(1+n))*a*(1-a*x)^(1/2*(1-n))*hypergeom([1/2*(1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/((1-n)*sqrt(1-a^2*x^2))],
[exp(n*arctanh(a*x))*(c-a^2*c*x^2)^(3/2),x,3,-2^(1/2*(5+n))*c*(1-a*x)^(1/2*(5-n))*hypergeom([1/2*(-3-n),1/2*(5-n)],[1/2*(7-n)],1/2*(1-a*x))*sqrt(c-a^2*c*x^2)/(a*(5-n)*sqrt(1-a^2*x^2))],

# p<0
[exp(n*arctanh(a*x))*x^3/sqrt(c-a^2*c*x^2),x,5,-1/3*x^2*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a^2*c*x^2))-1/6*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*(4+n+n^2+a*(1-n)*n*x)*sqrt(1-a^2*x^2)/(a^4*(1-n)*sqrt(c-a^2*c*x^2))-1/3*2^(1/2*(-1+n))*n*(5+n^2)*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(1-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a^4*(3-4*n+n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x^2/sqrt(c-a^2*c*x^2),x,5,1/2*(1-n)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(a^3*(1+n)*sqrt(c-a^2*c*x^2))-1/2*x*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(a^2*sqrt(c-a^2*c*x^2))-2^(1/2*(1+n))*(1+n^2)*(1-a*x)^(1/2*(1-n))*hypergeom([1/2*(-1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a^3*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x/sqrt(c-a^2*c*x^2),x,4,-(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(a^2*(1+n)*sqrt(c-a^2*c*x^2))-2^(1/2*(3+n))*n*(1-a*x)^(1/2*(1-n))*hypergeom([1/2*(-1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a^2*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/sqrt(c-a^2*c*x^2),x,3,-2^(1/2*(1+n))*(1-a*x)^(1/2*(1-n))*hypergeom([1/2*(1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x*sqrt(c-a^2*c*x^2)),x,3,-2*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(1-n)],[1/2*(3-n)],(1-a*x)/(1+a*x))*sqrt(1-a^2*x^2)/((1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x^2*sqrt(c-a^2*c*x^2)),x,4,-(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(x*sqrt(c-a^2*c*x^2))-2*a*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(1-n)],[1/2*(3-n)],(1-a*x)/(1+a*x))*sqrt(1-a^2*x^2)/((1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x^3*sqrt(c-a^2*c*x^2)),x,6,-1/2*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(x^2*sqrt(c-a^2*c*x^2))-1/2*a*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(1+n))*sqrt(1-a^2*x^2)/(x*sqrt(c-a^2*c*x^2))-a^2*(1+n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(1-n)],[1/2*(3-n)],(1-a*x)/(1+a*x))*sqrt(1-a^2*x^2)/((1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x^3/(c-a^2*c*x^2)^(3/2),x,5,-x^2*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(a^2*c*sqrt(c-a^2*c*x^2))+(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*(2+2*n+n^2-a*n*(3+2*n)*x)*sqrt(1-a^2*x^2)/(a^4*c*(1-n^2)*sqrt(c-a^2*c*x^2))-2^(1/2*(-1+n))*n*(1-a*x)^(1/2*(3-n))*hypergeom([1/2*(3-n),1/2*(3-n)],[1/2*(5-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a^4*c*(3-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(3/2),x,4,-exp(n*arctanh(a*x))*(n-a*x)/(a^3*c*(1-n^2)*sqrt(c-a^2*c*x^2))+2^(1/2*(1+n))*(1-a*x)^(1/2*(1-n))*hypergeom([1/2*(1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(1-a*x))*sqrt(1-a^2*x^2)/(a^3*c*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x/(c-a^2*c*x^2)^(3/2),x,1,exp(n*arctanh(a*x))*(1-a*n*x)/(a^2*c*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2)^(3/2),x,1,-exp(n*arctanh(a*x))*(n-a*x)/(a*c*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x*(c-a^2*c*x^2)^(3/2)),x,6,(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*(1+n)*sqrt(c-a^2*c*x^2))-(2+n)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*(1-n^2)*sqrt(c-a^2*c*x^2))+2*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(1-a^2*x^2)/(c*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(3/2)),x,7,a*(2+n)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*(1+n)*sqrt(c-a^2*c*x^2))-(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*x*sqrt(c-a^2*c*x^2))-a*(2+2*n+n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*(1-n^2)*sqrt(c-a^2*c*x^2))+2*a*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(1-a^2*x^2)/(c*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x^3*(c-a^2*c*x^2)^(3/2)),x,8,1/2*a^2*(3+2*n+n^2)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*(1+n)*sqrt(c-a^2*c*x^2))-1/2*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*x^2*sqrt(c-a^2*c*x^2))-1/2*a*n*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*x*sqrt(c-a^2*c*x^2))-1/2*a^2*(6+5*n+2*n^2+n^3)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(c*(1-n^2)*sqrt(c-a^2*c*x^2))+a^2*(3+n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(1-a^2*x^2)/(c*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x^3/(c-a^2*c*x^2)^(5/2),x,7,x^3*(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(a*c^2*(3+n)*sqrt(c-a^2*c*x^2))-3*(2-n)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(a^4*c^2*(9-n^2)*sqrt(c-a^2*c*x^2))-3*x*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(a^3*c^2*(3+n)*sqrt(c-a^2*c*x^2))+3*(1+2*n-n^2)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(a^4*c^2*(9+9*n-n^2-n^3)*sqrt(c-a^2*c*x^2))-3*(1+2*n-n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*sqrt(1-a^2*x^2)/(a^4*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(5/2),x,2,-exp(n*arctanh(a*x))*(n-3*a*x)/(a^3*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))+exp(n*arctanh(a*x))*(3-n^2)*(n-a*x)/(a^3*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))*x/(c-a^2*c*x^2)^(5/2),x,3,1/3*exp(n*arctanh(a*x))/(a^2*c*(c-a^2*c*x^2)^(3/2))+1/3*exp(n*arctanh(a*x))*n*(n-3*a*x)/(a^2*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))+2*exp(n*arctanh(a*x))*n*(n-a*x)/(a^2*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2)^(5/2),x,2,-exp(n*arctanh(a*x))*(n-3*a*x)/(a*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))-6*exp(n*arctanh(a*x))*(n-a*x)/(a*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x*(c-a^2*c*x^2)^(5/2)),x,8,(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+n)*sqrt(c-a^2*c*x^2))+(6+n)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+4*n+n^2)*sqrt(c-a^2*c*x^2))-(15+6*n+n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+n-3*n^2-n^3)*sqrt(c-a^2*c*x^2))+(18+7*n-2*n^2-n^3)*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))+2*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(1-a^2*x^2)/(c^2*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x^2*(c-a^2*c*x^2)^(5/2)),x,9,a*(4+n)*(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+n)*sqrt(c-a^2*c*x^2))-(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*x*sqrt(c-a^2*c*x^2))+a*(12+6*n+n^2)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+4*n+n^2)*sqrt(c-a^2*c*x^2))-a*(24+15*n+6*n^2+n^3)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+n-3*n^2-n^3)*sqrt(c-a^2*c*x^2))+a*(24+18*n+7*n^2-2*n^3-n^4)*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))+2*a*n*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(1-a^2*x^2)/(c^2*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(x^3*(c-a^2*c*x^2)^(5/2)),x,10,1/2*a^2*(5+4*n+n^2)*(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+n)*sqrt(c-a^2*c*x^2))-1/2*(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*x^2*sqrt(c-a^2*c*x^2))-1/2*a*n*(1-a*x)^(1/2*(-3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*x*sqrt(c-a^2*c*x^2))+1/2*a^2*(30+17*n+6*n^2+n^3)*(1-a*x)^(1/2*(-1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+4*n+n^2)*sqrt(c-a^2*c*x^2))-1/2*a^2*(75+54*n+20*n^2+6*n^3+n^4)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(3+n-3*n^2-n^3)*sqrt(c-a^2*c*x^2))+1/2*a^2*(90+59*n+8*n^2+2*n^3-2*n^4-n^5)*(1-a*x)^(1/2*(3-n))*(1+a*x)^(1/2*(-3+n))*sqrt(1-a^2*x^2)/(c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))+a^2*(5+n^2)*(1-a*x)^(1/2*(1-n))*(1+a*x)^(1/2*(-1+n))*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(1+a*x)/(1-a*x))*sqrt(1-a^2*x^2)/(c^2*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arctanh(a*x))/(c-a^2*c*x^2)^(7/2),x,3,-exp(n*arctanh(a*x))*(n-5*a*x)/(a*c*(25-n^2)*(c-a^2*c*x^2)^(5/2))-20*exp(n*arctanh(a*x))*(n-3*a*x)/(a*c^2*(225-34*n^2+n^4)*(c-a^2*c*x^2)^(3/2))-120*exp(n*arctanh(a*x))*(n-a*x)/(a*c^3*(25-n^2)*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^p with m symbolic
[exp(n*arctanh(a*x))*x^m*(c-a^2*c*x^2)^2,x,2,c^2*x^(1+m)*AppellF1(1+m,1/2*(-4+n),-2-1/2*n,2+m,a*x,-a*x)/(1+m)],
[exp(n*arctanh(a*x))*x^m*(c-a^2*c*x^2),x,2,c*x^(1+m)*AppellF1(1+m,1/2*(-2+n),-1-1/2*n,2+m,a*x,-a*x)/(1+m)],
[exp(n*arctanh(a*x))*x^m/(c-a^2*c*x^2),x,2,x^(1+m)*AppellF1(1+m,1/2*(2+n),1-1/2*n,2+m,a*x,-a*x)/(c*(1+m))],
[exp(n*arctanh(a*x))*x^m/(c-a^2*c*x^2)^2,x,2,x^(1+m)*AppellF1(1+m,1/2*(4+n),2-1/2*n,2+m,a*x,-a*x)/(c^2*(1+m))],

# Integrands of the form x^m E^(n ArcTanh[a x]) (c-a^2 c x^2)^p with p symbolic
[exp(n*arctanh(a*x))*x^m*(c-a^2*c*x^2)^p,x,3,x^(1+m)*(c-a^2*c*x^2)^p*AppellF1(1+m,1/2*(n-2*p),-1/2*n-p,2+m,a*x,-a*x)/((1+m)*(1-a^2*x^2)^p)],
[exp(n*arctanh(a*x))*x*(c-a^2*c*x^2)^p,x,4,-1/2*(1-a*x)^(1-1/2*n+p)*(1+a*x)^(1+1/2*n+p)*(c-a^2*c*x^2)^p/(a^2*(1+p)*(1-a^2*x^2)^p)-2^(1/2*n+p)*n*(1-a*x)^(1-1/2*n+p)*(c-a^2*c*x^2)^p*hypergeom([-1/2*n-p,1-1/2*n+p],[2-1/2*n+p],1/2*(1-a*x))/(a^2*(1+p)*(2-n+2*p)*(1-a^2*x^2)^p)],
[exp(n*arctanh(a*x))*(c-a^2*c*x^2)^p,x,3,-2^(1+1/2*n+p)*(1-a*x)^(1-1/2*n+p)*(c-a^2*c*x^2)^p*hypergeom([-1/2*n-p,1-1/2*n+p],[2-1/2*n+p],1/2*(1-a*x))/(a*(2-n+2*p)*(1-a^2*x^2)^p)],
[exp(2*(1+p)*arctanh(a*x))/(1-a^2*x^2)^p,x,3,(1-a*x)^(1-2*p)/(a*(1-2*p))+1/(a*p*(1-a*x)^(2*p))],
[exp(2*(1+p)*arctanh(a*x))/(c-a^2*c*x^2)^p,x,4,(1-a*x)^(1-2*p)*(1-a^2*x^2)^p/(a*(1-2*p)*(c-a^2*c*x^2)^p)+(1-a^2*x^2)^p/(a*p*(1-a*x)^(2*p)*(c-a^2*c*x^2)^p)],
[exp(2*p*arctanh(a*x))*(c-a^2*c*x^2)^p,x,3,(1+a*x)^(1+2*p)*(c-a^2*c*x^2)^p/(a*(1+2*p)*(1-a^2*x^2)^p)],
[(c-a^2*c*x^2)^p/exp(2*p*arctanh(a*x)),x,3,-(1-a*x)^(1+2*p)*(c-a^2*c*x^2)^p/(a*(1+2*p)*(1-a^2*x^2)^p)],
[exp(n*arctanh(a*x))*x^2*(c-a^2*c*x^2)^(-1-1/2*n^2),x,1,exp(n*arctanh(a*x))*(1-a*n*x)/(a^3*c*n*(1-n^2)*(c-a^2*c*x^2)^(1/2*n^2))],
[exp(6*arctanh(a*x))*x^2/(c-a^2*c*x^2)^19,x,2,1/210*(-1+6*a*x)/(a^3*c^19*(1-a*x)^21*(1+a*x)^15)],
[exp(4*arctanh(a*x))*x^2/(c-a^2*c*x^2)^9,x,2,1/60*(-1+4*a*x)/(a^3*c^9*(1-a*x)^10*(1+a*x)^6)],
[exp(2*arctanh(a*x))*x^2/(c-a^2*c*x^2)^3,x,2,1/6*(-1+2*a*x)/(a^3*c^3*(1-a*x)^3*(1+a*x))],
[x^2/(exp(2*arctanh(a*x))*(c-a^2*c*x^2)^3),x,2,1/6*(1+2*a*x)/(a^3*c^3*(1-a*x)*(1+a*x)^3)],
[x^2/(exp(4*arctanh(a*x))*(c-a^2*c*x^2)^9),x,2,1/60*(1+4*a*x)/(a^3*c^9*(1-a*x)^6*(1+a*x)^10)],
[exp(5*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(27/2),x,3,-1/120*(1-5*a*x)*sqrt(1-a^2*x^2)/(a^3*c^13*(1-a*x)^15*(1+a*x)^10*sqrt(c-a^2*c*x^2))],
[exp(3*arctanh(a*x))*x^2/(c-a^2*c*x^2)^(11/2),x,3,-1/24*(1-3*a*x)*sqrt(1-a^2*x^2)/(a^3*c^5*(1-a*x)^6*(1+a*x)^3*sqrt(c-a^2*c*x^2))],
[exp(arctanh(a*x))*x^2/(c-a^2*c*x^2)^(3/2),x,4,1/2*sqrt(1-a^2*x^2)/(a^3*c*(1-a*x)*sqrt(c-a^2*c*x^2))+3/4*log(1-a*x)*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))+1/4*log(1+a*x)*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))],
[x^2/(exp(arctanh(a*x))*(c-a^2*c*x^2)^(3/2)),x,4,-1/2*sqrt(1-a^2*x^2)/(a^3*c*(1+a*x)*sqrt(c-a^2*c*x^2))-1/4*log(1-a*x)*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))-3/4*log(1+a*x)*sqrt(1-a^2*x^2)/(a^3*c*sqrt(c-a^2*c*x^2))],
[x^2/(exp(3*arctanh(a*x))*(c-a^2*c*x^2)^(11/2)),x,3,1/24*(1+3*a*x)*sqrt(1-a^2*x^2)/(a^3*c^5*(1-a*x)^3*(1+a*x)^6*sqrt(c-a^2*c*x^2))],
[x^2/(exp(5*arctanh(a*x))*(c-a^2*c*x^2)^(27/2)),x,3,1/120*(1+5*a*x)*sqrt(1-a^2*x^2)/(a^3*c^13*(1-a*x)^10*(1+a*x)^15*sqrt(c-a^2*c*x^2))]]:
