# Maple integration test file: "7 Inverse hyperbolic functions\7.3 Inverse hyperbolic tangent\7.3.4 u (a+b arctanh(c x))^p.txt"

lst:=[

# Integrands of the form (f x)^m (d+e x)^q (a+b ArcTanh[c x])^p

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^p when c^2 d^2-e^2=0

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^1 when c^2 d^2 - e^2=0

# q>0
[x^3*(d+c*d*x)*(a+b*arctanh(c*x)),x,7,1/4*b*d*x/c^3+1/10*b*d*x^2/c^2+1/12*b*d*x^3/c+1/20*b*d*x^4+1/4*d*x^4*(a+b*arctanh(c*x))+1/5*c*d*x^5*(a+b*arctanh(c*x))+9/40*b*d*log(1-c*x)/c^4-1/40*b*d*log(1+c*x)/c^4],
[x^2*(d+c*d*x)*(a+b*arctanh(c*x)),x,7,1/4*b*d*x/c^2+1/6*b*d*x^2/c+1/12*b*d*x^3+1/3*d*x^3*(a+b*arctanh(c*x))+1/4*c*d*x^4*(a+b*arctanh(c*x))+7/24*b*d*log(1-c*x)/c^3+1/24*b*d*log(1+c*x)/c^3],
[x*(d+c*d*x)*(a+b*arctanh(c*x)),x,7,1/2*b*d*x/c+1/6*b*d*x^2+1/2*d*x^2*(a+b*arctanh(c*x))+1/3*c*d*x^3*(a+b*arctanh(c*x))+5/12*b*d*log(1-c*x)/c^2-1/12*b*d*log(1+c*x)/c^2],
[(d+c*d*x)*(a+b*arctanh(c*x)),x,4,1/2*b*d*x+1/2*d*(1+c*x)^2*(a+b*arctanh(c*x))/c+b*d*log(1-c*x)/c],
[(d+c*d*x)*(a+b*arctanh(c*x))/x,x,6,a*c*d*x+b*c*d*x*arctanh(c*x)+a*d*log(x)+1/2*b*d*log(1-c^2*x^2)-1/2*b*d*polylog(2,-c*x)+1/2*b*d*polylog(2,c*x)],
[(d+c*d*x)*(a+b*arctanh(c*x))/x^2,x,8,-d*(a+b*arctanh(c*x))/x+a*c*d*log(x)+b*c*d*log(x)-1/2*b*c*d*log(1-c^2*x^2)-1/2*b*c*d*polylog(2,-c*x)+1/2*b*c*d*polylog(2,c*x)],
[(d+c*d*x)*(a+b*arctanh(c*x))/x^3,x,4,-1/2*b*c*d/x-1/2*d*(1+c*x)^2*(a+b*arctanh(c*x))/x^2+b*c^2*d*log(x)-b*c^2*d*log(1-c*x)],
[(d+c*d*x)*(a+b*arctanh(c*x))/x^4,x,4,-1/6*b*c*d/x^2-1/2*b*c^2*d/x-1/3*d*(a+b*arctanh(c*x))/x^3-1/2*c*d*(a+b*arctanh(c*x))/x^2+1/3*b*c^3*d*log(x)-5/12*b*c^3*d*log(1-c*x)+1/12*b*c^3*d*log(1+c*x)],
[(d+c*d*x)*(a+b*arctanh(c*x))/x^5,x,4,-1/12*b*c*d/x^3-1/6*b*c^2*d/x^2-1/4*b*c^3*d/x-1/4*d*(a+b*arctanh(c*x))/x^4-1/3*c*d*(a+b*arctanh(c*x))/x^3+1/3*b*c^4*d*log(x)-7/24*b*c^4*d*log(1-c*x)-1/24*b*c^4*d*log(1+c*x)],
[x^3*(d+c*d*x)^2*(a+b*arctanh(c*x)),x,7,5/12*b*d^2*x/c^3+1/5*b*d^2*x^2/c^2+5/36*b*d^2*x^3/c+1/10*b*d^2*x^4+1/30*b*c*d^2*x^5+1/4*d^2*x^4*(a+b*arctanh(c*x))+2/5*c*d^2*x^5*(a+b*arctanh(c*x))+1/6*c^2*d^2*x^6*(a+b*arctanh(c*x))+49/120*b*d^2*log(1-c*x)/c^4-1/120*b*d^2*log(1+c*x)/c^4],
[x^2*(d+c*d*x)^2*(a+b*arctanh(c*x)),x,7,1/2*b*d^2*x/c^2+4/15*b*d^2*x^2/c+1/6*b*d^2*x^3+1/20*b*c*d^2*x^4+1/3*d^2*x^3*(a+b*arctanh(c*x))+1/2*c*d^2*x^4*(a+b*arctanh(c*x))+1/5*c^2*d^2*x^5*(a+b*arctanh(c*x))+31/60*b*d^2*log(1-c*x)/c^3+1/60*b*d^2*log(1+c*x)/c^3],
[x*(d+c*d*x)^2*(a+b*arctanh(c*x)),x,7,3/4*b*d^2*x/c+1/3*b*d^2*x^2+1/12*b*c*d^2*x^3+1/2*d^2*x^2*(a+b*arctanh(c*x))+2/3*c*d^2*x^3*(a+b*arctanh(c*x))+1/4*c^2*d^2*x^4*(a+b*arctanh(c*x))+17/24*b*d^2*log(1-c*x)/c^2-1/24*b*d^2*log(1+c*x)/c^2],
[(d+c*d*x)^2*(a+b*arctanh(c*x)),x,4,2/3*b*d^2*x+1/6*b*d^2*(1+c*x)^2/c+1/3*d^2*(1+c*x)^3*(a+b*arctanh(c*x))/c+4/3*b*d^2*log(1-c*x)/c],
[(d+c*d*x)^2*(a+b*arctanh(c*x))/x,x,9,2*a*c*d^2*x+1/2*b*c*d^2*x-1/2*b*d^2*arctanh(c*x)+2*b*c*d^2*x*arctanh(c*x)+1/2*c^2*d^2*x^2*(a+b*arctanh(c*x))+a*d^2*log(x)+b*d^2*log(1-c^2*x^2)-1/2*b*d^2*polylog(2,-c*x)+1/2*b*d^2*polylog(2,c*x)],
[(d+c*d*x)^2*(a+b*arctanh(c*x))/x^2,x,11,d^2*(-1+c^2*x^2)*(a+b*arctanh(c*x))/x+(2*a+b)*c*d^2*log(x)-b*c*d^2*polylog(2,-c*x)+b*c*d^2*polylog(2,c*x),a*c^2*d^2*x+b*c^2*d^2*x*arctanh(c*x)-d^2*(a+b*arctanh(c*x))/x+2*a*c*d^2*log(x)+b*c*d^2*log(x)-b*c*d^2*polylog(2,-c*x)+b*c*d^2*polylog(2,c*x)],
[(d+c*d*x)^2*(a+b*arctanh(c*x))/x^3,x,11,-1/2*b*c*d^2/x+1/2*b*c^2*d^2*arctanh(c*x)-1/2*d^2*(a+b*arctanh(c*x))/x^2-2*c*d^2*(a+b*arctanh(c*x))/x+a*c^2*d^2*log(x)+2*b*c^2*d^2*log(x)-b*c^2*d^2*log(1-c^2*x^2)-1/2*b*c^2*d^2*polylog(2,-c*x)+1/2*b*c^2*d^2*polylog(2,c*x)],
[(d+c*d*x)^2*(a+b*arctanh(c*x))/x^4,x,4,-1/6*b*c*d^2/x^2-b*c^2*d^2/x-1/3*d^2*(1+c*x)^3*(a+b*arctanh(c*x))/x^3+4/3*b*c^3*d^2*log(x)-4/3*b*c^3*d^2*log(1-c*x)],
[(d+c*d*x)^2*(a+b*arctanh(c*x))/x^5,x,4,-1/12*b*c*d^2/x^3-1/3*b*c^2*d^2/x^2-3/4*b*c^3*d^2/x-1/4*d^2*(a+b*arctanh(c*x))/x^4-2/3*c*d^2*(a+b*arctanh(c*x))/x^3-1/2*c^2*d^2*(a+b*arctanh(c*x))/x^2+2/3*b*c^4*d^2*log(x)-17/24*b*c^4*d^2*log(1-c*x)+1/24*b*c^4*d^2*log(1+c*x)],
[(d+c*d*x)^2*(a+b*arctanh(c*x))/x^6,x,4,-1/20*b*c*d^2/x^4-1/6*b*c^2*d^2/x^3-4/15*b*c^3*d^2/x^2-1/2*b*c^4*d^2/x-1/5*d^2*(a+b*arctanh(c*x))/x^5-1/2*c*d^2*(a+b*arctanh(c*x))/x^4-1/3*c^2*d^2*(a+b*arctanh(c*x))/x^3+8/15*b*c^5*d^2*log(x)-31/60*b*c^5*d^2*log(1-c*x)-1/60*b*c^5*d^2*log(1+c*x)],
[x^3*(d+c*d*x)^3*(a+b*arctanh(c*x)),x,7,3/4*b*d^3*x/c^3+13/35*b*d^3*x^2/c^2+1/4*b*d^3*x^3/c+13/70*b*d^3*x^4+1/10*b*c*d^3*x^5+1/42*b*c^2*d^3*x^6+1/4*d^3*x^4*(a+b*arctanh(c*x))+3/5*c*d^3*x^5*(a+b*arctanh(c*x))+1/2*c^2*d^3*x^6*(a+b*arctanh(c*x))+1/7*c^3*d^3*x^7*(a+b*arctanh(c*x))+209/280*b*d^3*log(1-c*x)/c^4-1/280*b*d^3*log(1+c*x)/c^4],
[x^2*(d+c*d*x)^3*(a+b*arctanh(c*x)),x,7,11/12*b*d^3*x/c^2+7/15*b*d^3*x^2/c+11/36*b*d^3*x^3+3/20*b*c*d^3*x^4+1/30*b*c^2*d^3*x^5+1/3*d^3*x^3*(a+b*arctanh(c*x))+3/4*c*d^3*x^4*(a+b*arctanh(c*x))+3/5*c^2*d^3*x^5*(a+b*arctanh(c*x))+1/6*c^3*d^3*x^6*(a+b*arctanh(c*x))+37/40*b*d^3*log(1-c*x)/c^3+1/120*b*d^3*log(1+c*x)/c^3],
[x*(d+c*d*x)^3*(a+b*arctanh(c*x)),x,4,3/5*b*d^3*x/c+3/20*b*d^3*(1+c*x)^2/c^2+1/20*b*d^3*(1+c*x)^3/c^2+1/20*b*d^3*(1+c*x)^4/c^2-1/4*d^3*(1+c*x)^4*(a+b*arctanh(c*x))/c^2+1/5*d^3*(1+c*x)^5*(a+b*arctanh(c*x))/c^2+6/5*b*d^3*log(1-c*x)/c^2],
[(d+c*d*x)^3*(a+b*arctanh(c*x)),x,4,b*d^3*x+1/4*b*d^3*(1+c*x)^2/c+1/12*b*d^3*(1+c*x)^3/c+1/4*d^3*(1+c*x)^4*(a+b*arctanh(c*x))/c+2*b*d^3*log(1-c*x)/c],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x,x,13,3*a*c*d^3*x+3/2*b*c*d^3*x+1/6*b*c^2*d^3*x^2-3/2*b*d^3*arctanh(c*x)+3*b*c*d^3*x*arctanh(c*x)+3/2*c^2*d^3*x^2*(a+b*arctanh(c*x))+1/3*c^3*d^3*x^3*(a+b*arctanh(c*x))+a*d^3*log(x)+5/3*b*d^3*log(1-c^2*x^2)-1/2*b*d^3*polylog(2,-c*x)+1/2*b*d^3*polylog(2,c*x)],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x^2,x,14,3*a*c^2*d^3*x+1/2*b*c^2*d^3*x-1/2*b*c*d^3*arctanh(c*x)+3*b*c^2*d^3*x*arctanh(c*x)-d^3*(a+b*arctanh(c*x))/x+1/2*c^3*d^3*x^2*(a+b*arctanh(c*x))+3*a*c*d^3*log(x)+b*c*d^3*log(x)+b*c*d^3*log(1-c^2*x^2)-3/2*b*c*d^3*polylog(2,-c*x)+3/2*b*c*d^3*polylog(2,c*x)],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x^3,x,14,-1/2*b*c*d^3/x+a*c^3*d^3*x+1/2*b*c^2*d^3*arctanh(c*x)+b*c^3*d^3*x*arctanh(c*x)-1/2*d^3*(a+b*arctanh(c*x))/x^2-3*c*d^3*(a+b*arctanh(c*x))/x+3*a*c^2*d^3*log(x)+3*b*c^2*d^3*log(x)-b*c^2*d^3*log(1-c^2*x^2)-3/2*b*c^2*d^3*polylog(2,-c*x)+3/2*b*c^2*d^3*polylog(2,c*x)],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x^4,x,15,-1/6*b*c*d^3/x^2-3/2*b*c^2*d^3/x+3/2*b*c^3*d^3*arctanh(c*x)-1/3*d^3*(a+b*arctanh(c*x))/x^3-3/2*c*d^3*(a+b*arctanh(c*x))/x^2-3*c^2*d^3*(a+b*arctanh(c*x))/x+a*c^3*d^3*log(x)+10/3*b*c^3*d^3*log(x)-5/3*b*c^3*d^3*log(1-c^2*x^2)-1/2*b*c^3*d^3*polylog(2,-c*x)+1/2*b*c^3*d^3*polylog(2,c*x)],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x^5,x,4,-1/12*b*c*d^3/x^3-1/2*b*c^2*d^3/x^2-7/4*b*c^3*d^3/x-1/4*d^3*(1+c*x)^4*(a+b*arctanh(c*x))/x^4+2*b*c^4*d^3*log(x)-2*b*c^4*d^3*log(1-c*x)],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x^6,x,4,-1/20*b*c*d^3/x^4-1/4*b*c^2*d^3/x^3-3/5*b*c^3*d^3/x^2-5/4*b*c^4*d^3/x-1/5*d^3*(1+c*x)^4*(a+b*arctanh(c*x))/x^5+1/20*c*d^3*(1+c*x)^4*(a+b*arctanh(c*x))/x^4+6/5*b*c^5*d^3*log(x)-6/5*b*c^5*d^3*log(1-c*x)],
[(d+c*d*x)^3*(a+b*arctanh(c*x))/x^7,x,4,-1/30*b*c*d^3/x^5-3/20*b*c^2*d^3/x^4-11/36*b*c^3*d^3/x^3-7/15*b*c^4*d^3/x^2-11/12*b*c^5*d^3/x-1/6*d^3*(a+b*arctanh(c*x))/x^6-3/5*c*d^3*(a+b*arctanh(c*x))/x^5-3/4*c^2*d^3*(a+b*arctanh(c*x))/x^4-1/3*c^3*d^3*(a+b*arctanh(c*x))/x^3+14/15*b*c^6*d^3*log(x)-37/40*b*c^6*d^3*log(1-c*x)-1/120*b*c^6*d^3*log(1+c*x)],
[x^3*(d+c*d*x)^4*(a+b*arctanh(c*x)),x,7,11/8*b*d^4*x/c^3+24/35*b*d^4*x^2/c^2+11/24*b*d^4*x^3/c+12/35*b*d^4*x^4+9/40*b*c*d^4*x^5+2/21*b*c^2*d^4*x^6+1/56*b*c^3*d^4*x^7+1/4*d^4*x^4*(a+b*arctanh(c*x))+4/5*c*d^4*x^5*(a+b*arctanh(c*x))+c^2*d^4*x^6*(a+b*arctanh(c*x))+4/7*c^3*d^4*x^7*(a+b*arctanh(c*x))+1/8*c^4*d^4*x^8*(a+b*arctanh(c*x))+769/560*b*d^4*log(1-c*x)/c^4-1/560*b*d^4*log(1+c*x)/c^4],
[x^2*(d+c*d*x)^4*(a+b*arctanh(c*x)),x,4,5/3*b*d^4*x/c^2+88/105*b*d^4*x^2/c+5/9*b*d^4*x^3+47/140*b*c*d^4*x^4+2/15*b*c^2*d^4*x^5+1/42*b*c^3*d^4*x^6+1/5*d^4*(1+c*x)^5*(a+b*arctanh(c*x))/c^3-1/3*d^4*(1+c*x)^6*(a+b*arctanh(c*x))/c^3+1/7*d^4*(1+c*x)^7*(a+b*arctanh(c*x))/c^3+176/105*b*d^4*log(1-c*x)/c^3],
[x*(d+c*d*x)^4*(a+b*arctanh(c*x)),x,4,16/15*b*d^4*x/c+4/15*b*d^4*(1+c*x)^2/c^2+4/45*b*d^4*(1+c*x)^3/c^2+1/30*b*d^4*(1+c*x)^4/c^2+1/30*b*d^4*(1+c*x)^5/c^2-1/5*d^4*(1+c*x)^5*(a+b*arctanh(c*x))/c^2+1/6*d^4*(1+c*x)^6*(a+b*arctanh(c*x))/c^2+32/15*b*d^4*log(1-c*x)/c^2],
[(d+c*d*x)^4*(a+b*arctanh(c*x)),x,4,8/5*b*d^4*x+2/5*b*d^4*(1+c*x)^2/c+2/15*b*d^4*(1+c*x)^3/c+1/20*b*d^4*(1+c*x)^4/c+1/5*d^4*(1+c*x)^5*(a+b*arctanh(c*x))/c+16/5*b*d^4*log(1-c*x)/c],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x,x,17,4*a*c*d^4*x+13/4*b*c*d^4*x+2/3*b*c^2*d^4*x^2+1/12*b*c^3*d^4*x^3-13/4*b*d^4*arctanh(c*x)+4*b*c*d^4*x*arctanh(c*x)+3*c^2*d^4*x^2*(a+b*arctanh(c*x))+4/3*c^3*d^4*x^3*(a+b*arctanh(c*x))+1/4*c^4*d^4*x^4*(a+b*arctanh(c*x))+a*d^4*log(x)+8/3*b*d^4*log(1-c^2*x^2)-1/2*b*d^4*polylog(2,-c*x)+1/2*b*d^4*polylog(2,c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^2,x,18,6*a*c^2*d^4*x+2*b*c^2*d^4*x+1/6*b*c^3*d^4*x^2-2*b*c*d^4*arctanh(c*x)+6*b*c^2*d^4*x*arctanh(c*x)-d^4*(a+b*arctanh(c*x))/x+2*c^3*d^4*x^2*(a+b*arctanh(c*x))+1/3*c^4*d^4*x^3*(a+b*arctanh(c*x))+4*a*c*d^4*log(x)+b*c*d^4*log(x)+8/3*b*c*d^4*log(1-c^2*x^2)-2*b*c*d^4*polylog(2,-c*x)+2*b*c*d^4*polylog(2,c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^3,x,17,-1/2*b*c*d^4/x+4*a*c^3*d^4*x+1/2*b*c^3*d^4*x+4*b*c^3*d^4*x*arctanh(c*x)-1/2*d^4*(a+b*arctanh(c*x))/x^2-4*c*d^4*(a+b*arctanh(c*x))/x+1/2*c^4*d^4*x^2*(a+b*arctanh(c*x))+6*a*c^2*d^4*log(x)+4*b*c^2*d^4*log(x)-3*b*c^2*d^4*polylog(2,-c*x)+3*b*c^2*d^4*polylog(2,c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^4,x,18,-1/6*b*c*d^4/x^2-2*b*c^2*d^4/x+a*c^4*d^4*x+2*b*c^3*d^4*arctanh(c*x)+b*c^4*d^4*x*arctanh(c*x)-1/3*d^4*(a+b*arctanh(c*x))/x^3-2*c*d^4*(a+b*arctanh(c*x))/x^2-6*c^2*d^4*(a+b*arctanh(c*x))/x+4*a*c^3*d^4*log(x)+19/3*b*c^3*d^4*log(x)-8/3*b*c^3*d^4*log(1-c^2*x^2)-2*b*c^3*d^4*polylog(2,-c*x)+2*b*c^3*d^4*polylog(2,c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^5,x,19,-1/12*b*c*d^4/x^3-2/3*b*c^2*d^4/x^2-13/4*b*c^3*d^4/x+13/4*b*c^4*d^4*arctanh(c*x)-1/4*d^4*(a+b*arctanh(c*x))/x^4-4/3*c*d^4*(a+b*arctanh(c*x))/x^3-3*c^2*d^4*(a+b*arctanh(c*x))/x^2-4*c^3*d^4*(a+b*arctanh(c*x))/x+a*c^4*d^4*log(x)+16/3*b*c^4*d^4*log(x)-8/3*b*c^4*d^4*log(1-c^2*x^2)-1/2*b*c^4*d^4*polylog(2,-c*x)+1/2*b*c^4*d^4*polylog(2,c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^6,x,4,-1/20*b*c*d^4/x^4-1/3*b*c^2*d^4/x^3-11/10*b*c^3*d^4/x^2-3*b*c^4*d^4/x-1/5*d^4*(1+c*x)^5*(a+b*arctanh(c*x))/x^5+16/5*b*c^5*d^4*log(x)-16/5*b*c^5*d^4*log(1-c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^7,x,4,-1/30*b*c*d^4/x^5-1/5*b*c^2*d^4/x^4-5/9*b*c^3*d^4/x^3-16/15*b*c^4*d^4/x^2-13/6*b*c^5*d^4/x-1/6*d^4*(1+c*x)^5*(a+b*arctanh(c*x))/x^6+1/30*c*d^4*(1+c*x)^5*(a+b*arctanh(c*x))/x^5+32/15*b*c^6*d^4*log(x)-32/15*b*c^6*d^4*log(1-c*x)],
[(d+c*d*x)^4*(a+b*arctanh(c*x))/x^8,x,4,-1/42*b*c*d^4/x^6-2/15*b*c^2*d^4/x^5-47/140*b*c^3*d^4/x^4-5/9*b*c^4*d^4/x^3-88/105*b*c^5*d^4/x^2-5/3*b*c^6*d^4/x-1/7*d^4*(a+b*arctanh(c*x))/x^7-2/3*c*d^4*(a+b*arctanh(c*x))/x^6-6/5*c^2*d^4*(a+b*arctanh(c*x))/x^5-c^3*d^4*(a+b*arctanh(c*x))/x^4-1/3*c^4*d^4*(a+b*arctanh(c*x))/x^3+176/105*b*c^7*d^4*log(x)-117/70*b*c^7*d^4*log(1-c*x)-1/210*b*c^7*d^4*log(1+c*x)],

# q<0
[x^3*(a+b*arctanh(c*x))/(d+c*d*x),x,16,a*x/(c^3*d)-1/2*b*x/(c^3*d)+1/6*b*x^2/(c^2*d)+1/2*b*arctanh(c*x)/(c^4*d)+b*x*arctanh(c*x)/(c^3*d)-1/2*x^2*(a+b*arctanh(c*x))/(c^2*d)+1/3*x^3*(a+b*arctanh(c*x))/(c*d)+(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^4*d)+2/3*b*log(1-c^2*x^2)/(c^4*d)-1/2*b*polylog(2,1+(-2)/(1+c*x))/(c^4*d)],
[x^2*(a+b*arctanh(c*x))/(d+c*d*x),x,11,-a*x/(c^2*d)+1/2*b*x/(c^2*d)-1/2*b*arctanh(c*x)/(c^3*d)-b*x*arctanh(c*x)/(c^2*d)+1/2*x^2*(a+b*arctanh(c*x))/(c*d)-(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^3*d)-1/2*b*log(1-c^2*x^2)/(c^3*d)+1/2*b*polylog(2,1+(-2)/(1+c*x))/(c^3*d)],
[x*(a+b*arctanh(c*x))/(d+c*d*x),x,7,a*x/(c*d)+b*x*arctanh(c*x)/(c*d)+(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^2*d)+1/2*b*log(1-c^2*x^2)/(c^2*d)-1/2*b*polylog(2,1+(-2)/(1+c*x))/(c^2*d)],
[(a+b*arctanh(c*x))/(d+c*d*x),x,3,-(a+b*arctanh(c*x))*log(2/(1+c*x))/(c*d)+1/2*b*polylog(2,1+(-2)/(1+c*x))/(c*d)],
[(a+b*arctanh(c*x))/(x*(d+c*d*x)),x,2,(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d-1/2*b*polylog(2,-1+2/(1+c*x))/d],
[(a+b*arctanh(c*x))/(x^2*(d+c*d*x)),x,8,(-a-b*arctanh(c*x))/(d*x)+b*c*log(x)/d-1/2*b*c*log(1-c^2*x^2)/d-c*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d+1/2*b*c*polylog(2,-1+2/(1+c*x))/d],
[(a+b*arctanh(c*x))/(x^3*(d+c*d*x)),x,12,-1/2*b*c/(d*x)+1/2*b*c^2*arctanh(c*x)/d+1/2*(-a-b*arctanh(c*x))/(d*x^2)+c*(a+b*arctanh(c*x))/(d*x)-b*c^2*log(x)/d+1/2*b*c^2*log(1-c^2*x^2)/d+c^2*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d-1/2*b*c^2*polylog(2,-1+2/(1+c*x))/d],
[(a+b*arctanh(c*x))/(x^4*(d+c*d*x)),x,17,-1/6*b*c/(d*x^2)+1/2*b*c^2/(d*x)-1/2*b*c^3*arctanh(c*x)/d+1/3*(-a-b*arctanh(c*x))/(d*x^3)+1/2*c*(a+b*arctanh(c*x))/(d*x^2)-c^2*(a+b*arctanh(c*x))/(d*x)+4/3*b*c^3*log(x)/d-2/3*b*c^3*log(1-c^2*x^2)/d-c^3*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d+1/2*b*c^3*polylog(2,-1+2/(1+c*x))/d],
[x^3*(a+b*arctanh(c*x))/(d+c*d*x)^2,x,16,-2*a*x/(c^3*d^2)+1/2*b*x/(c^3*d^2)+1/2*b/(c^4*d^2*(1+c*x))-b*arctanh(c*x)/(c^4*d^2)-2*b*x*arctanh(c*x)/(c^3*d^2)+1/2*x^2*(a+b*arctanh(c*x))/(c^2*d^2)+(a+b*arctanh(c*x))/(c^4*d^2*(1+c*x))-3*(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^4*d^2)-b*log(1-c^2*x^2)/(c^4*d^2)+3/2*b*polylog(2,1+(-2)/(1+c*x))/(c^4*d^2)],
[x^2*(a+b*arctanh(c*x))/(d+c*d*x)^2,x,13,a*x/(c^2*d^2)-1/2*b/(c^3*d^2*(1+c*x))+1/2*b*arctanh(c*x)/(c^3*d^2)+b*x*arctanh(c*x)/(c^2*d^2)+(-a-b*arctanh(c*x))/(c^3*d^2*(1+c*x))+2*(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^3*d^2)+1/2*b*log(1-c^2*x^2)/(c^3*d^2)-b*polylog(2,1+(-2)/(1+c*x))/(c^3*d^2)],
[x*(a+b*arctanh(c*x))/(d+c*d*x)^2,x,10,1/2*b/(c^2*d^2*(1+c*x))-1/2*b*arctanh(c*x)/(c^2*d^2)+(a+b*arctanh(c*x))/(c^2*d^2*(1+c*x))-(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^2*d^2)+1/2*b*polylog(2,1+(-2)/(1+c*x))/(c^2*d^2)],
[(a+b*arctanh(c*x))/(d+c*d*x)^2,x,5,-1/2*b/(c*d^2*(1+c*x))+1/2*b*arctanh(c*x)/(c*d^2)+(-a-b*arctanh(c*x))/(c*d^2*(1+c*x))],
[(a+b*arctanh(c*x))/(x*(d+c*d*x)^2),x,11,1/2*b/(d^2*(1+c*x))-1/2*b*arctanh(c*x)/d^2+(a+b*arctanh(c*x))/(d^2*(1+c*x))+a*log(x)/d^2+(a+b*arctanh(c*x))*log(2/(1+c*x))/d^2-1/2*b*polylog(2,-c*x)/d^2+1/2*b*polylog(2,c*x)/d^2-1/2*b*polylog(2,1+(-2)/(1+c*x))/d^2],
[(a+b*arctanh(c*x))/(x^2*(d+c*d*x)^2),x,16,-1/2*b*c/(d^2*(1+c*x))+1/2*b*c*arctanh(c*x)/d^2+(-a-b*arctanh(c*x))/(d^2*x)-c*(a+b*arctanh(c*x))/(d^2*(1+c*x))-2*a*c*log(x)/d^2+b*c*log(x)/d^2-2*c*(a+b*arctanh(c*x))*log(2/(1+c*x))/d^2-1/2*b*c*log(1-c^2*x^2)/d^2+b*c*polylog(2,-c*x)/d^2-b*c*polylog(2,c*x)/d^2+b*c*polylog(2,1+(-2)/(1+c*x))/d^2],
[(a+b*arctanh(c*x))/(x^3*(d+c*d*x)^2),x,19,-1/2*b*c/(d^2*x)+1/2*b*c^2/(d^2*(1+c*x))+1/2*(-a-b*arctanh(c*x))/(d^2*x^2)+2*c*(a+b*arctanh(c*x))/(d^2*x)+c^2*(a+b*arctanh(c*x))/(d^2*(1+c*x))+3*a*c^2*log(x)/d^2-2*b*c^2*log(x)/d^2+3*c^2*(a+b*arctanh(c*x))*log(2/(1+c*x))/d^2+b*c^2*log(1-c^2*x^2)/d^2-3/2*b*c^2*polylog(2,-c*x)/d^2+3/2*b*c^2*polylog(2,c*x)/d^2-3/2*b*c^2*polylog(2,1+(-2)/(1+c*x))/d^2],
[x^4*(a+b*arctanh(c*x))/(d+c*d*x)^3,x,21,-3*a*x/(c^4*d^3)+1/2*b*x/(c^4*d^3)-1/8*b/(c^5*d^3*(1+c*x)^2)+15/8*b/(c^5*d^3*(1+c*x))-19/8*b*arctanh(c*x)/(c^5*d^3)-3*b*x*arctanh(c*x)/(c^4*d^3)+1/2*x^2*(a+b*arctanh(c*x))/(c^3*d^3)+1/2*(-a-b*arctanh(c*x))/(c^5*d^3*(1+c*x)^2)+4*(a+b*arctanh(c*x))/(c^5*d^3*(1+c*x))-6*(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^5*d^3)-3/2*b*log(1-c^2*x^2)/(c^5*d^3)+3*b*polylog(2,1+(-2)/(1+c*x))/(c^5*d^3)],
[x^3*(a+b*arctanh(c*x))/(d+c*d*x)^3,x,18,a*x/(c^3*d^3)+1/8*b/(c^4*d^3*(1+c*x)^2)-11/8*b/(c^4*d^3*(1+c*x))+11/8*b*arctanh(c*x)/(c^4*d^3)+b*x*arctanh(c*x)/(c^3*d^3)+1/2*(a+b*arctanh(c*x))/(c^4*d^3*(1+c*x)^2)-3*(a+b*arctanh(c*x))/(c^4*d^3*(1+c*x))+3*(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^4*d^3)+1/2*b*log(1-c^2*x^2)/(c^4*d^3)-3/2*b*polylog(2,1+(-2)/(1+c*x))/(c^4*d^3)],
[x^2*(a+b*arctanh(c*x))/(d+c*d*x)^3,x,15,-1/8*b/(c^3*d^3*(1+c*x)^2)+7/8*b/(c^3*d^3*(1+c*x))-7/8*b*arctanh(c*x)/(c^3*d^3)+1/2*(-a-b*arctanh(c*x))/(c^3*d^3*(1+c*x)^2)+2*(a+b*arctanh(c*x))/(c^3*d^3*(1+c*x))-(a+b*arctanh(c*x))*log(2/(1+c*x))/(c^3*d^3)+1/2*b*polylog(2,1+(-2)/(1+c*x))/(c^3*d^3)],
[x*(a+b*arctanh(c*x))/(d+c*d*x)^3,x,5,1/8*b/(c^2*d^3*(1+c*x)^2)-3/8*b/(c^2*d^3*(1+c*x))-1/8*b*arctanh(c*x)/(c^2*d^3)+1/2*x^2*(a+b*arctanh(c*x))/(d^3*(1+c*x)^2)],
[(a+b*arctanh(c*x))/(d+c*d*x)^3,x,5,-1/8*b/(c*d^3*(1+c*x)^2)-1/8*b/(c*d^3*(1+c*x))+1/8*b*arctanh(c*x)/(c*d^3)+1/2*(-a-b*arctanh(c*x))/(c*d^3*(1+c*x)^2)],
[(a+b*arctanh(c*x))/(x*(d+c*d*x)^3),x,16,1/8*b/(d^3*(1+c*x)^2)+5/8*b/(d^3*(1+c*x))-5/8*b*arctanh(c*x)/d^3+1/2*(a+b*arctanh(c*x))/(d^3*(1+c*x)^2)+(a+b*arctanh(c*x))/(d^3*(1+c*x))+a*log(x)/d^3+(a+b*arctanh(c*x))*log(2/(1+c*x))/d^3-1/2*b*polylog(2,-c*x)/d^3+1/2*b*polylog(2,c*x)/d^3-1/2*b*polylog(2,1+(-2)/(1+c*x))/d^3],
[(a+b*arctanh(c*x))/(x^2*(d+c*d*x)^3),x,21,-1/8*b*c/(d^3*(1+c*x)^2)-9/8*b*c/(d^3*(1+c*x))+9/8*b*c*arctanh(c*x)/d^3+(-a-b*arctanh(c*x))/(d^3*x)-1/2*c*(a+b*arctanh(c*x))/(d^3*(1+c*x)^2)-2*c*(a+b*arctanh(c*x))/(d^3*(1+c*x))-3*a*c*log(x)/d^3+b*c*log(x)/d^3-3*c*(a+b*arctanh(c*x))*log(2/(1+c*x))/d^3-1/2*b*c*log(1-c^2*x^2)/d^3+3/2*b*c*polylog(2,-c*x)/d^3-3/2*b*c*polylog(2,c*x)/d^3+3/2*b*c*polylog(2,1+(-2)/(1+c*x))/d^3],
[(a+b*arctanh(c*x))/(x^3*(d+c*d*x)^3),x,24,-1/2*b*c/(d^3*x)+1/8*b*c^2/(d^3*(1+c*x)^2)+13/8*b*c^2/(d^3*(1+c*x))-9/8*b*c^2*arctanh(c*x)/d^3+1/2*(-a-b*arctanh(c*x))/(d^3*x^2)+3*c*(a+b*arctanh(c*x))/(d^3*x)+1/2*c^2*(a+b*arctanh(c*x))/(d^3*(1+c*x)^2)+3*c^2*(a+b*arctanh(c*x))/(d^3*(1+c*x))+6*a*c^2*log(x)/d^3-3*b*c^2*log(x)/d^3+6*c^2*(a+b*arctanh(c*x))*log(2/(1+c*x))/d^3+3/2*b*c^2*log(1-c^2*x^2)/d^3-3*b*c^2*polylog(2,-c*x)/d^3+3*b*c^2*polylog(2,c*x)/d^3-3*b*c^2*polylog(2,1+(-2)/(1+c*x))/d^3],
[(a+b*arctanh(c*x))/(1+c*x)^4,x,5,-1/18*b/(c*(1+c*x)^3)-1/24*b/(c*(1+c*x)^2)-1/24*b/(c*(1+c*x))+1/24*b*arctanh(c*x)/c+1/3*(-a-b*arctanh(c*x))/(c*(1+c*x)^3)],
[arctanh(a*x)/(c*x+a*c*x^2),x,3,arctanh(a*x)*log(2+(-2)/(1+a*x))/c-1/2*polylog(2,-1+2/(1+a*x))/c],

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^2 when c^2 d^2 - e^2=0

# q>0
[x^3*(d+c*d*x)*(a+b*arctanh(c*x))^2,x,27,1/2*a*b*d*x/c^3+3/10*b^2*d*x/c^3+1/12*b^2*d*x^2/c^2+1/30*b^2*d*x^3/c-3/10*b^2*d*arctanh(c*x)/c^4+1/2*b^2*d*x*arctanh(c*x)/c^3+1/5*b*d*x^2*(a+b*arctanh(c*x))/c^2+1/6*b*d*x^3*(a+b*arctanh(c*x))/c+1/10*b*d*x^4*(a+b*arctanh(c*x))-1/20*d*(a+b*arctanh(c*x))^2/c^4+1/4*d*x^4*(a+b*arctanh(c*x))^2+1/5*c*d*x^5*(a+b*arctanh(c*x))^2-2/5*b*d*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^4+1/3*b^2*d*log(1-c^2*x^2)/c^4-1/5*b^2*d*polylog(2,1+(-2)/(1-c*x))/c^4],
[x^2*(d+c*d*x)*(a+b*arctanh(c*x))^2,x,22,1/2*a*b*d*x/c^2+1/3*b^2*d*x/c^2+1/12*b^2*d*x^2/c-1/3*b^2*d*arctanh(c*x)/c^3+1/2*b^2*d*x*arctanh(c*x)/c^2+1/3*b*d*x^2*(a+b*arctanh(c*x))/c+1/6*b*d*x^3*(a+b*arctanh(c*x))+1/12*d*(a+b*arctanh(c*x))^2/c^3+1/3*d*x^3*(a+b*arctanh(c*x))^2+1/4*c*d*x^4*(a+b*arctanh(c*x))^2-2/3*b*d*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^3+1/3*b^2*d*log(1-c^2*x^2)/c^3-1/3*b^2*d*polylog(2,1+(-2)/(1-c*x))/c^3],
[x*(d+c*d*x)*(a+b*arctanh(c*x))^2,x,17,a*b*d*x/c+1/3*b^2*d*x/c-1/3*b^2*d*arctanh(c*x)/c^2+b^2*d*x*arctanh(c*x)/c+1/3*b*d*x^2*(a+b*arctanh(c*x))-1/6*d*(a+b*arctanh(c*x))^2/c^2+1/2*d*x^2*(a+b*arctanh(c*x))^2+1/3*c*d*x^3*(a+b*arctanh(c*x))^2-2/3*b*d*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^2+1/2*b^2*d*log(1-c^2*x^2)/c^2-1/3*b^2*d*polylog(2,1+(-2)/(1-c*x))/c^2],
[(d+c*d*x)*(a+b*arctanh(c*x))^2,x,9,a*b*d*x+b^2*d*x*arctanh(c*x)+1/2*d*(1+c*x)^2*(a+b*arctanh(c*x))^2/c-2*b*d*(a+b*arctanh(c*x))*log(2/(1-c*x))/c+1/2*b^2*d*log(1-c^2*x^2)/c-b^2*d*polylog(2,1+(-2)/(1-c*x))/c],
[(d+c*d*x)*(a+b*arctanh(c*x))^2/x,x,13,d*(a+b*arctanh(c*x))^2+c*d*x*(a+b*arctanh(c*x))^2+2*d*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))-2*b*d*(a+b*arctanh(c*x))*log(2/(1-c*x))-b^2*d*polylog(2,1+(-2)/(1-c*x))-b*d*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+b*d*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))+1/2*b^2*d*polylog(3,1+(-2)/(1-c*x))-1/2*b^2*d*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)*(a+b*arctanh(c*x))^2/x^2,x,12,c*d*(a+b*arctanh(c*x))^2-d*(a+b*arctanh(c*x))^2/x+2*c*d*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))+2*b*c*d*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-b*c*d*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+b*c*d*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))-b^2*c*d*polylog(2,-1+2/(1+c*x))+1/2*b^2*c*d*polylog(3,1+(-2)/(1-c*x))-1/2*b^2*c*d*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)*(a+b*arctanh(c*x))^2/x^3,x,14,-b*c*d*(a+b*arctanh(c*x))/x+3/2*c^2*d*(a+b*arctanh(c*x))^2-1/2*d*(a+b*arctanh(c*x))^2/x^2-c*d*(a+b*arctanh(c*x))^2/x+b^2*c^2*d*log(x)-1/2*b^2*c^2*d*log(1-c^2*x^2)+2*b*c^2*d*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-b^2*c^2*d*polylog(2,-1+2/(1+c*x))],
[(d+c*d*x)*(a+b*arctanh(c*x))^2/x^4,x,18,-1/3*b^2*c^2*d/x+1/3*b^2*c^3*d*arctanh(c*x)-1/3*b*c*d*(a+b*arctanh(c*x))/x^2-b*c^2*d*(a+b*arctanh(c*x))/x+5/6*c^3*d*(a+b*arctanh(c*x))^2-1/3*d*(a+b*arctanh(c*x))^2/x^3-1/2*c*d*(a+b*arctanh(c*x))^2/x^2+b^2*c^3*d*log(x)-1/2*b^2*c^3*d*log(1-c^2*x^2)+2/3*b*c^3*d*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-1/3*b^2*c^3*d*polylog(2,-1+2/(1+c*x))],
[x^3*(d+c*d*x)^2*(a+b*arctanh(c*x))^2,x,43,5/6*a*b*d^2*x/c^3+3/5*b^2*d^2*x/c^3+31/180*b^2*d^2*x^2/c^2+1/15*b^2*d^2*x^3/c+1/60*b^2*d^2*x^4-3/5*b^2*d^2*arctanh(c*x)/c^4+5/6*b^2*d^2*x*arctanh(c*x)/c^3+2/5*b*d^2*x^2*(a+b*arctanh(c*x))/c^2+5/18*b*d^2*x^3*(a+b*arctanh(c*x))/c+1/5*b*d^2*x^4*(a+b*arctanh(c*x))+1/15*b*c*d^2*x^5*(a+b*arctanh(c*x))-1/60*d^2*(a+b*arctanh(c*x))^2/c^4+1/4*d^2*x^4*(a+b*arctanh(c*x))^2+2/5*c*d^2*x^5*(a+b*arctanh(c*x))^2+1/6*c^2*d^2*x^6*(a+b*arctanh(c*x))^2-4/5*b*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^4+53/90*b^2*d^2*log(1-c^2*x^2)/c^4-2/5*b^2*d^2*polylog(2,1+(-2)/(1-c*x))/c^4],
[x^2*(d+c*d*x)^2*(a+b*arctanh(c*x))^2,x,36,a*b*d^2*x/c^2+19/30*b^2*d^2*x/c^2+1/6*b^2*d^2*x^2/c+1/30*b^2*d^2*x^3-19/30*b^2*d^2*arctanh(c*x)/c^3+b^2*d^2*x*arctanh(c*x)/c^2+8/15*b*d^2*x^2*(a+b*arctanh(c*x))/c+1/3*b*d^2*x^3*(a+b*arctanh(c*x))+1/10*b*c*d^2*x^4*(a+b*arctanh(c*x))+1/30*d^2*(a+b*arctanh(c*x))^2/c^3+1/3*d^2*x^3*(a+b*arctanh(c*x))^2+1/2*c*d^2*x^4*(a+b*arctanh(c*x))^2+1/5*c^2*d^2*x^5*(a+b*arctanh(c*x))^2-16/15*b*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^3+2/3*b^2*d^2*log(1-c^2*x^2)/c^3-8/15*b^2*d^2*polylog(2,1+(-2)/(1-c*x))/c^3],
[x*(d+c*d*x)^2*(a+b*arctanh(c*x))^2,x,28,3/2*a*b*d^2*x/c+2/3*b^2*d^2*x/c+1/12*b^2*d^2*x^2-2/3*b^2*d^2*arctanh(c*x)/c^2+3/2*b^2*d^2*x*arctanh(c*x)/c+2/3*b*d^2*x^2*(a+b*arctanh(c*x))+1/6*b*c*d^2*x^3*(a+b*arctanh(c*x))-1/12*d^2*(a+b*arctanh(c*x))^2/c^2+1/2*d^2*x^2*(a+b*arctanh(c*x))^2+2/3*c*d^2*x^3*(a+b*arctanh(c*x))^2+1/4*c^2*d^2*x^4*(a+b*arctanh(c*x))^2-4/3*b*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^2+5/6*b^2*d^2*log(1-c^2*x^2)/c^2-2/3*b^2*d^2*polylog(2,1+(-2)/(1-c*x))/c^2],
[(d+c*d*x)^2*(a+b*arctanh(c*x))^2,x,12,2*a*b*d^2*x+1/3*b^2*d^2*x-1/3*b^2*d^2*arctanh(c*x)/c+2*b^2*d^2*x*arctanh(c*x)+1/3*b*c*d^2*x^2*(a+b*arctanh(c*x))+1/3*d^2*(1+c*x)^3*(a+b*arctanh(c*x))^2/c-8/3*b*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c+b^2*d^2*log(1-c^2*x^2)/c-4/3*b^2*d^2*polylog(2,1+(-2)/(1-c*x))/c],
[(d+c*d*x)^2*(a+b*arctanh(c*x))^2/x,x,19,a*b*c*d^2*x+b^2*c*d^2*x*arctanh(c*x)+3/2*d^2*(a+b*arctanh(c*x))^2+2*c*d^2*x*(a+b*arctanh(c*x))^2+1/2*c^2*d^2*x^2*(a+b*arctanh(c*x))^2+2*d^2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))-4*b*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))+1/2*b^2*d^2*log(1-c^2*x^2)-2*b^2*d^2*polylog(2,1+(-2)/(1-c*x))-b*d^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+b*d^2*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))+1/2*b^2*d^2*polylog(3,1+(-2)/(1-c*x))-1/2*b^2*d^2*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^2*(a+b*arctanh(c*x))^2/x^2,x,17,2*c*d^2*(a+b*arctanh(c*x))^2-d^2*(a+b*arctanh(c*x))^2/x+c^2*d^2*x*(a+b*arctanh(c*x))^2+4*c*d^2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))-2*b*c*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))+2*b*c*d^2*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-b^2*c*d^2*polylog(2,1+(-2)/(1-c*x))-2*b*c*d^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+2*b*c*d^2*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))-b^2*c*d^2*polylog(2,-1+2/(1+c*x))+b^2*c*d^2*polylog(3,1+(-2)/(1-c*x))-b^2*c*d^2*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^2*(a+b*arctanh(c*x))^2/x^3,x,20,-b*c*d^2*(a+b*arctanh(c*x))/x+5/2*c^2*d^2*(a+b*arctanh(c*x))^2-1/2*d^2*(a+b*arctanh(c*x))^2/x^2-2*c*d^2*(a+b*arctanh(c*x))^2/x+2*c^2*d^2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))+b^2*c^2*d^2*log(x)-1/2*b^2*c^2*d^2*log(1-c^2*x^2)+4*b*c^2*d^2*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-b*c^2*d^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+b*c^2*d^2*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))-2*b^2*c^2*d^2*polylog(2,-1+2/(1+c*x))+1/2*b^2*c^2*d^2*polylog(3,1+(-2)/(1-c*x))-1/2*b^2*c^2*d^2*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^2*(a+b*arctanh(c*x))^2/x^4,x,14,-1/3*b^2*c^2*d^2/x+1/3*b^2*c^3*d^2*arctanh(c*x)-1/3*b*c*d^2*(a+b*arctanh(c*x))/x^2-2*b*c^2*d^2*(a+b*arctanh(c*x))/x-1/3*d^2*(1+c*x)^3*(a+b*arctanh(c*x))^2/x^3+8/3*a*b*c^3*d^2*log(x)+2*b^2*c^3*d^2*log(x)+8/3*b*c^3*d^2*(a+b*arctanh(c*x))*log(2/(1-c*x))-b^2*c^3*d^2*log(1-c^2*x^2)-4/3*b^2*c^3*d^2*polylog(2,-c*x)+4/3*b^2*c^3*d^2*polylog(2,c*x)+4/3*b^2*c^3*d^2*polylog(2,1+(-2)/(1-c*x))],
[x^3*(d+c*d*x)^3*(a+b*arctanh(c*x))^2,x,62,3/2*a*b*d^3*x/c^3+122/105*b^2*d^3*x/c^3+7/20*b^2*d^3*x^2/c^2+44/315*b^2*d^3*x^3/c+1/20*b^2*d^3*x^4+1/105*b^2*c*d^3*x^5-122/105*b^2*d^3*arctanh(c*x)/c^4+3/2*b^2*d^3*x*arctanh(c*x)/c^3+26/35*b*d^3*x^2*(a+b*arctanh(c*x))/c^2+1/2*b*d^3*x^3*(a+b*arctanh(c*x))/c+13/35*b*d^3*x^4*(a+b*arctanh(c*x))+1/5*b*c*d^3*x^5*(a+b*arctanh(c*x))+1/21*b*c^2*d^3*x^6*(a+b*arctanh(c*x))-1/140*d^3*(a+b*arctanh(c*x))^2/c^4+1/4*d^3*x^4*(a+b*arctanh(c*x))^2+3/5*c*d^3*x^5*(a+b*arctanh(c*x))^2+1/2*c^2*d^3*x^6*(a+b*arctanh(c*x))^2+1/7*c^3*d^3*x^7*(a+b*arctanh(c*x))^2-52/35*b*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^4+11/10*b^2*d^3*log(1-c^2*x^2)/c^4-26/35*b^2*d^3*polylog(2,1+(-2)/(1-c*x))/c^4],
[x^2*(d+c*d*x)^3*(a+b*arctanh(c*x))^2,x,52,11/6*a*b*d^3*x/c^2+37/30*b^2*d^3*x/c^2+61/180*b^2*d^3*x^2/c+1/10*b^2*d^3*x^3+1/60*b^2*c*d^3*x^4-37/30*b^2*d^3*arctanh(c*x)/c^3+11/6*b^2*d^3*x*arctanh(c*x)/c^2+14/15*b*d^3*x^2*(a+b*arctanh(c*x))/c+11/18*b*d^3*x^3*(a+b*arctanh(c*x))+3/10*b*c*d^3*x^4*(a+b*arctanh(c*x))+1/15*b*c^2*d^3*x^5*(a+b*arctanh(c*x))+1/60*d^3*(a+b*arctanh(c*x))^2/c^3+1/3*d^3*x^3*(a+b*arctanh(c*x))^2+3/4*c*d^3*x^4*(a+b*arctanh(c*x))^2+3/5*c^2*d^3*x^5*(a+b*arctanh(c*x))^2+1/6*c^3*d^3*x^6*(a+b*arctanh(c*x))^2-28/15*b*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^3+113/90*b^2*d^3*log(1-c^2*x^2)/c^3-14/15*b^2*d^3*polylog(2,1+(-2)/(1-c*x))/c^3],
[x*(d+c*d*x)^3*(a+b*arctanh(c*x))^2,x,38,5/2*a*b*d^3*x/c+13/10*b^2*d^3*x/c+1/4*b^2*d^3*x^2+1/30*b^2*c*d^3*x^3-13/10*b^2*d^3*arctanh(c*x)/c^2+5/2*b^2*d^3*x*arctanh(c*x)/c+6/5*b*d^3*x^2*(a+b*arctanh(c*x))+1/2*b*c*d^3*x^3*(a+b*arctanh(c*x))+1/10*b*c^2*d^3*x^4*(a+b*arctanh(c*x))-1/4*d^3*(1+c*x)^4*(a+b*arctanh(c*x))^2/c^2+1/5*d^3*(1+c*x)^5*(a+b*arctanh(c*x))^2/c^2-12/5*b*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))/c^2+3/2*b^2*d^3*log(1-c^2*x^2)/c^2-6/5*b^2*d^3*polylog(2,1+(-2)/(1-c*x))/c^2],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2,x,16,7/2*a*b*d^3*x+b^2*d^3*x+1/12*b^2*c*d^3*x^2-b^2*d^3*arctanh(c*x)/c+7/2*b^2*d^3*x*arctanh(c*x)+b*c*d^3*x^2*(a+b*arctanh(c*x))+1/6*b*c^2*d^3*x^3*(a+b*arctanh(c*x))+1/4*d^3*(1+c*x)^4*(a+b*arctanh(c*x))^2/c-4*b*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))/c+11/6*b^2*d^3*log(1-c^2*x^2)/c-2*b^2*d^3*polylog(2,1+(-2)/(1-c*x))/c],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x,x,28,3*a*b*c*d^3*x+1/3*b^2*c*d^3*x-1/3*b^2*d^3*arctanh(c*x)+3*b^2*c*d^3*x*arctanh(c*x)+1/3*b*c^2*d^3*x^2*(a+b*arctanh(c*x))+11/6*d^3*(a+b*arctanh(c*x))^2+3*c*d^3*x*(a+b*arctanh(c*x))^2+3/2*c^2*d^3*x^2*(a+b*arctanh(c*x))^2+1/3*c^3*d^3*x^3*(a+b*arctanh(c*x))^2+2*d^3*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))-20/3*b*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))+3/2*b^2*d^3*log(1-c^2*x^2)-10/3*b^2*d^3*polylog(2,1+(-2)/(1-c*x))-b*d^3*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+b*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))+1/2*b^2*d^3*polylog(3,1+(-2)/(1-c*x))-1/2*b^2*d^3*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x^2,x,23,a*b*c^2*d^3*x+b^2*c^2*d^3*x*arctanh(c*x)+7/2*c*d^3*(a+b*arctanh(c*x))^2-d^3*(a+b*arctanh(c*x))^2/x+3*c^2*d^3*x*(a+b*arctanh(c*x))^2+1/2*c^3*d^3*x^2*(a+b*arctanh(c*x))^2+6*c*d^3*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))-6*b*c*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))+1/2*b^2*c*d^3*log(1-c^2*x^2)+2*b*c*d^3*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-3*b^2*c*d^3*polylog(2,1+(-2)/(1-c*x))-3*b*c*d^3*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+3*b*c*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))-b^2*c*d^3*polylog(2,-1+2/(1+c*x))+3/2*b^2*c*d^3*polylog(3,1+(-2)/(1-c*x))-3/2*b^2*c*d^3*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x^3,x,25,-b*c*d^3*(a+b*arctanh(c*x))/x+9/2*c^2*d^3*(a+b*arctanh(c*x))^2-1/2*d^3*(a+b*arctanh(c*x))^2/x^2-3*c*d^3*(a+b*arctanh(c*x))^2/x+c^3*d^3*x*(a+b*arctanh(c*x))^2+6*c^2*d^3*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))+b^2*c^2*d^3*log(x)-2*b*c^2*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))-1/2*b^2*c^2*d^3*log(1-c^2*x^2)+6*b*c^2*d^3*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-b^2*c^2*d^3*polylog(2,1+(-2)/(1-c*x))-3*b*c^2*d^3*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+3*b*c^2*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))-3*b^2*c^2*d^3*polylog(2,-1+2/(1+c*x))+3/2*b^2*c^2*d^3*polylog(3,1+(-2)/(1-c*x))-3/2*b^2*c^2*d^3*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x^4,x,28,-1/3*b^2*c^2*d^3/x+1/3*b^2*c^3*d^3*arctanh(c*x)-1/3*b*c*d^3*(a+b*arctanh(c*x))/x^2-3*b*c^2*d^3*(a+b*arctanh(c*x))/x+29/6*c^3*d^3*(a+b*arctanh(c*x))^2-1/3*d^3*(a+b*arctanh(c*x))^2/x^3-3/2*c*d^3*(a+b*arctanh(c*x))^2/x^2-3*c^2*d^3*(a+b*arctanh(c*x))^2/x+2*c^3*d^3*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))+3*b^2*c^3*d^3*log(x)-3/2*b^2*c^3*d^3*log(1-c^2*x^2)+20/3*b*c^3*d^3*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))-b*c^3*d^3*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))+b*c^3*d^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))-10/3*b^2*c^3*d^3*polylog(2,-1+2/(1+c*x))+1/2*b^2*c^3*d^3*polylog(3,1+(-2)/(1-c*x))-1/2*b^2*c^3*d^3*polylog(3,-1+2/(1-c*x))],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x^5,x,18,-1/12*b^2*c^2*d^3/x^2-b^2*c^3*d^3/x+b^2*c^4*d^3*arctanh(c*x)-1/6*b*c*d^3*(a+b*arctanh(c*x))/x^3-b*c^2*d^3*(a+b*arctanh(c*x))/x^2-7/2*b*c^3*d^3*(a+b*arctanh(c*x))/x-1/4*d^3*(1+c*x)^4*(a+b*arctanh(c*x))^2/x^4+4*a*b*c^4*d^3*log(x)+11/3*b^2*c^4*d^3*log(x)+4*b*c^4*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))-11/6*b^2*c^4*d^3*log(1-c^2*x^2)-2*b^2*c^4*d^3*polylog(2,-c*x)+2*b^2*c^4*d^3*polylog(2,c*x)+2*b^2*c^4*d^3*polylog(2,1+(-2)/(1-c*x))],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x^6,x,22,-1/30*b^2*c^2*d^3/x^3-1/4*b^2*c^3*d^3/x^2-13/10*b^2*c^4*d^3/x+13/10*b^2*c^5*d^3*arctanh(c*x)-1/10*b*c*d^3*(a+b*arctanh(c*x))/x^4-1/2*b*c^2*d^3*(a+b*arctanh(c*x))/x^3-6/5*b*c^3*d^3*(a+b*arctanh(c*x))/x^2-5/2*b*c^4*d^3*(a+b*arctanh(c*x))/x-1/5*d^3*(1+c*x)^4*(a+b*arctanh(c*x))^2/x^5+1/20*c*d^3*(1+c*x)^4*(a+b*arctanh(c*x))^2/x^4+12/5*a*b*c^5*d^3*log(x)+3*b^2*c^5*d^3*log(x)+12/5*b*c^5*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))-3/2*b^2*c^5*d^3*log(1-c^2*x^2)-6/5*b^2*c^5*d^3*polylog(2,-c*x)+6/5*b^2*c^5*d^3*polylog(2,c*x)+6/5*b^2*c^5*d^3*polylog(2,1+(-2)/(1-c*x))],
[(d+c*d*x)^3*(a+b*arctanh(c*x))^2/x^7,x,29,-1/60*b^2*c^2*d^3/x^4-1/10*b^2*c^3*d^3/x^3-61/180*b^2*c^4*d^3/x^2-37/30*b^2*c^5*d^3/x+37/30*b^2*c^6*d^3*arctanh(c*x)-1/15*b*c*d^3*(a+b*arctanh(c*x))/x^5-3/10*b*c^2*d^3*(a+b*arctanh(c*x))/x^4-11/18*b*c^3*d^3*(a+b*arctanh(c*x))/x^3-14/15*b*c^4*d^3*(a+b*arctanh(c*x))/x^2-11/6*b*c^5*d^3*(a+b*arctanh(c*x))/x-1/6*d^3*(a+b*arctanh(c*x))^2/x^6-3/5*c*d^3*(a+b*arctanh(c*x))^2/x^5-3/4*c^2*d^3*(a+b*arctanh(c*x))^2/x^4-1/3*c^3*d^3*(a+b*arctanh(c*x))^2/x^3+28/15*a*b*c^6*d^3*log(x)+113/45*b^2*c^6*d^3*log(x)+37/20*b*c^6*d^3*(a+b*arctanh(c*x))*log(2/(1-c*x))+1/60*b*c^6*d^3*(a+b*arctanh(c*x))*log(2/(1+c*x))-113/90*b^2*c^6*d^3*log(1-c^2*x^2)-14/15*b^2*c^6*d^3*polylog(2,-c*x)+14/15*b^2*c^6*d^3*polylog(2,c*x)+37/40*b^2*c^6*d^3*polylog(2,1+(-2)/(1-c*x))-1/120*b^2*c^6*d^3*polylog(2,1+(-2)/(1+c*x))],

# q<0
[x^3*(a+b*arctanh(c*x))^2/(d+c*d*x),x,26,-a*b*x/(c^3*d)+1/3*b^2*x/(c^3*d)-1/3*b^2*arctanh(c*x)/(c^4*d)-b^2*x*arctanh(c*x)/(c^3*d)+1/3*b*x^2*(a+b*arctanh(c*x))/(c^2*d)+11/6*(a+b*arctanh(c*x))^2/(c^4*d)+x*(a+b*arctanh(c*x))^2/(c^3*d)-1/2*x^2*(a+b*arctanh(c*x))^2/(c^2*d)+1/3*x^3*(a+b*arctanh(c*x))^2/(c*d)-8/3*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^4*d)+(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^4*d)-1/2*b^2*log(1-c^2*x^2)/(c^4*d)-4/3*b^2*polylog(2,1+(-2)/(1-c*x))/(c^4*d)-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^4*d)-1/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^4*d)],
[x^2*(a+b*arctanh(c*x))^2/(d+c*d*x),x,16,a*b*x/(c^2*d)+b^2*x*arctanh(c*x)/(c^2*d)-3/2*(a+b*arctanh(c*x))^2/(c^3*d)-x*(a+b*arctanh(c*x))^2/(c^2*d)+1/2*x^2*(a+b*arctanh(c*x))^2/(c*d)+2*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^3*d)-(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^3*d)+1/2*b^2*log(1-c^2*x^2)/(c^3*d)+b^2*polylog(2,1+(-2)/(1-c*x))/(c^3*d)+b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^3*d)+1/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^3*d)],
[x*(a+b*arctanh(c*x))^2/(d+c*d*x),x,9,(a+b*arctanh(c*x))^2/(c^2*d)+x*(a+b*arctanh(c*x))^2/(c*d)-2*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^2*d)+(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^2*d)-b^2*polylog(2,1+(-2)/(1-c*x))/(c^2*d)-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^2*d)-1/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^2*d)],
[(a+b*arctanh(c*x))^2/(d+c*d*x),x,3,-(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c*d)+b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c*d)+1/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c*d)],
[(a+b*arctanh(c*x))^2/(x*(d+c*d*x)),x,3,(a+b*arctanh(c*x))^2*log(2+(-2)/(1+c*x))/d-b*(a+b*arctanh(c*x))*polylog(2,-1+2/(1+c*x))/d-1/2*b^2*polylog(3,-1+2/(1+c*x))/d],
[(a+b*arctanh(c*x))^2/(x^2*(d+c*d*x)),x,8,c*(a+b*arctanh(c*x))^2/d-(a+b*arctanh(c*x))^2/(d*x)+2*b*c*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d-c*(a+b*arctanh(c*x))^2*log(2+(-2)/(1+c*x))/d-b^2*c*polylog(2,-1+2/(1+c*x))/d+b*c*(a+b*arctanh(c*x))*polylog(2,-1+2/(1+c*x))/d+1/2*b^2*c*polylog(3,-1+2/(1+c*x))/d],
[(a+b*arctanh(c*x))^2/(x^3*(d+c*d*x)),x,17,-b*c*(a+b*arctanh(c*x))/(d*x)-1/2*c^2*(a+b*arctanh(c*x))^2/d-1/2*(a+b*arctanh(c*x))^2/(d*x^2)+c*(a+b*arctanh(c*x))^2/(d*x)+b^2*c^2*log(x)/d-1/2*b^2*c^2*log(1-c^2*x^2)/d-2*b*c^2*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d+c^2*(a+b*arctanh(c*x))^2*log(2+(-2)/(1+c*x))/d+b^2*c^2*polylog(2,-1+2/(1+c*x))/d-b*c^2*(a+b*arctanh(c*x))*polylog(2,-1+2/(1+c*x))/d-1/2*b^2*c^2*polylog(3,-1+2/(1+c*x))/d],
[(a+b*arctanh(c*x))^2/(x^4*(d+c*d*x)),x,26,-1/3*b^2*c^2/(d*x)+1/3*b^2*c^3*arctanh(c*x)/d-1/3*b*c*(a+b*arctanh(c*x))/(d*x^2)+b*c^2*(a+b*arctanh(c*x))/(d*x)+5/6*c^3*(a+b*arctanh(c*x))^2/d-1/3*(a+b*arctanh(c*x))^2/(d*x^3)+1/2*c*(a+b*arctanh(c*x))^2/(d*x^2)-c^2*(a+b*arctanh(c*x))^2/(d*x)-b^2*c^3*log(x)/d+1/2*b^2*c^3*log(1-c^2*x^2)/d+8/3*b*c^3*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d-c^3*(a+b*arctanh(c*x))^2*log(2+(-2)/(1+c*x))/d-4/3*b^2*c^3*polylog(2,-1+2/(1+c*x))/d+b*c^3*(a+b*arctanh(c*x))*polylog(2,-1+2/(1+c*x))/d+1/2*b^2*c^3*polylog(3,-1+2/(1+c*x))/d],
[x^4*(a+b*arctanh(c*x))^2/(d+c*d*x)^2,x,33,-2*a*b*x/(c^4*d^2)+1/3*b^2*x/(c^4*d^2)-1/2*b^2/(c^5*d^2*(1+c*x))+1/6*b^2*arctanh(c*x)/(c^5*d^2)-2*b^2*x*arctanh(c*x)/(c^4*d^2)+1/3*b*x^2*(a+b*arctanh(c*x))/(c^3*d^2)-b*(a+b*arctanh(c*x))/(c^5*d^2*(1+c*x))+29/6*(a+b*arctanh(c*x))^2/(c^5*d^2)+3*x*(a+b*arctanh(c*x))^2/(c^4*d^2)-x^2*(a+b*arctanh(c*x))^2/(c^3*d^2)+1/3*x^3*(a+b*arctanh(c*x))^2/(c^2*d^2)-(a+b*arctanh(c*x))^2/(c^5*d^2*(1+c*x))-20/3*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^5*d^2)+4*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^5*d^2)-b^2*log(1-c^2*x^2)/(c^5*d^2)-10/3*b^2*polylog(2,1+(-2)/(1-c*x))/(c^5*d^2)-4*b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^5*d^2)-2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^5*d^2)],
[x^3*(a+b*arctanh(c*x))^2/(d+c*d*x)^2,x,24,a*b*x/(c^3*d^2)+1/2*b^2/(c^4*d^2*(1+c*x))-1/2*b^2*arctanh(c*x)/(c^4*d^2)+b^2*x*arctanh(c*x)/(c^3*d^2)+b*(a+b*arctanh(c*x))/(c^4*d^2*(1+c*x))-3*(a+b*arctanh(c*x))^2/(c^4*d^2)-2*x*(a+b*arctanh(c*x))^2/(c^3*d^2)+1/2*x^2*(a+b*arctanh(c*x))^2/(c^2*d^2)+(a+b*arctanh(c*x))^2/(c^4*d^2*(1+c*x))+4*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^4*d^2)-3*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^4*d^2)+1/2*b^2*log(1-c^2*x^2)/(c^4*d^2)+2*b^2*polylog(2,1+(-2)/(1-c*x))/(c^4*d^2)+3*b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^4*d^2)+3/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^4*d^2)],
[x^2*(a+b*arctanh(c*x))^2/(d+c*d*x)^2,x,18,-1/2*b^2/(c^3*d^2*(1+c*x))+1/2*b^2*arctanh(c*x)/(c^3*d^2)-b*(a+b*arctanh(c*x))/(c^3*d^2*(1+c*x))+3/2*(a+b*arctanh(c*x))^2/(c^3*d^2)+x*(a+b*arctanh(c*x))^2/(c^2*d^2)-(a+b*arctanh(c*x))^2/(c^3*d^2*(1+c*x))-2*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^3*d^2)+2*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^3*d^2)-b^2*polylog(2,1+(-2)/(1-c*x))/(c^3*d^2)-2*b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^3*d^2)-b^2*polylog(3,1+(-2)/(1+c*x))/(c^3*d^2)],
[x*(a+b*arctanh(c*x))^2/(d+c*d*x)^2,x,13,1/2*b^2/(c^2*d^2*(1+c*x))-1/2*b^2*arctanh(c*x)/(c^2*d^2)+b*(a+b*arctanh(c*x))/(c^2*d^2*(1+c*x))-1/2*(a+b*arctanh(c*x))^2/(c^2*d^2)+(a+b*arctanh(c*x))^2/(c^2*d^2*(1+c*x))-(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^2*d^2)+b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^2*d^2)+1/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^2*d^2)],
[(a+b*arctanh(c*x))^2/(d+c*d*x)^2,x,8,-1/2*b^2/(c*d^2*(1+c*x))+1/2*b^2*arctanh(c*x)/(c*d^2)-b*(a+b*arctanh(c*x))/(c*d^2*(1+c*x))+1/2*(a+b*arctanh(c*x))^2/(c*d^2)-(a+b*arctanh(c*x))^2/(c*d^2*(1+c*x))],
[(a+b*arctanh(c*x))^2/(x*(d+c*d*x)^2),x,19,1/2*b^2/(d^2*(1+c*x))-1/2*b^2*arctanh(c*x)/d^2+b*(a+b*arctanh(c*x))/(d^2*(1+c*x))-1/2*(a+b*arctanh(c*x))^2/d^2+(a+b*arctanh(c*x))^2/(d^2*(1+c*x))+2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d^2+(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d^2-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d^2+b*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d^2-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d^2+1/2*b^2*polylog(3,1+(-2)/(1-c*x))/d^2-1/2*b^2*polylog(3,-1+2/(1-c*x))/d^2-1/2*b^2*polylog(3,1+(-2)/(1+c*x))/d^2],
[(a+b*arctanh(c*x))^2/(x^2*(d+c*d*x)^2),x,23,-1/2*b^2*c/(d^2*(1+c*x))+1/2*b^2*c*arctanh(c*x)/d^2-b*c*(a+b*arctanh(c*x))/(d^2*(1+c*x))+3/2*c*(a+b*arctanh(c*x))^2/d^2-(a+b*arctanh(c*x))^2/(d^2*x)-c*(a+b*arctanh(c*x))^2/(d^2*(1+c*x))-4*c*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d^2-2*c*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d^2+2*b*c*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d^2+2*b*c*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d^2-2*b*c*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d^2+2*b*c*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d^2-b^2*c*polylog(2,-1+2/(1+c*x))/d^2-b^2*c*polylog(3,1+(-2)/(1-c*x))/d^2+b^2*c*polylog(3,-1+2/(1-c*x))/d^2+b^2*c*polylog(3,1+(-2)/(1+c*x))/d^2],
[(a+b*arctanh(c*x))^2/(x^3*(d+c*d*x)^2),x,31,1/2*b^2*c^2/(d^2*(1+c*x))-1/2*b^2*c^2*arctanh(c*x)/d^2-b*c*(a+b*arctanh(c*x))/(d^2*x)+b*c^2*(a+b*arctanh(c*x))/(d^2*(1+c*x))-2*c^2*(a+b*arctanh(c*x))^2/d^2-1/2*(a+b*arctanh(c*x))^2/(d^2*x^2)+2*c*(a+b*arctanh(c*x))^2/(d^2*x)+c^2*(a+b*arctanh(c*x))^2/(d^2*(1+c*x))+6*c^2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d^2+b^2*c^2*log(x)/d^2+3*c^2*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d^2-1/2*b^2*c^2*log(1-c^2*x^2)/d^2-4*b*c^2*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d^2-3*b*c^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d^2+3*b*c^2*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d^2-3*b*c^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d^2+2*b^2*c^2*polylog(2,-1+2/(1+c*x))/d^2+3/2*b^2*c^2*polylog(3,1+(-2)/(1-c*x))/d^2-3/2*b^2*c^2*polylog(3,-1+2/(1-c*x))/d^2-3/2*b^2*c^2*polylog(3,1+(-2)/(1+c*x))/d^2],
[x^4*(a+b*arctanh(c*x))^2/(d+c*d*x)^3,x,37,a*b*x/(c^4*d^3)-1/16*b^2/(c^5*d^3*(1+c*x)^2)+29/16*b^2/(c^5*d^3*(1+c*x))-29/16*b^2*arctanh(c*x)/(c^5*d^3)+b^2*x*arctanh(c*x)/(c^4*d^3)-1/4*b*(a+b*arctanh(c*x))/(c^5*d^3*(1+c*x)^2)+15/4*b*(a+b*arctanh(c*x))/(c^5*d^3*(1+c*x))-43/8*(a+b*arctanh(c*x))^2/(c^5*d^3)-3*x*(a+b*arctanh(c*x))^2/(c^4*d^3)+1/2*x^2*(a+b*arctanh(c*x))^2/(c^3*d^3)-1/2*(a+b*arctanh(c*x))^2/(c^5*d^3*(1+c*x)^2)+4*(a+b*arctanh(c*x))^2/(c^5*d^3*(1+c*x))+6*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^5*d^3)-6*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^5*d^3)+1/2*b^2*log(1-c^2*x^2)/(c^5*d^3)+3*b^2*polylog(2,1+(-2)/(1-c*x))/(c^5*d^3)+6*b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^5*d^3)+3*b^2*polylog(3,1+(-2)/(1+c*x))/(c^5*d^3)],
[x^3*(a+b*arctanh(c*x))^2/(d+c*d*x)^3,x,31,1/16*b^2/(c^4*d^3*(1+c*x)^2)-21/16*b^2/(c^4*d^3*(1+c*x))+21/16*b^2*arctanh(c*x)/(c^4*d^3)+1/4*b*(a+b*arctanh(c*x))/(c^4*d^3*(1+c*x)^2)-11/4*b*(a+b*arctanh(c*x))/(c^4*d^3*(1+c*x))+19/8*(a+b*arctanh(c*x))^2/(c^4*d^3)+x*(a+b*arctanh(c*x))^2/(c^3*d^3)+1/2*(a+b*arctanh(c*x))^2/(c^4*d^3*(1+c*x)^2)-3*(a+b*arctanh(c*x))^2/(c^4*d^3*(1+c*x))-2*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c^4*d^3)+3*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^4*d^3)-b^2*polylog(2,1+(-2)/(1-c*x))/(c^4*d^3)-3*b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^4*d^3)-3/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^4*d^3)],
[x^2*(a+b*arctanh(c*x))^2/(d+c*d*x)^3,x,26,-1/16*b^2/(c^3*d^3*(1+c*x)^2)+13/16*b^2/(c^3*d^3*(1+c*x))-13/16*b^2*arctanh(c*x)/(c^3*d^3)-1/4*b*(a+b*arctanh(c*x))/(c^3*d^3*(1+c*x)^2)+7/4*b*(a+b*arctanh(c*x))/(c^3*d^3*(1+c*x))-7/8*(a+b*arctanh(c*x))^2/(c^3*d^3)-1/2*(a+b*arctanh(c*x))^2/(c^3*d^3*(1+c*x)^2)+2*(a+b*arctanh(c*x))^2/(c^3*d^3*(1+c*x))-(a+b*arctanh(c*x))^2*log(2/(1+c*x))/(c^3*d^3)+b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/(c^3*d^3)+1/2*b^2*polylog(3,1+(-2)/(1+c*x))/(c^3*d^3)],
[x*(a+b*arctanh(c*x))^2/(d+c*d*x)^3,x,13,1/16*b^2/(c^2*d^3*(1+c*x)^2)-5/16*b^2/(c^2*d^3*(1+c*x))+5/16*b^2*arctanh(c*x)/(c^2*d^3)+1/4*b*(a+b*arctanh(c*x))/(c^2*d^3*(1+c*x)^2)-3/4*b*(a+b*arctanh(c*x))/(c^2*d^3*(1+c*x))-1/8*(a+b*arctanh(c*x))^2/(c^2*d^3)+1/2*x^2*(a+b*arctanh(c*x))^2/(d^3*(1+c*x)^2)],
[(a+b*arctanh(c*x))^2/(d+c*d*x)^3,x,13,-1/16*b^2/(c*d^3*(1+c*x)^2)-3/16*b^2/(c*d^3*(1+c*x))+3/16*b^2*arctanh(c*x)/(c*d^3)-1/4*b*(a+b*arctanh(c*x))/(c*d^3*(1+c*x)^2)-1/4*b*(a+b*arctanh(c*x))/(c*d^3*(1+c*x))+1/8*(a+b*arctanh(c*x))^2/(c*d^3)-1/2*(a+b*arctanh(c*x))^2/(c*d^3*(1+c*x)^2)],
[(a+b*arctanh(c*x))^2/(x*(d+c*d*x)^3),x,32,1/16*b^2/(d^3*(1+c*x)^2)+11/16*b^2/(d^3*(1+c*x))-11/16*b^2*arctanh(c*x)/d^3+1/4*b*(a+b*arctanh(c*x))/(d^3*(1+c*x)^2)+5/4*b*(a+b*arctanh(c*x))/(d^3*(1+c*x))-5/8*(a+b*arctanh(c*x))^2/d^3+1/2*(a+b*arctanh(c*x))^2/(d^3*(1+c*x)^2)+(a+b*arctanh(c*x))^2/(d^3*(1+c*x))+2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d^3+(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d^3-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d^3+b*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d^3-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d^3+1/2*b^2*polylog(3,1+(-2)/(1-c*x))/d^3-1/2*b^2*polylog(3,-1+2/(1-c*x))/d^3-1/2*b^2*polylog(3,1+(-2)/(1+c*x))/d^3],
[(a+b*arctanh(c*x))^2/(x^2*(d+c*d*x)^3),x,36,-1/16*b^2*c/(d^3*(1+c*x)^2)-19/16*b^2*c/(d^3*(1+c*x))+19/16*b^2*c*arctanh(c*x)/d^3-1/4*b*c*(a+b*arctanh(c*x))/(d^3*(1+c*x)^2)-9/4*b*c*(a+b*arctanh(c*x))/(d^3*(1+c*x))+17/8*c*(a+b*arctanh(c*x))^2/d^3-(a+b*arctanh(c*x))^2/(d^3*x)-1/2*c*(a+b*arctanh(c*x))^2/(d^3*(1+c*x)^2)-2*c*(a+b*arctanh(c*x))^2/(d^3*(1+c*x))-6*c*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d^3-3*c*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d^3+2*b*c*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d^3+3*b*c*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d^3-3*b*c*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d^3+3*b*c*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d^3-b^2*c*polylog(2,-1+2/(1+c*x))/d^3-3/2*b^2*c*polylog(3,1+(-2)/(1-c*x))/d^3+3/2*b^2*c*polylog(3,-1+2/(1-c*x))/d^3+3/2*b^2*c*polylog(3,1+(-2)/(1+c*x))/d^3],
[(a+b*arctanh(c*x))^2/(1+c*x)^4,x,18,-1/54*b^2/(c*(1+c*x)^3)-5/144*b^2/(c*(1+c*x)^2)-11/144*b^2/(c*(1+c*x))+11/144*b^2*arctanh(c*x)/c-1/9*b*(a+b*arctanh(c*x))/(c*(1+c*x)^3)-1/12*b*(a+b*arctanh(c*x))/(c*(1+c*x)^2)-1/12*b*(a+b*arctanh(c*x))/(c*(1+c*x))+1/24*(a+b*arctanh(c*x))^2/c-1/3*(a+b*arctanh(c*x))^2/(c*(1+c*x)^3)],
[arctanh(a*x)^2/(c*x-a*c*x^2),x,4,arctanh(a*x)^2*log(2+(-2)/(1-a*x))/c+arctanh(a*x)*polylog(2,-1+2/(1-a*x))/c-1/2*polylog(3,-1+2/(1-a*x))/c],

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^3 when c^2 d^2 - e^2=0
[(1+c*x)^3*(a+b*arctanh(c*x))^3,x,26,3*a*b^2*x+1/4*b^3*x-1/4*b^3*arctanh(c*x)/c+3*b^3*x*arctanh(c*x)+1/4*b^2*c*x^2*(a+b*arctanh(c*x))+4*b*(a+b*arctanh(c*x))^2/c+21/4*b*x*(a+b*arctanh(c*x))^2+3/2*b*c*x^2*(a+b*arctanh(c*x))^2+1/4*b*c^2*x^3*(a+b*arctanh(c*x))^2+1/4*(1+c*x)^4*(a+b*arctanh(c*x))^3/c-11*b^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c-6*b*(a+b*arctanh(c*x))^2*log(2/(1-c*x))/c+3/2*b^3*log(1-c^2*x^2)/c-11/2*b^3*polylog(2,1+(-2)/(1-c*x))/c-6*b^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/c+3*b^3*polylog(3,1+(-2)/(1-c*x))/c],
[(1+c*x)^2*(a+b*arctanh(c*x))^3,x,17,a*b^2*x+b^3*x*arctanh(c*x)+5/2*b*(a+b*arctanh(c*x))^2/c+3*b*x*(a+b*arctanh(c*x))^2+1/2*b*c*x^2*(a+b*arctanh(c*x))^2+1/3*(1+c*x)^3*(a+b*arctanh(c*x))^3/c-6*b^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c-4*b*(a+b*arctanh(c*x))^2*log(2/(1-c*x))/c+1/2*b^3*log(1-c^2*x^2)/c-3*b^3*polylog(2,1+(-2)/(1-c*x))/c-4*b^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/c+2*b^3*polylog(3,1+(-2)/(1-c*x))/c],
[(1+c*x)*(a+b*arctanh(c*x))^3,x,11,3/2*b*(a+b*arctanh(c*x))^2/c+3/2*b*x*(a+b*arctanh(c*x))^2+1/2*(1+c*x)^2*(a+b*arctanh(c*x))^3/c-3*b^2*(a+b*arctanh(c*x))*log(2/(1-c*x))/c-3*b*(a+b*arctanh(c*x))^2*log(2/(1-c*x))/c-3/2*b^3*polylog(2,1+(-2)/(1-c*x))/c-3*b^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/c+3/2*b^3*polylog(3,1+(-2)/(1-c*x))/c],
[(a+b*arctanh(c*x))^3/(1+c*x),x,4,-(a+b*arctanh(c*x))^3*log(2/(1+c*x))/c+3/2*b*(a+b*arctanh(c*x))^2*polylog(2,1+(-2)/(1+c*x))/c+3/2*b^2*(a+b*arctanh(c*x))*polylog(3,1+(-2)/(1+c*x))/c+3/4*b^3*polylog(4,1+(-2)/(1+c*x))/c],
[(a+b*arctanh(c*x))^3/(1+c*x)^2,x,11,-3/4*b^3/(c*(1+c*x))+3/4*b^3*arctanh(c*x)/c-3/2*b^2*(a+b*arctanh(c*x))/(c*(1+c*x))+3/4*b*(a+b*arctanh(c*x))^2/c-3/2*b*(a+b*arctanh(c*x))^2/(c*(1+c*x))+1/2*(a+b*arctanh(c*x))^3/c-(a+b*arctanh(c*x))^3/(c*(1+c*x))],
[(a+b*arctanh(c*x))^3/(1+c*x)^3,x,24,-3/64*b^3/(c*(1+c*x)^2)-21/64*b^3/(c*(1+c*x))+21/64*b^3*arctanh(c*x)/c-3/16*b^2*(a+b*arctanh(c*x))/(c*(1+c*x)^2)-9/16*b^2*(a+b*arctanh(c*x))/(c*(1+c*x))+9/32*b*(a+b*arctanh(c*x))^2/c-3/8*b*(a+b*arctanh(c*x))^2/(c*(1+c*x)^2)-3/8*b*(a+b*arctanh(c*x))^2/(c*(1+c*x))+1/8*(a+b*arctanh(c*x))^3/c-1/2*(a+b*arctanh(c*x))^3/(c*(1+c*x)^2)],
[(a+b*arctanh(c*x))^3/(1+c*x)^4,x,42,-1/108*b^3/(c*(1+c*x)^3)-19/576*b^3/(c*(1+c*x)^2)-85/576*b^3/(c*(1+c*x))+85/576*b^3*arctanh(c*x)/c-1/18*b^2*(a+b*arctanh(c*x))/(c*(1+c*x)^3)-5/48*b^2*(a+b*arctanh(c*x))/(c*(1+c*x)^2)-11/48*b^2*(a+b*arctanh(c*x))/(c*(1+c*x))+11/96*b*(a+b*arctanh(c*x))^2/c-1/6*b*(a+b*arctanh(c*x))^2/(c*(1+c*x)^3)-1/8*b*(a+b*arctanh(c*x))^2/(c*(1+c*x)^2)-1/8*b*(a+b*arctanh(c*x))^2/(c*(1+c*x))+1/24*(a+b*arctanh(c*x))^3/c-1/3*(a+b*arctanh(c*x))^3/(c*(1+c*x)^3)],
[x^2*arctanh(a*x)^3/(c+a*c*x),x,19,3/2*arctanh(a*x)^2/(a^3*c)+3/2*x*arctanh(a*x)^2/(a^2*c)-3/2*arctanh(a*x)^3/(a^3*c)-x*arctanh(a*x)^3/(a^2*c)+1/2*x^2*arctanh(a*x)^3/(a*c)-3*arctanh(a*x)*log(2/(1-a*x))/(a^3*c)+3*arctanh(a*x)^2*log(2/(1-a*x))/(a^3*c)-arctanh(a*x)^3*log(2/(1+a*x))/(a^3*c)-3/2*polylog(2,1+(-2)/(1-a*x))/(a^3*c)+3*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/(a^3*c)+3/2*arctanh(a*x)^2*polylog(2,1+(-2)/(1+a*x))/(a^3*c)-3/2*polylog(3,1+(-2)/(1-a*x))/(a^3*c)+3/2*arctanh(a*x)*polylog(3,1+(-2)/(1+a*x))/(a^3*c)+3/4*polylog(4,1+(-2)/(1+a*x))/(a^3*c)],
[x*arctanh(a*x)^3/(c+a*c*x),x,10,arctanh(a*x)^3/(a^2*c)+x*arctanh(a*x)^3/(a*c)-3*arctanh(a*x)^2*log(2/(1-a*x))/(a^2*c)+arctanh(a*x)^3*log(2/(1+a*x))/(a^2*c)-3*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/(a^2*c)-3/2*arctanh(a*x)^2*polylog(2,1+(-2)/(1+a*x))/(a^2*c)+3/2*polylog(3,1+(-2)/(1-a*x))/(a^2*c)-3/2*arctanh(a*x)*polylog(3,1+(-2)/(1+a*x))/(a^2*c)-3/4*polylog(4,1+(-2)/(1+a*x))/(a^2*c)],
[arctanh(a*x)^3/(c+a*c*x),x,4,-arctanh(a*x)^3*log(2/(1+a*x))/(a*c)+3/2*arctanh(a*x)^2*polylog(2,1+(-2)/(1+a*x))/(a*c)+3/2*arctanh(a*x)*polylog(3,1+(-2)/(1+a*x))/(a*c)+3/4*polylog(4,1+(-2)/(1+a*x))/(a*c)],
[arctanh(a*x)^3/(x*(c+a*c*x)),x,4,arctanh(a*x)^3*log(2+(-2)/(1+a*x))/c-3/2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))/c-3/2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))/c-3/4*polylog(4,-1+2/(1+a*x))/c],
[arctanh(a*x)^3/(c*x+a*c*x^2),x,5,arctanh(a*x)^3*log(2+(-2)/(1+a*x))/c-3/2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))/c-3/2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))/c-3/4*polylog(4,-1+2/(1+a*x))/c],
[arctanh(a*x)^3/(x^2*(c+a*c*x)),x,10,a*arctanh(a*x)^3/c-arctanh(a*x)^3/(c*x)+3*a*arctanh(a*x)^2*log(2+(-2)/(1+a*x))/c-a*arctanh(a*x)^3*log(2+(-2)/(1+a*x))/c-3*a*arctanh(a*x)*polylog(2,-1+2/(1+a*x))/c+3/2*a*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))/c-3/2*a*polylog(3,-1+2/(1+a*x))/c+3/2*a*arctanh(a*x)*polylog(3,-1+2/(1+a*x))/c+3/4*a*polylog(4,-1+2/(1+a*x))/c],
[arctanh(a*x)^3/(x^3*(c+a*c*x)),x,18,3/2*a^2*arctanh(a*x)^2/c-3/2*a*arctanh(a*x)^2/(c*x)-1/2*a^2*arctanh(a*x)^3/c-1/2*arctanh(a*x)^3/(c*x^2)+a*arctanh(a*x)^3/(c*x)+3*a^2*arctanh(a*x)*log(2+(-2)/(1+a*x))/c-3*a^2*arctanh(a*x)^2*log(2+(-2)/(1+a*x))/c+a^2*arctanh(a*x)^3*log(2+(-2)/(1+a*x))/c-3/2*a^2*polylog(2,-1+2/(1+a*x))/c+3*a^2*arctanh(a*x)*polylog(2,-1+2/(1+a*x))/c-3/2*a^2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))/c+3/2*a^2*polylog(3,-1+2/(1+a*x))/c-3/2*a^2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))/c-3/4*a^2*polylog(4,-1+2/(1+a*x))/c],

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^4 when c^2 d^2 - e^2=0
[x^2*arctanh(a*x)^4/(c-a*c*x),x,21,-2*arctanh(a*x)^3/(a^3*c)-2*x*arctanh(a*x)^3/(a^2*c)-1/2*arctanh(a*x)^4/(a^3*c)-x*arctanh(a*x)^4/(a^2*c)-1/2*x^2*arctanh(a*x)^4/(a*c)+6*arctanh(a*x)^2*log(2/(1-a*x))/(a^3*c)+4*arctanh(a*x)^3*log(2/(1-a*x))/(a^3*c)+arctanh(a*x)^4*log(2/(1-a*x))/(a^3*c)+6*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/(a^3*c)+6*arctanh(a*x)^2*polylog(2,1+(-2)/(1-a*x))/(a^3*c)+2*arctanh(a*x)^3*polylog(2,1+(-2)/(1-a*x))/(a^3*c)-3*polylog(3,1+(-2)/(1-a*x))/(a^3*c)-6*arctanh(a*x)*polylog(3,1+(-2)/(1-a*x))/(a^3*c)-3*arctanh(a*x)^2*polylog(3,1+(-2)/(1-a*x))/(a^3*c)+3*polylog(4,1+(-2)/(1-a*x))/(a^3*c)+3*arctanh(a*x)*polylog(4,1+(-2)/(1-a*x))/(a^3*c)-3/2*polylog(5,1+(-2)/(1-a*x))/(a^3*c)],
[x*arctanh(a*x)^4/(c-a*c*x),x,12,-arctanh(a*x)^4/(a^2*c)-x*arctanh(a*x)^4/(a*c)+4*arctanh(a*x)^3*log(2/(1-a*x))/(a^2*c)+arctanh(a*x)^4*log(2/(1-a*x))/(a^2*c)+6*arctanh(a*x)^2*polylog(2,1+(-2)/(1-a*x))/(a^2*c)+2*arctanh(a*x)^3*polylog(2,1+(-2)/(1-a*x))/(a^2*c)-6*arctanh(a*x)*polylog(3,1+(-2)/(1-a*x))/(a^2*c)-3*arctanh(a*x)^2*polylog(3,1+(-2)/(1-a*x))/(a^2*c)+3*polylog(4,1+(-2)/(1-a*x))/(a^2*c)+3*arctanh(a*x)*polylog(4,1+(-2)/(1-a*x))/(a^2*c)-3/2*polylog(5,1+(-2)/(1-a*x))/(a^2*c)],
[arctanh(a*x)^4/(c-a*c*x),x,5,arctanh(a*x)^4*log(2/(1-a*x))/(a*c)+2*arctanh(a*x)^3*polylog(2,1+(-2)/(1-a*x))/(a*c)-3*arctanh(a*x)^2*polylog(3,1+(-2)/(1-a*x))/(a*c)+3*arctanh(a*x)*polylog(4,1+(-2)/(1-a*x))/(a*c)-3/2*polylog(5,1+(-2)/(1-a*x))/(a*c)],
[arctanh(a*x)^4/(x*(c-a*c*x)),x,5,arctanh(a*x)^4*log(2+(-2)/(1-a*x))/c+2*arctanh(a*x)^3*polylog(2,-1+2/(1-a*x))/c-3*arctanh(a*x)^2*polylog(3,-1+2/(1-a*x))/c+3*arctanh(a*x)*polylog(4,-1+2/(1-a*x))/c-3/2*polylog(5,-1+2/(1-a*x))/c],
[arctanh(a*x)^4/(c*x-a*c*x^2),x,6,arctanh(a*x)^4*log(2+(-2)/(1-a*x))/c+2*arctanh(a*x)^3*polylog(2,-1+2/(1-a*x))/c-3*arctanh(a*x)^2*polylog(3,-1+2/(1-a*x))/c+3*arctanh(a*x)*polylog(4,-1+2/(1-a*x))/c-3/2*polylog(5,-1+2/(1-a*x))/c],
[arctanh(a*x)^4/(x^2*(c-a*c*x)),x,12,a*arctanh(a*x)^4/c-arctanh(a*x)^4/(c*x)+a*arctanh(a*x)^4*log(2+(-2)/(1-a*x))/c+4*a*arctanh(a*x)^3*log(2+(-2)/(1+a*x))/c+2*a*arctanh(a*x)^3*polylog(2,-1+2/(1-a*x))/c-6*a*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))/c-3*a*arctanh(a*x)^2*polylog(3,-1+2/(1-a*x))/c-6*a*arctanh(a*x)*polylog(3,-1+2/(1+a*x))/c+3*a*arctanh(a*x)*polylog(4,-1+2/(1-a*x))/c-3*a*polylog(4,-1+2/(1+a*x))/c-3/2*a*polylog(5,-1+2/(1-a*x))/c],
[arctanh(a*x)^4/(x^3*(c-a*c*x)),x,21,2*a^2*arctanh(a*x)^3/c-2*a*arctanh(a*x)^3/(c*x)+3/2*a^2*arctanh(a*x)^4/c-1/2*arctanh(a*x)^4/(c*x^2)-a*arctanh(a*x)^4/(c*x)+a^2*arctanh(a*x)^4*log(2+(-2)/(1-a*x))/c+6*a^2*arctanh(a*x)^2*log(2+(-2)/(1+a*x))/c+4*a^2*arctanh(a*x)^3*log(2+(-2)/(1+a*x))/c+2*a^2*arctanh(a*x)^3*polylog(2,-1+2/(1-a*x))/c-6*a^2*arctanh(a*x)*polylog(2,-1+2/(1+a*x))/c-6*a^2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))/c-3*a^2*arctanh(a*x)^2*polylog(3,-1+2/(1-a*x))/c-3*a^2*polylog(3,-1+2/(1+a*x))/c-6*a^2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))/c+3*a^2*arctanh(a*x)*polylog(4,-1+2/(1-a*x))/c-3*a^2*polylog(4,-1+2/(1+a*x))/c-3/2*a^2*polylog(5,-1+2/(1-a*x))/c],

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^-1 when c^2 d^2 - e^2=0
[x/((c+a*c*x)*arctanh(a*x)),x,0,Unintegrable(x/((c+a*c*x)*arctanh(a*x)),x)],
[1/((c+a*c*x)*arctanh(a*x)),x,0,Unintegrable(1/((c+a*c*x)*arctanh(a*x)),x)],
[1/(x*(c+a*c*x)*arctanh(a*x)),x,0,Unintegrable(1/(x*(c+a*c*x)*arctanh(a*x)),x)],

# Integrands of the form x^m (d+e x)^q (a+b ArcTanh[c x])^-2 when c^2 d^2 - e^2=0
[x/((c+a*c*x)*arctanh(a*x)^2),x,0,Unintegrable(x/((c+a*c*x)*arctanh(a*x)^2),x)],
[1/((c+a*c*x)*arctanh(a*x)^2),x,0,Unintegrable(1/((c+a*c*x)*arctanh(a*x)^2),x)],
[1/(x*(c+a*c*x)*arctanh(a*x)^2),x,0,Unintegrable(1/(x*(c+a*c*x)*arctanh(a*x)^2),x)],

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

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

# p>0
[x^3*(a+b*arctanh(c*x))/(d+e*x),x,16,a*d^2*x/e^3-1/2*b*d*x/(c*e^2)+1/6*b*x^2/(c*e)+1/2*b*d*arctanh(c*x)/(c^2*e^2)+b*d^2*x*arctanh(c*x)/e^3-1/2*d*x^2*(a+b*arctanh(c*x))/e^2+1/3*x^3*(a+b*arctanh(c*x))/e+d^3*(a+b*arctanh(c*x))*log(2/(1+c*x))/e^4-d^3*(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^4+1/2*b*d^2*log(1-c^2*x^2)/(c*e^3)+1/6*b*log(1-c^2*x^2)/(c^3*e)-1/2*b*d^3*polylog(2,1+(-2)/(1+c*x))/e^4+1/2*b*d^3*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^4],
[x^2*(a+b*arctanh(c*x))/(d+e*x),x,12,-a*d*x/e^2+1/2*b*x/(c*e)-1/2*b*arctanh(c*x)/(c^2*e)-b*d*x*arctanh(c*x)/e^2+1/2*x^2*(a+b*arctanh(c*x))/e-d^2*(a+b*arctanh(c*x))*log(2/(1+c*x))/e^3+d^2*(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^3-1/2*b*d*log(1-c^2*x^2)/(c*e^2)+1/2*b*d^2*polylog(2,1+(-2)/(1+c*x))/e^3-1/2*b*d^2*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^3],
[x*(a+b*arctanh(c*x))/(d+e*x),x,9,a*x/e+b*x*arctanh(c*x)/e+d*(a+b*arctanh(c*x))*log(2/(1+c*x))/e^2-d*(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^2+1/2*b*log(1-c^2*x^2)/(c*e)-1/2*b*d*polylog(2,1+(-2)/(1+c*x))/e^2+1/2*b*d*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^2],
[(a+b*arctanh(c*x))/(d+e*x),x,4,-(a+b*arctanh(c*x))*log(2/(1+c*x))/e+(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e+1/2*b*polylog(2,1+(-2)/(1+c*x))/e-1/2*b*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e],
[(a+b*arctanh(c*x))/(x*(d+e*x)),x,7,a*log(x)/d+(a+b*arctanh(c*x))*log(2/(1+c*x))/d-(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d-1/2*b*polylog(2,-c*x)/d+1/2*b*polylog(2,c*x)/d-1/2*b*polylog(2,1+(-2)/(1+c*x))/d+1/2*b*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d],
[(a+b*arctanh(c*x))/(x^2*(d+e*x)),x,12,(-a-b*arctanh(c*x))/(d*x)+b*c*log(x)/d-a*e*log(x)/d^2-e*(a+b*arctanh(c*x))*log(2/(1+c*x))/d^2+e*(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^2-1/2*b*c*log(1-c^2*x^2)/d+1/2*b*e*polylog(2,-c*x)/d^2-1/2*b*e*polylog(2,c*x)/d^2+1/2*b*e*polylog(2,1+(-2)/(1+c*x))/d^2-1/2*b*e*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^2],
[(a+b*arctanh(c*x))/(x^3*(d+e*x)),x,15,-1/2*b*c/(d*x)+1/2*b*c^2*arctanh(c*x)/d+1/2*(-a-b*arctanh(c*x))/(d*x^2)+e*(a+b*arctanh(c*x))/(d^2*x)-b*c*e*log(x)/d^2+a*e^2*log(x)/d^3+e^2*(a+b*arctanh(c*x))*log(2/(1+c*x))/d^3-e^2*(a+b*arctanh(c*x))*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^3+1/2*b*c*e*log(1-c^2*x^2)/d^2-1/2*b*e^2*polylog(2,-c*x)/d^3+1/2*b*e^2*polylog(2,c*x)/d^3-1/2*b*e^2*polylog(2,1+(-2)/(1+c*x))/d^3+1/2*b*e^2*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^3],
[x^2*(a+b*arctanh(c*x))^2/(d+e*x),x,14,a*b*x/(c*e)+b^2*x*arctanh(c*x)/(c*e)-d*(a+b*arctanh(c*x))^2/(c*e^2)-1/2*(a+b*arctanh(c*x))^2/(c^2*e)-d*x*(a+b*arctanh(c*x))^2/e^2+1/2*x^2*(a+b*arctanh(c*x))^2/e+2*b*d*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c*e^2)-d^2*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/e^3+d^2*(a+b*arctanh(c*x))^2*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^3+1/2*b^2*log(1-c^2*x^2)/(c^2*e)+b^2*d*polylog(2,1+(-2)/(1-c*x))/(c*e^2)+b*d^2*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/e^3-b*d^2*(a+b*arctanh(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^3+1/2*b^2*d^2*polylog(3,1+(-2)/(1+c*x))/e^3-1/2*b^2*d^2*polylog(3,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^3],
[x*(a+b*arctanh(c*x))^2/(d+e*x),x,8,(a+b*arctanh(c*x))^2/(c*e)+x*(a+b*arctanh(c*x))^2/e-2*b*(a+b*arctanh(c*x))*log(2/(1-c*x))/(c*e)+d*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/e^2-d*(a+b*arctanh(c*x))^2*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^2-b^2*polylog(2,1+(-2)/(1-c*x))/(c*e)-b*d*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/e^2+b*d*(a+b*arctanh(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^2-1/2*b^2*d*polylog(3,1+(-2)/(1+c*x))/e^2+1/2*b^2*d*polylog(3,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e^2],
[(a+b*arctanh(c*x))^2/(d+e*x),x,1,-(a+b*arctanh(c*x))^2*log(2/(1+c*x))/e+(a+b*arctanh(c*x))^2*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e+b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/e-b*(a+b*arctanh(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e+1/2*b^2*polylog(3,1+(-2)/(1+c*x))/e-1/2*b^2*polylog(3,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/e],
[(a+b*arctanh(c*x))^2/(x*(d+e*x)),x,9,2*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d+(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d-(a+b*arctanh(c*x))^2*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d+b*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d-b*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d+b*(a+b*arctanh(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d+1/2*b^2*polylog(3,1+(-2)/(1-c*x))/d-1/2*b^2*polylog(3,-1+2/(1-c*x))/d-1/2*b^2*polylog(3,1+(-2)/(1+c*x))/d+1/2*b^2*polylog(3,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d],
[(a+b*arctanh(c*x))^2/(x^2*(d+e*x)),x,13,c*(a+b*arctanh(c*x))^2/d-(a+b*arctanh(c*x))^2/(d*x)-2*e*(a+b*arctanh(c*x))^2*arctanh(1+(-2)/(1-c*x))/d^2-e*(a+b*arctanh(c*x))^2*log(2/(1+c*x))/d^2+e*(a+b*arctanh(c*x))^2*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^2+2*b*c*(a+b*arctanh(c*x))*log(2+(-2)/(1+c*x))/d+b*e*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/d^2-b*e*(a+b*arctanh(c*x))*polylog(2,-1+2/(1-c*x))/d^2+b*e*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1+c*x))/d^2-b^2*c*polylog(2,-1+2/(1+c*x))/d-b*e*(a+b*arctanh(c*x))*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^2-1/2*b^2*e*polylog(3,1+(-2)/(1-c*x))/d^2+1/2*b^2*e*polylog(3,-1+2/(1-c*x))/d^2+1/2*b^2*e*polylog(3,1+(-2)/(1+c*x))/d^2-1/2*b^2*e*polylog(3,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d^2],
[arctanh(c*x)^2/(x*(d+e*x)),x,9,2*arctanh(c*x)^2*arctanh(1+(-2)/(1-c*x))/d+arctanh(c*x)^2*log(2/(1+c*x))/d-arctanh(c*x)^2*log(2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d-arctanh(c*x)*polylog(2,1+(-2)/(1-c*x))/d+arctanh(c*x)*polylog(2,-1+2/(1-c*x))/d-arctanh(c*x)*polylog(2,1+(-2)/(1+c*x))/d+arctanh(c*x)*polylog(2,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d+1/2*polylog(3,1+(-2)/(1-c*x))/d-1/2*polylog(3,-1+2/(1-c*x))/d-1/2*polylog(3,1+(-2)/(1+c*x))/d+1/2*polylog(3,1-2*c*(d+e*x)/((c*d+e)*(1+c*x)))/d],

# p<0
[1/((d+e*x)*(a+b*arctan(c*x))),x,0,Unintegrable(1/((d+e*x)*(a+b*arctan(c*x))),x)],

# Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTanh[c x])^p with c^2 d+e=0

# Integrands of the form x^m (d+e x^2)^q (a+b ArcTanh[c x])^p

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

# p>0
[x^4*(1-a^2*x^2)*arctanh(a*x),x,9,1/35*x^2/a^3+1/70*x^4/a-1/42*a*x^6+1/5*x^5*arctanh(a*x)-1/7*a^2*x^7*arctanh(a*x)+1/35*log(1-a^2*x^2)/a^5],
[x^3*(1-a^2*x^2)*arctanh(a*x),x,9,1/12*x/a^3+1/36*x^3/a-1/30*a*x^5-1/12*arctanh(a*x)/a^4+1/4*x^4*arctanh(a*x)-1/6*a^2*x^6*arctanh(a*x)],
[x^2*(1-a^2*x^2)*arctanh(a*x),x,9,1/15*x^2/a-1/20*a*x^4+1/3*x^3*arctanh(a*x)-1/5*a^2*x^5*arctanh(a*x)+1/15*log(1-a^2*x^2)/a^3],
[x*(1-a^2*x^2)*arctanh(a*x),x,2,1/4*x/a-1/12*a*x^3-1/4*(1-a^2*x^2)^2*arctanh(a*x)/a^2],
[(1-a^2*x^2)*arctanh(a*x),x,3,1/6*(1-a^2*x^2)/a+2/3*x*arctanh(a*x)+1/3*x*(1-a^2*x^2)*arctanh(a*x)+1/3*log(1-a^2*x^2)/a],
[(1-a^2*x^2)*arctanh(a*x)/x,x,5,-1/2*a*x+1/2*arctanh(a*x)-1/2*a^2*x^2*arctanh(a*x)-1/2*polylog(2,-a*x)+1/2*polylog(2,a*x)],
[(1-a^2*x^2)*arctanh(a*x)/x^2,x,8,-arctanh(a*x)/x-a^2*x*arctanh(a*x)+a*log(x)-a*log(1-a^2*x^2)],
[(1-a^2*x^2)*arctanh(a*x)/x^3,x,5,-1/2*a/x+1/2*a^2*arctanh(a*x)-1/2*arctanh(a*x)/x^2+1/2*a^2*polylog(2,-a*x)-1/2*a^2*polylog(2,a*x)],
[(1-a^2*x^2)*arctanh(a*x)/x^4,x,10,-1/6*a/x^2-1/3*arctanh(a*x)/x^3+a^2*arctanh(a*x)/x-2/3*a^3*log(x)+1/3*a^3*log(1-a^2*x^2)],
[(1-a^2*x^2)*arctanh(a*x)/x^5,x,3,-1/12*a/x^3+1/4*a^3/x-1/4*(1-a^2*x^2)^2*arctanh(a*x)/x^4],
[(1-a^2*x^2)*arctanh(a*x)/x^6,x,9,-1/20*a/x^4+1/15*a^3/x^2-1/5*arctanh(a*x)/x^5+1/3*a^2*arctanh(a*x)/x^3-2/15*a^5*log(x)+1/15*a^5*log(1-a^2*x^2)],
[x^4*(1-a^2*x^2)*arctanh(a*x)^2,x,34,4/105*x/a^4-2/315*x^3/a^2-1/105*x^5-4/105*arctanh(a*x)/a^5+2/35*x^2*arctanh(a*x)/a^3+1/35*x^4*arctanh(a*x)/a-1/21*a*x^6*arctanh(a*x)+2/35*arctanh(a*x)^2/a^5+1/5*x^5*arctanh(a*x)^2-1/7*a^2*x^7*arctanh(a*x)^2-4/35*arctanh(a*x)*log(2/(1-a*x))/a^5-2/35*polylog(2,1+(-2)/(1-a*x))/a^5],
[x^3*(1-a^2*x^2)*arctanh(a*x)^2,x,26,-1/180*x^2/a^2-1/60*x^4+1/6*x*arctanh(a*x)/a^3+1/18*x^3*arctanh(a*x)/a-1/15*a*x^5*arctanh(a*x)-1/12*arctanh(a*x)^2/a^4+1/4*x^4*arctanh(a*x)^2-1/6*a^2*x^6*arctanh(a*x)^2+7/90*log(1-a^2*x^2)/a^4],
[x^2*(1-a^2*x^2)*arctanh(a*x)^2,x,24,1/30*x/a^2-1/30*x^3-1/30*arctanh(a*x)/a^3+2/15*x^2*arctanh(a*x)/a-1/10*a*x^4*arctanh(a*x)+2/15*arctanh(a*x)^2/a^3+1/3*x^3*arctanh(a*x)^2-1/5*a^2*x^5*arctanh(a*x)^2-4/15*arctanh(a*x)*log(2/(1-a*x))/a^3-2/15*polylog(2,1+(-2)/(1-a*x))/a^3],
[x*(1-a^2*x^2)*arctanh(a*x)^2,x,4,1/12*(1-a^2*x^2)/a^2+1/3*x*arctanh(a*x)/a+1/6*x*(1-a^2*x^2)*arctanh(a*x)/a-1/4*(1-a^2*x^2)^2*arctanh(a*x)^2/a^2+1/6*log(1-a^2*x^2)/a^2],
[(1-a^2*x^2)*arctanh(a*x)^2,x,7,-1/3*x+1/3*(1-a^2*x^2)*arctanh(a*x)/a+2/3*arctanh(a*x)^2/a+2/3*x*arctanh(a*x)^2+1/3*x*(1-a^2*x^2)*arctanh(a*x)^2-4/3*arctanh(a*x)*log(2/(1-a*x))/a-2/3*polylog(2,1+(-2)/(1-a*x))/a],
[(1-a^2*x^2)*arctanh(a*x)^2/x,x,12,-a*x*arctanh(a*x)+1/2*arctanh(a*x)^2-1/2*a^2*x^2*arctanh(a*x)^2+2*arctanh(a*x)^2*arctanh(1+(-2)/(1-a*x))-1/2*log(1-a^2*x^2)-arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))+arctanh(a*x)*polylog(2,-1+2/(1-a*x))+1/2*polylog(3,1+(-2)/(1-a*x))-1/2*polylog(3,-1+2/(1-a*x))],
[(1-a^2*x^2)*arctanh(a*x)^2/x^2,x,10,-arctanh(a*x)^2/x-a^2*x*arctanh(a*x)^2+2*a*arctanh(a*x)*log(2/(1-a*x))+2*a*arctanh(a*x)*log(2+(-2)/(1+a*x))+a*polylog(2,1+(-2)/(1-a*x))-a*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)*arctanh(a*x)^2/x^3,x,15,-a*arctanh(a*x)/x+1/2*a^2*arctanh(a*x)^2-1/2*arctanh(a*x)^2/x^2-2*a^2*arctanh(a*x)^2*arctanh(1+(-2)/(1-a*x))+a^2*log(x)-1/2*a^2*log(1-a^2*x^2)+a^2*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))-a^2*arctanh(a*x)*polylog(2,-1+2/(1-a*x))-1/2*a^2*polylog(3,1+(-2)/(1-a*x))+1/2*a^2*polylog(3,-1+2/(1-a*x))],
[(1-a^2*x^2)*arctanh(a*x)^2/x^4,x,13,-1/3*a^2/x+1/3*a^3*arctanh(a*x)-1/3*a*arctanh(a*x)/x^2-2/3*a^3*arctanh(a*x)^2-1/3*arctanh(a*x)^2/x^3+a^2*arctanh(a*x)^2/x-4/3*a^3*arctanh(a*x)*log(2+(-2)/(1+a*x))+2/3*a^3*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)*arctanh(a*x)^2/x^5,x,11,-1/12*a^2/x^2-1/6*a*arctanh(a*x)/x^3+1/2*a^3*arctanh(a*x)/x-1/4*(1-a^2*x^2)^2*arctanh(a*x)^2/x^4-1/3*a^4*log(x)+1/6*a^4*log(1-a^2*x^2)],
[(1-a^2*x^2)*arctanh(a*x)^2/x^6,x,22,-1/30*a^2/x^3+1/30*a^4/x-1/30*a^5*arctanh(a*x)-1/10*a*arctanh(a*x)/x^4+2/15*a^3*arctanh(a*x)/x^2-2/15*a^5*arctanh(a*x)^2-1/5*arctanh(a*x)^2/x^5+1/3*a^2*arctanh(a*x)^2/x^3-4/15*a^5*arctanh(a*x)*log(2+(-2)/(1+a*x))+2/15*a^5*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)*arctanh(a*x)^3,x,8,-x*arctanh(a*x)+1/2*(1-a^2*x^2)*arctanh(a*x)^2/a+2/3*arctanh(a*x)^3/a+2/3*x*arctanh(a*x)^3+1/3*x*(1-a^2*x^2)*arctanh(a*x)^3-2*arctanh(a*x)^2*log(2/(1-a*x))/a-1/2*log(1-a^2*x^2)/a-2*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a+polylog(3,1+(-2)/(1-a*x))/a],
[x*arctanh(x/sqrt(2))/(1-x^2),x,10,-1/2*arctanh(x/sqrt(2))*log(-4*(1-x)/((2-sqrt(2))*(x+sqrt(2))))+arctanh(x/sqrt(2))*log(2*sqrt(2)/(x+sqrt(2)))-1/2*arctanh(x/sqrt(2))*log(4*(1+x)/((2+sqrt(2))*(x+sqrt(2))))+1/4*polylog(2,1+4*(1-x)/((2-sqrt(2))*(x+sqrt(2))))-1/2*polylog(2,1-2*sqrt(2)/(x+sqrt(2)))+1/4*polylog(2,1-4*(1+x)/((2+sqrt(2))*(x+sqrt(2))))],

# p<0
[x*(1-a^2*x^2)/arctanh(a*x),x,0,Unintegrable(x*(1-a^2*x^2)/arctanh(a*x),x)],
[(1-a^2*x^2)/arctanh(a*x),x,0,Unintegrable((1-a^2*x^2)/arctanh(a*x),x)],
[(1-a^2*x^2)/(x*arctanh(a*x)),x,0,Unintegrable((1-a^2*x^2)/(x*arctanh(a*x)),x)],
[x*(1-a^2*x^2)/arctanh(a*x)^2,x,0,Unintegrable(x*(1-a^2*x^2)/arctanh(a*x)^2,x)],
[(1-a^2*x^2)/arctanh(a*x)^2,x,0,Unintegrable((1-a^2*x^2)/arctanh(a*x)^2,x)],
[(1-a^2*x^2)/(x*arctanh(a*x)^2),x,0,Unintegrable((1-a^2*x^2)/(x*arctanh(a*x)^2),x)],
[(1-a^2*x^2)/arctanh(a*x)^3,x,0,Unintegrable((1-a^2*x^2)/arctanh(a*x)^3,x)],

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

# p>0
[x^4*(1-a^2*x^2)^2*arctanh(a*x),x,14,4/315*x^2/a^3+2/315*x^4/a-11/378*a*x^6+1/72*a^3*x^8+1/5*x^5*arctanh(a*x)-2/7*a^2*x^7*arctanh(a*x)+1/9*a^4*x^9*arctanh(a*x)+4/315*log(1-a^2*x^2)/a^5],
[x^3*(1-a^2*x^2)^2*arctanh(a*x),x,14,1/24*x/a^3+1/72*x^3/a-1/24*a*x^5+1/56*a^3*x^7-1/24*arctanh(a*x)/a^4+1/4*x^4*arctanh(a*x)-1/3*a^2*x^6*arctanh(a*x)+1/8*a^4*x^8*arctanh(a*x)],
[x^2*(1-a^2*x^2)^2*arctanh(a*x),x,14,4/105*x^2/a-9/140*a*x^4+1/42*a^3*x^6+1/3*x^3*arctanh(a*x)-2/5*a^2*x^5*arctanh(a*x)+1/7*a^4*x^7*arctanh(a*x)+4/105*log(1-a^2*x^2)/a^3],
[x*(1-a^2*x^2)^2*arctanh(a*x),x,3,1/6*x/a-1/9*a*x^3+1/30*a^3*x^5-1/6*(1-a^2*x^2)^3*arctanh(a*x)/a^2],
[(1-a^2*x^2)^2*arctanh(a*x),x,4,2/15*(1-a^2*x^2)/a+1/20*(1-a^2*x^2)^2/a+8/15*x*arctanh(a*x)+4/15*x*(1-a^2*x^2)*arctanh(a*x)+1/5*x*(1-a^2*x^2)^2*arctanh(a*x)+4/15*log(1-a^2*x^2)/a],
[(1-a^2*x^2)^2*arctanh(a*x)/x,x,10,-3/4*a*x+1/12*a^3*x^3+3/4*arctanh(a*x)-a^2*x^2*arctanh(a*x)+1/4*a^4*x^4*arctanh(a*x)-1/2*polylog(2,-a*x)+1/2*polylog(2,a*x)],
[(1-a^2*x^2)^2*arctanh(a*x)/x^2,x,13,1/6*a^3*x^2-arctanh(a*x)/x-2*a^2*x*arctanh(a*x)+1/3*a^4*x^3*arctanh(a*x)+a*log(x)-4/3*a*log(1-a^2*x^2)],
[(1-a^2*x^2)^2*arctanh(a*x)/x^3,x,9,-1/2*a/x+1/2*a^3*x-1/2*arctanh(a*x)/x^2+1/2*a^4*x^2*arctanh(a*x)+a^2*polylog(2,-a*x)-a^2*polylog(2,a*x)],
[(1-a^2*x^2)^2*arctanh(a*x)/x^4,x,13,-1/6*a/x^2-1/3*arctanh(a*x)/x^3+2*a^2*arctanh(a*x)/x+a^4*x*arctanh(a*x)-5/3*a^3*log(x)+4/3*a^3*log(1-a^2*x^2)],
[(1-a^2*x^2)^2*arctanh(a*x)/x^5,x,10,-1/12*a/x^3+3/4*a^3/x-3/4*a^4*arctanh(a*x)-1/4*arctanh(a*x)/x^4+a^2*arctanh(a*x)/x^2-1/2*a^4*polylog(2,-a*x)+1/2*a^4*polylog(2,a*x)],
[(1-a^2*x^2)^2*arctanh(a*x)/x^6,x,15,-1/20*a/x^4+7/30*a^3/x^2-1/5*arctanh(a*x)/x^5+2/3*a^2*arctanh(a*x)/x^3-a^4*arctanh(a*x)/x+8/15*a^5*log(x)-4/15*a^5*log(1-a^2*x^2)],
[x^4*(1-a^2*x^2)^2*arctanh(a*x)^2,x,59,29/3780*x/a^4-67/11340*x^3/a^2-23/3780*x^5+1/252*a^2*x^7-29/3780*arctanh(a*x)/a^5+8/315*x^2*arctanh(a*x)/a^3+4/315*x^4*arctanh(a*x)/a-11/189*a*x^6*arctanh(a*x)+1/36*a^3*x^8*arctanh(a*x)+8/315*arctanh(a*x)^2/a^5+1/5*x^5*arctanh(a*x)^2-2/7*a^2*x^7*arctanh(a*x)^2+1/9*a^4*x^9*arctanh(a*x)^2-16/315*arctanh(a*x)*log(2/(1-a*x))/a^5-8/315*polylog(2,1+(-2)/(1-a*x))/a^5],
[x^3*(1-a^2*x^2)^2*arctanh(a*x)^2,x,47,-5/504*x^2/a^2-1/84*x^4+1/168*a^2*x^6+1/12*x*arctanh(a*x)/a^3+1/36*x^3*arctanh(a*x)/a-1/12*a*x^5*arctanh(a*x)+1/28*a^3*x^7*arctanh(a*x)-1/24*arctanh(a*x)^2/a^4+1/4*x^4*arctanh(a*x)^2-1/3*a^2*x^6*arctanh(a*x)^2+1/8*a^4*x^8*arctanh(a*x)^2+2/63*log(1-a^2*x^2)/a^4],
[x^2*(1-a^2*x^2)^2*arctanh(a*x)^2,x,44,-1/210*x/a^2-17/630*x^3+1/105*a^2*x^5+1/210*arctanh(a*x)/a^3+8/105*x^2*arctanh(a*x)/a-9/70*a*x^4*arctanh(a*x)+1/21*a^3*x^6*arctanh(a*x)+8/105*arctanh(a*x)^2/a^3+1/3*x^3*arctanh(a*x)^2-2/5*a^2*x^5*arctanh(a*x)^2+1/7*a^4*x^7*arctanh(a*x)^2-16/105*arctanh(a*x)*log(2/(1-a*x))/a^3-8/105*polylog(2,1+(-2)/(1-a*x))/a^3],
[x*(1-a^2*x^2)^2*arctanh(a*x)^2,x,5,2/45*(1-a^2*x^2)/a^2+1/60*(1-a^2*x^2)^2/a^2+8/45*x*arctanh(a*x)/a+4/45*x*(1-a^2*x^2)*arctanh(a*x)/a+1/15*x*(1-a^2*x^2)^2*arctanh(a*x)/a-1/6*(1-a^2*x^2)^3*arctanh(a*x)^2/a^2+4/45*log(1-a^2*x^2)/a^2],
[(1-a^2*x^2)^2*arctanh(a*x)^2,x,9,-11/30*x+1/30*a^2*x^3+4/15*(1-a^2*x^2)*arctanh(a*x)/a+1/10*(1-a^2*x^2)^2*arctanh(a*x)/a+8/15*arctanh(a*x)^2/a+8/15*x*arctanh(a*x)^2+4/15*x*(1-a^2*x^2)*arctanh(a*x)^2+1/5*x*(1-a^2*x^2)^2*arctanh(a*x)^2-16/15*arctanh(a*x)*log(2/(1-a*x))/a-8/15*polylog(2,1+(-2)/(1-a*x))/a],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x,x,23,1/12*a^2*x^2-3/2*a*x*arctanh(a*x)+1/6*a^3*x^3*arctanh(a*x)+3/4*arctanh(a*x)^2-a^2*x^2*arctanh(a*x)^2+1/4*a^4*x^4*arctanh(a*x)^2+2*arctanh(a*x)^2*arctanh(1+(-2)/(1-a*x))-2/3*log(1-a^2*x^2)-arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))+arctanh(a*x)*polylog(2,-1+2/(1-a*x))+1/2*polylog(3,1+(-2)/(1-a*x))-1/2*polylog(3,-1+2/(1-a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^2,x,20,1/3*a^2*x-1/3*a*arctanh(a*x)+1/3*a^3*x^2*arctanh(a*x)-2/3*a*arctanh(a*x)^2-arctanh(a*x)^2/x-2*a^2*x*arctanh(a*x)^2+1/3*a^4*x^3*arctanh(a*x)^2+10/3*a*arctanh(a*x)*log(2/(1-a*x))+2*a*arctanh(a*x)*log(2+(-2)/(1+a*x))+5/3*a*polylog(2,1+(-2)/(1-a*x))-a*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^3,x,21,-a*arctanh(a*x)/x+a^3*x*arctanh(a*x)-1/2*arctanh(a*x)^2/x^2+1/2*a^4*x^2*arctanh(a*x)^2-4*a^2*arctanh(a*x)^2*arctanh(1+(-2)/(1-a*x))+a^2*log(x)+2*a^2*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))-2*a^2*arctanh(a*x)*polylog(2,-1+2/(1-a*x))-a^2*polylog(3,1+(-2)/(1-a*x))+a^2*polylog(3,-1+2/(1-a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^4,x,19,-1/3*a^2/x+1/3*a^3*arctanh(a*x)-1/3*a*arctanh(a*x)/x^2-2/3*a^3*arctanh(a*x)^2-1/3*arctanh(a*x)^2/x^3+2*a^2*arctanh(a*x)^2/x+a^4*x*arctanh(a*x)^2-2*a^3*arctanh(a*x)*log(2/(1-a*x))-10/3*a^3*arctanh(a*x)*log(2+(-2)/(1+a*x))-a^3*polylog(2,1+(-2)/(1-a*x))+5/3*a^3*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^5,x,29,-1/12*a^2/x^2-1/6*a*arctanh(a*x)/x^3+3/2*a^3*arctanh(a*x)/x-3/4*a^4*arctanh(a*x)^2-1/4*arctanh(a*x)^2/x^4+a^2*arctanh(a*x)^2/x^2+2*a^4*arctanh(a*x)^2*arctanh(1+(-2)/(1-a*x))-4/3*a^4*log(x)+2/3*a^4*log(1-a^2*x^2)-a^4*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))+a^4*arctanh(a*x)*polylog(2,-1+2/(1-a*x))+1/2*a^4*polylog(3,1+(-2)/(1-a*x))-1/2*a^4*polylog(3,-1+2/(1-a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^6,x,27,-1/30*a^2/x^3+11/30*a^4/x-11/30*a^5*arctanh(a*x)-1/10*a*arctanh(a*x)/x^4+7/15*a^3*arctanh(a*x)/x^2+8/15*a^5*arctanh(a*x)^2-1/5*arctanh(a*x)^2/x^5+2/3*a^2*arctanh(a*x)^2/x^3-a^4*arctanh(a*x)^2/x+16/15*a^5*arctanh(a*x)*log(2+(-2)/(1+a*x))-8/15*a^5*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^7,x,16,-1/60*a^2/x^4+7/90*a^4/x^2-1/15*a*arctanh(a*x)/x^5+2/9*a^3*arctanh(a*x)/x^3-1/3*a^5*arctanh(a*x)/x-1/6*(1-a^2*x^2)^3*arctanh(a*x)^2/x^6+8/45*a^6*log(x)-4/45*a^6*log(1-a^2*x^2)],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^8,x,42,-1/105*a^2/x^5+17/630*a^4/x^3+1/210*a^6/x-1/210*a^7*arctanh(a*x)-1/21*a*arctanh(a*x)/x^6+9/70*a^3*arctanh(a*x)/x^4-8/105*a^5*arctanh(a*x)/x^2+8/105*a^7*arctanh(a*x)^2-1/7*arctanh(a*x)^2/x^7+2/5*a^2*arctanh(a*x)^2/x^5-1/3*a^4*arctanh(a*x)^2/x^3+16/105*a^7*arctanh(a*x)*log(2+(-2)/(1+a*x))-8/105*a^7*polylog(2,-1+2/(1+a*x))],
[(1-a^2*x^2)^2*arctanh(a*x)^2/x^9,x,56,-1/168*a^2/x^6+1/84*a^4/x^4+5/504*a^6/x^2-1/28*a*arctanh(a*x)/x^7+1/12*a^3*arctanh(a*x)/x^5-1/36*a^5*arctanh(a*x)/x^3-1/12*a^7*arctanh(a*x)/x+1/24*a^8*arctanh(a*x)^2-1/8*arctanh(a*x)^2/x^8+1/3*a^2*arctanh(a*x)^2/x^6-1/4*a^4*arctanh(a*x)^2/x^4+4/63*a^8*log(x)-2/63*a^8*log(1-a^2*x^2)],
[(1-a^2*x^2)^2*arctanh(a*x)^3,x,12,1/20*(-1+a^2*x^2)/a-x*arctanh(a*x)-1/10*x*(1-a^2*x^2)*arctanh(a*x)+2/5*(1-a^2*x^2)*arctanh(a*x)^2/a+3/20*(1-a^2*x^2)^2*arctanh(a*x)^2/a+8/15*arctanh(a*x)^3/a+8/15*x*arctanh(a*x)^3+4/15*x*(1-a^2*x^2)*arctanh(a*x)^3+1/5*x*(1-a^2*x^2)^2*arctanh(a*x)^3-8/5*arctanh(a*x)^2*log(2/(1-a*x))/a-1/2*log(1-a^2*x^2)/a-8/5*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a+4/5*polylog(3,1+(-2)/(1-a*x))/a],

# p<0
[x*(1-a^2*x^2)^2/arctanh(a*x),x,0,Unintegrable(x*(1-a^2*x^2)^2/arctanh(a*x),x)],
[(1-a^2*x^2)^2/arctanh(a*x),x,0,Unintegrable((1-a^2*x^2)^2/arctanh(a*x),x)],
[(1-a^2*x^2)^2/(x*arctanh(a*x)),x,0,Unintegrable((1-a^2*x^2)^2/(x*arctanh(a*x)),x)],
[x*(1-a^2*x^2)^2/arctanh(a*x)^2,x,0,Unintegrable(x*(1-a^2*x^2)^2/arctanh(a*x)^2,x)],
[(1-a^2*x^2)^2/arctanh(a*x)^2,x,0,Unintegrable((1-a^2*x^2)^2/arctanh(a*x)^2,x)],
[(1-a^2*x^2)^2/(x*arctanh(a*x)^2),x,0,Unintegrable((1-a^2*x^2)^2/(x*arctanh(a*x)^2),x)],

# Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)^3

# p>0
[(1-a^2*x^2)^3*arctanh(a*x),x,5,4/35*(1-a^2*x^2)/a+3/70*(1-a^2*x^2)^2/a+1/42*(1-a^2*x^2)^3/a+16/35*x*arctanh(a*x)+8/35*x*(1-a^2*x^2)*arctanh(a*x)+6/35*x*(1-a^2*x^2)^2*arctanh(a*x)+1/7*x*(1-a^2*x^2)^3*arctanh(a*x)+8/35*log(1-a^2*x^2)/a],
[(1-a^2*x^2)^3*arctanh(a*x)^2,x,12,-38/105*x+19/315*a^2*x^3-1/105*a^4*x^5+8/35*(1-a^2*x^2)*arctanh(a*x)/a+3/35*(1-a^2*x^2)^2*arctanh(a*x)/a+1/21*(1-a^2*x^2)^3*arctanh(a*x)/a+16/35*arctanh(a*x)^2/a+16/35*x*arctanh(a*x)^2+8/35*x*(1-a^2*x^2)*arctanh(a*x)^2+6/35*x*(1-a^2*x^2)^2*arctanh(a*x)^2+1/7*x*(1-a^2*x^2)^3*arctanh(a*x)^2-32/35*arctanh(a*x)*log(2/(1-a*x))/a-16/35*polylog(2,1+(-2)/(1-a*x))/a],
[(1-a^2*x^2)^3*arctanh(a*x)^3,x,17,-13/210*(1-a^2*x^2)/a-1/140*(1-a^2*x^2)^2/a-14/15*x*arctanh(a*x)-13/105*x*(1-a^2*x^2)*arctanh(a*x)-1/35*x*(1-a^2*x^2)^2*arctanh(a*x)+12/35*(1-a^2*x^2)*arctanh(a*x)^2/a+9/70*(1-a^2*x^2)^2*arctanh(a*x)^2/a+1/14*(1-a^2*x^2)^3*arctanh(a*x)^2/a+16/35*arctanh(a*x)^3/a+16/35*x*arctanh(a*x)^3+8/35*x*(1-a^2*x^2)*arctanh(a*x)^3+6/35*x*(1-a^2*x^2)^2*arctanh(a*x)^3+1/7*x*(1-a^2*x^2)^3*arctanh(a*x)^3-48/35*arctanh(a*x)^2*log(2/(1-a*x))/a-7/15*log(1-a^2*x^2)/a-48/35*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a+24/35*polylog(3,1+(-2)/(1-a*x))/a],

# p<0

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

# p>0
[x^3*arctanh(a*x)/(1-a^2*x^2),x,8,-1/2*x/a^3+1/2*arctanh(a*x)/a^4-1/2*x^2*arctanh(a*x)/a^2-1/2*arctanh(a*x)^2/a^4+arctanh(a*x)*log(2/(1-a*x))/a^4+1/2*polylog(2,1+(-2)/(1-a*x))/a^4],
[x^2*arctanh(a*x)/(1-a^2*x^2),x,4,-x*arctanh(a*x)/a^2+1/2*arctanh(a*x)^2/a^3-1/2*log(1-a^2*x^2)/a^3],
[x*arctanh(a*x)/(1-a^2*x^2),x,4,-1/2*arctanh(a*x)^2/a^2+arctanh(a*x)*log(2/(1-a*x))/a^2+1/2*polylog(2,1+(-2)/(1-a*x))/a^2],
[arctanh(a*x)/(1-a^2*x^2),x,1,1/2*arctanh(a*x)^2/a],
[arctanh(a*x)/(x*(1-a^2*x^2)),x,3,1/2*arctanh(a*x)^2+arctanh(a*x)*log(2+(-2)/(1+a*x))-1/2*polylog(2,-1+2/(1+a*x))],
[arctanh(a*x)/(x^2*(1-a^2*x^2)),x,7,-arctanh(a*x)/x+1/2*a*arctanh(a*x)^2+a*log(x)-1/2*a*log(1-a^2*x^2)],
[arctanh(a*x)/(x^3*(1-a^2*x^2)),x,7,-1/2*a/x+1/2*a^2*arctanh(a*x)-1/2*arctanh(a*x)/x^2+1/2*a^2*arctanh(a*x)^2+a^2*arctanh(a*x)*log(2+(-2)/(1+a*x))-1/2*a^2*polylog(2,-1+2/(1+a*x))],
[x^3*arctanh(a*x)^2/(1-a^2*x^2),x,10,-x*arctanh(a*x)/a^3+1/2*arctanh(a*x)^2/a^4-1/2*x^2*arctanh(a*x)^2/a^2-1/3*arctanh(a*x)^3/a^4+arctanh(a*x)^2*log(2/(1-a*x))/a^4-1/2*log(1-a^2*x^2)/a^4+arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a^4-1/2*polylog(3,1+(-2)/(1-a*x))/a^4],
[x^2*arctanh(a*x)^2/(1-a^2*x^2),x,7,-arctanh(a*x)^2/a^3-x*arctanh(a*x)^2/a^2+1/3*arctanh(a*x)^3/a^3+2*arctanh(a*x)*log(2/(1-a*x))/a^3+polylog(2,1+(-2)/(1-a*x))/a^3],
[x*arctanh(a*x)^2/(1-a^2*x^2),x,4,-1/3*arctanh(a*x)^3/a^2+arctanh(a*x)^2*log(2/(1-a*x))/a^2+arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a^2-1/2*polylog(3,1+(-2)/(1-a*x))/a^2],
[arctanh(a*x)^2/(1-a^2*x^2),x,1,1/3*arctanh(a*x)^3/a],
[arctanh(a*x)^2/(x*(1-a^2*x^2)),x,4,1/3*arctanh(a*x)^3+arctanh(a*x)^2*log(2+(-2)/(1+a*x))-arctanh(a*x)*polylog(2,-1+2/(1+a*x))-1/2*polylog(3,-1+2/(1+a*x))],
[arctanh(a*x)^2/(x^2*(1-a^2*x^2)),x,6,a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/3*a*arctanh(a*x)^3+2*a*arctanh(a*x)*log(2+(-2)/(1+a*x))-a*polylog(2,-1+2/(1+a*x))],
[arctanh(a*x)^2/(x^3*(1-a^2*x^2)),x,13,-a*arctanh(a*x)/x+1/2*a^2*arctanh(a*x)^2-1/2*arctanh(a*x)^2/x^2+1/3*a^2*arctanh(a*x)^3+a^2*log(x)-1/2*a^2*log(1-a^2*x^2)+a^2*arctanh(a*x)^2*log(2+(-2)/(1+a*x))-a^2*arctanh(a*x)*polylog(2,-1+2/(1+a*x))-1/2*a^2*polylog(3,-1+2/(1+a*x))],
[x^3*arctanh(a*x)^3/(1-a^2*x^2),x,14,-3/2*arctanh(a*x)^2/a^4-3/2*x*arctanh(a*x)^2/a^3+1/2*arctanh(a*x)^3/a^4-1/2*x^2*arctanh(a*x)^3/a^2-1/4*arctanh(a*x)^4/a^4+3*arctanh(a*x)*log(2/(1-a*x))/a^4+arctanh(a*x)^3*log(2/(1-a*x))/a^4+3/2*polylog(2,1+(-2)/(1-a*x))/a^4+3/2*arctanh(a*x)^2*polylog(2,1+(-2)/(1-a*x))/a^4-3/2*arctanh(a*x)*polylog(3,1+(-2)/(1-a*x))/a^4+3/4*polylog(4,1+(-2)/(1-a*x))/a^4],
[x^2*arctanh(a*x)^3/(1-a^2*x^2),x,7,-arctanh(a*x)^3/a^3-x*arctanh(a*x)^3/a^2+1/4*arctanh(a*x)^4/a^3+3*arctanh(a*x)^2*log(2/(1-a*x))/a^3+3*arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a^3-3/2*polylog(3,1+(-2)/(1-a*x))/a^3],
[x*arctanh(a*x)^3/(1-a^2*x^2),x,5,-1/4*arctanh(a*x)^4/a^2+arctanh(a*x)^3*log(2/(1-a*x))/a^2+3/2*arctanh(a*x)^2*polylog(2,1+(-2)/(1-a*x))/a^2-3/2*arctanh(a*x)*polylog(3,1+(-2)/(1-a*x))/a^2+3/4*polylog(4,1+(-2)/(1-a*x))/a^2],
[arctanh(a*x)^3/(1-a^2*x^2),x,1,1/4*arctanh(a*x)^4/a],
[arctanh(a*x)^3/(x*(1-a^2*x^2)),x,5,1/4*arctanh(a*x)^4+arctanh(a*x)^3*log(2+(-2)/(1+a*x))-3/2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))-3/2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))-3/4*polylog(4,-1+2/(1+a*x))],
[arctanh(a*x)^3/(x^2*(1-a^2*x^2)),x,7,a*arctanh(a*x)^3-arctanh(a*x)^3/x+1/4*a*arctanh(a*x)^4+3*a*arctanh(a*x)^2*log(2+(-2)/(1+a*x))-3*a*arctanh(a*x)*polylog(2,-1+2/(1+a*x))-3/2*a*polylog(3,-1+2/(1+a*x))],
[arctanh(a*x)^3/(x^3*(1-a^2*x^2)),x,13,3/2*a^2*arctanh(a*x)^2-3/2*a*arctanh(a*x)^2/x+1/2*a^2*arctanh(a*x)^3-1/2*arctanh(a*x)^3/x^2+1/4*a^2*arctanh(a*x)^4+3*a^2*arctanh(a*x)*log(2+(-2)/(1+a*x))+a^2*arctanh(a*x)^3*log(2+(-2)/(1+a*x))-3/2*a^2*polylog(2,-1+2/(1+a*x))-3/2*a^2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))-3/2*a^2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))-3/4*a^2*polylog(4,-1+2/(1+a*x))],
[arctanh(a*x)^(1/2)/(1-a^2*x^2),x,1,2/3*arctanh(a*x)^(3/2)/a],

# p<0
[x/((1-a^2*x^2)*arctanh(a*x)),x,0,Unintegrable(x/((1-a^2*x^2)*arctanh(a*x)),x)],
[1/((1-a^2*x^2)*arctanh(a*x)),x,1,log(arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)*arctanh(a*x)),x,0,Unintegrable(1/(x*(1-a^2*x^2)*arctanh(a*x)),x)],
[x/((1-a^2*x^2)*arctanh(a*x)^2),x,1,-x/(a*arctanh(a*x))+Unintegrable(1/arctanh(a*x),x)/a],
[1/((1-a^2*x^2)*arctanh(a*x)^2),x,1,(-1)/(a*arctanh(a*x))],
[1/(x*(1-a^2*x^2)*arctanh(a*x)^2),x,1,(-1)/(a*x*arctanh(a*x))-Unintegrable(1/(x^2*arctanh(a*x)),x)/a],
[x/((1-a^2*x^2)*arctanh(a*x)^3),x,1,-1/2*x/(a*arctanh(a*x)^2)+1/2*Unintegrable(1/arctanh(a*x)^2,x)/a],
[1/((1-a^2*x^2)*arctanh(a*x)^3),x,1,(-1/2)/(a*arctanh(a*x)^2)],
[1/(x*(1-a^2*x^2)*arctanh(a*x)^3),x,1,(-1/2)/(a*x*arctanh(a*x)^2)-1/2*Unintegrable(1/(x^2*arctanh(a*x)^2),x)/a],

# p symbolic
[arctanh(a*x)^p/(1-a^2*x^2),x,1,arctanh(a*x)^(1+p)/(a*(1+p))],

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

# p>0
[x^3*arctanh(a*x)/(1-a^2*x^2)^2,x,8,-1/4*x/(a^3*(1-a^2*x^2))-1/4*arctanh(a*x)/a^4+1/2*arctanh(a*x)/(a^4*(1-a^2*x^2))+1/2*arctanh(a*x)^2/a^4-arctanh(a*x)*log(2/(1-a*x))/a^4-1/2*polylog(2,1+(-2)/(1-a*x))/a^4],
[x^2*arctanh(a*x)/(1-a^2*x^2)^2,x,2,(-1/4)/(a^3*(1-a^2*x^2))+1/2*x*arctanh(a*x)/(a^2*(1-a^2*x^2))-1/4*arctanh(a*x)^2/a^3],
[x*arctanh(a*x)/(1-a^2*x^2)^2,x,3,-1/4*x/(a*(1-a^2*x^2))-1/4*arctanh(a*x)/a^2+1/2*arctanh(a*x)/(a^2*(1-a^2*x^2))],
[arctanh(a*x)/(1-a^2*x^2)^2,x,2,(-1/4)/(a*(1-a^2*x^2))+1/2*x*arctanh(a*x)/(1-a^2*x^2)+1/4*arctanh(a*x)^2/a],
[arctanh(a*x)/(x*(1-a^2*x^2)^2),x,7,-1/4*a*x/(1-a^2*x^2)-1/4*arctanh(a*x)+1/2*arctanh(a*x)/(1-a^2*x^2)+1/2*arctanh(a*x)^2+arctanh(a*x)*log(2+(-2)/(1+a*x))-1/2*polylog(2,-1+2/(1+a*x))],
[arctanh(a*x)/(x^2*(1-a^2*x^2)^2),x,10,-1/4*a/(1-a^2*x^2)-arctanh(a*x)/x+1/2*a^2*x*arctanh(a*x)/(1-a^2*x^2)+3/4*a*arctanh(a*x)^2+a*log(x)-1/2*a*log(1-a^2*x^2)],
[arctanh(a*x)/(x^3*(1-a^2*x^2)^2),x,15,-1/2*a/x-1/4*a^3*x/(1-a^2*x^2)+1/4*a^2*arctanh(a*x)-1/2*arctanh(a*x)/x^2+1/2*a^2*arctanh(a*x)/(1-a^2*x^2)+a^2*arctanh(a*x)^2+2*a^2*arctanh(a*x)*log(2+(-2)/(1+a*x))-a^2*polylog(2,-1+2/(1+a*x))],
[x^3*arctanh(a*x)^2/(1-a^2*x^2)^2,x,8,1/4/(a^4*(1-a^2*x^2))-1/2*x*arctanh(a*x)/(a^3*(1-a^2*x^2))-1/4*arctanh(a*x)^2/a^4+1/2*arctanh(a*x)^2/(a^4*(1-a^2*x^2))+1/3*arctanh(a*x)^3/a^4-arctanh(a*x)^2*log(2/(1-a*x))/a^4-arctanh(a*x)*polylog(2,1+(-2)/(1-a*x))/a^4+1/2*polylog(3,1+(-2)/(1-a*x))/a^4],
[x^2*arctanh(a*x)^2/(1-a^2*x^2)^2,x,4,1/4*x/(a^2*(1-a^2*x^2))+1/4*arctanh(a*x)/a^3-1/2*arctanh(a*x)/(a^3*(1-a^2*x^2))+1/2*x*arctanh(a*x)^2/(a^2*(1-a^2*x^2))-1/6*arctanh(a*x)^3/a^3],
[x*arctanh(a*x)^2/(1-a^2*x^2)^2,x,3,1/4/(a^2*(1-a^2*x^2))-1/2*x*arctanh(a*x)/(a*(1-a^2*x^2))-1/4*arctanh(a*x)^2/a^2+1/2*arctanh(a*x)^2/(a^2*(1-a^2*x^2))],
[arctanh(a*x)^2/(1-a^2*x^2)^2,x,4,1/4*x/(1-a^2*x^2)+1/4*arctanh(a*x)/a-1/2*arctanh(a*x)/(a*(1-a^2*x^2))+1/2*x*arctanh(a*x)^2/(1-a^2*x^2)+1/6*arctanh(a*x)^3/a],
[arctanh(a*x)^2/(x*(1-a^2*x^2)^2),x,8,1/4/(1-a^2*x^2)-1/2*a*x*arctanh(a*x)/(1-a^2*x^2)-1/4*arctanh(a*x)^2+1/2*arctanh(a*x)^2/(1-a^2*x^2)+1/3*arctanh(a*x)^3+arctanh(a*x)^2*log(2+(-2)/(1+a*x))-arctanh(a*x)*polylog(2,-1+2/(1+a*x))-1/2*polylog(3,-1+2/(1+a*x))],
[arctanh(a*x)^2/(x^2*(1-a^2*x^2)^2),x,11,1/4*a^2*x/(1-a^2*x^2)+1/4*a*arctanh(a*x)-1/2*a*arctanh(a*x)/(1-a^2*x^2)+a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/2*a^2*x*arctanh(a*x)^2/(1-a^2*x^2)+1/2*a*arctanh(a*x)^3+2*a*arctanh(a*x)*log(2+(-2)/(1+a*x))-a*polylog(2,-1+2/(1+a*x))],
[arctanh(a*x)^2/(x^3*(1-a^2*x^2)^2),x,22,1/4*a^2/(1-a^2*x^2)-a*arctanh(a*x)/x-1/2*a^3*x*arctanh(a*x)/(1-a^2*x^2)+1/4*a^2*arctanh(a*x)^2-1/2*arctanh(a*x)^2/x^2+1/2*a^2*arctanh(a*x)^2/(1-a^2*x^2)+2/3*a^2*arctanh(a*x)^3+a^2*log(x)-1/2*a^2*log(1-a^2*x^2)+2*a^2*arctanh(a*x)^2*log(2+(-2)/(1+a*x))-2*a^2*arctanh(a*x)*polylog(2,-1+2/(1+a*x))-a^2*polylog(3,-1+2/(1+a*x))],
[x^3*arctanh(a*x)^3/(1-a^2*x^2)^2,x,11,-3/8*x/(a^3*(1-a^2*x^2))-3/8*arctanh(a*x)/a^4+3/4*arctanh(a*x)/(a^4*(1-a^2*x^2))-3/4*x*arctanh(a*x)^2/(a^3*(1-a^2*x^2))-1/4*arctanh(a*x)^3/a^4+1/2*arctanh(a*x)^3/(a^4*(1-a^2*x^2))+1/4*arctanh(a*x)^4/a^4-arctanh(a*x)^3*log(2/(1-a*x))/a^4-3/2*arctanh(a*x)^2*polylog(2,1+(-2)/(1-a*x))/a^4+3/2*arctanh(a*x)*polylog(3,1+(-2)/(1-a*x))/a^4-3/4*polylog(4,1+(-2)/(1-a*x))/a^4],
[x^2*arctanh(a*x)^3/(1-a^2*x^2)^2,x,4,(-3/8)/(a^3*(1-a^2*x^2))+3/4*x*arctanh(a*x)/(a^2*(1-a^2*x^2))+3/8*arctanh(a*x)^2/a^3-3/4*arctanh(a*x)^2/(a^3*(1-a^2*x^2))+1/2*x*arctanh(a*x)^3/(a^2*(1-a^2*x^2))-1/8*arctanh(a*x)^4/a^3],
[x*arctanh(a*x)^3/(1-a^2*x^2)^2,x,5,-3/8*x/(a*(1-a^2*x^2))-3/8*arctanh(a*x)/a^2+3/4*arctanh(a*x)/(a^2*(1-a^2*x^2))-3/4*x*arctanh(a*x)^2/(a*(1-a^2*x^2))-1/4*arctanh(a*x)^3/a^2+1/2*arctanh(a*x)^3/(a^2*(1-a^2*x^2))],
[arctanh(a*x)^3/(1-a^2*x^2)^2,x,4,(-3/8)/(a*(1-a^2*x^2))+3/4*x*arctanh(a*x)/(1-a^2*x^2)+3/8*arctanh(a*x)^2/a-3/4*arctanh(a*x)^2/(a*(1-a^2*x^2))+1/2*x*arctanh(a*x)^3/(1-a^2*x^2)+1/8*arctanh(a*x)^4/a],
[arctanh(a*x)^3/(x*(1-a^2*x^2)^2),x,11,-3/8*a*x/(1-a^2*x^2)-3/8*arctanh(a*x)+3/4*arctanh(a*x)/(1-a^2*x^2)-3/4*a*x*arctanh(a*x)^2/(1-a^2*x^2)-1/4*arctanh(a*x)^3+1/2*arctanh(a*x)^3/(1-a^2*x^2)+1/4*arctanh(a*x)^4+arctanh(a*x)^3*log(2+(-2)/(1+a*x))-3/2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))-3/2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))-3/4*polylog(4,-1+2/(1+a*x))],
[arctanh(a*x)^3/(x^2*(1-a^2*x^2)^2),x,12,-3/8*a/(1-a^2*x^2)+3/4*a^2*x*arctanh(a*x)/(1-a^2*x^2)+3/8*a*arctanh(a*x)^2-3/4*a*arctanh(a*x)^2/(1-a^2*x^2)+a*arctanh(a*x)^3-arctanh(a*x)^3/x+1/2*a^2*x*arctanh(a*x)^3/(1-a^2*x^2)+3/8*a*arctanh(a*x)^4+3*a*arctanh(a*x)^2*log(2+(-2)/(1+a*x))-3*a*arctanh(a*x)*polylog(2,-1+2/(1+a*x))-3/2*a*polylog(3,-1+2/(1+a*x))],
[arctanh(a*x)^3/(x^3*(1-a^2*x^2)^2),x,25,-3/8*a^3*x/(1-a^2*x^2)-3/8*a^2*arctanh(a*x)+3/4*a^2*arctanh(a*x)/(1-a^2*x^2)+3/2*a^2*arctanh(a*x)^2-3/2*a*arctanh(a*x)^2/x-3/4*a^3*x*arctanh(a*x)^2/(1-a^2*x^2)+1/4*a^2*arctanh(a*x)^3-1/2*arctanh(a*x)^3/x^2+1/2*a^2*arctanh(a*x)^3/(1-a^2*x^2)+1/2*a^2*arctanh(a*x)^4+3*a^2*arctanh(a*x)*log(2+(-2)/(1+a*x))+2*a^2*arctanh(a*x)^3*log(2+(-2)/(1+a*x))-3/2*a^2*polylog(2,-1+2/(1+a*x))-3*a^2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))-3*a^2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))-3/2*a^2*polylog(4,-1+2/(1+a*x))],
[sqrt(arctanh(a*x))/(1-a^2*x^2)^2,x,9,1/3*arctanh(a*x)^(3/2)/a+1/16*erf(sqrt(2)*sqrt(arctanh(a*x)))*sqrt(1/2*Pi)/a-1/16*erfi(sqrt(2)*sqrt(arctanh(a*x)))*sqrt(1/2*Pi)/a+1/2*x*sqrt(arctanh(a*x))/(1-a^2*x^2)],

# p<0
[x^4/((1-a^2*x^2)^2*arctanh(a*x)),x,0,1/2*Chi(2*arctanh(a*x))/a^5-3/2*log(arctanh(a*x))/a^5+Unintegrable(1/arctanh(a*x),x)/a^4,Unintegrable(x^4/((1-a^2*x^2)^2*arctanh(a*x)),x)],
[x^3/((1-a^2*x^2)^2*arctanh(a*x)),x,0,1/2*Shi(2*arctanh(a*x))/a^4-Unintegrable(x/((1-a^2*x^2)*arctanh(a*x)),x)/a^2,Unintegrable(x^3/((1-a^2*x^2)^2*arctanh(a*x)),x)],
[x^2/((1-a^2*x^2)^2*arctanh(a*x)),x,4,1/2*Chi(2*arctanh(a*x))/a^3-1/2*log(arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^2*arctanh(a*x)),x,4,1/2*Shi(2*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^2*arctanh(a*x)),x,4,1/2*Chi(2*arctanh(a*x))/a+1/2*log(arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^2*arctanh(a*x)),x,0,1/2*Shi(2*arctanh(a*x))+Unintegrable(1/(x*(1-a^2*x^2)*arctanh(a*x)),x),Unintegrable(1/(x*(1-a^2*x^2)^2*arctanh(a*x)),x)],
[x^3/((1-a^2*x^2)^2*arctanh(a*x)^2),x,11,x/(a^3*arctanh(a*x))-x/(a^3*(1-a^2*x^2)*arctanh(a*x))+Chi(2*arctanh(a*x))/a^4-Unintegrable(1/arctanh(a*x),x)/a^3],
[x^2/((1-a^2*x^2)^2*arctanh(a*x)^2),x,5,-x^2/(a*(1-a^2*x^2)*arctanh(a*x))+Shi(2*arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^2*arctanh(a*x)^2),x,9,-x/(a*(1-a^2*x^2)*arctanh(a*x))+Chi(2*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^2*arctanh(a*x)^2),x,5,(-1)/(a*(1-a^2*x^2)*arctanh(a*x))+Shi(2*arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^2*arctanh(a*x)^2),x,11,(-1)/(a*x*arctanh(a*x))-a*x/((1-a^2*x^2)*arctanh(a*x))+Chi(2*arctanh(a*x))-Unintegrable(1/(x^2*arctanh(a*x)),x)/a],
[x^3/((1-a^2*x^2)^2*arctanh(a*x)^3),x,7,1/2*x/(a^3*arctanh(a*x)^2)-1/2*x/(a^3*(1-a^2*x^2)*arctanh(a*x)^2)+1/2*(-1-a^2*x^2)/(a^4*(1-a^2*x^2)*arctanh(a*x))+Shi(2*arctanh(a*x))/a^4-1/2*Unintegrable(1/arctanh(a*x)^2,x)/a^3],
[x^2/((1-a^2*x^2)^2*arctanh(a*x)^3),x,10,-1/2*x^2/(a*(1-a^2*x^2)*arctanh(a*x)^2)-x/(a^2*(1-a^2*x^2)*arctanh(a*x))+Chi(2*arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^2*arctanh(a*x)^3),x,5,-1/2*x/(a*(1-a^2*x^2)*arctanh(a*x)^2)+1/2*(-1-a^2*x^2)/(a^2*(1-a^2*x^2)*arctanh(a*x))+Shi(2*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^2*arctanh(a*x)^3),x,10,(-1/2)/(a*(1-a^2*x^2)*arctanh(a*x)^2)-x/((1-a^2*x^2)*arctanh(a*x))+Chi(2*arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^2*arctanh(a*x)^3),x,7,(-1/2)/(a*x*arctanh(a*x)^2)-1/2*a*x/((1-a^2*x^2)*arctanh(a*x)^2)+1/2*(-1-a^2*x^2)/((1-a^2*x^2)*arctanh(a*x))+Shi(2*arctanh(a*x))-1/2*Unintegrable(1/(x^2*arctanh(a*x)^2),x)/a],
[1/((1-a^2*x^2)^2*arctanh(a*x)^4),x,6,(-1/3)/(a*(1-a^2*x^2)*arctanh(a*x)^3)-1/3*x/((1-a^2*x^2)*arctanh(a*x)^2)+1/3*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x))+2/3*Shi(2*arctanh(a*x))/a],
[1/((1-a^2*x^2)^2*arctanh(a*x)^5),x,11,(-1/4)/(a*(1-a^2*x^2)*arctanh(a*x)^4)-1/6*x/((1-a^2*x^2)*arctanh(a*x)^3)+1/12*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x)^2)-1/3*x/((1-a^2*x^2)*arctanh(a*x))+1/3*Chi(2*arctanh(a*x))/a],
[1/((1-a^2*x^2)^2*arctanh(a*x)^6),x,7,(-1/5)/(a*(1-a^2*x^2)*arctanh(a*x)^5)-1/10*x/((1-a^2*x^2)*arctanh(a*x)^4)+1/30*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x)^3)-1/15*x/((1-a^2*x^2)*arctanh(a*x)^2)+1/15*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x))+2/15*Shi(2*arctanh(a*x))/a],
[1/((1-a^2*x^2)^2*arctanh(a*x)^7),x,12,(-1/6)/(a*(1-a^2*x^2)*arctanh(a*x)^6)-1/15*x/((1-a^2*x^2)*arctanh(a*x)^5)+1/60*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x)^4)-1/45*x/((1-a^2*x^2)*arctanh(a*x)^3)+1/90*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x)^2)-2/45*x/((1-a^2*x^2)*arctanh(a*x))+2/45*Chi(2*arctanh(a*x))/a],
[1/((1-a^2*x^2)^2*arctanh(a*x)^8),x,8,(-1/7)/(a*(1-a^2*x^2)*arctanh(a*x)^7)-1/21*x/((1-a^2*x^2)*arctanh(a*x)^6)+1/105*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x)^5)-1/105*x/((1-a^2*x^2)*arctanh(a*x)^4)+1/315*(-1-a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x)^3)-2/315*x/((1-a^2*x^2)*arctanh(a*x)^2)-2/315*(1+a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x))+4/315*Shi(2*arctanh(a*x))/a],

# Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^3

# p>0
[x^3*arctanh(a*x)/(1-a^2*x^2)^3,x,4,-1/16*x^3/(a*(1-a^2*x^2)^2)+3/32*x/(a^3*(1-a^2*x^2))-3/32*arctanh(a*x)/a^4+1/4*x^4*arctanh(a*x)/(1-a^2*x^2)^2],
[x^2*arctanh(a*x)/(1-a^2*x^2)^3,x,3,(-1/16)/(a^3*(1-a^2*x^2)^2)+1/16/(a^3*(1-a^2*x^2))+1/4*x*arctanh(a*x)/(a^2*(1-a^2*x^2)^2)-1/8*x*arctanh(a*x)/(a^2*(1-a^2*x^2))-1/16*arctanh(a*x)^2/a^3],
[x*arctanh(a*x)/(1-a^2*x^2)^3,x,4,-1/16*x/(a*(1-a^2*x^2)^2)-3/32*x/(a*(1-a^2*x^2))-3/32*arctanh(a*x)/a^2+1/4*arctanh(a*x)/(a^2*(1-a^2*x^2)^2)],
[arctanh(a*x)/(1-a^2*x^2)^3,x,3,(-1/16)/(a*(1-a^2*x^2)^2)+(-3/16)/(a*(1-a^2*x^2))+1/4*x*arctanh(a*x)/(1-a^2*x^2)^2+3/8*x*arctanh(a*x)/(1-a^2*x^2)+3/16*arctanh(a*x)^2/a],
[arctanh(a*x)/(x*(1-a^2*x^2)^3),x,12,-1/16*a*x/(1-a^2*x^2)^2-11/32*a*x/(1-a^2*x^2)-11/32*arctanh(a*x)+1/4*arctanh(a*x)/(1-a^2*x^2)^2+1/2*arctanh(a*x)/(1-a^2*x^2)+1/2*arctanh(a*x)^2+arctanh(a*x)*log(2+(-2)/(1+a*x))-1/2*polylog(2,-1+2/(1+a*x))],
[arctanh(a*x)/(x^2*(1-a^2*x^2)^3),x,14,-1/16*a/(1-a^2*x^2)^2-7/16*a/(1-a^2*x^2)-arctanh(a*x)/x+1/4*a^2*x*arctanh(a*x)/(1-a^2*x^2)^2+7/8*a^2*x*arctanh(a*x)/(1-a^2*x^2)+15/16*a*arctanh(a*x)^2+a*log(x)-1/2*a*log(1-a^2*x^2)],
[x^3*arctanh(a*x)^2/(1-a^2*x^2)^3,x,4,1/32*x^4/(1-a^2*x^2)^2+(-3/32)/(a^4*(1-a^2*x^2))-1/8*x^3*arctanh(a*x)/(a*(1-a^2*x^2)^2)+3/16*x*arctanh(a*x)/(a^3*(1-a^2*x^2))-3/32*arctanh(a*x)^2/a^4+1/4*x^4*arctanh(a*x)^2/(1-a^2*x^2)^2],
[x^2*arctanh(a*x)^2/(1-a^2*x^2)^3,x,13,1/32*x/(a^2*(1-a^2*x^2)^2)-1/64*x/(a^2*(1-a^2*x^2))-1/64*arctanh(a*x)/a^3-1/8*arctanh(a*x)/(a^3*(1-a^2*x^2)^2)+1/8*arctanh(a*x)/(a^3*(1-a^2*x^2))+1/4*x*arctanh(a*x)^2/(a^2*(1-a^2*x^2)^2)-1/8*x*arctanh(a*x)^2/(a^2*(1-a^2*x^2))-1/24*arctanh(a*x)^3/a^3],
[x*arctanh(a*x)^2/(1-a^2*x^2)^3,x,4,1/32/(a^2*(1-a^2*x^2)^2)+3/32/(a^2*(1-a^2*x^2))-1/8*x*arctanh(a*x)/(a*(1-a^2*x^2)^2)-3/16*x*arctanh(a*x)/(a*(1-a^2*x^2))-3/32*arctanh(a*x)^2/a^2+1/4*arctanh(a*x)^2/(a^2*(1-a^2*x^2)^2)],
[arctanh(a*x)^2/(1-a^2*x^2)^3,x,8,1/32*x/(1-a^2*x^2)^2+15/64*x/(1-a^2*x^2)+15/64*arctanh(a*x)/a-1/8*arctanh(a*x)/(a*(1-a^2*x^2)^2)-3/8*arctanh(a*x)/(a*(1-a^2*x^2))+1/4*x*arctanh(a*x)^2/(1-a^2*x^2)^2+3/8*x*arctanh(a*x)^2/(1-a^2*x^2)+1/8*arctanh(a*x)^3/a],
[arctanh(a*x)^2/(x*(1-a^2*x^2)^3),x,13,1/32/(1-a^2*x^2)^2+11/32/(1-a^2*x^2)-1/8*a*x*arctanh(a*x)/(1-a^2*x^2)^2-11/16*a*x*arctanh(a*x)/(1-a^2*x^2)-11/32*arctanh(a*x)^2+1/4*arctanh(a*x)^2/(1-a^2*x^2)^2+1/2*arctanh(a*x)^2/(1-a^2*x^2)+1/3*arctanh(a*x)^3+arctanh(a*x)^2*log(2+(-2)/(1+a*x))-arctanh(a*x)*polylog(2,-1+2/(1+a*x))-1/2*polylog(3,-1+2/(1+a*x))],
[arctanh(a*x)^2/(x^2*(1-a^2*x^2)^3),x,20,1/32*a^2*x/(1-a^2*x^2)^2+31/64*a^2*x/(1-a^2*x^2)+31/64*a*arctanh(a*x)-1/8*a*arctanh(a*x)/(1-a^2*x^2)^2-7/8*a*arctanh(a*x)/(1-a^2*x^2)+a*arctanh(a*x)^2-arctanh(a*x)^2/x+1/4*a^2*x*arctanh(a*x)^2/(1-a^2*x^2)^2+7/8*a^2*x*arctanh(a*x)^2/(1-a^2*x^2)+5/8*a*arctanh(a*x)^3+2*a*arctanh(a*x)*log(2+(-2)/(1+a*x))-a*polylog(2,-1+2/(1+a*x))],
[x^3*arctanh(a*x)^3/(1-a^2*x^2)^3,x,9,-3/128*x^3/(a*(1-a^2*x^2)^2)+45/256*x/(a^3*(1-a^2*x^2))+27/256*arctanh(a*x)/a^4+3/32*x^4*arctanh(a*x)/(1-a^2*x^2)^2-9/32*arctanh(a*x)/(a^4*(1-a^2*x^2))-3/16*x^3*arctanh(a*x)^2/(a*(1-a^2*x^2)^2)+9/32*x*arctanh(a*x)^2/(a^3*(1-a^2*x^2))-3/32*arctanh(a*x)^3/a^4+1/4*x^4*arctanh(a*x)^3/(1-a^2*x^2)^2],
[x^2*arctanh(a*x)^3/(1-a^2*x^2)^3,x,13,(-3/128)/(a^3*(1-a^2*x^2)^2)+3/128/(a^3*(1-a^2*x^2))+3/32*x*arctanh(a*x)/(a^2*(1-a^2*x^2)^2)-3/64*x*arctanh(a*x)/(a^2*(1-a^2*x^2))-3/128*arctanh(a*x)^2/a^3-3/16*arctanh(a*x)^2/(a^3*(1-a^2*x^2)^2)+3/16*arctanh(a*x)^2/(a^3*(1-a^2*x^2))+1/4*x*arctanh(a*x)^3/(a^2*(1-a^2*x^2)^2)-1/8*x*arctanh(a*x)^3/(a^2*(1-a^2*x^2))-1/32*arctanh(a*x)^4/a^3],
[x*arctanh(a*x)^3/(1-a^2*x^2)^3,x,9,-3/128*x/(a*(1-a^2*x^2)^2)-45/256*x/(a*(1-a^2*x^2))-45/256*arctanh(a*x)/a^2+3/32*arctanh(a*x)/(a^2*(1-a^2*x^2)^2)+9/32*arctanh(a*x)/(a^2*(1-a^2*x^2))-3/16*x*arctanh(a*x)^2/(a*(1-a^2*x^2)^2)-9/32*x*arctanh(a*x)^2/(a*(1-a^2*x^2))-3/32*arctanh(a*x)^3/a^2+1/4*arctanh(a*x)^3/(a^2*(1-a^2*x^2)^2)],
[arctanh(a*x)^3/(1-a^2*x^2)^3,x,8,(-3/128)/(a*(1-a^2*x^2)^2)+(-45/128)/(a*(1-a^2*x^2))+3/32*x*arctanh(a*x)/(1-a^2*x^2)^2+45/64*x*arctanh(a*x)/(1-a^2*x^2)+45/128*arctanh(a*x)^2/a-3/16*arctanh(a*x)^2/(a*(1-a^2*x^2)^2)-9/16*arctanh(a*x)^2/(a*(1-a^2*x^2))+1/4*x*arctanh(a*x)^3/(1-a^2*x^2)^2+3/8*x*arctanh(a*x)^3/(1-a^2*x^2)+3/32*arctanh(a*x)^4/a],
[arctanh(a*x)^3/(x*(1-a^2*x^2)^3),x,21,-3/128*a*x/(1-a^2*x^2)^2-141/256*a*x/(1-a^2*x^2)-141/256*arctanh(a*x)+3/32*arctanh(a*x)/(1-a^2*x^2)^2+33/32*arctanh(a*x)/(1-a^2*x^2)-3/16*a*x*arctanh(a*x)^2/(1-a^2*x^2)^2-33/32*a*x*arctanh(a*x)^2/(1-a^2*x^2)-11/32*arctanh(a*x)^3+1/4*arctanh(a*x)^3/(1-a^2*x^2)^2+1/2*arctanh(a*x)^3/(1-a^2*x^2)+1/4*arctanh(a*x)^4+arctanh(a*x)^3*log(2+(-2)/(1+a*x))-3/2*arctanh(a*x)^2*polylog(2,-1+2/(1+a*x))-3/2*arctanh(a*x)*polylog(3,-1+2/(1+a*x))-3/4*polylog(4,-1+2/(1+a*x))],
[arctanh(a*x)^3/(x^2*(1-a^2*x^2)^3),x,21,-3/128*a/(1-a^2*x^2)^2-93/128*a/(1-a^2*x^2)+3/32*a^2*x*arctanh(a*x)/(1-a^2*x^2)^2+93/64*a^2*x*arctanh(a*x)/(1-a^2*x^2)+93/128*a*arctanh(a*x)^2-3/16*a*arctanh(a*x)^2/(1-a^2*x^2)^2-21/16*a*arctanh(a*x)^2/(1-a^2*x^2)+a*arctanh(a*x)^3-arctanh(a*x)^3/x+1/4*a^2*x*arctanh(a*x)^3/(1-a^2*x^2)^2+7/8*a^2*x*arctanh(a*x)^3/(1-a^2*x^2)+15/32*a*arctanh(a*x)^4+3*a*arctanh(a*x)^2*log(2+(-2)/(1+a*x))-3*a*arctanh(a*x)*polylog(2,-1+2/(1+a*x))-3/2*a*polylog(3,-1+2/(1+a*x))],
[sqrt(arctanh(a*x))/(1-a^2*x^2)^3,x,15,1/4*arctanh(a*x)^(3/2)/a+1/16*erf(sqrt(2)*sqrt(arctanh(a*x)))*sqrt(1/2*Pi)/a-1/16*erfi(sqrt(2)*sqrt(arctanh(a*x)))*sqrt(1/2*Pi)/a+1/256*erf(2*sqrt(arctanh(a*x)))*sqrt(Pi)/a-1/256*erfi(2*sqrt(arctanh(a*x)))*sqrt(Pi)/a+1/4*sinh(2*arctanh(a*x))*sqrt(arctanh(a*x))/a+1/32*sinh(4*arctanh(a*x))*sqrt(arctanh(a*x))/a],

# p<0
[x^6/((1-a^2*x^2)^3*arctanh(a*x)),x,0,Unintegrable(x^6/((1-a^2*x^2)^3*arctanh(a*x)),x)],
[x^5/((1-a^2*x^2)^3*arctanh(a*x)),x,0,Unintegrable(x^5/((1-a^2*x^2)^3*arctanh(a*x)),x)],
[x^4/((1-a^2*x^2)^3*arctanh(a*x)),x,5,-1/2*Chi(2*arctanh(a*x))/a^5+1/8*Chi(4*arctanh(a*x))/a^5+3/8*log(arctanh(a*x))/a^5],
[x^3/((1-a^2*x^2)^3*arctanh(a*x)),x,5,-1/4*Shi(2*arctanh(a*x))/a^4+1/8*Shi(4*arctanh(a*x))/a^4],
[x^2/((1-a^2*x^2)^3*arctanh(a*x)),x,4,1/8*Chi(4*arctanh(a*x))/a^3-1/8*log(arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^3*arctanh(a*x)),x,5,1/4*Shi(2*arctanh(a*x))/a^2+1/8*Shi(4*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^3*arctanh(a*x)),x,5,1/2*Chi(2*arctanh(a*x))/a+1/8*Chi(4*arctanh(a*x))/a+3/8*log(arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^3*arctanh(a*x)),x,0,3/4*Shi(2*arctanh(a*x))+1/8*Shi(4*arctanh(a*x))-Unintegrable(1/(x*(-1+a^2*x^2)*arctanh(a*x)),x),Unintegrable(1/(x*(1-a^2*x^2)^3*arctanh(a*x)),x)],
[x^5/((1-a^2*x^2)^3*arctanh(a*x)^2),x,32,-x/(a^5*arctanh(a*x))-x/(a^5*(1-a^2*x^2)^2*arctanh(a*x))+2*x/(a^5*(1-a^2*x^2)*arctanh(a*x))-3/2*Chi(2*arctanh(a*x))/a^6+1/2*Chi(4*arctanh(a*x))/a^6+Unintegrable(1/arctanh(a*x),x)/a^5],
[x^4/((1-a^2*x^2)^3*arctanh(a*x)^2),x,6,-x^4/(a*(1-a^2*x^2)^2*arctanh(a*x))-Shi(2*arctanh(a*x))/a^5+1/2*Shi(4*arctanh(a*x))/a^5],
[x^3/((1-a^2*x^2)^3*arctanh(a*x)^2),x,20,-x^3/(a*(1-a^2*x^2)^2*arctanh(a*x))-1/2*Chi(2*arctanh(a*x))/a^4+1/2*Chi(4*arctanh(a*x))/a^4,-x/(a^3*(1-a^2*x^2)^2*arctanh(a*x))+x/(a^3*(1-a^2*x^2)*arctanh(a*x))-1/2*Chi(2*arctanh(a*x))/a^4+1/2*Chi(4*arctanh(a*x))/a^4],
[x^2/((1-a^2*x^2)^3*arctanh(a*x)^2),x,12,-x^2/(a*(1-a^2*x^2)^2*arctanh(a*x))+1/2*Shi(4*arctanh(a*x))/a^3,(-1)/(a^3*(1-a^2*x^2)^2*arctanh(a*x))+1/(a^3*(1-a^2*x^2)*arctanh(a*x))+1/2*Shi(4*arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^3*arctanh(a*x)^2),x,10,-x/(a*(1-a^2*x^2)^2*arctanh(a*x))+1/2*Chi(2*arctanh(a*x))/a^2+1/2*Chi(4*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^3*arctanh(a*x)^2),x,6,(-1)/(a*(1-a^2*x^2)^2*arctanh(a*x))+Shi(2*arctanh(a*x))/a+1/2*Shi(4*arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^3*arctanh(a*x)^2),x,22,(-1)/(a*x*arctanh(a*x))-a*x/((1-a^2*x^2)^2*arctanh(a*x))-a*x/((1-a^2*x^2)*arctanh(a*x))+3/2*Chi(2*arctanh(a*x))+1/2*Chi(4*arctanh(a*x))-Unintegrable(1/(x^2*arctanh(a*x)),x)/a],
[x^4/((1-a^2*x^2)^3*arctanh(a*x)^3),x,21,-1/2*x^4/(a*(1-a^2*x^2)^2*arctanh(a*x)^2)-2*x/(a^4*(1-a^2*x^2)^2*arctanh(a*x))+2*x/(a^4*(1-a^2*x^2)*arctanh(a*x))-Chi(2*arctanh(a*x))/a^5+Chi(4*arctanh(a*x))/a^5],
[x^3/((1-a^2*x^2)^3*arctanh(a*x)^3),x,25,-1/2*x^3/(a*(1-a^2*x^2)^2*arctanh(a*x)^2)-3/2*x^2/(a^2*(1-a^2*x^2)^2*arctanh(a*x))-1/2*x^4/((1-a^2*x^2)^2*arctanh(a*x))-1/2*Shi(2*arctanh(a*x))/a^4+Shi(4*arctanh(a*x))/a^4,-1/2*x/(a^3*(1-a^2*x^2)^2*arctanh(a*x)^2)+1/2*x/(a^3*(1-a^2*x^2)*arctanh(a*x)^2)+(-2)/(a^4*(1-a^2*x^2)^2*arctanh(a*x))+3/2/(a^4*(1-a^2*x^2)*arctanh(a*x))+1/2*(1+a^2*x^2)/(a^4*(1-a^2*x^2)*arctanh(a*x))-1/2*Shi(2*arctanh(a*x))/a^4+Shi(4*arctanh(a*x))/a^4],
[x^2/((1-a^2*x^2)^3*arctanh(a*x)^3),x,22,-1/2*x^2/(a*(-1+a^2*x^2)^2*arctanh(a*x)^2)-2*x/(a^2*(1-a^2*x^2)^2*arctanh(a*x))+x/(a^2*(1-a^2*x^2)*arctanh(a*x))+Chi(4*arctanh(a*x))/a^3,(-1/2)/(a^3*(1-a^2*x^2)^2*arctanh(a*x)^2)+1/2/(a^3*(1-a^2*x^2)*arctanh(a*x)^2)-2*x/(a^2*(1-a^2*x^2)^2*arctanh(a*x))+x/(a^2*(1-a^2*x^2)*arctanh(a*x))+Chi(4*arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^3*arctanh(a*x)^3),x,19,-1/2*x/(a*(1-a^2*x^2)^2*arctanh(a*x)^2)+(-2)/(a^2*(1-a^2*x^2)^2*arctanh(a*x))+3/2/(a^2*(1-a^2*x^2)*arctanh(a*x))+1/2*Shi(2*arctanh(a*x))/a^2+Shi(4*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^3*arctanh(a*x)^3),x,11,(-1/2)/(a*(1-a^2*x^2)^2*arctanh(a*x)^2)-2*x/((1-a^2*x^2)^2*arctanh(a*x))+Chi(2*arctanh(a*x))/a+Chi(4*arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^3*arctanh(a*x)^3),x,27,(-1/2)/(a*x*arctanh(a*x)^2)-1/2*a*x/((1-a^2*x^2)^2*arctanh(a*x)^2)-1/2*a*x/((1-a^2*x^2)*arctanh(a*x)^2)+(-2)/((1-a^2*x^2)^2*arctanh(a*x))+3/2/((1-a^2*x^2)*arctanh(a*x))+1/2*(-1-a^2*x^2)/((1-a^2*x^2)*arctanh(a*x))+3/2*Shi(2*arctanh(a*x))+Shi(4*arctanh(a*x))-1/2*Unintegrable(1/(x^2*arctanh(a*x)^2),x)/a],
[1/((1-a^2*x^2)^3*arctanh(a*x)^4),x,20,(-1/3)/(a*(1-a^2*x^2)^2*arctanh(a*x)^3)-2/3*x/((1-a^2*x^2)^2*arctanh(a*x)^2)+(-8/3)/(a*(1-a^2*x^2)^2*arctanh(a*x))+2/(a*(1-a^2*x^2)*arctanh(a*x))+2/3*Shi(2*arctanh(a*x))/a+4/3*Shi(4*arctanh(a*x))/a],
[1/((1-a^2*x^2)^3*arctanh(a*x)^5),x,35,(-1/4)/(a*(1-a^2*x^2)^2*arctanh(a*x)^4)-1/3*x/((1-a^2*x^2)^2*arctanh(a*x)^3)+(-2/3)/(a*(1-a^2*x^2)^2*arctanh(a*x)^2)+1/2/(a*(1-a^2*x^2)*arctanh(a*x)^2)-8/3*x/((1-a^2*x^2)^2*arctanh(a*x))+x/((1-a^2*x^2)*arctanh(a*x))+1/3*Chi(2*arctanh(a*x))/a+4/3*Chi(4*arctanh(a*x))/a],
[1/((1-a^2*x^2)^3*arctanh(a*x)^6),x,49,(-1/5)/(a*(1-a^2*x^2)^2*arctanh(a*x)^5)-1/5*x/((1-a^2*x^2)^2*arctanh(a*x)^4)+(-4/15)/(a*(1-a^2*x^2)^2*arctanh(a*x)^3)+1/5/(a*(1-a^2*x^2)*arctanh(a*x)^3)-8/15*x/((1-a^2*x^2)^2*arctanh(a*x)^2)+1/5*x/((1-a^2*x^2)*arctanh(a*x)^2)+(-32/15)/(a*(1-a^2*x^2)^2*arctanh(a*x))+8/5/(a*(1-a^2*x^2)*arctanh(a*x))+1/5*(1+a^2*x^2)/(a*(1-a^2*x^2)*arctanh(a*x))+2/15*Shi(2*arctanh(a*x))/a+16/15*Shi(4*arctanh(a*x))/a],

# Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^4

# p>0
[arctanh(a*x)/(1-a^2*x^2)^4,x,4,(-1/36)/(a*(1-a^2*x^2)^3)+(-5/96)/(a*(1-a^2*x^2)^2)+(-5/32)/(a*(1-a^2*x^2))+1/6*x*arctanh(a*x)/(1-a^2*x^2)^3+5/24*x*arctanh(a*x)/(1-a^2*x^2)^2+5/16*x*arctanh(a*x)/(1-a^2*x^2)+5/32*arctanh(a*x)^2/a],
[arctanh(a*x)^2/(1-a^2*x^2)^4,x,13,1/108*x/(1-a^2*x^2)^3+65/1728*x/(1-a^2*x^2)^2+245/1152*x/(1-a^2*x^2)+245/1152*arctanh(a*x)/a-1/18*arctanh(a*x)/(a*(1-a^2*x^2)^3)-5/48*arctanh(a*x)/(a*(1-a^2*x^2)^2)-5/16*arctanh(a*x)/(a*(1-a^2*x^2))+1/6*x*arctanh(a*x)^2/(1-a^2*x^2)^3+5/24*x*arctanh(a*x)^2/(1-a^2*x^2)^2+5/16*x*arctanh(a*x)^2/(1-a^2*x^2)+5/48*arctanh(a*x)^3/a],
[arctanh(a*x)^3/(1-a^2*x^2)^4,x,13,(-1/216)/(a*(1-a^2*x^2)^3)+(-65/2304)/(a*(1-a^2*x^2)^2)+(-245/768)/(a*(1-a^2*x^2))+1/36*x*arctanh(a*x)/(1-a^2*x^2)^3+65/576*x*arctanh(a*x)/(1-a^2*x^2)^2+245/384*x*arctanh(a*x)/(1-a^2*x^2)+245/768*arctanh(a*x)^2/a-1/12*arctanh(a*x)^2/(a*(1-a^2*x^2)^3)-5/32*arctanh(a*x)^2/(a*(1-a^2*x^2)^2)-15/32*arctanh(a*x)^2/(a*(1-a^2*x^2))+1/6*x*arctanh(a*x)^3/(1-a^2*x^2)^3+5/24*x*arctanh(a*x)^3/(1-a^2*x^2)^2+5/16*x*arctanh(a*x)^3/(1-a^2*x^2)+5/64*arctanh(a*x)^4/a],
[sqrt(arctanh(a*x))/(1-a^2*x^2)^4,x,21,5/24*arctanh(a*x)^(3/2)/a+1/768*erf(sqrt(6)*sqrt(arctanh(a*x)))*sqrt(1/6*Pi)/a-1/768*erfi(sqrt(6)*sqrt(arctanh(a*x)))*sqrt(1/6*Pi)/a+15/256*erf(sqrt(2)*sqrt(arctanh(a*x)))*sqrt(1/2*Pi)/a-15/256*erfi(sqrt(2)*sqrt(arctanh(a*x)))*sqrt(1/2*Pi)/a+3/512*erf(2*sqrt(arctanh(a*x)))*sqrt(Pi)/a-3/512*erfi(2*sqrt(arctanh(a*x)))*sqrt(Pi)/a+15/64*sinh(2*arctanh(a*x))*sqrt(arctanh(a*x))/a+3/64*sinh(4*arctanh(a*x))*sqrt(arctanh(a*x))/a+1/192*sinh(6*arctanh(a*x))*sqrt(arctanh(a*x))/a],

# p<0
[x^8/((1-a^2*x^2)^4*arctanh(a*x)),x,0,Unintegrable(x^8/((1-a^2*x^2)^4*arctanh(a*x)),x)],
[x^7/((1-a^2*x^2)^4*arctanh(a*x)),x,0,Unintegrable(x^7/((1-a^2*x^2)^4*arctanh(a*x)),x)],
[x^6/((1-a^2*x^2)^4*arctanh(a*x)),x,6,15/32*Chi(2*arctanh(a*x))/a^7-3/16*Chi(4*arctanh(a*x))/a^7+1/32*Chi(6*arctanh(a*x))/a^7-5/16*log(arctanh(a*x))/a^7],
[x^5/((1-a^2*x^2)^4*arctanh(a*x)),x,6,5/32*Shi(2*arctanh(a*x))/a^6-1/8*Shi(4*arctanh(a*x))/a^6+1/32*Shi(6*arctanh(a*x))/a^6],
[x^4/((1-a^2*x^2)^4*arctanh(a*x)),x,6,-1/32*Chi(2*arctanh(a*x))/a^5-1/16*Chi(4*arctanh(a*x))/a^5+1/32*Chi(6*arctanh(a*x))/a^5+1/16*log(arctanh(a*x))/a^5],
[x^3/((1-a^2*x^2)^4*arctanh(a*x)),x,5,-3/32*Shi(2*arctanh(a*x))/a^4+1/32*Shi(6*arctanh(a*x))/a^4],
[x^2/((1-a^2*x^2)^4*arctanh(a*x)),x,6,-1/32*Chi(2*arctanh(a*x))/a^3+1/16*Chi(4*arctanh(a*x))/a^3+1/32*Chi(6*arctanh(a*x))/a^3-1/16*log(arctanh(a*x))/a^3],
[x/((1-a^2*x^2)^4*arctanh(a*x)),x,6,5/32*Shi(2*arctanh(a*x))/a^2+1/8*Shi(4*arctanh(a*x))/a^2+1/32*Shi(6*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^4*arctanh(a*x)),x,6,15/32*Chi(2*arctanh(a*x))/a+3/16*Chi(4*arctanh(a*x))/a+1/32*Chi(6*arctanh(a*x))/a+5/16*log(arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^4*arctanh(a*x)),x,0,Unintegrable(1/(x*(1-a^2*x^2)^4*arctanh(a*x)),x)],
[1/(x^2*(1-a^2*x^2)^4*arctanh(a*x)),x,0,Unintegrable(1/(x^2*(1-a^2*x^2)^4*arctanh(a*x)),x)],
[x/((1-a^2*x^2)^4*arctanh(a*x)^2),x,13,-x/(a*(1-a^2*x^2)^3*arctanh(a*x))+5/16*Chi(2*arctanh(a*x))/a^2+1/2*Chi(4*arctanh(a*x))/a^2+3/16*Chi(6*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^4*arctanh(a*x)^2),x,7,(-1)/(a*(1-a^2*x^2)^3*arctanh(a*x))+15/16*Shi(2*arctanh(a*x))/a+3/4*Shi(4*arctanh(a*x))/a+3/16*Shi(6*arctanh(a*x))/a],
[x/((1-a^2*x^2)^4*arctanh(a*x)^3),x,22,-1/2*x/(a*(1-a^2*x^2)^3*arctanh(a*x)^2)+(-3)/(a^2*(1-a^2*x^2)^3*arctanh(a*x))+5/2/(a^2*(1-a^2*x^2)^2*arctanh(a*x))+5/16*Shi(2*arctanh(a*x))/a^2+Shi(4*arctanh(a*x))/a^2+9/16*Shi(6*arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^4*arctanh(a*x)^3),x,14,(-1/2)/(a*(1-a^2*x^2)^3*arctanh(a*x)^2)-3*x/((1-a^2*x^2)^3*arctanh(a*x))+15/16*Chi(2*arctanh(a*x))/a+3/2*Chi(4*arctanh(a*x))/a+9/16*Chi(6*arctanh(a*x))/a],

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

# p>0
[x^5*arctanh(a*x)/(1-a^2*x^2)^(1/2),x,9,89/120*arcsin(a*x)/a^6-5/24*x*sqrt(1-a^2*x^2)/a^5-1/20*x^3*sqrt(1-a^2*x^2)/a^3-8/15*arctanh(a*x)*sqrt(1-a^2*x^2)/a^6-4/15*x^2*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/5*x^4*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2],
[x^4*arctanh(a*x)/(1-a^2*x^2)^(1/2),x,7,1/12*(1-a^2*x^2)^(3/2)/a^5-3/4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^5-3/8*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^5+3/8*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^5-5/8*sqrt(1-a^2*x^2)/a^5-3/8*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/4*x^3*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2],
[x^3*arctanh(a*x)/(1-a^2*x^2)^(1/2),x,5,5/6*arcsin(a*x)/a^4-1/6*x*sqrt(1-a^2*x^2)/a^3-2/3*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/3*x^2*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2],
[x^2*arctanh(a*x)/(1-a^2*x^2)^(1/2),x,3,-arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^3-1/2*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+1/2*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^3-1/2*sqrt(1-a^2*x^2)/a^3-1/2*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2],
[x*arctanh(a*x)/(1-a^2*x^2)^(1/2),x,2,arcsin(a*x)/a^2-arctanh(a*x)*sqrt(1-a^2*x^2)/a^2],
[arctanh(a*x)/(1-a^2*x^2)^(1/2),x,1,-2*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a-I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a+I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a],
[arctanh(a*x)/(x*(1-a^2*x^2)^(1/2)),x,1,-2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-polylog(2,sqrt(1-a*x)/sqrt(1+a*x))],
[arctanh(a*x)/(x^2*(1-a^2*x^2)^(1/2)),x,4,-a*arctanh(sqrt(1-a^2*x^2))-arctanh(a*x)*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)/(x^3*(1-a^2*x^2)^(1/2)),x,3,-a^2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+1/2*a^2*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-1/2*a^2*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/2*a*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[x^3*arctanh(a*x)^2/(1-a^2*x^2)^(1/2),x,6,-10/3*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^4-5/3*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^4+5/3*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^4-1/3*sqrt(1-a^2*x^2)/a^4-1/3*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^3-2/3*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^4-1/3*x^2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^2],
[x^2*arctanh(a*x)^2/(1-a^2*x^2)^(1/2),x,11,arcsin(a*x)/a^3+arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^3-I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^3+I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^3+I*polylog(3,-I*exp(arctanh(a*x)))/a^3-I*polylog(3,I*exp(arctanh(a*x)))/a^3-arctanh(a*x)*sqrt(1-a^2*x^2)/a^3-1/2*x*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^2],
[x*arctanh(a*x)^2/(1-a^2*x^2)^(1/2),x,2,-4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^2-2*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^2+2*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^2-arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^2],
[arctanh(a*x)^2/(1-a^2*x^2)^(1/2),x,8,2*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a-2*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a+2*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a+2*I*polylog(3,-I*exp(arctanh(a*x)))/a-2*I*polylog(3,I*exp(arctanh(a*x)))/a],
[arctanh(a*x)^2/(x*(1-a^2*x^2)^(1/2)),x,8,-2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-2*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+2*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+2*polylog(3,-exp(arctanh(a*x)))-2*polylog(3,exp(arctanh(a*x)))],
[arctanh(a*x)^2/(x^2*(1-a^2*x^2)^(1/2)),x,2,-4*a*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+2*a*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-2*a*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-arctanh(a*x)^2*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)^2/(x^3*(1-a^2*x^2)^(1/2)),x,13,-a^2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-a^2*arctanh(sqrt(1-a^2*x^2))-a^2*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+a^2*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+a^2*polylog(3,-exp(arctanh(a*x)))-a^2*polylog(3,exp(arctanh(a*x)))-a*arctanh(a*x)*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/x^2],
[x^3*arctanh(a*x)^3/(1-a^2*x^2)^(1/2),x,21,arcsin(a*x)/a^4+5*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^4-5*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^4+5*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^4+5*I*polylog(3,-I*exp(arctanh(a*x)))/a^4-5*I*polylog(3,I*exp(arctanh(a*x)))/a^4-arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/2*x*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^3-2/3*arctanh(a*x)^3*sqrt(1-a^2*x^2)/a^4-1/3*x^2*arctanh(a*x)^3*sqrt(1-a^2*x^2)/a^2],
[x^2*arctanh(a*x)^3/(1-a^2*x^2)^(1/2),x,13,-6*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^3+arctan(exp(arctanh(a*x)))*arctanh(a*x)^3/a^3-3/2*I*arctanh(a*x)^2*polylog(2,-I*exp(arctanh(a*x)))/a^3+3/2*I*arctanh(a*x)^2*polylog(2,I*exp(arctanh(a*x)))/a^3-3*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+3*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+3*I*arctanh(a*x)*polylog(3,-I*exp(arctanh(a*x)))/a^3-3*I*arctanh(a*x)*polylog(3,I*exp(arctanh(a*x)))/a^3-3*I*polylog(4,-I*exp(arctanh(a*x)))/a^3+3*I*polylog(4,I*exp(arctanh(a*x)))/a^3-3/2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^3-1/2*x*arctanh(a*x)^3*sqrt(1-a^2*x^2)/a^2],
[x*arctanh(a*x)^3/(1-a^2*x^2)^(1/2),x,9,6*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^2-6*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^2+6*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^2+6*I*polylog(3,-I*exp(arctanh(a*x)))/a^2-6*I*polylog(3,I*exp(arctanh(a*x)))/a^2-arctanh(a*x)^3*sqrt(1-a^2*x^2)/a^2],
[arctanh(a*x)^3/(1-a^2*x^2)^(1/2),x,10,2*arctan(exp(arctanh(a*x)))*arctanh(a*x)^3/a-3*I*arctanh(a*x)^2*polylog(2,-I*exp(arctanh(a*x)))/a+3*I*arctanh(a*x)^2*polylog(2,I*exp(arctanh(a*x)))/a+6*I*arctanh(a*x)*polylog(3,-I*exp(arctanh(a*x)))/a-6*I*arctanh(a*x)*polylog(3,I*exp(arctanh(a*x)))/a-6*I*polylog(4,-I*exp(arctanh(a*x)))/a+6*I*polylog(4,I*exp(arctanh(a*x)))/a],
[arctanh(a*x)^3/(x*(1-a^2*x^2)^(1/2)),x,10,-2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^3-3*arctanh(a*x)^2*polylog(2,-exp(arctanh(a*x)))+3*arctanh(a*x)^2*polylog(2,exp(arctanh(a*x)))+6*arctanh(a*x)*polylog(3,-exp(arctanh(a*x)))-6*arctanh(a*x)*polylog(3,exp(arctanh(a*x)))-6*polylog(4,-exp(arctanh(a*x)))+6*polylog(4,exp(arctanh(a*x)))],
[arctanh(a*x)^3/(x^2*(1-a^2*x^2)^(1/2)),x,9,-6*a*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-6*a*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+6*a*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+6*a*polylog(3,-exp(arctanh(a*x)))-6*a*polylog(3,exp(arctanh(a*x)))-arctanh(a*x)^3*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)^3/(x^3*(1-a^2*x^2)^(1/2)),x,13,-a^2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^3-6*a^2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-3/2*a^2*arctanh(a*x)^2*polylog(2,-exp(arctanh(a*x)))+3/2*a^2*arctanh(a*x)^2*polylog(2,exp(arctanh(a*x)))+3*a^2*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-3*a^2*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))+3*a^2*arctanh(a*x)*polylog(3,-exp(arctanh(a*x)))-3*a^2*arctanh(a*x)*polylog(3,exp(arctanh(a*x)))-3*a^2*polylog(4,-exp(arctanh(a*x)))+3*a^2*polylog(4,exp(arctanh(a*x)))-3/2*a*arctanh(a*x)^2*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)^3*sqrt(1-a^2*x^2)/x^2],

# p<0

# Integrands of the form x^m ArcTanh[c x]^p / (1-c^2 x^2)^(3/2)

# p>0
[x^m*arctanh(a*x)/(1-a^2*x^2)^(3/2),x,0,Unintegrable(x^m*arctanh(a*x)/(1-a^2*x^2)^(3/2),x)],
[x^3*arctanh(a*x)/(1-a^2*x^2)^(3/2),x,5,-arcsin(a*x)/a^4-x/(a^3*sqrt(1-a^2*x^2))+arctanh(a*x)/(a^4*sqrt(1-a^2*x^2))+arctanh(a*x)*sqrt(1-a^2*x^2)/a^4],
[x^2*arctanh(a*x)/(1-a^2*x^2)^(3/2),x,2,2*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^3+I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^3-I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+(-1)/(a^3*sqrt(1-a^2*x^2))+x*arctanh(a*x)/(a^2*sqrt(1-a^2*x^2))],
[x*arctanh(a*x)/(1-a^2*x^2)^(3/2),x,2,-x/(a*sqrt(1-a^2*x^2))+arctanh(a*x)/(a^2*sqrt(1-a^2*x^2))],
[arctanh(a*x)/(1-a^2*x^2)^(3/2),x,1,(-1)/(a*sqrt(1-a^2*x^2))+x*arctanh(a*x)/sqrt(1-a^2*x^2)],
[arctanh(a*x)/(x*(1-a^2*x^2)^(3/2)),x,4,-2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-a*x/sqrt(1-a^2*x^2)+arctanh(a*x)/sqrt(1-a^2*x^2)],
[arctanh(a*x)/(x^2*(1-a^2*x^2)^(3/2)),x,6,-a*arctanh(sqrt(1-a^2*x^2))-a/sqrt(1-a^2*x^2)+a^2*x*arctanh(a*x)/sqrt(1-a^2*x^2)-arctanh(a*x)*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)/(x^3*(1-a^2*x^2)^(3/2)),x,8,-3*a^2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+3/2*a^2*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-3/2*a^2*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-a^3*x/sqrt(1-a^2*x^2)+a^2*arctanh(a*x)/sqrt(1-a^2*x^2)-1/2*a*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[x^m*arctanh(a*x)^2/(1-a^2*x^2)^(3/2),x,0,Unintegrable(x^m*arctanh(a*x)^2/(1-a^2*x^2)^(3/2),x)],
[x^3*arctanh(a*x)^2/(1-a^2*x^2)^(3/2),x,5,4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^4+2*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^4-2*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^4+2/(a^4*sqrt(1-a^2*x^2))-2*x*arctanh(a*x)/(a^3*sqrt(1-a^2*x^2))+arctanh(a*x)^2/(a^4*sqrt(1-a^2*x^2))+arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^4],
[x^2*arctanh(a*x)^2/(1-a^2*x^2)^(3/2),x,11,-2*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^3+2*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^3-2*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^3-2*I*polylog(3,-I*exp(arctanh(a*x)))/a^3+2*I*polylog(3,I*exp(arctanh(a*x)))/a^3+2*x/(a^2*sqrt(1-a^2*x^2))-2*arctanh(a*x)/(a^3*sqrt(1-a^2*x^2))+x*arctanh(a*x)^2/(a^2*sqrt(1-a^2*x^2))],
[x*arctanh(a*x)^2/(1-a^2*x^2)^(3/2),x,2,2/(a^2*sqrt(1-a^2*x^2))-2*x*arctanh(a*x)/(a*sqrt(1-a^2*x^2))+arctanh(a*x)^2/(a^2*sqrt(1-a^2*x^2))],
[arctanh(a*x)^2/(1-a^2*x^2)^(3/2),x,2,2*x/sqrt(1-a^2*x^2)-2*arctanh(a*x)/(a*sqrt(1-a^2*x^2))+x*arctanh(a*x)^2/sqrt(1-a^2*x^2)],
[arctanh(a*x)^2/(x*(1-a^2*x^2)^(3/2)),x,11,-2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-2*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+2*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+2*polylog(3,-exp(arctanh(a*x)))-2*polylog(3,exp(arctanh(a*x)))+2/sqrt(1-a^2*x^2)-2*a*x*arctanh(a*x)/sqrt(1-a^2*x^2)+arctanh(a*x)^2/sqrt(1-a^2*x^2)],
[arctanh(a*x)^2/(x^2*(1-a^2*x^2)^(3/2)),x,5,-4*a*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+2*a*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-2*a*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))+2*a^2*x/sqrt(1-a^2*x^2)-2*a*arctanh(a*x)/sqrt(1-a^2*x^2)+a^2*x*arctanh(a*x)^2/sqrt(1-a^2*x^2)-arctanh(a*x)^2*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)^2/(x^3*(1-a^2*x^2)^(3/2)),x,25,-3*a^2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-a^2*arctanh(sqrt(1-a^2*x^2))-3*a^2*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+3*a^2*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+3*a^2*polylog(3,-exp(arctanh(a*x)))-3*a^2*polylog(3,exp(arctanh(a*x)))+2*a^2/sqrt(1-a^2*x^2)-2*a^3*x*arctanh(a*x)/sqrt(1-a^2*x^2)+a^2*arctanh(a*x)^2/sqrt(1-a^2*x^2)-a*arctanh(a*x)*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/x^2],
[x^m*arctanh(a*x)^3/(1-a^2*x^2)^(3/2),x,0,Unintegrable(x^m*arctanh(a*x)^3/(1-a^2*x^2)^(3/2),x)],
[x^3*arctanh(a*x)^3/(1-a^2*x^2)^(3/2),x,13,-6*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^4+6*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^4-6*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^4-6*I*polylog(3,-I*exp(arctanh(a*x)))/a^4+6*I*polylog(3,I*exp(arctanh(a*x)))/a^4-6*x/(a^3*sqrt(1-a^2*x^2))+6*arctanh(a*x)/(a^4*sqrt(1-a^2*x^2))-3*x*arctanh(a*x)^2/(a^3*sqrt(1-a^2*x^2))+arctanh(a*x)^3/(a^4*sqrt(1-a^2*x^2))+arctanh(a*x)^3*sqrt(1-a^2*x^2)/a^4],
[x^2*arctanh(a*x)^3/(1-a^2*x^2)^(3/2),x,13,-2*arctan(exp(arctanh(a*x)))*arctanh(a*x)^3/a^3+3*I*arctanh(a*x)^2*polylog(2,-I*exp(arctanh(a*x)))/a^3-3*I*arctanh(a*x)^2*polylog(2,I*exp(arctanh(a*x)))/a^3-6*I*arctanh(a*x)*polylog(3,-I*exp(arctanh(a*x)))/a^3+6*I*arctanh(a*x)*polylog(3,I*exp(arctanh(a*x)))/a^3+6*I*polylog(4,-I*exp(arctanh(a*x)))/a^3-6*I*polylog(4,I*exp(arctanh(a*x)))/a^3+(-6)/(a^3*sqrt(1-a^2*x^2))+6*x*arctanh(a*x)/(a^2*sqrt(1-a^2*x^2))-3*arctanh(a*x)^2/(a^3*sqrt(1-a^2*x^2))+x*arctanh(a*x)^3/(a^2*sqrt(1-a^2*x^2))],
[x*arctanh(a*x)^3/(1-a^2*x^2)^(3/2),x,3,-6*x/(a*sqrt(1-a^2*x^2))+6*arctanh(a*x)/(a^2*sqrt(1-a^2*x^2))-3*x*arctanh(a*x)^2/(a*sqrt(1-a^2*x^2))+arctanh(a*x)^3/(a^2*sqrt(1-a^2*x^2))],
[arctanh(a*x)^3/(1-a^2*x^2)^(3/2),x,2,(-6)/(a*sqrt(1-a^2*x^2))+6*x*arctanh(a*x)/sqrt(1-a^2*x^2)-3*arctanh(a*x)^2/(a*sqrt(1-a^2*x^2))+x*arctanh(a*x)^3/sqrt(1-a^2*x^2)],
[arctanh(a*x)^3/(x*(1-a^2*x^2)^(3/2)),x,14,-2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^3-3*arctanh(a*x)^2*polylog(2,-exp(arctanh(a*x)))+3*arctanh(a*x)^2*polylog(2,exp(arctanh(a*x)))+6*arctanh(a*x)*polylog(3,-exp(arctanh(a*x)))-6*arctanh(a*x)*polylog(3,exp(arctanh(a*x)))-6*polylog(4,-exp(arctanh(a*x)))+6*polylog(4,exp(arctanh(a*x)))-6*a*x/sqrt(1-a^2*x^2)+6*arctanh(a*x)/sqrt(1-a^2*x^2)-3*a*x*arctanh(a*x)^2/sqrt(1-a^2*x^2)+arctanh(a*x)^3/sqrt(1-a^2*x^2)],
[arctanh(a*x)^3/(x^2*(1-a^2*x^2)^(3/2)),x,12,-6*a*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-6*a*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+6*a*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+6*a*polylog(3,-exp(arctanh(a*x)))-6*a*polylog(3,exp(arctanh(a*x)))-6*a/sqrt(1-a^2*x^2)+6*a^2*x*arctanh(a*x)/sqrt(1-a^2*x^2)-3*a*arctanh(a*x)^2/sqrt(1-a^2*x^2)+a^2*x*arctanh(a*x)^3/sqrt(1-a^2*x^2)-arctanh(a*x)^3*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)^3/(x^3*(1-a^2*x^2)^(3/2)),x,28,-3*a^2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^3-6*a^2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-9/2*a^2*arctanh(a*x)^2*polylog(2,-exp(arctanh(a*x)))+9/2*a^2*arctanh(a*x)^2*polylog(2,exp(arctanh(a*x)))+3*a^2*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-3*a^2*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))+9*a^2*arctanh(a*x)*polylog(3,-exp(arctanh(a*x)))-9*a^2*arctanh(a*x)*polylog(3,exp(arctanh(a*x)))-9*a^2*polylog(4,-exp(arctanh(a*x)))+9*a^2*polylog(4,exp(arctanh(a*x)))-6*a^3*x/sqrt(1-a^2*x^2)+6*a^2*arctanh(a*x)/sqrt(1-a^2*x^2)-3*a^3*x*arctanh(a*x)^2/sqrt(1-a^2*x^2)+a^2*arctanh(a*x)^3/sqrt(1-a^2*x^2)-3/2*a*arctanh(a*x)^2*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)^3*sqrt(1-a^2*x^2)/x^2],

# p<0
[x^m/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x,0,Unintegrable(x^m/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x)],
[x^2/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x,0,Unintegrable(x^2/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x)],
[x/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x,2,Shi(arctanh(a*x))/a^2],
[1/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x,2,Chi(arctanh(a*x))/a],
[1/(x*(1-a^2*x^2)^(3/2)*arctanh(a*x)),x,0,Unintegrable(1/(x*(1-a^2*x^2)^(3/2)*arctanh(a*x)),x)],
[x^m/((1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x,0,Unintegrable(x^m/((1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x)],
[x^2/((1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x,4,Shi(arctanh(a*x))/a^3+(-1)/(a^3*arctanh(a*x)*sqrt(1-a^2*x^2))-Unintegrable(1/(arctanh(a*x)^2*sqrt(1-a^2*x^2)),x)/a^2],
[x/((1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x,3,Chi(arctanh(a*x))/a^2-x/(a*arctanh(a*x)*sqrt(1-a^2*x^2))],
[1/((1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x,3,Shi(arctanh(a*x))/a+(-1)/(a*arctanh(a*x)*sqrt(1-a^2*x^2))],
[1/(x*(1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x,5,Chi(arctanh(a*x))-a*x/(arctanh(a*x)*sqrt(1-a^2*x^2))-sqrt(1-a^2*x^2)/(a*x*arctanh(a*x))-Unintegrable(1/(x^2*arctanh(a*x)*sqrt(1-a^2*x^2)),x)/a],
[x^m/((1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x,0,Unintegrable(x^m/((1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x)],
[x^2/((1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x,5,1/2*Chi(arctanh(a*x))/a^3+(-1/2)/(a^3*arctanh(a*x)^2*sqrt(1-a^2*x^2))-1/2*x/(a^2*arctanh(a*x)*sqrt(1-a^2*x^2))-Unintegrable(1/(arctanh(a*x)^3*sqrt(1-a^2*x^2)),x)/a^2],
[x/((1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x,4,1/2*Shi(arctanh(a*x))/a^2-1/2*x/(a*arctanh(a*x)^2*sqrt(1-a^2*x^2))+(-1/2)/(a^2*arctanh(a*x)*sqrt(1-a^2*x^2))],
[1/((1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x,4,1/2*Chi(arctanh(a*x))/a+(-1/2)/(a*arctanh(a*x)^2*sqrt(1-a^2*x^2))-1/2*x/(arctanh(a*x)*sqrt(1-a^2*x^2))],
[1/(x*(1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x,6,1/2*Shi(arctanh(a*x))-1/2*a*x/(arctanh(a*x)^2*sqrt(1-a^2*x^2))+(-1/2)/(arctanh(a*x)*sqrt(1-a^2*x^2))-1/2*sqrt(1-a^2*x^2)/(a*x*arctanh(a*x)^2)-1/2*Unintegrable(1/(x^2*arctanh(a*x)^2*sqrt(1-a^2*x^2)),x)/a],

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

# p>0
[x^4*arctanh(a*x)*sqrt(1-a^2*x^2),x,11,-7/72*(1-a^2*x^2)^(3/2)/a^5+1/30*(1-a^2*x^2)^(5/2)/a^5-1/8*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^5-1/16*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^5+1/16*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^5+1/16*sqrt(1-a^2*x^2)/a^5-1/16*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/24*x^3*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2+1/6*x^5*arctanh(a*x)*sqrt(1-a^2*x^2)],
[x^3*arctanh(a*x)*sqrt(1-a^2*x^2),x,9,11/120*arcsin(a*x)/a^4+1/24*x*sqrt(1-a^2*x^2)/a^3+1/20*x^3*sqrt(1-a^2*x^2)/a-2/15*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/15*x^2*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2+1/5*x^4*arctanh(a*x)*sqrt(1-a^2*x^2)],
[x^2*arctanh(a*x)*sqrt(1-a^2*x^2),x,7,-1/12*(1-a^2*x^2)^(3/2)/a^3-1/4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^3-1/8*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+1/8*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+1/8*sqrt(1-a^2*x^2)/a^3-1/8*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2+1/4*x^3*arctanh(a*x)*sqrt(1-a^2*x^2)],
[x*arctanh(a*x)*sqrt(1-a^2*x^2),x,3,1/6*arcsin(a*x)/a^2-1/3*(1-a^2*x^2)^(3/2)*arctanh(a*x)/a^2+1/6*x*sqrt(1-a^2*x^2)/a],
[arctanh(a*x)*sqrt(1-a^2*x^2),x,2,-arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a-1/2*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a+1/2*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a+1/2*sqrt(1-a^2*x^2)/a+1/2*x*arctanh(a*x)*sqrt(1-a^2*x^2)],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x,x,3,-arcsin(a*x)-2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-polylog(2,sqrt(1-a*x)/sqrt(1+a*x))+arctanh(a*x)*sqrt(1-a^2*x^2)],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x^2,x,6,2*a*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)-a*arctanh(sqrt(1-a^2*x^2))+I*a*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))-I*a*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))-arctanh(a*x)*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x^3,x,5,a^2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-1/2*a^2*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))+1/2*a^2*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/2*a*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x^4,x,5,-1/3*(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^3+1/6*a^3*arctanh(sqrt(1-a^2*x^2))-1/6*a*sqrt(1-a^2*x^2)/x^2],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x^5,x,9,1/4*a^4*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-1/8*a^4*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))+1/8*a^4*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/12*a*sqrt(1-a^2*x^2)/x^3-1/24*a^3*sqrt(1-a^2*x^2)/x-1/4*arctanh(a*x)*sqrt(1-a^2*x^2)/x^4+1/8*a^2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x^6,x,21,11/120*a^5*arctanh(sqrt(1-a^2*x^2))-1/20*a*sqrt(1-a^2*x^2)/x^4-1/24*a^3*sqrt(1-a^2*x^2)/x^2-1/5*arctanh(a*x)*sqrt(1-a^2*x^2)/x^5+1/15*a^2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^3+2/15*a^4*arctanh(a*x)*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)*sqrt(1-a^2*x^2)/x^7,x,14,1/8*a^6*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-1/16*a^6*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))+1/16*a^6*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/30*a*sqrt(1-a^2*x^2)/x^5-11/360*a^3*sqrt(1-a^2*x^2)/x^3+1/720*a^5*sqrt(1-a^2*x^2)/x-1/6*arctanh(a*x)*sqrt(1-a^2*x^2)/x^6+1/24*a^2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^4+1/16*a^4*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[x^4*arctanh(a*x)^2*sqrt(1-a^2*x^2),x,45,-19/360*arcsin(a*x)/a^5+1/8*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^5-1/8*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^5+1/8*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^5+1/8*I*polylog(3,-I*exp(arctanh(a*x)))/a^5-1/8*I*polylog(3,I*exp(arctanh(a*x)))/a^5+1/18*x*sqrt(1-a^2*x^2)/a^4+1/60*x^3*sqrt(1-a^2*x^2)/a^2-1/360*arctanh(a*x)*sqrt(1-a^2*x^2)/a^5+11/180*x^2*arctanh(a*x)*sqrt(1-a^2*x^2)/a^3+1/15*x^4*arctanh(a*x)*sqrt(1-a^2*x^2)/a-1/16*x*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^4-1/24*x^3*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^2+1/6*x^5*arctanh(a*x)^2*sqrt(1-a^2*x^2)],
[x^3*arctanh(a*x)^2*sqrt(1-a^2*x^2),x,21,-1/30*(1-a^2*x^2)^(3/2)/a^4-11/30*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^4-11/60*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^4+11/60*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^4+11/60*sqrt(1-a^2*x^2)/a^4+1/12*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^3+1/10*x^3*arctanh(a*x)*sqrt(1-a^2*x^2)/a-2/15*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^4-1/15*x^2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^2+1/5*x^4*arctanh(a*x)^2*sqrt(1-a^2*x^2)],
[x^2*arctanh(a*x)^2*sqrt(1-a^2*x^2),x,29,-1/6*arcsin(a*x)/a^3+1/4*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a^3-1/4*I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a^3+1/4*I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a^3+1/4*I*polylog(3,-I*exp(arctanh(a*x)))/a^3-1/4*I*polylog(3,I*exp(arctanh(a*x)))/a^3+1/12*x*sqrt(1-a^2*x^2)/a^2+1/12*arctanh(a*x)*sqrt(1-a^2*x^2)/a^3+1/6*x^2*arctanh(a*x)*sqrt(1-a^2*x^2)/a-1/8*x*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a^2+1/4*x^3*arctanh(a*x)^2*sqrt(1-a^2*x^2)],
[x*arctanh(a*x)^2*sqrt(1-a^2*x^2),x,3,-2/3*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^2-1/3*(1-a^2*x^2)^(3/2)*arctanh(a*x)^2/a^2-1/3*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^2+1/3*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^2+1/3*sqrt(1-a^2*x^2)/a^2+1/3*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a],
[arctanh(a*x)^2*sqrt(1-a^2*x^2),x,10,-arcsin(a*x)/a+arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a-I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a+I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a+I*polylog(3,-I*exp(arctanh(a*x)))/a-I*polylog(3,I*exp(arctanh(a*x)))/a+arctanh(a*x)*sqrt(1-a^2*x^2)/a+1/2*x*arctanh(a*x)^2*sqrt(1-a^2*x^2)],
[arctanh(a*x)^2*sqrt(1-a^2*x^2)/x,x,11,4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)-2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-2*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))+2*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))+2*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))-2*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))+2*polylog(3,-exp(arctanh(a*x)))-2*polylog(3,exp(arctanh(a*x)))+arctanh(a*x)^2*sqrt(1-a^2*x^2)],
[arctanh(a*x)^2*sqrt(1-a^2*x^2)/x^2,x,11,-2*a*arctan(exp(arctanh(a*x)))*arctanh(a*x)^2-4*a*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+2*I*a*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))-2*I*a*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))+2*a*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-2*a*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-2*I*a*polylog(3,-I*exp(arctanh(a*x)))+2*I*a*polylog(3,I*exp(arctanh(a*x)))-arctanh(a*x)^2*sqrt(1-a^2*x^2)/x],
[arctanh(a*x)^2*sqrt(1-a^2*x^2)/x^3,x,22,a^2*arctanh(exp(arctanh(a*x)))*arctanh(a*x)^2-a^2*arctanh(sqrt(1-a^2*x^2))+a^2*arctanh(a*x)*polylog(2,-exp(arctanh(a*x)))-a^2*arctanh(a*x)*polylog(2,exp(arctanh(a*x)))-a^2*polylog(3,-exp(arctanh(a*x)))+a^2*polylog(3,exp(arctanh(a*x)))-a*arctanh(a*x)*sqrt(1-a^2*x^2)/x-1/2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/x^2],
[arctanh(a*x)^2*sqrt(1-a^2*x^2)/x^4,x,6,-1/3*(1-a^2*x^2)^(3/2)*arctanh(a*x)^2/x^3+2/3*a^3*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-1/3*a^3*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))+1/3*a^3*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/3*a^2*sqrt(1-a^2*x^2)/x-1/3*a*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],

# p<0

# Integrands of the form x^m ArcTanh[c x]^p (1-c^2 x^2)^(3/2)

# p>0
[x^4*(1-a^2*x^2)^(3/2)*arctanh(a*x),x,27,1/192*(1-a^2*x^2)^(3/2)/a^5-3/80*(1-a^2*x^2)^(5/2)/a^5+1/56*(1-a^2*x^2)^(7/2)/a^5-3/64*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^5-3/128*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^5+3/128*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^5+3/128*sqrt(1-a^2*x^2)/a^5-3/128*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/64*x^3*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2+3/16*x^5*arctanh(a*x)*sqrt(1-a^2*x^2)-1/8*a^2*x^7*arctanh(a*x)*sqrt(1-a^2*x^2)],
[x^3*(1-a^2*x^2)^(3/2)*arctanh(a*x),x,24,17/560*arcsin(a*x)/a^4+3/112*x*sqrt(1-a^2*x^2)/a^3+23/840*x^3*sqrt(1-a^2*x^2)/a-1/42*a*x^5*sqrt(1-a^2*x^2)-2/35*arctanh(a*x)*sqrt(1-a^2*x^2)/a^4-1/35*x^2*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2+8/35*x^4*arctanh(a*x)*sqrt(1-a^2*x^2)-1/7*a^2*x^6*arctanh(a*x)*sqrt(1-a^2*x^2)],
[x^2*(1-a^2*x^2)^(3/2)*arctanh(a*x),x,19,1/72*(1-a^2*x^2)^(3/2)/a^3-1/30*(1-a^2*x^2)^(5/2)/a^3-1/8*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a^3-1/16*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+1/16*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a^3+1/16*sqrt(1-a^2*x^2)/a^3-1/16*x*arctanh(a*x)*sqrt(1-a^2*x^2)/a^2+7/24*x^3*arctanh(a*x)*sqrt(1-a^2*x^2)-1/6*a^2*x^5*arctanh(a*x)*sqrt(1-a^2*x^2)],
[x*(1-a^2*x^2)^(3/2)*arctanh(a*x),x,4,1/20*x*(1-a^2*x^2)^(3/2)/a+3/40*arcsin(a*x)/a^2-1/5*(1-a^2*x^2)^(5/2)*arctanh(a*x)/a^2+3/40*x*sqrt(1-a^2*x^2)/a],
[(1-a^2*x^2)^(3/2)*arctanh(a*x),x,3,1/12*(1-a^2*x^2)^(3/2)/a+1/4*x*(1-a^2*x^2)^(3/2)*arctanh(a*x)-3/4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a-3/8*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a+3/8*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a+3/8*sqrt(1-a^2*x^2)/a+3/8*x*arctanh(a*x)*sqrt(1-a^2*x^2)],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x,x,7,-7/6*arcsin(a*x)+1/3*(1-a^2*x^2)^(3/2)*arctanh(a*x)-2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/6*a*x*sqrt(1-a^2*x^2)+arctanh(a*x)*sqrt(1-a^2*x^2)],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^2,x,9,3*a*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)-a*arctanh(sqrt(1-a^2*x^2))+3/2*I*a*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))-3/2*I*a*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))-1/2*a*sqrt(1-a^2*x^2)-arctanh(a*x)*sqrt(1-a^2*x^2)/x-1/2*a^2*x*arctanh(a*x)*sqrt(1-a^2*x^2)],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^3,x,9,a^2*arcsin(a*x)+3*a^2*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))-3/2*a^2*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))+3/2*a^2*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/2*a*sqrt(1-a^2*x^2)/x-a^2*arctanh(a*x)*sqrt(1-a^2*x^2)-1/2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^4,x,12,-1/3*(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^3-2*a^3*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)+7/6*a^3*arctanh(sqrt(1-a^2*x^2))-I*a^3*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))+I*a^3*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))-1/6*a*sqrt(1-a^2*x^2)/x^2+a^2*arctanh(a*x)*sqrt(1-a^2*x^2)/x],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^5,x,15,-3/4*a^4*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+3/8*a^4*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-3/8*a^4*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/12*a*sqrt(1-a^2*x^2)/x^3+11/24*a^3*sqrt(1-a^2*x^2)/x-1/4*arctanh(a*x)*sqrt(1-a^2*x^2)/x^4+5/8*a^2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^6,x,6,-1/20*a*(1-a^2*x^2)^(3/2)/x^4-1/5*(1-a^2*x^2)^(5/2)*arctanh(a*x)/x^5-3/40*a^5*arctanh(sqrt(1-a^2*x^2))+3/40*a^3*sqrt(1-a^2*x^2)/x^2],
[(1-a^2*x^2)^(3/2)*arctanh(a*x)/x^7,x,24,-1/8*a^6*arctanh(a*x)*arctanh(sqrt(1-a*x)/sqrt(1+a*x))+1/16*a^6*polylog(2,-sqrt(1-a*x)/sqrt(1+a*x))-1/16*a^6*polylog(2,sqrt(1-a*x)/sqrt(1+a*x))-1/30*a*sqrt(1-a^2*x^2)/x^5+19/360*a^3*sqrt(1-a^2*x^2)/x^3+31/720*a^5*sqrt(1-a^2*x^2)/x-1/6*arctanh(a*x)*sqrt(1-a^2*x^2)/x^6+7/24*a^2*arctanh(a*x)*sqrt(1-a^2*x^2)/x^4-1/16*a^4*arctanh(a*x)*sqrt(1-a^2*x^2)/x^2],

# p<0

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

# p>0
[(1-a^2*x^2)^(5/2)*arctanh(a*x),x,4,5/72*(1-a^2*x^2)^(3/2)/a+1/30*(1-a^2*x^2)^(5/2)/a+5/24*x*(1-a^2*x^2)^(3/2)*arctanh(a*x)+1/6*x*(1-a^2*x^2)^(5/2)*arctanh(a*x)-5/8*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a-5/16*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a+5/16*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a+5/16*sqrt(1-a^2*x^2)/a+5/16*x*arctanh(a*x)*sqrt(1-a^2*x^2)],
[(1-a^2*x^2)^(3/2)*arctanh(a*x),x,3,1/12*(1-a^2*x^2)^(3/2)/a+1/4*x*(1-a^2*x^2)^(3/2)*arctanh(a*x)-3/4*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a-3/8*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a+3/8*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a+3/8*sqrt(1-a^2*x^2)/a+3/8*x*arctanh(a*x)*sqrt(1-a^2*x^2)],
[(1-a^2*x^2)^(1/2)*arctanh(a*x),x,2,-arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a-1/2*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a+1/2*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a+1/2*sqrt(1-a^2*x^2)/a+1/2*x*arctanh(a*x)*sqrt(1-a^2*x^2)],
[arctanh(a*x)/(1-a^2*x^2)^(5/2),x,2,(-1/9)/(a*(1-a^2*x^2)^(3/2))+1/3*x*arctanh(a*x)/(1-a^2*x^2)^(3/2)+(-2/3)/(a*sqrt(1-a^2*x^2))+2/3*x*arctanh(a*x)/sqrt(1-a^2*x^2)],
[arctanh(a*x)/(1-a^2*x^2)^(7/2),x,3,(-1/25)/(a*(1-a^2*x^2)^(5/2))+(-4/45)/(a*(1-a^2*x^2)^(3/2))+1/5*x*arctanh(a*x)/(1-a^2*x^2)^(5/2)+4/15*x*arctanh(a*x)/(1-a^2*x^2)^(3/2)+(-8/15)/(a*sqrt(1-a^2*x^2))+8/15*x*arctanh(a*x)/sqrt(1-a^2*x^2)],
[arctanh(a*x)/(1-a^2*x^2)^(9/2),x,4,(-1/49)/(a*(1-a^2*x^2)^(7/2))+(-6/175)/(a*(1-a^2*x^2)^(5/2))+(-8/105)/(a*(1-a^2*x^2)^(3/2))+1/7*x*arctanh(a*x)/(1-a^2*x^2)^(7/2)+6/35*x*arctanh(a*x)/(1-a^2*x^2)^(5/2)+8/35*x*arctanh(a*x)/(1-a^2*x^2)^(3/2)+(-16/35)/(a*sqrt(1-a^2*x^2))+16/35*x*arctanh(a*x)/sqrt(1-a^2*x^2)],
[(c-a^2*c*x^2)^(3/2)*arctanh(a*x),x,4,1/12*(c-a^2*c*x^2)^(3/2)/a+1/4*x*(c-a^2*c*x^2)^(3/2)*arctanh(a*x)-3/4*c^2*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-3/8*I*c^2*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))+3/8*I*c^2*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))+3/8*c*sqrt(c-a^2*c*x^2)/a+3/8*c*x*arctanh(a*x)*sqrt(c-a^2*c*x^2)],
[(c-a^2*c*x^2)^(1/2)*arctanh(a*x),x,3,-c*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-1/2*I*c*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))+1/2*I*c*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))+1/2*sqrt(c-a^2*c*x^2)/a+1/2*x*arctanh(a*x)*sqrt(c-a^2*c*x^2)],
[arctanh(a*x)/(c-a^2*c*x^2)^(1/2),x,2,-2*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))-I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))+I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))*sqrt(1-a^2*x^2)/(a*sqrt(c-a^2*c*x^2))],
[arctanh(a*x)/(c-a^2*c*x^2)^(3/2),x,1,(-1)/(a*c*sqrt(c-a^2*c*x^2))+x*arctanh(a*x)/(c*sqrt(c-a^2*c*x^2))],
[arctanh(a*x)/(c-a^2*c*x^2)^(5/2),x,2,(-1/9)/(a*c*(c-a^2*c*x^2)^(3/2))+1/3*x*arctanh(a*x)/(c*(c-a^2*c*x^2)^(3/2))+(-2/3)/(a*c^2*sqrt(c-a^2*c*x^2))+2/3*x*arctanh(a*x)/(c^2*sqrt(c-a^2*c*x^2))],
[arctanh(a*x)/(c-a^2*c*x^2)^(7/2),x,3,(-1/25)/(a*c*(c-a^2*c*x^2)^(5/2))+(-4/45)/(a*c^2*(c-a^2*c*x^2)^(3/2))+1/5*x*arctanh(a*x)/(c*(c-a^2*c*x^2)^(5/2))+4/15*x*arctanh(a*x)/(c^2*(c-a^2*c*x^2)^(3/2))+(-8/15)/(a*c^3*sqrt(c-a^2*c*x^2))+8/15*x*arctanh(a*x)/(c^3*sqrt(c-a^2*c*x^2))],
[(1-a^2*x^2)^(1/2)*arctanh(a*x)^2,x,10,-arcsin(a*x)/a+arctan(exp(arctanh(a*x)))*arctanh(a*x)^2/a-I*arctanh(a*x)*polylog(2,-I*exp(arctanh(a*x)))/a+I*arctanh(a*x)*polylog(2,I*exp(arctanh(a*x)))/a+I*polylog(3,-I*exp(arctanh(a*x)))/a-I*polylog(3,I*exp(arctanh(a*x)))/a+arctanh(a*x)*sqrt(1-a^2*x^2)/a+1/2*x*arctanh(a*x)^2*sqrt(1-a^2*x^2)],
[arctanh(a*x)^2/(1-a^2*x^2)^(5/2),x,5,2/27*x/(1-a^2*x^2)^(3/2)-2/9*arctanh(a*x)/(a*(1-a^2*x^2)^(3/2))+1/3*x*arctanh(a*x)^2/(1-a^2*x^2)^(3/2)+40/27*x/sqrt(1-a^2*x^2)-4/3*arctanh(a*x)/(a*sqrt(1-a^2*x^2))+2/3*x*arctanh(a*x)^2/sqrt(1-a^2*x^2)],
[arctanh(a*x)^2/(1-a^2*x^2)^(7/2),x,9,2/125*x/(1-a^2*x^2)^(5/2)+272/3375*x/(1-a^2*x^2)^(3/2)-2/25*arctanh(a*x)/(a*(1-a^2*x^2)^(5/2))-8/45*arctanh(a*x)/(a*(1-a^2*x^2)^(3/2))+1/5*x*arctanh(a*x)^2/(1-a^2*x^2)^(5/2)+4/15*x*arctanh(a*x)^2/(1-a^2*x^2)^(3/2)+4144/3375*x/sqrt(1-a^2*x^2)-16/15*arctanh(a*x)/(a*sqrt(1-a^2*x^2))+8/15*x*arctanh(a*x)^2/sqrt(1-a^2*x^2)],
[arctanh(a*x)^2/(1-a^2*x^2)^(9/2),x,14,2/343*x/(1-a^2*x^2)^(7/2)+888/42875*x/(1-a^2*x^2)^(5/2)+30256/385875*x/(1-a^2*x^2)^(3/2)-2/49*arctanh(a*x)/(a*(1-a^2*x^2)^(7/2))-12/175*arctanh(a*x)/(a*(1-a^2*x^2)^(5/2))-16/105*arctanh(a*x)/(a*(1-a^2*x^2)^(3/2))+1/7*x*arctanh(a*x)^2/(1-a^2*x^2)^(7/2)+6/35*x*arctanh(a*x)^2/(1-a^2*x^2)^(5/2)+8/35*x*arctanh(a*x)^2/(1-a^2*x^2)^(3/2)+413312/385875*x/sqrt(1-a^2*x^2)-32/35*arctanh(a*x)/(a*sqrt(1-a^2*x^2))+16/35*x*arctanh(a*x)^2/sqrt(1-a^2*x^2)],
[(1-a^2*x^2)^(1/2)*arctanh(a*x)^3,x,12,6*arctan(sqrt(1-a*x)/sqrt(1+a*x))*arctanh(a*x)/a+arctan(exp(arctanh(a*x)))*arctanh(a*x)^3/a-3/2*I*arctanh(a*x)^2*polylog(2,-I*exp(arctanh(a*x)))/a+3/2*I*arctanh(a*x)^2*polylog(2,I*exp(arctanh(a*x)))/a+3*I*polylog(2,-I*sqrt(1-a*x)/sqrt(1+a*x))/a-3*I*polylog(2,I*sqrt(1-a*x)/sqrt(1+a*x))/a+3*I*arctanh(a*x)*polylog(3,-I*exp(arctanh(a*x)))/a-3*I*arctanh(a*x)*polylog(3,I*exp(arctanh(a*x)))/a-3*I*polylog(4,-I*exp(arctanh(a*x)))/a+3*I*polylog(4,I*exp(arctanh(a*x)))/a+3/2*arctanh(a*x)^2*sqrt(1-a^2*x^2)/a+1/2*x*arctanh(a*x)^3*sqrt(1-a^2*x^2)],
[arctanh(a*x)^3/(1-a^2*x^2)^(5/2),x,5,(-2/27)/(a*(1-a^2*x^2)^(3/2))+2/9*x*arctanh(a*x)/(1-a^2*x^2)^(3/2)-1/3*arctanh(a*x)^2/(a*(1-a^2*x^2)^(3/2))+1/3*x*arctanh(a*x)^3/(1-a^2*x^2)^(3/2)+(-40/9)/(a*sqrt(1-a^2*x^2))+40/9*x*arctanh(a*x)/sqrt(1-a^2*x^2)-2*arctanh(a*x)^2/(a*sqrt(1-a^2*x^2))+2/3*x*arctanh(a*x)^3/sqrt(1-a^2*x^2)],
[arctanh(a*x)^3/(1-a^2*x^2)^(7/2),x,9,(-6/625)/(a*(1-a^2*x^2)^(5/2))+(-272/3375)/(a*(1-a^2*x^2)^(3/2))+6/125*x*arctanh(a*x)/(1-a^2*x^2)^(5/2)+272/1125*x*arctanh(a*x)/(1-a^2*x^2)^(3/2)-3/25*arctanh(a*x)^2/(a*(1-a^2*x^2)^(5/2))-4/15*arctanh(a*x)^2/(a*(1-a^2*x^2)^(3/2))+1/5*x*arctanh(a*x)^3/(1-a^2*x^2)^(5/2)+4/15*x*arctanh(a*x)^3/(1-a^2*x^2)^(3/2)+(-4144/1125)/(a*sqrt(1-a^2*x^2))+4144/1125*x*arctanh(a*x)/sqrt(1-a^2*x^2)-8/5*arctanh(a*x)^2/(a*sqrt(1-a^2*x^2))+8/15*x*arctanh(a*x)^3/sqrt(1-a^2*x^2)],
[arctanh(a*x)^3/(1-a^2*x^2)^(9/2),x,14,(-6/2401)/(a*(1-a^2*x^2)^(7/2))+(-2664/214375)/(a*(1-a^2*x^2)^(5/2))+(-30256/385875)/(a*(1-a^2*x^2)^(3/2))+6/343*x*arctanh(a*x)/(1-a^2*x^2)^(7/2)+2664/42875*x*arctanh(a*x)/(1-a^2*x^2)^(5/2)+30256/128625*x*arctanh(a*x)/(1-a^2*x^2)^(3/2)-3/49*arctanh(a*x)^2/(a*(1-a^2*x^2)^(7/2))-18/175*arctanh(a*x)^2/(a*(1-a^2*x^2)^(5/2))-8/35*arctanh(a*x)^2/(a*(1-a^2*x^2)^(3/2))+1/7*x*arctanh(a*x)^3/(1-a^2*x^2)^(7/2)+6/35*x*arctanh(a*x)^3/(1-a^2*x^2)^(5/2)+8/35*x*arctanh(a*x)^3/(1-a^2*x^2)^(3/2)+(-413312/128625)/(a*sqrt(1-a^2*x^2))+413312/128625*x*arctanh(a*x)/sqrt(1-a^2*x^2)-48/35*arctanh(a*x)^2/(a*sqrt(1-a^2*x^2))+16/35*x*arctanh(a*x)^3/sqrt(1-a^2*x^2)],

# p<0
[(1-a^2*x^2)^(1/2)/arctanh(a*x),x,0,Unintegrable(sqrt(1-a^2*x^2)/arctanh(a*x),x)],
[1/((1-a^2*x^2)^(1/2)*arctanh(a*x)),x,0,Unintegrable(1/(arctanh(a*x)*sqrt(1-a^2*x^2)),x)],
[1/((1-a^2*x^2)^(3/2)*arctanh(a*x)),x,2,Chi(arctanh(a*x))/a],
[1/((1-a^2*x^2)^(5/2)*arctanh(a*x)),x,5,3/4*Chi(arctanh(a*x))/a+1/4*Chi(3*arctanh(a*x))/a],
[1/((1-a^2*x^2)^(7/2)*arctanh(a*x)),x,6,5/8*Chi(arctanh(a*x))/a+5/16*Chi(3*arctanh(a*x))/a+1/16*Chi(5*arctanh(a*x))/a],
[1/((1-a^2*x^2)^(9/2)*arctanh(a*x)),x,7,35/64*Chi(arctanh(a*x))/a+21/64*Chi(3*arctanh(a*x))/a+7/64*Chi(5*arctanh(a*x))/a+1/64*Chi(7*arctanh(a*x))/a],
[(1-a^2*x^2)^(1/2)/arctanh(a*x)^2,x,0,Unintegrable(sqrt(1-a^2*x^2)/arctanh(a*x)^2,x)],
[1/((1-a^2*x^2)^(1/2)*arctanh(a*x)^2),x,0,Unintegrable(1/(arctanh(a*x)^2*sqrt(1-a^2*x^2)),x)],
[1/((1-a^2*x^2)^(3/2)*arctanh(a*x)^2),x,3,Shi(arctanh(a*x))/a+(-1)/(a*arctanh(a*x)*sqrt(1-a^2*x^2))],
[1/((1-a^2*x^2)^(5/2)*arctanh(a*x)^2),x,6,(-1)/(a*(1-a^2*x^2)^(3/2)*arctanh(a*x))+3/4*Shi(arctanh(a*x))/a+3/4*Shi(3*arctanh(a*x))/a],
[1/((1-a^2*x^2)^(7/2)*arctanh(a*x)^2),x,7,(-1)/(a*(1-a^2*x^2)^(5/2)*arctanh(a*x))+5/8*Shi(arctanh(a*x))/a+15/16*Shi(3*arctanh(a*x))/a+5/16*Shi(5*arctanh(a*x))/a],
[1/((1-a^2*x^2)^(9/2)*arctanh(a*x)^2),x,8,(-1)/(a*(1-a^2*x^2)^(7/2)*arctanh(a*x))+35/64*Shi(arctanh(a*x))/a+63/64*Shi(3*arctanh(a*x))/a+35/64*Shi(5*arctanh(a*x))/a+7/64*Shi(7*arctanh(a*x))/a],
[(1-a^2*x^2)^(1/2)/arctanh(a*x)^3,x,0,Unintegrable(sqrt(1-a^2*x^2)/arctanh(a*x)^3,x)],
[1/((1-a^2*x^2)^(1/2)*arctanh(a*x)^3),x,0,Unintegrable(1/(arctanh(a*x)^3*sqrt(1-a^2*x^2)),x)],
[1/((1-a^2*x^2)^(3/2)*arctanh(a*x)^3),x,4,1/2*Chi(arctanh(a*x))/a+(-1/2)/(a*arctanh(a*x)^2*sqrt(1-a^2*x^2))-1/2*x/(arctanh(a*x)*sqrt(1-a^2*x^2))],
[1/((1-a^2*x^2)^(5/2)*arctanh(a*x)^3),x,12,(-1/2)/(a*(1-a^2*x^2)^(3/2)*arctanh(a*x)^2)-3/2*x/((1-a^2*x^2)^(3/2)*arctanh(a*x))+3/8*Chi(arctanh(a*x))/a+9/8*Chi(3*arctanh(a*x))/a],
[1/((1-a^2*x^2)^(7/2)*arctanh(a*x)^3),x,14,(-1/2)/(a*(1-a^2*x^2)^(5/2)*arctanh(a*x)^2)-5/2*x/((1-a^2*x^2)^(5/2)*arctanh(a*x))+5/16*Chi(arctanh(a*x))/a+45/32*Chi(3*arctanh(a*x))/a+25/32*Chi(5*arctanh(a*x))/a],
[1/((1-a^2*x^2)^(9/2)*arctanh(a*x)^3),x,16,(-1/2)/(a*(1-a^2*x^2)^(7/2)*arctanh(a*x)^2)-7/2*x/((1-a^2*x^2)^(7/2)*arctanh(a*x))+35/128*Chi(arctanh(a*x))/a+189/128*Chi(3*arctanh(a*x))/a+175/128*Chi(5*arctanh(a*x))/a+49/128*Chi(7*arctanh(a*x))/a],

# Integrands of the form (f+g x)^m (d+e x^2)^q (a+b ArcTanh[c x])^p with c^2 d+e=0
[(d+e*x)*(a+b*arctanh(c*x))^2/(1-c^2*x^2),x,7,1/3*d*(a+b*arctanh(c*x))^3/(b*c)-1/3*e*(a+b*arctanh(c*x))^3/(b*c^2)+e*(a+b*arctanh(c*x))^2*log(2/(1-c*x))/c^2+b*e*(a+b*arctanh(c*x))*polylog(2,1+(-2)/(1-c*x))/c^2-1/2*b^2*e*polylog(3,1+(-2)/(1-c*x))/c^2],

# Integrands of the form (f x)^m (d+e x^2)^q (a+b ArcTanh[c x])^p

# Integrands of the form (d+e x^2)^q (a+b ArcTanh[c x])^p

# Integrands of the form (d+e x^2)^q (a+b ArcTanh[c x])^p
[(c+d*x^2)^4*arctanh(a*x),x,4,1/630*d*(420*a^6*c^3+378*a^4*c^2*d+180*a^2*c*d^2+35*d^3)*x^2/a^7+1/1260*d^2*(378*a^4*c^2+180*a^2*c*d+35*d^2)*x^4/a^5+1/378*d^3*(36*a^2*c+7*d)*x^6/a^3+1/72*d^4*x^8/a+c^4*x*arctanh(a*x)+4/3*c^3*d*x^3*arctanh(a*x)+6/5*c^2*d^2*x^5*arctanh(a*x)+4/7*c*d^3*x^7*arctanh(a*x)+1/9*d^4*x^9*arctanh(a*x)+1/630*(315*a^8*c^4+420*a^6*c^3*d+378*a^4*c^2*d^2+180*a^2*c*d^3+35*d^4)*log(1-a^2*x^2)/a^9],
[(c+d*x^2)^3*arctanh(a*x),x,4,1/70*d*(35*a^4*c^2+21*a^2*c*d+5*d^2)*x^2/a^5+1/140*d^2*(21*a^2*c+5*d)*x^4/a^3+1/42*d^3*x^6/a+c^3*x*arctanh(a*x)+c^2*d*x^3*arctanh(a*x)+3/5*c*d^2*x^5*arctanh(a*x)+1/7*d^3*x^7*arctanh(a*x)+1/70*(35*a^6*c^3+35*a^4*c^2*d+21*a^2*c*d^2+5*d^3)*log(1-a^2*x^2)/a^7],
[(c+d*x^2)^2*arctanh(a*x),x,5,1/30*d*(10*a^2*c+3*d)*x^2/a^3+1/20*d^2*x^4/a+c^2*x*arctanh(a*x)+2/3*c*d*x^3*arctanh(a*x)+1/5*d^2*x^5*arctanh(a*x)+1/30*(15*a^4*c^2+10*a^2*c*d+3*d^2)*log(1-a^2*x^2)/a^5],
[(c+d*x^2)*arctanh(a*x),x,5,1/6*d*x^2/a+c*x*arctanh(a*x)+1/3*d*x^3*arctanh(a*x)+1/6*(3*a^2*c+d)*log(1-a^2*x^2)/a^3],
[arctanh(a*x)/(c+d*x^2),x,17,-1/4*log(1-a*x)*log(a*(sqrt(-c)-x*sqrt(d))/(a*sqrt(-c)-sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*log(1+a*x)*log(a*(sqrt(-c)-x*sqrt(d))/(a*sqrt(-c)+sqrt(d)))/(sqrt(-c)*sqrt(d))-1/4*log(1+a*x)*log(a*(sqrt(-c)+x*sqrt(d))/(a*sqrt(-c)-sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*log(1-a*x)*log(a*(sqrt(-c)+x*sqrt(d))/(a*sqrt(-c)+sqrt(d)))/(sqrt(-c)*sqrt(d))-1/4*polylog(2,-(1-a*x)*sqrt(d)/(a*sqrt(-c)-sqrt(d)))/(sqrt(-c)*sqrt(d))-1/4*polylog(2,-(1+a*x)*sqrt(d)/(a*sqrt(-c)-sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*polylog(2,(1-a*x)*sqrt(d)/(a*sqrt(-c)+sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*polylog(2,(1+a*x)*sqrt(d)/(a*sqrt(-c)+sqrt(d)))/(sqrt(-c)*sqrt(d))],
[arctanh(a*x)/(c+d*x^2)^2,x,24,1/2*x*arctanh(a*x)/(c*(c+d*x^2))+1/4*a*log(1-a^2*x^2)/(c*(a^2*c+d))-1/4*a*log(c+d*x^2)/(c*(a^2*c+d))+1/2*arctan(x*sqrt(d)/sqrt(c))*arctanh(a*x)/(c^(3/2)*sqrt(d))-1/8*I*log(-(1+a*x)*sqrt(d)/(I*a*sqrt(c)-sqrt(d)))*log(1-I*x*sqrt(d)/sqrt(c))/(c^(3/2)*sqrt(d))+1/8*I*log((1-a*x)*sqrt(d)/(I*a*sqrt(c)+sqrt(d)))*log(1-I*x*sqrt(d)/sqrt(c))/(c^(3/2)*sqrt(d))-1/8*I*log(-(1-a*x)*sqrt(d)/(I*a*sqrt(c)-sqrt(d)))*log(1+I*x*sqrt(d)/sqrt(c))/(c^(3/2)*sqrt(d))+1/8*I*log((1+a*x)*sqrt(d)/(I*a*sqrt(c)+sqrt(d)))*log(1+I*x*sqrt(d)/sqrt(c))/(c^(3/2)*sqrt(d))+1/8*I*polylog(2,a*(sqrt(c)-I*x*sqrt(d))/(a*sqrt(c)-I*sqrt(d)))/(c^(3/2)*sqrt(d))-1/8*I*polylog(2,a*(sqrt(c)-I*x*sqrt(d))/(a*sqrt(c)+I*sqrt(d)))/(c^(3/2)*sqrt(d))+1/8*I*polylog(2,a*(sqrt(c)+I*x*sqrt(d))/(a*sqrt(c)-I*sqrt(d)))/(c^(3/2)*sqrt(d))-1/8*I*polylog(2,a*(sqrt(c)+I*x*sqrt(d))/(a*sqrt(c)+I*sqrt(d)))/(c^(3/2)*sqrt(d))],
[arctanh(a*x)/(c+d*x^2)^3,x,23,1/8*a/(c*(a^2*c+d)*(c+d*x^2))+1/4*x*arctanh(a*x)/(c*(c+d*x^2)^2)+3/8*x*arctanh(a*x)/(c^2*(c+d*x^2))+1/16*a*(5*a^2*c+3*d)*log(1-a^2*x^2)/(c^2*(a^2*c+d)^2)-1/16*a*(5*a^2*c+3*d)*log(c+d*x^2)/(c^2*(a^2*c+d)^2)+3/8*arctan(x*sqrt(d)/sqrt(c))*arctanh(a*x)/(c^(5/2)*sqrt(d))-3/32*I*log(-(1+a*x)*sqrt(d)/(I*a*sqrt(c)-sqrt(d)))*log(1-I*x*sqrt(d)/sqrt(c))/(c^(5/2)*sqrt(d))+3/32*I*log((1-a*x)*sqrt(d)/(I*a*sqrt(c)+sqrt(d)))*log(1-I*x*sqrt(d)/sqrt(c))/(c^(5/2)*sqrt(d))-3/32*I*log(-(1-a*x)*sqrt(d)/(I*a*sqrt(c)-sqrt(d)))*log(1+I*x*sqrt(d)/sqrt(c))/(c^(5/2)*sqrt(d))+3/32*I*log((1+a*x)*sqrt(d)/(I*a*sqrt(c)+sqrt(d)))*log(1+I*x*sqrt(d)/sqrt(c))/(c^(5/2)*sqrt(d))+3/32*I*polylog(2,a*(sqrt(c)-I*x*sqrt(d))/(a*sqrt(c)-I*sqrt(d)))/(c^(5/2)*sqrt(d))-3/32*I*polylog(2,a*(sqrt(c)-I*x*sqrt(d))/(a*sqrt(c)+I*sqrt(d)))/(c^(5/2)*sqrt(d))+3/32*I*polylog(2,a*(sqrt(c)+I*x*sqrt(d))/(a*sqrt(c)-I*sqrt(d)))/(c^(5/2)*sqrt(d))-3/32*I*polylog(2,a*(sqrt(c)+I*x*sqrt(d))/(a*sqrt(c)+I*sqrt(d)))/(c^(5/2)*sqrt(d))],
[1/((a-a*x^2)*(b-2*b*arctanh(x))),x,1,-1/2*log(1-2*arctanh(x))/(a*b)],
[arctanh(b*x)/(1-x^2),x,17,1/4*log(-b*(1-x)/(1-b))*log(1-b*x)-1/4*log(b*(1+x)/(1+b))*log(1-b*x)-1/4*log(b*(1-x)/(1+b))*log(1+b*x)+1/4*log(-b*(1+x)/(1-b))*log(1+b*x)+1/4*polylog(2,(1-b*x)/(1-b))-1/4*polylog(2,(1-b*x)/(1+b))+1/4*polylog(2,(1+b*x)/(1-b))-1/4*polylog(2,(1+b*x)/(1+b))],
[arctanh(a+b*x)/(1-x^2),x,17,1/4*log(-b*(1-x)/(1-a-b))*log(1-a-b*x)-1/4*log(b*(1+x)/(1-a+b))*log(1-a-b*x)-1/4*log(b*(1-x)/(1+a+b))*log(1+a+b*x)+1/4*log(-b*(1+x)/(1+a-b))*log(1+a+b*x)+1/4*polylog(2,(1-a-b*x)/(1-a-b))-1/4*polylog(2,(1-a-b*x)/(1-a+b))+1/4*polylog(2,(1+a+b*x)/(1+a-b))-1/4*polylog(2,(1+a+b*x)/(1+a+b))],
[arctanh(x)/(a+b*x),x,4,-arctanh(x)*log(2/(1+x))/b+arctanh(x)*log(2*(a+b*x)/((a+b)*(1+x)))/b+1/2*polylog(2,1+(-2)/(1+x))/b-1/2*polylog(2,1-2*(a+b*x)/((a+b)*(1+x)))/b],
[arctanh(x)/(a+b*x^2),x,17,-1/4*log(1-x)*log((sqrt(-a)-x*sqrt(b))/(sqrt(-a)-sqrt(b)))/(sqrt(-a)*sqrt(b))+1/4*log(1+x)*log((sqrt(-a)-x*sqrt(b))/(sqrt(-a)+sqrt(b)))/(sqrt(-a)*sqrt(b))-1/4*log(1+x)*log((sqrt(-a)+x*sqrt(b))/(sqrt(-a)-sqrt(b)))/(sqrt(-a)*sqrt(b))+1/4*log(1-x)*log((sqrt(-a)+x*sqrt(b))/(sqrt(-a)+sqrt(b)))/(sqrt(-a)*sqrt(b))-1/4*polylog(2,-(1-x)*sqrt(b)/(sqrt(-a)-sqrt(b)))/(sqrt(-a)*sqrt(b))-1/4*polylog(2,-(1+x)*sqrt(b)/(sqrt(-a)-sqrt(b)))/(sqrt(-a)*sqrt(b))+1/4*polylog(2,(1-x)*sqrt(b)/(sqrt(-a)+sqrt(b)))/(sqrt(-a)*sqrt(b))+1/4*polylog(2,(1+x)*sqrt(b)/(sqrt(-a)+sqrt(b)))/(sqrt(-a)*sqrt(b))],
[arctanh(x)/(a+b*x+c*x^2),x,10,arctanh(x)*log(2*(b+2*c*x-sqrt(b^2-4*a*c))/((1+x)*(b+2*c-sqrt(b^2-4*a*c))))/sqrt(b^2-4*a*c)-arctanh(x)*log(2*(b+2*c*x+sqrt(b^2-4*a*c))/((1+x)*(b+2*c+sqrt(b^2-4*a*c))))/sqrt(b^2-4*a*c)-1/2*polylog(2,1-2*(b+2*c*x-sqrt(b^2-4*a*c))/((1+x)*(b+2*c-sqrt(b^2-4*a*c))))/sqrt(b^2-4*a*c)+1/2*polylog(2,1-2*(b+2*c*x+sqrt(b^2-4*a*c))/((1+x)*(b+2*c+sqrt(b^2-4*a*c))))/sqrt(b^2-4*a*c)],

# Integrands of the form (d+e x^2)^(q/2) (a+b ArcTanh[c x])^p
[(c+d*x^2)^(1/2)*arctanh(a*x),x,0,Unintegrable(arctanh(a*x)*sqrt(c+d*x^2),x)],
[arctanh(a*x)/(c+d*x^2)^(1/2),x,0,Unintegrable(arctanh(a*x)/sqrt(c+d*x^2),x)],
[arctanh(a*x)/(c+d*x^2)^(3/2),x,5,-arctanh(a*sqrt(c+d*x^2)/sqrt(a^2*c+d))/(c*sqrt(a^2*c+d))+x*arctanh(a*x)/(c*sqrt(c+d*x^2))],
[arctanh(a*x)/(c+d*x^2)^(5/2),x,7,1/3*x*arctanh(a*x)/(c*(c+d*x^2)^(3/2))-1/3*(3*a^2*c+2*d)*arctanh(a*sqrt(c+d*x^2)/sqrt(a^2*c+d))/(c^2*(a^2*c+d)^(3/2))+1/3*a/(c*(a^2*c+d)*sqrt(c+d*x^2))+2/3*x*arctanh(a*x)/(c^2*sqrt(c+d*x^2))],
[arctanh(a*x)/(c+d*x^2)^(7/2),x,8,1/15*a/(c*(a^2*c+d)*(c+d*x^2)^(3/2))+1/5*x*arctanh(a*x)/(c*(c+d*x^2)^(5/2))+4/15*x*arctanh(a*x)/(c^2*(c+d*x^2)^(3/2))-1/15*(15*a^4*c^2+20*a^2*c*d+8*d^2)*arctanh(a*sqrt(c+d*x^2)/sqrt(a^2*c+d))/(c^3*(a^2*c+d)^(5/2))+1/15*a*(7*a^2*c+4*d)/(c^2*(a^2*c+d)^2*sqrt(c+d*x^2))+8/15*x*arctanh(a*x)/(c^3*sqrt(c+d*x^2))],
[arctanh(a*x)/(c+d*x^2)^(9/2),x,8,1/35*a/(c*(a^2*c+d)*(c+d*x^2)^(5/2))+1/105*a*(11*a^2*c+6*d)/(c^2*(a^2*c+d)^2*(c+d*x^2)^(3/2))+1/7*x*arctanh(a*x)/(c*(c+d*x^2)^(7/2))+6/35*x*arctanh(a*x)/(c^2*(c+d*x^2)^(5/2))+8/35*x*arctanh(a*x)/(c^3*(c+d*x^2)^(3/2))-1/35*(35*a^6*c^3+70*a^4*c^2*d+56*a^2*c*d^2+16*d^3)*arctanh(a*sqrt(c+d*x^2)/sqrt(a^2*c+d))/(c^4*(a^2*c+d)^(7/2))+1/35*a*(19*a^4*c^2+22*a^2*c*d+8*d^2)/(c^3*(a^2*c+d)^3*sqrt(c+d*x^2))+16/35*x*arctanh(a*x)/(c^4*sqrt(c+d*x^2))],
[(a-a*x^2)^(1/2)*arctanh(x),x,3,-a*arctan(sqrt(1-x)/sqrt(1+x))*arctanh(x)*sqrt(1-x^2)/sqrt(a-a*x^2)-1/2*I*a*polylog(2,-I*sqrt(1-x)/sqrt(1+x))*sqrt(1-x^2)/sqrt(a-a*x^2)+1/2*I*a*polylog(2,I*sqrt(1-x)/sqrt(1+x))*sqrt(1-x^2)/sqrt(a-a*x^2)+1/2*sqrt(a-a*x^2)+1/2*x*arctanh(x)*sqrt(a-a*x^2)],
[arctanh(x)/(a-a*x^2)^(1/2),x,2,-2*arctan(sqrt(1-x)/sqrt(1+x))*arctanh(x)*sqrt(1-x^2)/sqrt(a-a*x^2)-I*polylog(2,-I*sqrt(1-x)/sqrt(1+x))*sqrt(1-x^2)/sqrt(a-a*x^2)+I*polylog(2,I*sqrt(1-x)/sqrt(1+x))*sqrt(1-x^2)/sqrt(a-a*x^2)],
[arctanh(x)/(a-a*x^2)^(3/2),x,1,(-1)/(a*sqrt(a-a*x^2))+x*arctanh(x)/(a*sqrt(a-a*x^2))],
[arctanh(x)/(a-a*x^2)^(5/2),x,2,(-1/9)/(a*(a-a*x^2)^(3/2))+1/3*x*arctanh(x)/(a*(a-a*x^2)^(3/2))+(-2/3)/(a^2*sqrt(a-a*x^2))+2/3*x*arctanh(x)/(a^2*sqrt(a-a*x^2))],
[arctanh(x)/(a-a*x^2)^(7/2),x,3,(-1/25)/(a*(a-a*x^2)^(5/2))+(-4/45)/(a^2*(a-a*x^2)^(3/2))+1/5*x*arctanh(x)/(a*(a-a*x^2)^(5/2))+4/15*x*arctanh(x)/(a^2*(a-a*x^2)^(3/2))+(-8/15)/(a^3*sqrt(a-a*x^2))+8/15*x*arctanh(x)/(a^3*sqrt(a-a*x^2))],

# Integrands of the form (h x)^m (d+e Log[f+g x^2]) (a+b ArcTanh[c x])^p

# Integrands of the form x^m (d+e Log[f+g x^2]) (a+b ArcTanh[c x])
[x^4*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2)),x,26,-2/5*a*e*x/c^4-77/300*b*e*x^2/c^3-2/15*a*e*x^3/c^2-9/200*b*e*x^4/c-2/25*a*e*x^5-2/5*b*e*x*arctanh(c*x)/c^4-2/15*b*e*x^3*arctanh(c*x)/c^2-2/25*b*e*x^5*arctanh(c*x)+1/5*b*e*arctanh(c*x)^2/c^5-1/20*(4*a+3*b)*e*log(1-c*x)/c^5+1/20*(4*a-3*b)*e*log(1+c*x)/c^5-23/75*b*e*log(1-c^2*x^2)/c^5-1/20*b*e*log(1-c^2*x^2)^2/c^5+1/10*b*x^2*(d+e*log(1-c^2*x^2))/c^3+1/20*b*x^4*(d+e*log(1-c^2*x^2))/c+1/5*x^5*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))+1/10*b*log(1-c^2*x^2)*(d+e*log(1-c^2*x^2))/c^5],
[x^3*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2)),x,14,1/8*b*(2*d-3*e)*x/c^3-2/3*b*e*x/c^3+1/24*b*(2*d-e)*x^3/c-1/18*b*e*x^3/c-1/8*b*(2*d-3*e)*arctanh(c*x)/c^4+2/3*b*e*arctanh(c*x)/c^4-1/4*e*x^2*(a+b*arctanh(c*x))/c^2-1/8*e*x^4*(a+b*arctanh(c*x))+1/4*b*e*x*log(1-c^2*x^2)/c^3+1/12*b*e*x^3*log(1-c^2*x^2)/c-1/4*e*(a+b*arctanh(c*x))*log(1-c^2*x^2)/c^4+1/4*x^4*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))],
[x^2*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2)),x,21,-2/3*a*e*x/c^2-5/18*b*e*x^2/c-2/9*a*e*x^3-2/3*b*e*x*arctanh(c*x)/c^2-2/9*b*e*x^3*arctanh(c*x)+1/3*b*e*arctanh(c*x)^2/c^3-1/6*(2*a+b)*e*log(1-c*x)/c^3+1/6*(2*a-b)*e*log(1+c*x)/c^3-4/9*b*e*log(1-c^2*x^2)/c^3-1/12*b*e*log(1-c^2*x^2)^2/c^3+1/6*b*x^2*(d+e*log(1-c^2*x^2))/c+1/3*x^3*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))+1/6*b*log(1-c^2*x^2)*(d+e*log(1-c^2*x^2))/c^3],
[x*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2)),x,7,1/2*b*(d-e)*x/c-b*e*x/c-1/2*b*(d-e)*arctanh(c*x)/c^2+b*e*arctanh(c*x)/c^2+1/2*d*x^2*(a+b*arctanh(c*x))-1/2*e*x^2*(a+b*arctanh(c*x))+1/2*b*e*x*log(1-c^2*x^2)/c-1/2*e*(1-c^2*x^2)*(a+b*arctanh(c*x))*log(1-c^2*x^2)/c^2],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2)),x,9,-2*a*e*x-2*b*e*x*arctanh(c*x)+e*(a+b*arctanh(c*x))^2/(b*c)-b*e*log(1-c^2*x^2)/c+x*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))+1/4*b*(d+e*log(1-c^2*x^2))^2/(c*e)],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x,x,14,a*d*log(x)-1/2*b*e*log(c*x)*log(1-c*x)^2+1/2*b*e*log(-c*x)*log(1+c*x)^2-1/2*b*d*polylog(2,-c*x)+1/2*b*e*(log(1-c*x)+log(1+c*x)-log(1-c^2*x^2))*polylog(2,-c*x)+1/2*b*d*polylog(2,c*x)-1/2*b*e*(log(1-c*x)+log(1+c*x)-log(1-c^2*x^2))*polylog(2,c*x)-1/2*a*e*polylog(2,c^2*x^2)-b*e*log(1-c*x)*polylog(2,1-c*x)+b*e*log(1+c*x)*polylog(2,1+c*x)+b*e*polylog(3,1-c*x)-b*e*polylog(3,1+c*x)],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^2,x,6,-c*e*(a+b*arctanh(c*x))^2/b-(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x+1/2*b*c*(d+e*log(1-c^2*x^2))*log(1+(-1)/(1-c^2*x^2))-1/2*b*c*e*polylog(2,1/(1-c^2*x^2))],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^3,x,5,-a*c^2*e*log(x)+1/2*(a+b)*c^2*e*log(1-c*x)+1/2*(a-b)*c^2*e*log(1+c*x)-1/2*b*c*(d+e*log(1-c^2*x^2))/x+1/2*b*c^2*arctanh(c*x)*(d+e*log(1-c^2*x^2))-1/2*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^2+1/2*b*c^2*e*polylog(2,-c*x)-1/2*b*c^2*e*polylog(2,c*x)],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^4,x,15,2/3*c^2*e*(a+b*arctanh(c*x))/x-1/3*c^3*e*(a+b*arctanh(c*x))^2/b-b*c^3*e*log(x)+1/3*b*c^3*e*log(1-c^2*x^2)-1/6*b*c*(1-c^2*x^2)*(d+e*log(1-c^2*x^2))/x^2-1/3*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^3+1/6*b*c^3*(d+e*log(1-c^2*x^2))*log(1+(-1)/(1-c^2*x^2))-1/6*b*c^3*e*polylog(2,1/(1-c^2*x^2))],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^5,x,10,1/4*a*c^2*e/x^2+5/12*b*c^3*e/x-1/4*b*c^4*e*arctanh(c*x)+1/4*b*c^2*e*arctanh(c*x)/x^2-1/2*a*c^4*e*log(x)+1/12*(3*a+4*b)*c^4*e*log(1-c*x)+1/12*(3*a-4*b)*c^4*e*log(1+c*x)-1/12*b*c*(d+e*log(1-c^2*x^2))/x^3-1/4*b*c^3*(d+e*log(1-c^2*x^2))/x+1/4*b*c^4*arctanh(c*x)*(d+e*log(1-c^2*x^2))-1/4*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^4+1/4*b*c^4*e*polylog(2,-c*x)-1/4*b*c^4*e*polylog(2,c*x)],
[(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^6,x,24,7/60*b*c^3*e/x^2+2/15*c^2*e*(a+b*arctanh(c*x))/x^3+2/5*c^4*e*(a+b*arctanh(c*x))/x-1/5*c^5*e*(a+b*arctanh(c*x))^2/b-5/6*b*c^5*e*log(x)+19/60*b*c^5*e*log(1-c^2*x^2)-1/20*b*c*(d+e*log(1-c^2*x^2))/x^4-1/10*b*c^3*(1-c^2*x^2)*(d+e*log(1-c^2*x^2))/x^2-1/5*(a+b*arctanh(c*x))*(d+e*log(1-c^2*x^2))/x^5+1/10*b*c^5*(d+e*log(1-c^2*x^2))*log(1+(-1)/(1-c^2*x^2))-1/10*b*c^5*e*polylog(2,1/(1-c^2*x^2))],
[x*(a+b*arctanh(c*x))*(d+e*log(f+g*x^2)),x,21,1/2*b*(d-e)*x/c-b*e*x/c-1/2*b*(d-e)*arctanh(c*x)/c^2+1/2*d*x^2*(a+b*arctanh(c*x))-1/2*e*x^2*(a+b*arctanh(c*x))-b*e*(c^2*f+g)*arctanh(c*x)*log(2/(1+c*x))/(c^2*g)+1/2*b*e*x*log(f+g*x^2)/c-1/2*b*e*(c^2*f+g)*arctanh(c*x)*log(f+g*x^2)/(c^2*g)+1/2*e*(f+g*x^2)*(a+b*arctanh(c*x))*log(f+g*x^2)/g+1/2*b*e*(c^2*f+g)*arctanh(c*x)*log(2*c*(sqrt(-f)-x*sqrt(g))/((1+c*x)*(c*sqrt(-f)-sqrt(g))))/(c^2*g)+1/2*b*e*(c^2*f+g)*arctanh(c*x)*log(2*c*(sqrt(-f)+x*sqrt(g))/((1+c*x)*(c*sqrt(-f)+sqrt(g))))/(c^2*g)+1/2*b*e*(c^2*f+g)*polylog(2,1+(-2)/(1+c*x))/(c^2*g)-1/4*b*e*(c^2*f+g)*polylog(2,1-2*c*(sqrt(-f)-x*sqrt(g))/((1+c*x)*(c*sqrt(-f)-sqrt(g))))/(c^2*g)-1/4*b*e*(c^2*f+g)*polylog(2,1-2*c*(sqrt(-f)+x*sqrt(g))/((1+c*x)*(c*sqrt(-f)+sqrt(g))))/(c^2*g)+b*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(f)/(c*sqrt(g))],
[(a+b*arctanh(c*x))*(d+e*log(f+g*x^2)),x,28,-2*a*e*x-2*b*e*x*arctanh(c*x)-b*e*log(1-c^2*x^2)/c+x*(a+b*arctanh(c*x))*(d+e*log(f+g*x^2))+1/2*b*log(g*(1-c^2*x^2)/(c^2*f+g))*(d+e*log(f+g*x^2))/c+1/2*b*e*polylog(2,c^2*(f+g*x^2)/(c^2*f+g))/c+1/2*b*e*log(1-c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)-sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*b*e*log(1+c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)+sqrt(g)))*sqrt(-f)/sqrt(g)+1/2*b*e*log(1+c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)-sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*b*e*log(1-c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)+sqrt(g)))*sqrt(-f)/sqrt(g)+1/2*b*e*polylog(2,-(1-c*x)*sqrt(g)/(c*sqrt(-f)-sqrt(g)))*sqrt(-f)/sqrt(g)+1/2*b*e*polylog(2,-(1+c*x)*sqrt(g)/(c*sqrt(-f)-sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*b*e*polylog(2,(1-c*x)*sqrt(g)/(c*sqrt(-f)+sqrt(g)))*sqrt(-f)/sqrt(g)-1/2*b*e*polylog(2,(1+c*x)*sqrt(g)/(c*sqrt(-f)+sqrt(g)))*sqrt(-f)/sqrt(g)+2*a*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(f)/sqrt(g)],
[(a+b*arctanh(c*x))*(d+e*log(f+g*x^2))/x,x,6,b*e*CannotIntegrate(arctanh(c*x)*log(f+g*x^2)/x,x)+a*d*log(x)+1/2*a*e*log(-g*x^2/f)*log(f+g*x^2)-1/2*b*d*polylog(2,-c*x)+1/2*b*d*polylog(2,c*x)+1/2*a*e*polylog(2,1+g*x^2/f)],
[(a+b*arctanh(c*x))*(d+e*log(f+g*x^2))/x^2,x,28,-(a+b*arctanh(c*x))*(d+e*log(f+g*x^2))/x+1/2*b*c*log(-g*x^2/f)*(d+e*log(f+g*x^2))-1/2*b*c*log(g*(1-c^2*x^2)/(c^2*f+g))*(d+e*log(f+g*x^2))-1/2*b*c*e*polylog(2,c^2*(f+g*x^2)/(c^2*f+g))+1/2*b*c*e*polylog(2,1+g*x^2/f)-1/2*b*e*log(1-c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)-sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*b*e*log(1+c*x)*log(c*(sqrt(-f)-x*sqrt(g))/(c*sqrt(-f)+sqrt(g)))*sqrt(g)/sqrt(-f)-1/2*b*e*log(1+c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)-sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*b*e*log(1-c*x)*log(c*(sqrt(-f)+x*sqrt(g))/(c*sqrt(-f)+sqrt(g)))*sqrt(g)/sqrt(-f)-1/2*b*e*polylog(2,-(1-c*x)*sqrt(g)/(c*sqrt(-f)-sqrt(g)))*sqrt(g)/sqrt(-f)-1/2*b*e*polylog(2,-(1+c*x)*sqrt(g)/(c*sqrt(-f)-sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*b*e*polylog(2,(1-c*x)*sqrt(g)/(c*sqrt(-f)+sqrt(g)))*sqrt(g)/sqrt(-f)+1/2*b*e*polylog(2,(1+c*x)*sqrt(g)/(c*sqrt(-f)+sqrt(g)))*sqrt(g)/sqrt(-f)+2*a*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(g)/sqrt(f)],
[(a+b*arctanh(c*x))*(d+e*log(f+g*x^2))/x^3,x,20,a*e*g*log(x)/f+b*e*(c^2*f+g)*arctanh(c*x)*log(2/(1+c*x))/f-1/2*a*e*g*log(f+g*x^2)/f-1/2*b*c*(d+e*log(f+g*x^2))/x+1/2*b*c^2*arctanh(c*x)*(d+e*log(f+g*x^2))-1/2*(a+b*arctanh(c*x))*(d+e*log(f+g*x^2))/x^2-1/2*b*e*(c^2*f+g)*arctanh(c*x)*log(2*c*(sqrt(-f)-x*sqrt(g))/((1+c*x)*(c*sqrt(-f)-sqrt(g))))/f-1/2*b*e*(c^2*f+g)*arctanh(c*x)*log(2*c*(sqrt(-f)+x*sqrt(g))/((1+c*x)*(c*sqrt(-f)+sqrt(g))))/f-1/2*b*e*g*polylog(2,-c*x)/f+1/2*b*e*g*polylog(2,c*x)/f-1/2*b*e*(c^2*f+g)*polylog(2,1+(-2)/(1+c*x))/f+1/4*b*e*(c^2*f+g)*polylog(2,1-2*c*(sqrt(-f)-x*sqrt(g))/((1+c*x)*(c*sqrt(-f)-sqrt(g))))/f+1/4*b*e*(c^2*f+g)*polylog(2,1-2*c*(sqrt(-f)+x*sqrt(g))/((1+c*x)*(c*sqrt(-f)+sqrt(g))))/f+b*c*e*arctan(x*sqrt(g)/sqrt(f))*sqrt(g)/sqrt(f)],

# Integrands of the form u (a+b ArcTanh[c x])^p
[arctanh(c*x)*(a+b*arctanh(c*x))/(1+c*x)^2,x,16,1/2*(-a-b)/(c*(1+c*x))+1/2*(a+b)*arctanh(c*x)/c-(a+b)*arctanh(c*x)/(c*(1+c*x))-1/2*b*(1-c*x)*arctanh(c*x)^2/(c*(1+c*x)),-1/2*a/(c*(1+c*x))-1/2*b/(c*(1+c*x))+1/2*a*arctanh(c*x)/c+1/2*b*arctanh(c*x)/c-a*arctanh(c*x)/(c*(1+c*x))-b*arctanh(c*x)/(c*(1+c*x))+1/2*b*arctanh(c*x)^2/c-b*arctanh(c*x)^2/(c*(1+c*x))]]:
