# Maple integration test file: "7 Inverse hyperbolic functions\7.3 Inverse hyperbolic tangent\7.3.5 u (a+b arctanh(c+d x))^p.txt"

lst:=[

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

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

# Integrands of the form x^m (a+b ArcTanh[c+d x])^p
[x^3*arctanh(a+b*x)^2,x,19,-a*x/b^3+1/12*(a+b*x)^2/b^4+a*arctanh(a+b*x)/b^4+1/2*(1+6*a^2)*(a+b*x)*arctanh(a+b*x)/b^4-a*(a+b*x)^2*arctanh(a+b*x)/b^4+1/6*(a+b*x)^3*arctanh(a+b*x)/b^4-a*(1+a^2)*arctanh(a+b*x)^2/b^4-1/4*(1+6*a^2+a^4)*arctanh(a+b*x)^2/b^4+1/4*x^4*arctanh(a+b*x)^2+2*a*(1+a^2)*arctanh(a+b*x)*log(2/(1-a-b*x))/b^4+1/12*log(1-(a+b*x)^2)/b^4+1/4*(1+6*a^2)*log(1-(a+b*x)^2)/b^4+a*(1+a^2)*polylog(2,(-1-a-b*x)/(1-a-b*x))/b^4],
[x^2*arctanh(a+b*x)^2,x,15,1/3*x/b^2-1/3*arctanh(a+b*x)/b^3-2*a*(a+b*x)*arctanh(a+b*x)/b^3+1/3*(a+b*x)^2*arctanh(a+b*x)/b^3+1/3*a*(3+a^2)*arctanh(a+b*x)^2/b^3+1/3*(1+3*a^2)*arctanh(a+b*x)^2/b^3+1/3*x^3*arctanh(a+b*x)^2-2/3*(1+3*a^2)*arctanh(a+b*x)*log(2/(1-a-b*x))/b^3-a*log(1-(a+b*x)^2)/b^3-1/3*(1+3*a^2)*polylog(2,(-1-a-b*x)/(1-a-b*x))/b^3],
[x*arctanh(a+b*x)^2,x,12,(a+b*x)*arctanh(a+b*x)/b^2-a*arctanh(a+b*x)^2/b^2-1/2*(1+a^2)*arctanh(a+b*x)^2/b^2+1/2*x^2*arctanh(a+b*x)^2+2*a*arctanh(a+b*x)*log(2/(1-a-b*x))/b^2+1/2*log(1-(a+b*x)^2)/b^2+a*polylog(2,(-1-a-b*x)/(1-a-b*x))/b^2],
[arctanh(a+b*x)^2,x,6,arctanh(a+b*x)^2/b+(a+b*x)*arctanh(a+b*x)^2/b-2*arctanh(a+b*x)*log(2/(1-a-b*x))/b-polylog(2,(-1-a-b*x)/(1-a-b*x))/b],
[arctanh(a+b*x)^2/x,x,2,-arctanh(a+b*x)^2*log(2/(1+a+b*x))+arctanh(a+b*x)^2*log(2*b*x/((1-a)*(1+a+b*x)))+arctanh(a+b*x)*polylog(2,1+(-2)/(1+a+b*x))-arctanh(a+b*x)*polylog(2,1-2*b*x/((1-a)*(1+a+b*x)))+1/2*polylog(3,1+(-2)/(1+a+b*x))-1/2*polylog(3,1-2*b*x/((1-a)*(1+a+b*x)))],
[arctanh(a+b*x)^2/x^2,x,17,-arctanh(a+b*x)^2/x+b*arctanh(a+b*x)*log(2/(1-a-b*x))/(1-a)+b*arctanh(a+b*x)*log(2/(1+a+b*x))/(1+a)-2*b*arctanh(a+b*x)*log(2/(1+a+b*x))/(1-a^2)+2*b*arctanh(a+b*x)*log(2*b*x/((1-a)*(1+a+b*x)))/(1-a^2)+1/2*b*polylog(2,(-1-a-b*x)/(1-a-b*x))/(1-a)-1/2*b*polylog(2,1+(-2)/(1+a+b*x))/(1+a)+b*polylog(2,1+(-2)/(1+a+b*x))/(1-a^2)-b*polylog(2,1-2*b*x/((1-a)*(1+a+b*x)))/(1-a^2)],
[arctanh(a+b*x)^2/x^3,x,21,-b*arctanh(a+b*x)/((1-a^2)*x)-1/2*arctanh(a+b*x)^2/x^2+b^2*log(x)/(1-a^2)^2+1/2*b^2*arctanh(a+b*x)*log(2/(1-a-b*x))/(1-a)^2-1/2*b^2*log(1-a-b*x)/((1-a)^2*(1+a))-1/2*b^2*arctanh(a+b*x)*log(2/(1+a+b*x))/(1+a)^2-2*a*b^2*arctanh(a+b*x)*log(2/(1+a+b*x))/(1-a^2)^2+2*a*b^2*arctanh(a+b*x)*log(2*b*x/((1-a)*(1+a+b*x)))/(1-a^2)^2-1/2*b^2*log(1+a+b*x)/((1-a)*(1+a)^2)+1/4*b^2*polylog(2,(-1-a-b*x)/(1-a-b*x))/(1-a)^2+1/4*b^2*polylog(2,1+(-2)/(1+a+b*x))/(1+a)^2+a*b^2*polylog(2,1+(-2)/(1+a+b*x))/(1-a^2)^2-a*b^2*polylog(2,1-2*b*x/((1-a)*(1+a+b*x)))/(1-a^2)^2],
[arctanh(1+b*x)^2/x,x,4,-arctanh(1+b*x)^2*log((-2)/(b*x))-arctanh(1+b*x)*polylog(2,1+2/(b*x))+1/2*polylog(3,1+2/(b*x))],

# Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p when d e-c f=0
[(c*e+d*e*x)^3*(a+b*arctanh(c+d*x)),x,6,1/4*b*e^3*x+1/12*b*e^3*(c+d*x)^3/d-1/4*b*e^3*arctanh(c+d*x)/d+1/4*e^3*(c+d*x)^4*(a+b*arctanh(c+d*x))/d],
[(c*e+d*e*x)^2*(a+b*arctanh(c+d*x)),x,6,1/6*b*e^2*(c+d*x)^2/d+1/3*e^2*(c+d*x)^3*(a+b*arctanh(c+d*x))/d+1/6*b*e^2*log(1-(c+d*x)^2)/d],
[(c*e+d*e*x)*(a+b*arctanh(c+d*x)),x,5,1/2*b*e*x-1/2*b*e*arctanh(c+d*x)/d+1/2*e*(c+d*x)^2*(a+b*arctanh(c+d*x))/d],
[(a+b*arctanh(c+d*x))/(c*e+d*e*x),x,3,a*log(c+d*x)/(d*e)-1/2*b*polylog(2,-c-d*x)/(d*e)+1/2*b*polylog(2,c+d*x)/(d*e)],
[(a+b*arctanh(c+d*x))/(c*e+d*e*x)^2,x,7,(-a-b*arctanh(c+d*x))/(d*e^2*(c+d*x))+b*log(c+d*x)/(d*e^2)-1/2*b*log(1-(c+d*x)^2)/(d*e^2)],
[(a+b*arctanh(c+d*x))/(c*e+d*e*x)^3,x,5,-1/2*b/(d*e^3*(c+d*x))+1/2*b*arctanh(c+d*x)/(d*e^3)+1/2*(-a-b*arctanh(c+d*x))/(d*e^3*(c+d*x)^2)],
[(c*e+d*e*x)^3*(a+b*arctanh(c+d*x))^2,x,13,1/2*a*b*e^3*x+1/12*b^2*e^3*(c+d*x)^2/d+1/2*b^2*e^3*(c+d*x)*arctanh(c+d*x)/d+1/6*b*e^3*(c+d*x)^3*(a+b*arctanh(c+d*x))/d-1/4*e^3*(a+b*arctanh(c+d*x))^2/d+1/4*e^3*(c+d*x)^4*(a+b*arctanh(c+d*x))^2/d+1/3*b^2*e^3*log(1-(c+d*x)^2)/d],
[(c*e+d*e*x)^2*(a+b*arctanh(c+d*x))^2,x,11,1/3*b^2*e^2*x-1/3*b^2*e^2*arctanh(c+d*x)/d+1/3*b*e^2*(c+d*x)^2*(a+b*arctanh(c+d*x))/d+1/3*e^2*(a+b*arctanh(c+d*x))^2/d+1/3*e^2*(c+d*x)^3*(a+b*arctanh(c+d*x))^2/d-2/3*b*e^2*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d-1/3*b^2*e^2*polylog(2,(-1-c-d*x)/(1-c-d*x))/d],
[(c*e+d*e*x)*(a+b*arctanh(c+d*x))^2,x,8,a*b*e*x+b^2*e*(c+d*x)*arctanh(c+d*x)/d-1/2*e*(a+b*arctanh(c+d*x))^2/d+1/2*e*(c+d*x)^2*(a+b*arctanh(c+d*x))^2/d+1/2*b^2*e*log(1-(c+d*x)^2)/d],
[(a+b*arctanh(c+d*x))^2/(c*e+d*e*x),x,8,2*(a+b*arctanh(c+d*x))^2*arctanh(1+(-2)/(1-c-d*x))/(d*e)-b*(a+b*arctanh(c+d*x))*polylog(2,1+(-2)/(1-c-d*x))/(d*e)+b*(a+b*arctanh(c+d*x))*polylog(2,-1+2/(1-c-d*x))/(d*e)+1/2*b^2*polylog(3,1+(-2)/(1-c-d*x))/(d*e)-1/2*b^2*polylog(3,-1+2/(1-c-d*x))/(d*e)],
[(a+b*arctanh(c+d*x))^2/(c*e+d*e*x)^2,x,6,(a+b*arctanh(c+d*x))^2/(d*e^2)-(a+b*arctanh(c+d*x))^2/(d*e^2*(c+d*x))+2*b*(a+b*arctanh(c+d*x))*log(2+(-2)/(1+c+d*x))/(d*e^2)-b^2*polylog(2,-1+2/(1+c+d*x))/(d*e^2)],
[(a+b*arctanh(c+d*x))^2/(c*e+d*e*x)^3,x,10,-b*(a+b*arctanh(c+d*x))/(d*e^3*(c+d*x))+1/2*(a+b*arctanh(c+d*x))^2/(d*e^3)-1/2*(a+b*arctanh(c+d*x))^2/(d*e^3*(c+d*x)^2)+b^2*log(c+d*x)/(d*e^3)-1/2*b^2*log(1-(c+d*x)^2)/(d*e^3)],
[(a+b*arctanh(c+d*x))^2/(c*e+d*e*x)^4,x,10,-1/3*b^2/(d*e^4*(c+d*x))+1/3*b^2*arctanh(c+d*x)/(d*e^4)-1/3*b*(a+b*arctanh(c+d*x))/(d*e^4*(c+d*x)^2)+1/3*(a+b*arctanh(c+d*x))^2/(d*e^4)-1/3*(a+b*arctanh(c+d*x))^2/(d*e^4*(c+d*x)^3)+2/3*b*(a+b*arctanh(c+d*x))*log(2+(-2)/(1+c+d*x))/(d*e^4)-1/3*b^2*polylog(2,-1+2/(1+c+d*x))/(d*e^4)],
[(a+b*arctanh(c+d*x))^2/(c*e+d*e*x)^5,x,15,-1/12*b^2/(d*e^5*(c+d*x)^2)-1/6*b*(a+b*arctanh(c+d*x))/(d*e^5*(c+d*x)^3)-1/2*b*(a+b*arctanh(c+d*x))/(d*e^5*(c+d*x))+1/4*(a+b*arctanh(c+d*x))^2/(d*e^5)-1/4*(a+b*arctanh(c+d*x))^2/(d*e^5*(c+d*x)^4)+2/3*b^2*log(c+d*x)/(d*e^5)-1/3*b^2*log(1-(c+d*x)^2)/(d*e^5)],
[(c*e+d*e*x)^2*(a+b*arctanh(c+d*x))^3,x,14,a*b^2*e^2*x+b^3*e^2*(c+d*x)*arctanh(c+d*x)/d-1/2*b*e^2*(a+b*arctanh(c+d*x))^2/d+1/2*b*e^2*(c+d*x)^2*(a+b*arctanh(c+d*x))^2/d+1/3*e^2*(a+b*arctanh(c+d*x))^3/d+1/3*e^2*(c+d*x)^3*(a+b*arctanh(c+d*x))^3/d-b*e^2*(a+b*arctanh(c+d*x))^2*log(2/(1-c-d*x))/d+1/2*b^3*e^2*log(1-(c+d*x)^2)/d-b^2*e^2*(a+b*arctanh(c+d*x))*polylog(2,1+(-2)/(1-c-d*x))/d+1/2*b^3*e^2*polylog(3,1+(-2)/(1-c-d*x))/d],
[(c*e+d*e*x)*(a+b*arctanh(c+d*x))^3,x,10,3/2*b*e*(a+b*arctanh(c+d*x))^2/d+3/2*b*e*(c+d*x)*(a+b*arctanh(c+d*x))^2/d-1/2*e*(a+b*arctanh(c+d*x))^3/d+1/2*e*(c+d*x)^2*(a+b*arctanh(c+d*x))^3/d-3*b^2*e*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d-3/2*b^3*e*polylog(2,(-1-c-d*x)/(1-c-d*x))/d],
[(a+b*arctanh(c+d*x))^3/(c*e+d*e*x),x,10,2*(a+b*arctanh(c+d*x))^3*arctanh(1+(-2)/(1-c-d*x))/(d*e)-3/2*b*(a+b*arctanh(c+d*x))^2*polylog(2,1+(-2)/(1-c-d*x))/(d*e)+3/2*b*(a+b*arctanh(c+d*x))^2*polylog(2,-1+2/(1-c-d*x))/(d*e)+3/2*b^2*(a+b*arctanh(c+d*x))*polylog(3,1+(-2)/(1-c-d*x))/(d*e)-3/2*b^2*(a+b*arctanh(c+d*x))*polylog(3,-1+2/(1-c-d*x))/(d*e)-3/4*b^3*polylog(4,1+(-2)/(1-c-d*x))/(d*e)+3/4*b^3*polylog(4,-1+2/(1-c-d*x))/(d*e)],
[(a+b*arctanh(c+d*x))^3/(c*e+d*e*x)^2,x,7,(a+b*arctanh(c+d*x))^3/(d*e^2)-(a+b*arctanh(c+d*x))^3/(d*e^2*(c+d*x))+3*b*(a+b*arctanh(c+d*x))^2*log(2+(-2)/(1+c+d*x))/(d*e^2)-3*b^2*(a+b*arctanh(c+d*x))*polylog(2,-1+2/(1+c+d*x))/(d*e^2)-3/2*b^3*polylog(3,-1+2/(1+c+d*x))/(d*e^2)],
[(a+b*arctanh(c+d*x))^3/(c*e+d*e*x)^3,x,9,3/2*b*(a+b*arctanh(c+d*x))^2/(d*e^3)-3/2*b*(a+b*arctanh(c+d*x))^2/(d*e^3*(c+d*x))+1/2*(a+b*arctanh(c+d*x))^3/(d*e^3)-1/2*(a+b*arctanh(c+d*x))^3/(d*e^3*(c+d*x)^2)+3*b^2*(a+b*arctanh(c+d*x))*log(2+(-2)/(1+c+d*x))/(d*e^3)-3/2*b^3*polylog(2,-1+2/(1+c+d*x))/(d*e^3)],
[(a+b*arctanh(c+d*x))^3/(c*e+d*e*x)^4,x,16,-b^2*(a+b*arctanh(c+d*x))/(d*e^4*(c+d*x))+1/2*b*(a+b*arctanh(c+d*x))^2/(d*e^4)-1/2*b*(a+b*arctanh(c+d*x))^2/(d*e^4*(c+d*x)^2)+1/3*(a+b*arctanh(c+d*x))^3/(d*e^4)-1/3*(a+b*arctanh(c+d*x))^3/(d*e^4*(c+d*x)^3)+b^3*log(c+d*x)/(d*e^4)-1/2*b^3*log(1-(c+d*x)^2)/(d*e^4)+b*(a+b*arctanh(c+d*x))^2*log(2+(-2)/(1+c+d*x))/(d*e^4)-b^2*(a+b*arctanh(c+d*x))*polylog(2,-1+2/(1+c+d*x))/(d*e^4)-1/2*b^3*polylog(3,-1+2/(1+c+d*x))/(d*e^4)],
[arctanh(1+x)/(2+2*x),x,3,-1/4*polylog(2,-1-x)+1/4*polylog(2,1+x)],
[arctanh(a+b*x)/(a*d/b+d*x),x,3,-1/2*polylog(2,-a-b*x)/d+1/2*polylog(2,a+b*x)/d],

# Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p
[(e+f*x)^3*(a+b*arctanh(c+d*x)),x,7,1/4*b*f*(6*d^2*e^2-12*c*d*e*f+(1+6*c^2)*f^2)*x/d^3+1/2*b*f^2*(d*e-c*f)*(c+d*x)^2/d^4+1/12*b*f^3*(c+d*x)^3/d^4+1/4*(e+f*x)^4*(a+b*arctanh(c+d*x))/f+1/8*b*(d*e+f-c*f)^4*log(1-c-d*x)/(d^4*f)-1/8*b*(d*e-f-c*f)^4*log(1+c+d*x)/(d^4*f)],
[(e+f*x)^2*(a+b*arctanh(c+d*x)),x,7,b*f*(d*e-c*f)*x/d^2+1/6*b*f^2*(c+d*x)^2/d^3+1/3*(e+f*x)^3*(a+b*arctanh(c+d*x))/f+1/6*b*(d*e+f-c*f)^3*log(1-c-d*x)/(d^3*f)-1/6*b*(d*e-(1+c)*f)^3*log(1+c+d*x)/(d^3*f)],
[(e+f*x)*(a+b*arctanh(c+d*x)),x,7,1/2*b*f*x/d+1/2*(e+f*x)^2*(a+b*arctanh(c+d*x))/f+1/4*b*(d*e+f-c*f)^2*log(1-c-d*x)/(d^2*f)-1/4*b*(d*e-(1+c)*f)^2*log(1+c+d*x)/(d^2*f)],
[a+b*arctanh(c+d*x),x,4,a*x+b*(c+d*x)*arctanh(c+d*x)/d+1/2*b*log(1-(c+d*x)^2)/d],
[(a+b*arctanh(c+d*x))/(e+f*x),x,5,-(a+b*arctanh(c+d*x))*log(2/(1+c+d*x))/f+(a+b*arctanh(c+d*x))*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f+1/2*b*polylog(2,1+(-2)/(1+c+d*x))/f-1/2*b*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f],
[(a+b*arctanh(c+d*x))/(e+f*x)^2,x,7,(-a-b*arctanh(c+d*x))/(f*(e+f*x))-1/2*b*d*log(1-c-d*x)/(f*(d*e+f-c*f))+1/2*b*d*log(1+c+d*x)/(f*(d*e-f-c*f))-b*d*log(e+f*x)/((d*e-f-c*f)*(d*e+f-c*f))],
[(a+b*arctanh(c+d*x))/(e+f*x)^3,x,5,1/2*b*d/((d*e-f-c*f)*(d*e+f-c*f)*(e+f*x))+1/2*(-a-b*arctanh(c+d*x))/(f*(e+f*x)^2)-1/4*b*d^2*log(1-c-d*x)/(f*(d*e+f-c*f)^2)+1/4*b*d^2*log(1+c+d*x)/(f*(d*e-f-c*f)^2)-b*d^2*(d*e-c*f)*log(e+f*x)/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)],
[(e+f*x)^3*(a+b*arctanh(c+d*x))^2,x,20,b^2*f^2*(d*e-c*f)*x/d^3+1/2*a*b*f*(6*d^2*e^2-12*c*d*e*f+(1+6*c^2)*f^2)*x/d^3+1/12*b^2*f^3*(c+d*x)^2/d^4-b^2*f^2*(d*e-c*f)*arctanh(c+d*x)/d^4+1/2*b^2*f*(6*d^2*e^2-12*c*d*e*f+(1+6*c^2)*f^2)*(c+d*x)*arctanh(c+d*x)/d^4+b*f^2*(d*e-c*f)*(c+d*x)^2*(a+b*arctanh(c+d*x))/d^4+1/6*b*f^3*(c+d*x)^3*(a+b*arctanh(c+d*x))/d^4+(d*e-c*f)*(d^2*e^2-2*c*d*e*f+(1+c^2)*f^2)*(a+b*arctanh(c+d*x))^2/d^4-1/4*(d^4*e^4-4*c*d^3*e^3*f+6*(1+c^2)*d^2*e^2*f^2-4*c*(3+c^2)*d*e*f^3+(1+6*c^2+c^4)*f^4)*(a+b*arctanh(c+d*x))^2/(d^4*f)+1/4*(e+f*x)^4*(a+b*arctanh(c+d*x))^2/f-2*b*(d*e-c*f)*(d^2*e^2-2*c*d*e*f+(1+c^2)*f^2)*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d^4+1/12*b^2*f^3*log(1-(c+d*x)^2)/d^4+1/4*b^2*f*(6*d^2*e^2-12*c*d*e*f+(1+6*c^2)*f^2)*log(1-(c+d*x)^2)/d^4-b^2*(d*e-c*f)*(d^2*e^2-2*c*d*e*f+(1+c^2)*f^2)*polylog(2,(-1-c-d*x)/(1-c-d*x))/d^4],
[(e+f*x)^2*(a+b*arctanh(c+d*x))^2,x,16,1/3*b^2*f^2*x/d^2+2*a*b*f*(d*e-c*f)*x/d^2-1/3*b^2*f^2*arctanh(c+d*x)/d^3+2*b^2*f*(d*e-c*f)*(c+d*x)*arctanh(c+d*x)/d^3+1/3*b*f^2*(c+d*x)^2*(a+b*arctanh(c+d*x))/d^3-1/3*(d*e-c*f)*(d^2*e^2-2*c*d*e*f+(3+c^2)*f^2)*(a+b*arctanh(c+d*x))^2/(d^3*f)+1/3*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*(a+b*arctanh(c+d*x))^2/d^3+1/3*(e+f*x)^3*(a+b*arctanh(c+d*x))^2/f-2/3*b*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d^3+b^2*f*(d*e-c*f)*log(1-(c+d*x)^2)/d^3-1/3*b^2*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*polylog(2,(-1-c-d*x)/(1-c-d*x))/d^3],
[(e+f*x)*(a+b*arctanh(c+d*x))^2,x,13,a*b*f*x/d+b^2*f*(c+d*x)*arctanh(c+d*x)/d^2+(d*e-c*f)*(a+b*arctanh(c+d*x))^2/d^2-1/2*(d^2*e^2-2*c*d*e*f+(1+c^2)*f^2)*(a+b*arctanh(c+d*x))^2/(d^2*f)+1/2*(e+f*x)^2*(a+b*arctanh(c+d*x))^2/f-2*b*(d*e-c*f)*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d^2+1/2*b^2*f*log(1-(c+d*x)^2)/d^2-b^2*(d*e-c*f)*polylog(2,(-1-c-d*x)/(1-c-d*x))/d^2],
[(a+b*arctanh(c+d*x))^2,x,6,(a+b*arctanh(c+d*x))^2/d+(c+d*x)*(a+b*arctanh(c+d*x))^2/d-2*b*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d-b^2*polylog(2,(-1-c-d*x)/(1-c-d*x))/d],
[(a+b*arctanh(c+d*x))^2/(e+f*x),x,2,-(a+b*arctanh(c+d*x))^2*log(2/(1+c+d*x))/f+(a+b*arctanh(c+d*x))^2*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f+b*(a+b*arctanh(c+d*x))*polylog(2,1+(-2)/(1+c+d*x))/f-b*(a+b*arctanh(c+d*x))*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f+1/2*b^2*polylog(3,1+(-2)/(1+c+d*x))/f-1/2*b^2*polylog(3,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f],
[(a+b*arctanh(c+d*x))^2/(e+f*x)^2,x,24,-(a+b*arctanh(c+d*x))^2/(f*(e+f*x))+b^2*d*arctanh(c+d*x)*log(2/(1-c-d*x))/(f*(d*e+f-c*f))-a*b*d*log(1-c-d*x)/(f*(d*e+f-c*f))-b^2*d*arctanh(c+d*x)*log(2/(1+c+d*x))/(f*(d*e-f-c*f))+2*b^2*d*arctanh(c+d*x)*log(2/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))+a*b*d*log(1+c+d*x)/(f*(d*e-f-c*f))+2*a*b*d*log(e+f*x)/(f^2-(d*e-c*f)^2)-2*b^2*d*arctanh(c+d*x)*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))+1/2*b^2*d*polylog(2,(-1-c-d*x)/(1-c-d*x))/(f*(d*e+f-c*f))+1/2*b^2*d*polylog(2,1+(-2)/(1+c+d*x))/(f*(d*e-f-c*f))-b^2*d*polylog(2,1+(-2)/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))+b^2*d*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))],
[(a+b*arctanh(c+d*x))^2/(e+f*x)^3,x,26,-a*b*d/((f^2-(d*e-c*f)^2)*(e+f*x))+b^2*d*arctanh(c+d*x)/((d*e-f-c*f)*(d*e+f-c*f)*(e+f*x))-1/2*(a+b*arctanh(c+d*x))^2/(f*(e+f*x)^2)+1/2*b^2*d^2*arctanh(c+d*x)*log(2/(1-c-d*x))/(f*(d*e+f-c*f)^2)-1/2*a*b*d^2*log(1-c-d*x)/(f*(d*e+f-c*f)^2)+1/2*b^2*d^2*log(1-c-d*x)/((d*e+f-c*f)^2*(d*e-(1+c)*f))-1/2*b^2*d^2*arctanh(c+d*x)*log(2/(1+c+d*x))/(f*(d*e-f-c*f)^2)+2*b^2*d^2*(d*e-c*f)*arctanh(c+d*x)*log(2/(1+c+d*x))/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)+1/2*a*b*d^2*log(1+c+d*x)/(f*(d*e-f-c*f)^2)-1/2*b^2*d^2*log(1+c+d*x)/((d*e+f-c*f)*(d*e-(1+c)*f)^2)+b^2*d^2*f*log(e+f*x)/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)-2*a*b*d^2*(d*e-c*f)*log(e+f*x)/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)-2*b^2*d^2*(d*e-c*f)*arctanh(c+d*x)*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)+1/4*b^2*d^2*polylog(2,(-1-c-d*x)/(1-c-d*x))/(f*(d*e+f-c*f)^2)+1/4*b^2*d^2*polylog(2,1+(-2)/(1+c+d*x))/(f*(d*e-f-c*f)^2)-b^2*d^2*(d*e-c*f)*polylog(2,1+(-2)/(1+c+d*x))/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)+b^2*d^2*(d*e-c*f)*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e+f-c*f)^2*(d*e-(1+c)*f)^2)],
[(e+f*x)^2*(a+b*arctanh(c+d*x))^3,x,21,a*b^2*f^2*x/d^2+b^3*f^2*(c+d*x)*arctanh(c+d*x)/d^3-1/2*b*f^2*(a+b*arctanh(c+d*x))^2/d^3+3*b*f*(d*e-c*f)*(a+b*arctanh(c+d*x))^2/d^3+3*b*f*(d*e-c*f)*(c+d*x)*(a+b*arctanh(c+d*x))^2/d^3+1/2*b*f^2*(c+d*x)^2*(a+b*arctanh(c+d*x))^2/d^3-1/3*(d*e-c*f)*(d^2*e^2-2*c*d*e*f+(3+c^2)*f^2)*(a+b*arctanh(c+d*x))^3/(d^3*f)+1/3*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*(a+b*arctanh(c+d*x))^3/d^3+1/3*(e+f*x)^3*(a+b*arctanh(c+d*x))^3/f-6*b^2*f*(d*e-c*f)*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d^3-b*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*(a+b*arctanh(c+d*x))^2*log(2/(1-c-d*x))/d^3+1/2*b^3*f^2*log(1-(c+d*x)^2)/d^3-3*b^3*f*(d*e-c*f)*polylog(2,(-1-c-d*x)/(1-c-d*x))/d^3-b^2*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*(a+b*arctanh(c+d*x))*polylog(2,1+(-2)/(1-c-d*x))/d^3+1/2*b^3*(3*d^2*e^2-6*c*d*e*f+(1+3*c^2)*f^2)*polylog(3,1+(-2)/(1-c-d*x))/d^3],
[(e+f*x)*(a+b*arctanh(c+d*x))^3,x,15,3/2*b*f*(a+b*arctanh(c+d*x))^2/d^2+3/2*b*f*(c+d*x)*(a+b*arctanh(c+d*x))^2/d^2+(d*e-c*f)*(a+b*arctanh(c+d*x))^3/d^2-1/2*(d^2*e^2-2*c*d*e*f+(1+c^2)*f^2)*(a+b*arctanh(c+d*x))^3/(d^2*f)+1/2*(e+f*x)^2*(a+b*arctanh(c+d*x))^3/f-3*b^2*f*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d^2-3*b*(d*e-c*f)*(a+b*arctanh(c+d*x))^2*log(2/(1-c-d*x))/d^2-3/2*b^3*f*polylog(2,(-1-c-d*x)/(1-c-d*x))/d^2-3*b^2*(d*e-c*f)*(a+b*arctanh(c+d*x))*polylog(2,1+(-2)/(1-c-d*x))/d^2+3/2*b^3*(d*e-c*f)*polylog(3,1+(-2)/(1-c-d*x))/d^2],
[(a+b*arctanh(c+d*x))^3,x,6,(a+b*arctanh(c+d*x))^3/d+(c+d*x)*(a+b*arctanh(c+d*x))^3/d-3*b*(a+b*arctanh(c+d*x))^2*log(2/(1-c-d*x))/d-3*b^2*(a+b*arctanh(c+d*x))*polylog(2,1+(-2)/(1-c-d*x))/d+3/2*b^3*polylog(3,1+(-2)/(1-c-d*x))/d],
[(a+b*arctanh(c+d*x))^3/(e+f*x),x,2,-(a+b*arctanh(c+d*x))^3*log(2/(1+c+d*x))/f+(a+b*arctanh(c+d*x))^3*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f+3/2*b*(a+b*arctanh(c+d*x))^2*polylog(2,1+(-2)/(1+c+d*x))/f-3/2*b*(a+b*arctanh(c+d*x))^2*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f+3/2*b^2*(a+b*arctanh(c+d*x))*polylog(3,1+(-2)/(1+c+d*x))/f-3/2*b^2*(a+b*arctanh(c+d*x))*polylog(3,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f+3/4*b^3*polylog(4,1+(-2)/(1+c+d*x))/f-3/4*b^3*polylog(4,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/f],
[(a+b*arctanh(c+d*x))^3/(e+f*x)^2,x,33,-(a+b*arctanh(c+d*x))^3/(f*(e+f*x))+3*a*b^2*d*arctanh(c+d*x)*log(2/(1-c-d*x))/(f*(d*e+f-c*f))+3/2*b^3*d*arctanh(c+d*x)^2*log(2/(1-c-d*x))/(f*(d*e+f-c*f))-3/2*a^2*b*d*log(1-c-d*x)/(f*(d*e+f-c*f))-3*a*b^2*d*arctanh(c+d*x)*log(2/(1+c+d*x))/(f*(d*e-f-c*f))+6*a*b^2*d*arctanh(c+d*x)*log(2/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))-3/2*b^3*d*arctanh(c+d*x)^2*log(2/(1+c+d*x))/(f*(d*e-f-c*f))+3*b^3*d*arctanh(c+d*x)^2*log(2/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))+3/2*a^2*b*d*log(1+c+d*x)/(f*(d*e-f-c*f))+3*a^2*b*d*log(e+f*x)/(f^2-(d*e-c*f)^2)-6*a*b^2*d*arctanh(c+d*x)*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))-3*b^3*d*arctanh(c+d*x)^2*log(2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))+3/2*a*b^2*d*polylog(2,(-1-c-d*x)/(1-c-d*x))/(f*(d*e+f-c*f))+3/2*b^3*d*arctanh(c+d*x)*polylog(2,1+(-2)/(1-c-d*x))/(f*(d*e+f-c*f))+3/2*a*b^2*d*polylog(2,1+(-2)/(1+c+d*x))/(f*(d*e-f-c*f))-3*a*b^2*d*polylog(2,1+(-2)/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))+3/2*b^3*d*arctanh(c+d*x)*polylog(2,1+(-2)/(1+c+d*x))/(f*(d*e-f-c*f))-3*b^3*d*arctanh(c+d*x)*polylog(2,1+(-2)/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))+3*a*b^2*d*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))+3*b^3*d*arctanh(c+d*x)*polylog(2,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))-3/4*b^3*d*polylog(3,1+(-2)/(1-c-d*x))/(f*(d*e+f-c*f))+3/4*b^3*d*polylog(3,1+(-2)/(1+c+d*x))/(f*(d*e-f-c*f))-3/2*b^3*d*polylog(3,1+(-2)/(1+c+d*x))/((d*e-f-c*f)*(d*e+f-c*f))+3/2*b^3*d*polylog(3,1-2*d*(e+f*x)/((d*e+f-c*f)*(1+c+d*x)))/((d*e-f-c*f)*(d*e+f-c*f))],

# Integrands of the form (e+f x)^m (a+b ArcTanh[c+d x])^p with m symbolic
[(e+f*x)^m*(a+b*arctanh(c+d*x))^3,x,1,Unintegrable((e+f*x)^m*(a+b*arctanh(c+d*x))^3,x)],
[(e+f*x)^m*(a+b*arctanh(c+d*x))^2,x,1,Unintegrable((e+f*x)^m*(a+b*arctanh(c+d*x))^2,x)],
[(e+f*x)^m*(a+b*arctanh(c+d*x)),x,6,(e+f*x)^(1+m)*(a+b*arctanh(c+d*x))/(f*(1+m))+1/2*b*d*(e+f*x)^(2+m)*hypergeom([1,2+m],[3+m],d*(e+f*x)/(d*e-f-c*f))/(f*(d*e-(1+c)*f)*(1+m)*(2+m))-1/2*b*d*(e+f*x)^(2+m)*hypergeom([1,2+m],[3+m],d*(e+f*x)/(d*e+f-c*f))/(f*(d*e+f-c*f)*(1+m)*(2+m))],

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

# Integrands of the form (e+f x^m)^q (a+b ArcTanh[c+d x])^p
[arctanh(a+b*x)/(c+d*x^3),x,23,-1/6*log(1-a-b*x)*log(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)+(1-a)*d^(1/3)))/(c^(2/3)*d^(1/3))+1/6*log(1+a+b*x)*log(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)-(1+a)*d^(1/3)))/(c^(2/3)*d^(1/3))-1/6*(-1)^(2/3)*log(1-a-b*x)*log(b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^(1/3)*(1-a)*d^(1/3)))/(c^(2/3)*d^(1/3))+1/6*(-1)^(2/3)*log(1+a+b*x)*log(b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(1/3)*(1+a)*d^(1/3)))/(c^(2/3)*d^(1/3))+1/6*(-1)^(1/3)*log(1-a-b*x)*log(b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(1-a)*d^(1/3)))/(c^(2/3)*d^(1/3))-1/6*(-1)^(1/3)*log(1+a+b*x)*log(b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^(2/3)*(1+a)*d^(1/3)))/(c^(2/3)*d^(1/3))-1/6*polylog(2,d^(1/3)*(1-a-b*x)/(b*c^(1/3)+(1-a)*d^(1/3)))/(c^(2/3)*d^(1/3))-1/6*(-1)^(2/3)*polylog(2,-(-1)^(1/3)*d^(1/3)*(1-a-b*x)/(b*c^(1/3)-(-1)^(1/3)*(1-a)*d^(1/3)))/(c^(2/3)*d^(1/3))+1/6*(-1)^(1/3)*polylog(2,(-1)^(2/3)*d^(1/3)*(1-a-b*x)/(b*c^(1/3)+(-1)^(2/3)*(1-a)*d^(1/3)))/(c^(2/3)*d^(1/3))+1/6*polylog(2,-d^(1/3)*(1+a+b*x)/(b*c^(1/3)-(1+a)*d^(1/3)))/(c^(2/3)*d^(1/3))+1/6*(-1)^(2/3)*polylog(2,(-1)^(1/3)*d^(1/3)*(1+a+b*x)/(b*c^(1/3)+(-1)^(1/3)*(1+a)*d^(1/3)))/(c^(2/3)*d^(1/3))-1/6*(-1)^(1/3)*polylog(2,-(-1)^(2/3)*d^(1/3)*(1+a+b*x)/(b*c^(1/3)-(-1)^(2/3)*(1+a)*d^(1/3)))/(c^(2/3)*d^(1/3))],
[arctanh(a+b*x)/(c+d*x^2),x,17,-1/4*log(1-a-b*x)*log(b*(sqrt(-c)-x*sqrt(d))/(b*sqrt(-c)-(1-a)*sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*log(1+a+b*x)*log(b*(sqrt(-c)-x*sqrt(d))/(b*sqrt(-c)+(1+a)*sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*log(1-a-b*x)*log(b*(sqrt(-c)+x*sqrt(d))/(b*sqrt(-c)+(1-a)*sqrt(d)))/(sqrt(-c)*sqrt(d))-1/4*log(1+a+b*x)*log(b*(sqrt(-c)+x*sqrt(d))/(b*sqrt(-c)-(1+a)*sqrt(d)))/(sqrt(-c)*sqrt(d))-1/4*polylog(2,-(1-a-b*x)*sqrt(d)/(b*sqrt(-c)-(1-a)*sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*polylog(2,(1-a-b*x)*sqrt(d)/(b*sqrt(-c)+(1-a)*sqrt(d)))/(sqrt(-c)*sqrt(d))-1/4*polylog(2,-(1+a+b*x)*sqrt(d)/(b*sqrt(-c)-(1+a)*sqrt(d)))/(sqrt(-c)*sqrt(d))+1/4*polylog(2,(1+a+b*x)*sqrt(d)/(b*sqrt(-c)+(1+a)*sqrt(d)))/(sqrt(-c)*sqrt(d))],
[arctanh(a+b*x)/(c+d*x),x,5,-arctanh(a+b*x)*log(2/(1+a+b*x))/d+arctanh(a+b*x)*log(2*b*(c+d*x)/((b*c+d-a*d)*(1+a+b*x)))/d+1/2*polylog(2,1+(-2)/(1+a+b*x))/d-1/2*polylog(2,1-2*b*(c+d*x)/((b*c+d-a*d)*(1+a+b*x)))/d],
[arctanh(a+b*x)/(c+d/x),x,15,1/2*(1-a-b*x)*log(1-a-b*x)/(b*c)+1/2*(1+a+b*x)*log(1+a+b*x)/(b*c)-1/2*d*log(1+a+b*x)*log(-b*(d+c*x)/(c+a*c-b*d))/c^2+1/2*d*log(1-a-b*x)*log(b*(d+c*x)/(c-a*c+b*d))/c^2+1/2*d*polylog(2,c*(1-a-b*x)/(c-a*c+b*d))/c^2-1/2*d*polylog(2,c*(1+a+b*x)/(c+a*c-b*d))/c^2],
[arctanh(a+b*x)/(c+d/x^2),x,25,1/2*(1-a-b*x)*log(1-a-b*x)/(b*c)+1/2*(1+a+b*x)*log(1+a+b*x)/(b*c)+1/4*log(1-a-b*x)*log(-b*(-x*sqrt(-c)+sqrt(d))/((1-a)*sqrt(-c)-b*sqrt(d)))*sqrt(d)/(-c)^(3/2)+1/4*log(1+a+b*x)*log(-b*(x*sqrt(-c)+sqrt(d))/((1+a)*sqrt(-c)-b*sqrt(d)))*sqrt(d)/(-c)^(3/2)-1/4*log(1-a-b*x)*log(b*(x*sqrt(-c)+sqrt(d))/((1-a)*sqrt(-c)+b*sqrt(d)))*sqrt(d)/(-c)^(3/2)-1/4*log(1+a+b*x)*log(b*(-x*sqrt(-c)+sqrt(d))/((1+a)*sqrt(-c)+b*sqrt(d)))*sqrt(d)/(-c)^(3/2)+1/4*polylog(2,(1-a-b*x)*sqrt(-c)/(sqrt(-c)-a*sqrt(-c)-b*sqrt(d)))*sqrt(d)/(-c)^(3/2)+1/4*polylog(2,(1+a+b*x)*sqrt(-c)/((1+a)*sqrt(-c)-b*sqrt(d)))*sqrt(d)/(-c)^(3/2)-1/4*polylog(2,(1-a-b*x)*sqrt(-c)/((1-a)*sqrt(-c)+b*sqrt(d)))*sqrt(d)/(-c)^(3/2)-1/4*polylog(2,(1+a+b*x)*sqrt(-c)/((1+a)*sqrt(-c)+b*sqrt(d)))*sqrt(d)/(-c)^(3/2)],
[arctanh(a+b*x)/(c+d/x^3),x,31,1/2*(1-a-b*x)*log(1-a-b*x)/(b*c)+1/2*(1+a+b*x)*log(1+a+b*x)/(b*c)-1/6*d^(1/3)*log(1+a+b*x)*log(-b*(d^(1/3)+c^(1/3)*x)/((1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*d^(1/3)*log(1-a-b*x)*log(b*(d^(1/3)+c^(1/3)*x)/((1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(2/3)*d^(1/3)*log(1-a-b*x)*log(-b*(d^(1/3)-(-1)^(1/3)*c^(1/3)*x)/((-1)^(1/3)*(1-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(2/3)*d^(1/3)*log(1+a+b*x)*log(b*(d^(1/3)-(-1)^(1/3)*c^(1/3)*x)/((-1)^(1/3)*(1+a)*c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/3)*d^(1/3)*log(1+a+b*x)*log(-b*(d^(1/3)+(-1)^(2/3)*c^(1/3)*x)/((-1)^(2/3)*(1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(1/3)*d^(1/3)*log(1-a-b*x)*log(b*(d^(1/3)+(-1)^(2/3)*c^(1/3)*x)/((-1)^(2/3)*(1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(2/3)*d^(1/3)*polylog(2,(-1)^(1/3)*c^(1/3)*(1-a-b*x)/((-1)^(1/3)*(1-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*d^(1/3)*polylog(2,c^(1/3)*(1-a-b*x)/((1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*(-1)^(1/3)*d^(1/3)*polylog(2,(-1)^(2/3)*c^(1/3)*(1-a-b*x)/((-1)^(2/3)*(1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*d^(1/3)*polylog(2,c^(1/3)*(1+a+b*x)/((1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/3)*d^(1/3)*polylog(2,(-1)^(2/3)*c^(1/3)*(1+a+b*x)/((-1)^(2/3)*(1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(2/3)*d^(1/3)*polylog(2,(-1)^(1/3)*c^(1/3)*(1+a+b*x)/((-1)^(1/3)*(1+a)*c^(1/3)+b*d^(1/3)))/c^(4/3)],

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

#  {ArcTanh[a + b*x]/(a + b*x^(3/2)), x, 41, (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] - Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[(Sqrt[1 - a] + Sqrt[b]*Sqrt[x])/(Sqrt[1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(1/3)*b^(2/3)) - (Log[-a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*Log[(-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*Log[-((-1)^(2/3)*a^(1/3)) - b^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, -((b^(1/6)*((-1)^(1/3)*a^(1/3) - b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - (-1)^(1/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - a^(1/3)*b^(1/6)))]/(3*a^(1/3)*b^(2/3)) - PolyLog[2, -((b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - a^(1/3)*b^(1/6)))]/(3*a^(1/3)*b^(2/3)) + PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + a^(1/3)*b^(1/6))]/(3*a^(1/3)*b^(2/3)) - PolyLog[2, (b^(1/6)*(a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + a^(1/3)*b^(1/6))]/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, -((b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] - (-1)^(2/3)*a^(1/3)*b^(1/6)))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(1/3)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(1/3)*PolyLog[2, (b^(1/6)*((-1)^(2/3)*a^(1/3) + b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + (-1)^(2/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) + ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[-1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3)) - ((-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*b^(1/6)*(a^(1/3) + (-1)^(2/3)*b^(1/3)*Sqrt[x]))/(Sqrt[1 - a] + (-1)^(1/3)*a^(1/3)*b^(1/6))])/(3*a^(1/3)*b^(2/3))} 
[arctanh(a+b*x)/(c+d*sqrt(x)),x,31,c*log(1-a-b*x)*log(c+d*sqrt(x))/d^2-c*log(1+a+b*x)*log(c+d*sqrt(x))/d^2+c*log(c+d*sqrt(x))*log(d*(sqrt(-1-a)-sqrt(b)*sqrt(x))/(d*sqrt(-1-a)+c*sqrt(b)))/d^2-c*log(c+d*sqrt(x))*log(d*(sqrt(1-a)-sqrt(b)*sqrt(x))/(d*sqrt(1-a)+c*sqrt(b)))/d^2+c*log(c+d*sqrt(x))*log(-d*(sqrt(-1-a)+sqrt(b)*sqrt(x))/(-d*sqrt(-1-a)+c*sqrt(b)))/d^2-c*log(c+d*sqrt(x))*log(-d*(sqrt(1-a)+sqrt(b)*sqrt(x))/(-d*sqrt(1-a)+c*sqrt(b)))/d^2+c*polylog(2,sqrt(b)*(c+d*sqrt(x))/(-d*sqrt(-1-a)+c*sqrt(b)))/d^2+c*polylog(2,sqrt(b)*(c+d*sqrt(x))/(d*sqrt(-1-a)+c*sqrt(b)))/d^2-c*polylog(2,sqrt(b)*(c+d*sqrt(x))/(-d*sqrt(1-a)+c*sqrt(b)))/d^2-c*polylog(2,sqrt(b)*(c+d*sqrt(x))/(d*sqrt(1-a)+c*sqrt(b)))/d^2-2*arctanh(sqrt(b)*sqrt(x)/sqrt(1-a))*sqrt(1-a)/(d*sqrt(b))+2*arctan(sqrt(b)*sqrt(x)/sqrt(1+a))*sqrt(1+a)/(d*sqrt(b))-log(1-a-b*x)*sqrt(x)/d+log(1+a+b*x)*sqrt(x)/d],
[arctanh(a+b*x)/(c+d/sqrt(x)),x,37,1/2*(1-a-b*x)*log(1-a-b*x)/(b*c)+1/2*(1+a+b*x)*log(1+a+b*x)/(b*c)-d^2*log(1-a-b*x)*log(d+c*sqrt(x))/c^3+d^2*log(1+a+b*x)*log(d+c*sqrt(x))/c^3-d^2*log(d+c*sqrt(x))*log(c*(sqrt(-1-a)-sqrt(b)*sqrt(x))/(c*sqrt(-1-a)+d*sqrt(b)))/c^3+d^2*log(d+c*sqrt(x))*log(c*(sqrt(1-a)-sqrt(b)*sqrt(x))/(c*sqrt(1-a)+d*sqrt(b)))/c^3-d^2*log(d+c*sqrt(x))*log(c*(sqrt(-1-a)+sqrt(b)*sqrt(x))/(c*sqrt(-1-a)-d*sqrt(b)))/c^3+d^2*log(d+c*sqrt(x))*log(c*(sqrt(1-a)+sqrt(b)*sqrt(x))/(c*sqrt(1-a)-d*sqrt(b)))/c^3-d^2*polylog(2,-sqrt(b)*(d+c*sqrt(x))/(c*sqrt(-1-a)-d*sqrt(b)))/c^3+d^2*polylog(2,-sqrt(b)*(d+c*sqrt(x))/(c*sqrt(1-a)-d*sqrt(b)))/c^3-d^2*polylog(2,sqrt(b)*(d+c*sqrt(x))/(c*sqrt(-1-a)+d*sqrt(b)))/c^3+d^2*polylog(2,sqrt(b)*(d+c*sqrt(x))/(c*sqrt(1-a)+d*sqrt(b)))/c^3+2*d*arctanh(sqrt(b)*sqrt(x)/sqrt(1-a))*sqrt(1-a)/(c^2*sqrt(b))-2*d*arctan(sqrt(b)*sqrt(x)/sqrt(1+a))*sqrt(1+a)/(c^2*sqrt(b))+d*log(1-a-b*x)*sqrt(x)/c^2-d*log(1+a+b*x)*sqrt(x)/c^2],

#  {ArcTanh[a + b*x]/(a + b/x^(3/2)), x, 49, -(b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] - Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[-1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[(a^(1/3)*(Sqrt[1 - a] + Sqrt[b]*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) + ((1 - a - b*x)*Log[1 - a - b*x])/(2*a*b) - (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 - a - b*x])/(3*a^(5/3)) + ((1 + a + b*x)*Log[1 + a + b*x])/(2*a*b) + (b^(2/3)*Log[-b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*Log[(-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*Log[-((-1)^(2/3)*b^(1/3)) - a^(1/3)*Sqrt[x]]*Log[1 + a + b*x])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(1/3)*b^(1/3) - a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(1/3)*b^(5/6)))])/(3*a^(5/3)) - (b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) + (b^(2/3)*PolyLog[2, -((Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - b^(5/6)))])/(3*a^(5/3)) - (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + (b^(2/3)*PolyLog[2, (Sqrt[b]*(b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + b^(5/6))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, -((Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) - (-1)^(2/3)*b^(5/6)))])/(3*a^(5/3)) + ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(1/3)*b^(2/3)*PolyLog[2, (Sqrt[b]*((-1)^(2/3)*b^(1/3) + a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(2/3)*b^(5/6))])/(3*a^(5/3)) - ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[-1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3)) + ((-1)^(2/3)*b^(2/3)*PolyLog[2, ((-1)^(1/3)*Sqrt[b]*(b^(1/3) + (-1)^(2/3)*a^(1/3)*Sqrt[x]))/(Sqrt[1 - a]*a^(1/3) + (-1)^(1/3)*b^(5/6))])/(3*a^(5/3))} 

# Integrands of the form (e+f x+g x^2)^q (a+b ArcTanh[c+d x])^p
[arctanh(d+e*x)/(a+b*x+c*x^2),x,12,arctanh(d+e*x)*log(2*e*(b+2*c*x-sqrt(b^2-4*a*c))/((1+d+e*x)*(2*c*(1-d)+e*(b-sqrt(b^2-4*a*c)))))/sqrt(b^2-4*a*c)-arctanh(d+e*x)*log(2*e*(b+2*c*x+sqrt(b^2-4*a*c))/((1+d+e*x)*(2*c*(1-d)+e*(b+sqrt(b^2-4*a*c)))))/sqrt(b^2-4*a*c)-1/2*polylog(2,1+2*(2*c*d-2*c*(d+e*x)-e*(b-sqrt(b^2-4*a*c)))/((1+d+e*x)*(2*c-2*c*d+b*e-e*sqrt(b^2-4*a*c))))/sqrt(b^2-4*a*c)+1/2*polylog(2,1+2*(2*c*d-2*c*(d+e*x)-e*(b+sqrt(b^2-4*a*c)))/((1+d+e*x)*(2*c*(1-d)+e*(b+sqrt(b^2-4*a*c)))))/sqrt(b^2-4*a*c)],

# Integrands of the form (c e+d e x)^m (a+b ArcTanh[c+d x])^p / (1-(c+d x)^2)
[(c*e+d*e*x)*(a+b*arctanh(c+d*x))/(1-(c+d*x)^2),x,6,-1/2*e*(a+b*arctanh(c+d*x))^2/(b*d)+e*(a+b*arctanh(c+d*x))*log(2/(1-c-d*x))/d+1/2*b*e*polylog(2,(-1-c-d*x)/(1-c-d*x))/d]]:
