# Maple integration test file: "7 Inverse hyperbolic functions\7.4 Inverse hyperbolic cotangent\7.4.2 Exponentials of inverse hyperbolic cotangent functions.txt"

lst:=[

# Integrands involving exponentials of inverse hyperbolic cotangents

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

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

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

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

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

# n>0
[exp(1/2*arccoth(a*x))*x^4,x,11,611/1920*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a^4+269/960*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^2/a^3+11/48*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^3/a^2+9/40*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^4/a+1/5*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^5+31/128*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5+31/128*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5],
[exp(1/2*arccoth(a*x))*x^3,x,10,83/192*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a^3+29/96*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^2/a^2+7/24*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^3/a+1/4*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^4+11/64*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4+11/64*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4],
[exp(1/2*arccoth(a*x))*x^2,x,9,11/24*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a^2+5/12*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^2/a+1/3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^3+3/8*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3+3/8*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3],
[exp(1/2*arccoth(a*x))*x,x,7,1/4*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a+1/2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(5/4)*x^2+1/4*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2+1/4*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2],
[exp(1/2*arccoth(a*x)),x,6,(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x+arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a+arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a],
[exp(1/2*arccoth(a*x))/x,x,17,2*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+2*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)+arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)],
[exp(1/2*arccoth(a*x))/x^2,x,13,a*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)-a*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+a*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+1/2*a*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-1/2*a*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(1/2*arccoth(a*x))/x^3,x,14,1/4*a^2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)+1/2*a^2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(5/4)-1/4*a^2*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+1/4*a^2*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+1/8*a^2*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-1/8*a^2*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(1/2*arccoth(a*x))/x^4,x,15,3/8*a^3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)+1/12*a^3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(5/4)+1/3*a^2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(5/4)/x-3/8*a^3*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+3/8*a^3*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+3/16*a^3*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-3/16*a^3*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(3/2*arccoth(a*x))*x^4,x,11,557/640*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a^4+157/320*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^2/a^3+5/16*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^3/a^2+11/40*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^4/a+1/5*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^5-237/128*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5+237/128*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5],
[exp(3/2*arccoth(a*x))*x^3,x,10,63/64*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a^3+15/32*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^2/a^2+3/8*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^3/a+1/4*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^4-123/64*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4+123/64*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4],
[exp(3/2*arccoth(a*x))*x^2,x,9,23/24*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a^2+7/12*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^2/a+1/3*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^3-17/8*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3+17/8*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3],
[exp(3/2*arccoth(a*x))*x,x,7,3/4*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a+1/2*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(7/4)*x^2-9/4*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2+9/4*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2],
[exp(3/2*arccoth(a*x)),x,6,(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x-3*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a+3*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a],
[exp(3/2*arccoth(a*x))/x,x,17,-2*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+2*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))-log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)+arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)],
[exp(3/2*arccoth(a*x))/x^2,x,13,a*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)-3*a*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+3*a*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-3/2*a*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+3/2*a*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(3/2*arccoth(a*x))/x^3,x,14,3/4*a^2*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)+1/2*a^2*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(7/4)-9/4*a^2*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+9/4*a^2*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-9/8*a^2*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+9/8*a^2*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(3/2*arccoth(a*x))/x^4,x,15,17/24*a^3*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)+1/4*a^3*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(7/4)+1/3*a^2*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(7/4)/x-17/8*a^3*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+17/8*a^3*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-17/16*a^3*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+17/16*a^3*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(5/2*arccoth(a*x))*x^4,x,12,-26111/1920*(1+1/(a*x))^(1/4)/(a^5*(1+(-1)/(a*x))^(1/4))+5533/1920*(1+1/(a*x))^(1/4)*x/(a^4*(1+(-1)/(a*x))^(1/4))+1189/960*(1+1/(a*x))^(1/4)*x^2/(a^3*(1+(-1)/(a*x))^(1/4))+181/240*(1+1/(a*x))^(1/4)*x^3/(a^2*(1+(-1)/(a*x))^(1/4))+21/40*(1+1/(a*x))^(1/4)*x^4/(a*(1+(-1)/(a*x))^(1/4))+1/5*(1+1/(a*x))^(1/4)*x^5/(1+(-1)/(a*x))^(1/4)+1003/128*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5+1003/128*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5],
[exp(5/2*arccoth(a*x))*x^3,x,11,-2467/192*(1+1/(a*x))^(1/4)/(a^4*(1+(-1)/(a*x))^(1/4))+521/192*(1+1/(a*x))^(1/4)*x/(a^3*(1+(-1)/(a*x))^(1/4))+113/96*(1+1/(a*x))^(1/4)*x^2/(a^2*(1+(-1)/(a*x))^(1/4))+17/24*(1+1/(a*x))^(1/4)*x^3/(a*(1+(-1)/(a*x))^(1/4))+1/4*(1+1/(a*x))^(1/4)*x^4/(1+(-1)/(a*x))^(1/4)+475/64*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4+475/64*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4],
[exp(5/2*arccoth(a*x))*x^2,x,10,-287/24*(1+1/(a*x))^(1/4)/(a^3*(1+(-1)/(a*x))^(1/4))+61/24*(1+1/(a*x))^(1/4)*x/(a^2*(1+(-1)/(a*x))^(1/4))+13/12*(1+1/(a*x))^(1/4)*x^2/(a*(1+(-1)/(a*x))^(1/4))+1/3*(1+1/(a*x))^(1/4)*x^3/(1+(-1)/(a*x))^(1/4)+55/8*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3+55/8*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3],
[exp(5/2*arccoth(a*x))*x,x,8,-25/2*(1+1/(a*x))^(1/4)/(a^2*(1+(-1)/(a*x))^(1/4))+5/4*(1+1/(a*x))^(5/4)*x/(a*(1+(-1)/(a*x))^(1/4))+1/2*(1+1/(a*x))^(9/4)*x^2/(1+(-1)/(a*x))^(1/4)+25/4*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2+25/4*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2],
[exp(5/2*arccoth(a*x)),x,7,-10*(1+1/(a*x))^(1/4)/(a*(1+(-1)/(a*x))^(1/4))+(1+1/(a*x))^(5/4)*x/(1+(-1)/(a*x))^(1/4)+5*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a+5*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a],
[exp(5/2*arccoth(a*x))/x,x,19,-8*(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)+2*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+2*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))-log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)-arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)],
[exp(5/2*arccoth(a*x))/x^2,x,14,-5*a*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)-4*a*(1+1/(a*x))^(5/4)/(1+(-1)/(a*x))^(1/4)+5*a*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-5*a*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-5/2*a*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+5/2*a*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(5/2*arccoth(a*x))/x^3,x,15,-25/4*a^2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)-5/2*a^2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(5/4)-2*a^2*(1+1/(a*x))^(9/4)/(1+(-1)/(a*x))^(1/4)+25/4*a^2*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-25/4*a^2*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-25/8*a^2*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+25/8*a^2*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[exp(5/2*arccoth(a*x))/x^4,x,16,-55/8*a^3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)-11/4*a^3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(5/4)-2*a^3*(1+1/(a*x))^(9/4)/(1+(-1)/(a*x))^(1/4)-1/3*a^3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(9/4)+55/8*a^3*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-55/8*a^3*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-55/16*a^3*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+55/16*a^3*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],

# n<0
[x^4/exp(1/2*arccoth(a*x)),x,11,611/1920*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a^4-269/960*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^2/a^3+11/48*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^3/a^2-9/40*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^4/a+1/5*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^5+31/128*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5-31/128*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5],
[x^3/exp(1/2*arccoth(a*x)),x,10,-83/192*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a^3+29/96*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^2/a^2-7/24*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^3/a+1/4*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^4-11/64*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4+11/64*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4],
[x^2/exp(1/2*arccoth(a*x)),x,9,11/24*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a^2-5/12*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^2/a+1/3*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x^3+3/8*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3-3/8*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3],
[x/exp(1/2*arccoth(a*x)),x,7,-1/4*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x/a+1/2*(1+(-1)/(a*x))^(5/4)*(1+1/(a*x))^(3/4)*x^2-1/4*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2+1/4*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2],
[exp(-1/2*arccoth(a*x)),x,6,(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)*x+arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a-arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a],
[1/(exp(1/2*arccoth(a*x))*x),x,17,-2*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+2*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)-arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)],
[1/(exp(1/2*arccoth(a*x))*x^2),x,13,-a*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)-a*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+a*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-1/2*a*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+1/2*a*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[1/(exp(1/2*arccoth(a*x))*x^3),x,14,1/4*a^2*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)+1/2*a^2*(1+(-1)/(a*x))^(5/4)*(1+1/(a*x))^(3/4)+1/4*a^2*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-1/4*a^2*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+1/8*a^2*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-1/8*a^2*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[1/(exp(1/2*arccoth(a*x))*x^4),x,15,-3/8*a^3*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)-1/12*a^3*(1+(-1)/(a*x))^(5/4)*(1+1/(a*x))^(3/4)+1/3*a^2*(1+(-1)/(a*x))^(5/4)*(1+1/(a*x))^(3/4)/x-3/8*a^3*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+3/8*a^3*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-3/16*a^3*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+3/16*a^3*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[x^4/exp(3/2*arccoth(a*x)),x,11,557/640*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a^4-157/320*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^2/a^3+5/16*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^3/a^2-11/40*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^4/a+1/5*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^5-237/128*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5-237/128*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5],
[x^3/exp(3/2*arccoth(a*x)),x,10,-63/64*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a^3+15/32*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^2/a^2-3/8*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^3/a+1/4*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^4+123/64*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4+123/64*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4],
[x^2/exp(3/2*arccoth(a*x)),x,9,23/24*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a^2-7/12*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^2/a+1/3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x^3-17/8*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3-17/8*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3],
[x/exp(3/2*arccoth(a*x)),x,7,-3/4*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x/a+1/2*(1+(-1)/(a*x))^(7/4)*(1+1/(a*x))^(1/4)*x^2+9/4*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2+9/4*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2],
[exp(-3/2*arccoth(a*x)),x,6,(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)*x-3*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a-3*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a],
[1/(exp(3/2*arccoth(a*x))*x),x,17,2*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+2*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))-log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)-arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)],
[1/(exp(3/2*arccoth(a*x))*x^2),x,13,-a*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)-3*a*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+3*a*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+3/2*a*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-3/2*a*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[1/(exp(3/2*arccoth(a*x))*x^3),x,14,3/4*a^2*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)+1/2*a^2*(1+(-1)/(a*x))^(7/4)*(1+1/(a*x))^(1/4)+9/4*a^2*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-9/4*a^2*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-9/8*a^2*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+9/8*a^2*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[1/(exp(3/2*arccoth(a*x))*x^4),x,15,-17/24*a^3*(1+(-1)/(a*x))^(3/4)*(1+1/(a*x))^(1/4)-1/4*a^3*(1+(-1)/(a*x))^(7/4)*(1+1/(a*x))^(1/4)+1/3*a^2*(1+(-1)/(a*x))^(7/4)*(1+1/(a*x))^(1/4)/x-17/8*a^3*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+17/8*a^3*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+17/16*a^3*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-17/16*a^3*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[x^4/exp(5/2*arccoth(a*x)),x,12,26111/1920*(1+(-1)/(a*x))^(1/4)/(a^5*(1+1/(a*x))^(1/4))+5533/1920*(1+(-1)/(a*x))^(1/4)*x/(a^4*(1+1/(a*x))^(1/4))-1189/960*(1+(-1)/(a*x))^(1/4)*x^2/(a^3*(1+1/(a*x))^(1/4))+181/240*(1+(-1)/(a*x))^(1/4)*x^3/(a^2*(1+1/(a*x))^(1/4))-21/40*(1+(-1)/(a*x))^(1/4)*x^4/(a*(1+1/(a*x))^(1/4))+1/5*(1+(-1)/(a*x))^(1/4)*x^5/(1+1/(a*x))^(1/4)+1003/128*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5-1003/128*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^5],
[x^3/exp(5/2*arccoth(a*x)),x,11,-2467/192*(1+(-1)/(a*x))^(1/4)/(a^4*(1+1/(a*x))^(1/4))-521/192*(1+(-1)/(a*x))^(1/4)*x/(a^3*(1+1/(a*x))^(1/4))+113/96*(1+(-1)/(a*x))^(1/4)*x^2/(a^2*(1+1/(a*x))^(1/4))-17/24*(1+(-1)/(a*x))^(1/4)*x^3/(a*(1+1/(a*x))^(1/4))+1/4*(1+(-1)/(a*x))^(1/4)*x^4/(1+1/(a*x))^(1/4)-475/64*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4+475/64*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^4],
[x^2/exp(5/2*arccoth(a*x)),x,10,287/24*(1+(-1)/(a*x))^(1/4)/(a^3*(1+1/(a*x))^(1/4))+61/24*(1+(-1)/(a*x))^(1/4)*x/(a^2*(1+1/(a*x))^(1/4))-13/12*(1+(-1)/(a*x))^(1/4)*x^2/(a*(1+1/(a*x))^(1/4))+1/3*(1+(-1)/(a*x))^(1/4)*x^3/(1+1/(a*x))^(1/4)+55/8*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3-55/8*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^3],
[x/exp(5/2*arccoth(a*x)),x,8,-25/2*(1+(-1)/(a*x))^(1/4)/(a^2*(1+1/(a*x))^(1/4))-5/4*(1+(-1)/(a*x))^(5/4)*x/(a*(1+1/(a*x))^(1/4))+1/2*(1+(-1)/(a*x))^(9/4)*x^2/(1+1/(a*x))^(1/4)-25/4*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2+25/4*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a^2],
[exp(-5/2*arccoth(a*x)),x,7,10*(1+(-1)/(a*x))^(1/4)/(a*(1+1/(a*x))^(1/4))+(1+(-1)/(a*x))^(5/4)*x/(1+1/(a*x))^(1/4)+5*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a-5*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))/a],
[1/(exp(5/2*arccoth(a*x))*x),x,19,-8*(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-2*arctan((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))+2*arctanh((1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4))-log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)+arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))*sqrt(2)],
[1/(exp(5/2*arccoth(a*x))*x^2),x,14,4*a*(1+(-1)/(a*x))^(5/4)/(1+1/(a*x))^(1/4)+5*a*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)+5*a*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-5*a*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+5/2*a*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-5/2*a*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[1/(exp(5/2*arccoth(a*x))*x^3),x,15,-2*a^2*(1+(-1)/(a*x))^(9/4)/(1+1/(a*x))^(1/4)-25/4*a^2*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)-5/2*a^2*(1+(-1)/(a*x))^(5/4)*(1+1/(a*x))^(3/4)-25/4*a^2*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+25/4*a^2*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-25/8*a^2*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)+25/8*a^2*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],
[1/(exp(5/2*arccoth(a*x))*x^4),x,16,2*a^3*(1+(-1)/(a*x))^(9/4)/(1+1/(a*x))^(1/4)+55/8*a^3*(1+(-1)/(a*x))^(1/4)*(1+1/(a*x))^(3/4)+11/4*a^3*(1+(-1)/(a*x))^(5/4)*(1+1/(a*x))^(3/4)+1/3*a^3*(1+(-1)/(a*x))^(9/4)*(1+1/(a*x))^(3/4)+55/8*a^3*arctan(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)-55/8*a^3*arctan(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4))/sqrt(2)+55/16*a^3*log(1-(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)-55/16*a^3*log(1+(1+(-1)/(a*x))^(1/4)*sqrt(2)/(1+1/(a*x))^(1/4)+sqrt(1+(-1)/(a*x))/sqrt(1+1/(a*x)))/sqrt(2)],

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

# n>0
[exp(1/3*arccoth(x))*x^2,x,16,11/27*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x+7/18*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x^2+1/3*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x^3+19/81*arctanh((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-19/324*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)-(1+1/x)^(1/6)/((-1+x)/x)^(1/6))+19/324*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)+(1+1/x)^(1/6)/((-1+x)/x)^(1/6))-19/54*arctan((1-2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))/sqrt(3)+19/54*arctan((1+2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))/sqrt(3)],
[exp(1/3*arccoth(x))*x,x,14,1/6*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x+1/2*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)*x^2+1/9*arctanh((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/36*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)-(1+1/x)^(1/6)/((-1+x)/x)^(1/6))+1/36*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)+(1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/6*arctan((1-2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))/sqrt(3)+1/6*arctan((1+2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))/sqrt(3)],
[exp(1/3*arccoth(x)),x,13,(1+1/x)^(1/6)*((-1+x)/x)^(5/6)*x+2/3*arctanh((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/6*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)-(1+1/x)^(1/6)/((-1+x)/x)^(1/6))+1/6*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)+(1+1/x)^(1/6)/((-1+x)/x)^(1/6))-arctan((1-2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))/sqrt(3)+arctan((1+2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))/sqrt(3)],
[exp(1/3*arccoth(x))/x,x,25,2*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))-arctan(-2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+2*arctanh((1+1/x)^(1/6)/((-1+x)/x)^(1/6))-1/2*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)-(1+1/x)^(1/6)/((-1+x)/x)^(1/6))+1/2*log(1+(1+1/x)^(1/3)/((-1+x)/x)^(1/3)+(1+1/x)^(1/6)/((-1+x)/x)^(1/6))-arctan((1-2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))*sqrt(3)+arctan((1+2*(1+1/x)^(1/6)/((-1+x)/x)^(1/6))/sqrt(3))*sqrt(3)+1/2*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))*sqrt(3)-1/2*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))*sqrt(3)],
[exp(1/3*arccoth(x))/x^2,x,14,(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+2/3*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))-1/3*arctan(-2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+1/3*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+1/2*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))/sqrt(3)-1/2*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))/sqrt(3)],
[exp(1/3*arccoth(x))/x^3,x,15,1/6*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+1/2*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)+1/9*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))-1/18*arctan(-2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+1/18*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+1/12*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))/sqrt(3)-1/12*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))/sqrt(3)],
[exp(1/3*arccoth(x))/x^4,x,16,19/54*(1+1/x)^(1/6)*((-1+x)/x)^(5/6)+1/18*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)+1/3*(1+1/x)^(7/6)*((-1+x)/x)^(5/6)/x+19/81*arctan(((-1+x)/x)^(1/6)/(1+1/x)^(1/6))-19/162*arctan(-2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+19/162*arctan(2*((-1+x)/x)^(1/6)/(1+1/x)^(1/6)+sqrt(3))+19/108*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)-((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))/sqrt(3)-19/108*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3)+((-1+x)/x)^(1/6)*sqrt(3)/(1+1/x)^(1/6))/sqrt(3)],
[exp(2/3*arccoth(x))*x^2,x,6,14/27*(1+1/x)^(1/3)*((-1+x)/x)^(2/3)*x+4/9*(1+1/x)^(1/3)*((-1+x)/x)^(2/3)*x^2+1/3*(1+1/x)^(1/3)*((-1+x)/x)^(2/3)*x^3-11/27*log((1+1/x)^(1/3)-((-1+x)/x)^(1/3))-11/81*log(x)-22/27*arctan(1/sqrt(3)+2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arccoth(x))*x,x,4,1/3*(1+1/x)^(1/3)*((-1+x)/x)^(2/3)*x+1/2*(1+1/x)^(4/3)*((-1+x)/x)^(2/3)*x^2-1/3*log((1+1/x)^(1/3)-((-1+x)/x)^(1/3))-1/9*log(x)-2/3*arctan(1/sqrt(3)+2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arccoth(x)),x,3,(1+1/x)^(1/3)*((-1+x)/x)^(2/3)*x-log((1+1/x)^(1/3)-((-1+x)/x)^(1/3))-1/3*log(x)-2*arctan(1/sqrt(3)+2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arccoth(x))/x,x,4,-3/2*log((1+1/x)^(1/3)-((-1+x)/x)^(1/3))-3/2*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3))-1/2*log(1+1/x)-1/2*log(x)-arctan(1/sqrt(3)-2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))*sqrt(3)-arctan(1/sqrt(3)+2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))*sqrt(3)],
[exp(2/3*arccoth(x))/x^2,x,3,(1+1/x)^(1/3)*((-1+x)/x)^(2/3)-log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3))-1/3*log(1+1/x)-2*arctan(1/sqrt(3)-2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))/sqrt(3)],
[exp(2/3*arccoth(x))/x^3,x,4,1/3*(1+1/x)^(1/3)*((-1+x)/x)^(2/3)+1/2*(1+1/x)^(4/3)*((-1+x)/x)^(2/3)-1/3*log(1+((-1+x)/x)^(1/3)/(1+1/x)^(1/3))-1/9*log(1+1/x)-2/3*arctan(1/sqrt(3)-2*((-1+x)/x)^(1/3)/((1+1/x)^(1/3)*sqrt(3)))/sqrt(3)],

# n<0

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

# n>0
[exp(1/4*arccoth(a*x))*x^2,x,19,37/96*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)*x/a^2+3/8*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)*x^2/a+1/3*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)*x^3+11/64*arctan((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))/a^3+11/64*arctanh((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))/a^3-11/64*arctan(1-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^3*sqrt(2))+11/64*arctan(1+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^3*sqrt(2))-11/128*log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^3*sqrt(2))+11/128*log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^3*sqrt(2))],
[exp(1/4*arccoth(a*x))*x,x,17,1/8*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)*x/a+1/2*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(9/8)*x^2+1/16*arctan((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))/a^2+1/16*arctanh((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))/a^2-1/16*arctan(1-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^2*sqrt(2))+1/16*arctan(1+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^2*sqrt(2))-1/32*log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^2*sqrt(2))+1/32*log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a^2*sqrt(2))],
[exp(1/4*arccoth(a*x)),x,16,(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)*x+1/2*arctan((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))/a+1/2*arctanh((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))/a-1/2*arctan(1-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a*sqrt(2))+1/2*arctan(1+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a*sqrt(2))-1/4*log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a*sqrt(2))+1/4*log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/(a*sqrt(2))],
[exp(1/4*arccoth(a*x))/x,x,39,2*arctan((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))+2*arctanh((1+1/(a*x))^(1/8)/(1+(-1)/(a*x))^(1/8))-log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/sqrt(2)+log(1+(1+1/(a*x))^(1/4)/(1+(-1)/(a*x))^(1/4)+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))/sqrt(2)-arctan(1-(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))*sqrt(2)+arctan(1+(1+1/(a*x))^(1/8)*sqrt(2)/(1+(-1)/(a*x))^(1/8))*sqrt(2)-arctan((-2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+arctan((2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-(1+(-1)/(a*x))^(1/8)*sqrt(2-sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2-sqrt(2))-1/2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)+(1+(-1)/(a*x))^(1/8)*sqrt(2-sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2-sqrt(2))-arctan((-2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+arctan((2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-(1+(-1)/(a*x))^(1/8)*sqrt(2+sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2+sqrt(2))-1/2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)+(1+(-1)/(a*x))^(1/8)*sqrt(2+sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2+sqrt(2))],
[exp(1/4*arccoth(a*x))/x^2,x,25,a*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)-1/4*a*arctan((-2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/4*a*arctan((2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/8*a*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-(1+(-1)/(a*x))^(1/8)*sqrt(2-sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2-sqrt(2))-1/8*a*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)+(1+(-1)/(a*x))^(1/8)*sqrt(2-sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2-sqrt(2))-1/4*a*arctan((-2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/4*a*arctan((2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/8*a*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-(1+(-1)/(a*x))^(1/8)*sqrt(2+sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2+sqrt(2))-1/8*a*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)+(1+(-1)/(a*x))^(1/8)*sqrt(2+sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2+sqrt(2))],
[exp(1/4*arccoth(a*x))/x^3,x,26,1/8*a^2*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(1/8)+1/2*a^2*(1+(-1)/(a*x))^(7/8)*(1+1/(a*x))^(9/8)-1/32*a^2*arctan((-2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/32*a^2*arctan((2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2+sqrt(2)))/sqrt(2-sqrt(2)))*sqrt(2-sqrt(2))+1/64*a^2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-(1+(-1)/(a*x))^(1/8)*sqrt(2-sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2-sqrt(2))-1/64*a^2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)+(1+(-1)/(a*x))^(1/8)*sqrt(2-sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2-sqrt(2))-1/32*a^2*arctan((-2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/32*a^2*arctan((2*(1+(-1)/(a*x))^(1/8)/(1+1/(a*x))^(1/8)+sqrt(2-sqrt(2)))/sqrt(2+sqrt(2)))*sqrt(2+sqrt(2))+1/64*a^2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)-(1+(-1)/(a*x))^(1/8)*sqrt(2+sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2+sqrt(2))-1/64*a^2*log(1+(1+(-1)/(a*x))^(1/4)/(1+1/(a*x))^(1/4)+(1+(-1)/(a*x))^(1/8)*sqrt(2+sqrt(2))/(1+1/(a*x))^(1/8))*sqrt(2+sqrt(2))],

# n<0

# Integrands of the form x^m E^(n ArcCoth[a x]) with m symbolic
[exp(4*arccoth(a*x))*x^m,x,5,x^(1+m)/(1+m)+4*x^(1+m)/(1-a*x)-4*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)],
[exp(3*arccoth(a*x))*x^m,x,9,-3*x^(1+m)*hypergeom([1/2,1/2*(-1-m)],[1/2*(1-m)],1/(a^2*x^2))/(1+m)-x^m*hypergeom([1/2,-1/2*m],[1-1/2*m],1/(a^2*x^2))/(a*m)+4*x^(1+m)*hypergeom([3/2,1/2*(-1-m)],[1/2*(1-m)],1/(a^2*x^2))/(1+m)+4*x^m*hypergeom([3/2,-1/2*m],[1-1/2*m],1/(a^2*x^2))/(a*m)],
[exp(2*arccoth(a*x))*x^m,x,4,x^(1+m)/(1+m)-2*x^(1+m)*hypergeom([1,1+m],[2+m],a*x)/(1+m)],
[exp(arccoth(a*x))*x^m,x,4,x^(1+m)*hypergeom([1/2,1/2*(-1-m)],[1/2*(1-m)],1/(a^2*x^2))/(1+m)+x^m*hypergeom([1/2,-1/2*m],[1-1/2*m],1/(a^2*x^2))/(a*m)],
[x^m/exp(arccoth(a*x)),x,4,x^(1+m)*hypergeom([1/2,1/2*(-1-m)],[1/2*(1-m)],1/(a^2*x^2))/(1+m)-x^m*hypergeom([1/2,-1/2*m],[1-1/2*m],1/(a^2*x^2))/(a*m)],
[x^m/exp(2*arccoth(a*x)),x,4,x^(1+m)/(1+m)-2*x^(1+m)*hypergeom([1,1+m],[2+m],-a*x)/(1+m)],
[x^m/exp(3*arccoth(a*x)),x,9,-3*x^(1+m)*hypergeom([1/2,1/2*(-1-m)],[1/2*(1-m)],1/(a^2*x^2))/(1+m)+x^m*hypergeom([1/2,-1/2*m],[1-1/2*m],1/(a^2*x^2))/(a*m)+4*x^(1+m)*hypergeom([3/2,1/2*(-1-m)],[1/2*(1-m)],1/(a^2*x^2))/(1+m)-4*x^m*hypergeom([3/2,-1/2*m],[1-1/2*m],1/(a^2*x^2))/(a*m)],
[exp(5/2*arccoth(a*x))*x^m,x,2,x^(1+m)*AppellF1(-1-m,5/4,-5/4,-m,1/(a*x),(-1)/(a*x))/(1+m)],
[exp(3/2*arccoth(a*x))*x^m,x,2,x^(1+m)*AppellF1(-1-m,3/4,-3/4,-m,1/(a*x),(-1)/(a*x))/(1+m)],
[exp(1/2*arccoth(a*x))*x^m,x,2,x^(1+m)*AppellF1(-1-m,1/4,-1/4,-m,1/(a*x),(-1)/(a*x))/(1+m)],
[x^m/exp(1/2*arccoth(a*x)),x,2,x^(1+m)*AppellF1(-1-m,-1/4,1/4,-m,1/(a*x),(-1)/(a*x))/(1+m)],
[x^m/exp(3/2*arccoth(a*x)),x,2,x^(1+m)*AppellF1(-1-m,-3/4,3/4,-m,1/(a*x),(-1)/(a*x))/(1+m)],
[x^m/exp(5/2*arccoth(a*x)),x,2,x^(1+m)*AppellF1(-1-m,-5/4,5/4,-m,1/(a*x),(-1)/(a*x))/(1+m)],
[exp(2/3*arccoth(x))*x^m,x,2,x^(1+m)*AppellF1(-1-m,1/3,-1/3,-m,1/x,(-1)/x)/(1+m)],
[exp(1/3*arccoth(x))*x^m,x,2,x^(1+m)*AppellF1(-1-m,1/6,-1/6,-m,1/x,(-1)/x)/(1+m)],
[exp(1/4*arccoth(a*x))*x^m,x,2,x^(1+m)*AppellF1(-1-m,1/8,-1/8,-m,1/(a*x),(-1)/(a*x))/(1+m)],

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

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

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

# n>0
[exp(arccoth(a*x))*(c-a*c*x)^p,x,4,((a+(-1)/x)/(a+1/x))^(1/2-p)*(c-a*c*x)^p*hypergeom([1/2-p,-p],[1-p],2/((a+1/x)*x))*sqrt(1+1/(a*x))/(a*p*(1+p)*sqrt(1+(-1)/(a*x)))+x*(c-a*c*x)^p*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/(1+p)],
[exp(arccoth(a*x))*(c-a*c*x)^4,x,9,17/15*a^2*c^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3-3/4*a^3*c^4*(1+(-1)/(a^2*x^2))^(3/2)*x^4+1/5*a^4*c^4*(1+(-1)/(a^2*x^2))^(3/2)*x^5+7/8*c^4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-7/8*a*c^4*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*(c-a*c*x)^3,x,8,2/3*a^2*c^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3-1/4*a^3*c^3*(1+(-1)/(a^2*x^2))^(3/2)*x^4+5/8*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-5/8*a*c^3*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*(c-a*c*x)^2,x,7,1/3*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3+1/2*c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-1/2*a*c^2*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*(c-a*c*x),x,6,1/2*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-1/2*a*c*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))/(c-a*c*x),x,7,-arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)+2*(a+1/x)/(a^2*c*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))/(c-a*c*x)^2,x,3,-1/3*a^2*(1+(-1)/(a^2*x^2))^(3/2)/(c^2*(a+(-1)/x)^3)],
[exp(arccoth(a*x))/(c-a*c*x)^3,x,4,1/5*a^3*(1+(-1)/(a^2*x^2))^(3/2)/(c^3*(a+(-1)/x)^4)-4/15*a^2*(1+(-1)/(a^2*x^2))^(3/2)/(c^3*(a+(-1)/x)^3)],
[exp(arccoth(a*x))/(c-a*c*x)^4,x,6,-1/7*a^4*(1+(-1)/(a^2*x^2))^(3/2)/(c^4*(a+(-1)/x)^5)+12/35*a^3*(1+(-1)/(a^2*x^2))^(3/2)/(c^4*(a+(-1)/x)^4)-23/105*a^2*(1+(-1)/(a^2*x^2))^(3/2)/(c^4*(a+(-1)/x)^3)],
[exp(arccoth(a*x))/(c-a*c*x)^5,x,8,1/9*a^5*(1+(-1)/(a^2*x^2))^(3/2)/(c^5*(a+(-1)/x)^6)-8/21*a^4*(1+(-1)/(a^2*x^2))^(3/2)/(c^5*(a+(-1)/x)^5)+47/105*a^3*(1+(-1)/(a^2*x^2))^(3/2)/(c^5*(a+(-1)/x)^4)-58/315*a^2*(1+(-1)/(a^2*x^2))^(3/2)/(c^5*(a+(-1)/x)^3)],
[exp(2*arccoth(a*x))*(c-a*c*x)^p,x,5,2*(c-a*c*x)^p/(a*p)-(c-a*c*x)^(1+p)/(a*c*(1+p))],
[exp(2*arccoth(a*x))*(c-a*c*x)^5,x,4,2/5*c^5*(1-a*x)^5/a-1/6*c^5*(1-a*x)^6/a],
[exp(2*arccoth(a*x))*(c-a*c*x)^4,x,4,1/2*c^4*(1-a*x)^4/a-1/5*c^4*(1-a*x)^5/a],
[exp(2*arccoth(a*x))*(c-a*c*x)^3,x,4,2/3*c^3*(1-a*x)^3/a-1/4*c^3*(1-a*x)^4/a],
[exp(2*arccoth(a*x))*(c-a*c*x)^2,x,4,-c^2*x+1/3*a^2*c^2*x^3],
[exp(2*arccoth(a*x))*(c-a*c*x),x,1,-c*x-1/2*a*c*x^2,1/2*c*exp(2*arccoth(a*x))*(1-a^2*x^2)/a],
[exp(2*arccoth(a*x))/(c-a*c*x),x,4,(-2)/(a*c*(1-a*x))-log(1-a*x)/(a*c)],
[exp(2*arccoth(a*x))/(c-a*c*x)^2,x,3,-x/(c^2*(1-a*x)^2)],
[exp(2*arccoth(a*x))/(c-a*c*x)^3,x,4,(-2/3)/(a*c^3*(1-a*x)^3)+1/2/(a*c^3*(1-a*x)^2)],
[exp(2*arccoth(a*x))/(c-a*c*x)^4,x,4,(-1/2)/(a*c^4*(1-a*x)^4)+1/3/(a*c^4*(1-a*x)^3)],
[exp(3*arccoth(a*x))*(c-a*c*x)^p,x,5,(1+1/(a*x))^(3/2)*x*(c-a*c*x)^p/((1+p)*sqrt(1+(-1)/(a*x)))-3*((a+(-1)/x)/(a+1/x))^(3/2-p)*(c-a*c*x)^p*hypergeom([1-p,3/2-p],[2-p],2/((a+1/x)*x))*sqrt(1+1/(a*x))/(a^2*p*(1-p^2)*(1+(-1)/(a*x))^(3/2)*x)+3*(c-a*c*x)^p*sqrt(1+1/(a*x))/(a*p*(1+p)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))*(c-a*c*x)^4,x,8,-1/4*a^3*c^4*(1+(-1)/(a^2*x^2))^(3/2)*x^4+1/5*a^4*c^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5-3/8*c^4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+3/8*a*c^4*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*(c-a*c*x)^3,x,7,-1/4*a^3*c^3*(1+(-1)/(a^2*x^2))^(3/2)*x^4-3/8*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+3/8*a*c^3*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*(c-a*c*x)^2,x,8,1/3*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3-1/2*c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+1/2*a*c^2*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*(c-a*c*x),x,8,-3/2*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-2*c*x*sqrt(1+(-1)/(a^2*x^2))-1/2*a*c*x^2*sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))/(c-a*c*x),x,9,8/3*(a+1/x)/(a^2*c*(1+(-1)/(a^2*x^2))^(3/2))-arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)+4/3/(a^2*c*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))/(c-a*c*x)^2,x,3,-1/5*a^4*(1+(-1)/(a^2*x^2))^(5/2)/(c^2*(a+(-1)/x)^5)],
[exp(3*arccoth(a*x))/(c-a*c*x)^3,x,4,1/7*a^5*(1+(-1)/(a^2*x^2))^(5/2)/(c^3*(a+(-1)/x)^6)-6/35*a^4*(1+(-1)/(a^2*x^2))^(5/2)/(c^3*(a+(-1)/x)^5)],
[exp(3*arccoth(a*x))/(c-a*c*x)^4,x,6,-47/315*(a+1/x)^5/(a^6*c^4*(1+(-1)/(a^2*x^2))^(5/2))+16/63*(a+1/x)^6/(a^7*c^4*(1+(-1)/(a^2*x^2))^(7/2))-1/9*(a+1/x)^7/(a^8*c^4*(1+(-1)/(a^2*x^2))^(9/2))],
[exp(3*arccoth(a*x))/(c-a*c*x)^5,x,7,-152/1155*(a+1/x)^5/(a^6*c^5*(1+(-1)/(a^2*x^2))^(5/2))+79/231*(a+1/x)^6/(a^7*c^5*(1+(-1)/(a^2*x^2))^(7/2))-10/33*(a+1/x)^7/(a^8*c^5*(1+(-1)/(a^2*x^2))^(9/2))+1/11*(a+1/x)^8/(a^9*c^5*(1+(-1)/(a^2*x^2))^(11/2))],
[exp(4*arccoth(a*x))*(c-a*c*x)^p,x,5,4*c*(c-a*c*x)^(-1+p)/(a*(1-p))+4*(c-a*c*x)^p/(a*p)-(c-a*c*x)^(1+p)/(a*c*(1+p))],
[exp(4*arccoth(a*x))*(c-a*c*x)^5,x,4,-c^5*(1-a*x)^4/a+4/5*c^5*(1-a*x)^5/a-1/6*c^5*(1-a*x)^6/a],
[exp(4*arccoth(a*x))*(c-a*c*x)^4,x,5,c^4*x-2/3*a^2*c^4*x^3+1/5*a^4*c^4*x^5],
[exp(4*arccoth(a*x))*(c-a*c*x)^3,x,4,2/3*c^3*(1+a*x)^3/a-1/4*c^3*(1+a*x)^4/a],
[exp(4*arccoth(a*x))*(c-a*c*x)^2,x,3,1/3*c^2*(1+a*x)^3/a],
[exp(4*arccoth(a*x))*(c-a*c*x),x,4,-3*c*x-1/2*a*c*x^2-4*c*log(1-a*x)/a],
[exp(4*arccoth(a*x))/(c-a*c*x),x,4,2/(a*c*(1-a*x)^2)+(-4)/(a*c*(1-a*x))-log(1-a*x)/(a*c)],
[exp(4*arccoth(a*x))/(c-a*c*x)^2,x,3,1/6*(1+a*x)^3/(a*c^2*(1-a*x)^3)],
[exp(4*arccoth(a*x))/(c-a*c*x)^3,x,4,1/(a*c^3*(1-a*x)^4)+(-4/3)/(a*c^3*(1-a*x)^3)+1/2/(a*c^3*(1-a*x)^2)],
[exp(4*arccoth(a*x))/(c-a*c*x)^4,x,4,4/5/(a*c^4*(1-a*x)^5)+(-1)/(a*c^4*(1-a*x)^4)+1/3/(a*c^4*(1-a*x)^3)],

# n<0
[(c-a*c*x)^p/exp(arccoth(a*x)),x,3,((a+(-1)/x)/(a+1/x))^(-1/2-p)*x*(c-a*c*x)^p*hypergeom([-1-p,-1/2-p],[-p],2/((a+1/x)*x))*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/(1+p)],
[(c-a*c*x)^3/exp(arccoth(a*x)),x,9,-35/8*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+20/3*c^3*x*sqrt(1+(-1)/(a^2*x^2))-27/8*a*c^3*x^2*sqrt(1+(-1)/(a^2*x^2))+4/3*a^2*c^3*x^3*sqrt(1+(-1)/(a^2*x^2))-1/4*a^3*c^3*x^4*sqrt(1+(-1)/(a^2*x^2))],
[(c-a*c*x)^2/exp(arccoth(a*x)),x,8,-5/2*c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+11/3*c^2*x*sqrt(1+(-1)/(a^2*x^2))-3/2*a*c^2*x^2*sqrt(1+(-1)/(a^2*x^2))+1/3*a^2*c^2*x^3*sqrt(1+(-1)/(a^2*x^2))],
[(c-a*c*x)/exp(arccoth(a*x)),x,7,-3/2*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+2*c*x*sqrt(1+(-1)/(a^2*x^2))-1/2*a*c*x^2*sqrt(1+(-1)/(a^2*x^2))],
[1/(exp(arccoth(a*x))*(c-a*c*x)),x,5,-arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)],
[1/(exp(arccoth(a*x))*(c-a*c*x)^2),x,3,-sqrt(1+(-1)/(a^2*x^2))/(c^2*(a+(-1)/x))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^3),x,4,1/3*a*sqrt(1+(-1)/(a^2*x^2))/(c^3*(a+(-1)/x)^2)-2/3*sqrt(1+(-1)/(a^2*x^2))/(c^3*(a+(-1)/x))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^4),x,6,-1/5*a^2*sqrt(1+(-1)/(a^2*x^2))/(c^4*(a+(-1)/x)^3)+8/15*a*sqrt(1+(-1)/(a^2*x^2))/(c^4*(a+(-1)/x)^2)-7/15*sqrt(1+(-1)/(a^2*x^2))/(c^4*(a+(-1)/x))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^5),x,7,1/7*a^3*sqrt(1+(-1)/(a^2*x^2))/(c^5*(a+(-1)/x)^4)-18/35*a^2*sqrt(1+(-1)/(a^2*x^2))/(c^5*(a+(-1)/x)^3)+23/35*a*sqrt(1+(-1)/(a^2*x^2))/(c^5*(a+(-1)/x)^2)-12/35*sqrt(1+(-1)/(a^2*x^2))/(c^5*(a+(-1)/x))],
[(c-a*c*x)^p/exp(2*arccoth(a*x)),x,4,1/2*(c-a*c*x)^(2+p)*hypergeom([1,2+p],[3+p],1/2*(1-a*x))/(a*c^2*(2+p))],
[(c-a*c*x)^4/exp(2*arccoth(a*x)),x,4,16*c^4*x-4*c^4*(1-a*x)^2/a-4/3*c^4*(1-a*x)^3/a-1/2*c^4*(1-a*x)^4/a-1/5*c^4*(1-a*x)^5/a-32*c^4*log(1+a*x)/a],
[(c-a*c*x)^3/exp(2*arccoth(a*x)),x,4,8*c^3*x-2*c^3*(1-a*x)^2/a-2/3*c^3*(1-a*x)^3/a-1/4*c^3*(1-a*x)^4/a-16*c^3*log(1+a*x)/a],
[(c-a*c*x)^2/exp(2*arccoth(a*x)),x,4,4*c^2*x-c^2*(1-a*x)^2/a-1/3*c^2*(1-a*x)^3/a-8*c^2*log(1+a*x)/a],
[(c-a*c*x)/exp(2*arccoth(a*x)),x,4,3*c*x-1/2*a*c*x^2-4*c*log(1+a*x)/a],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)),x,3,-log(1+a*x)/(a*c)],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^2),x,4,-arctanh(a*x)/(a*c^2)],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^3),x,5,(-1/2)/(a*c^3*(1-a*x))-1/2*arctanh(a*x)/(a*c^3)],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^4),x,5,(-1/4)/(a*c^4*(1-a*x)^2)+(-1/4)/(a*c^4*(1-a*x))-1/4*arctanh(a*x)/(a*c^4)],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^5),x,5,(-1/6)/(a*c^5*(1-a*x)^3)+(-1/8)/(a*c^5*(1-a*x)^2)+(-1/8)/(a*c^5*(1-a*x))-1/8*arctanh(a*x)/(a*c^5)],
[(c-a*c*x)^p/exp(3*arccoth(a*x)),x,3,((a+(-1)/x)/(a+1/x))^(-3/2-p)*(1+(-1)/(a*x))^(3/2)*x*(c-a*c*x)^p*hypergeom([-3/2-p,-1-p],[-p],2/((a+1/x)*x))/((1+p)*sqrt(1+1/(a*x)))],
[(c-a*c*x)^3/exp(3*arccoth(a*x)),x,10,-315/8*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+32*c^3*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+30*c^3*x*sqrt(1+(-1)/(a^2*x^2))-67/8*a*c^3*x^2*sqrt(1+(-1)/(a^2*x^2))+2*a^2*c^3*x^3*sqrt(1+(-1)/(a^2*x^2))-1/4*a^3*c^3*x^4*sqrt(1+(-1)/(a^2*x^2))],
[(c-a*c*x)^2/exp(3*arccoth(a*x)),x,9,-35/2*c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+16*c^2*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+35/3*c^2*x*sqrt(1+(-1)/(a^2*x^2))-5/2*a*c^2*x^2*sqrt(1+(-1)/(a^2*x^2))+1/3*a^2*c^2*x^3*sqrt(1+(-1)/(a^2*x^2))],
[(c-a*c*x)/exp(3*arccoth(a*x)),x,8,-15/2*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+8*c*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+4*c*x*sqrt(1+(-1)/(a^2*x^2))-1/2*a*c*x^2*sqrt(1+(-1)/(a^2*x^2))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)),x,6,-arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)+2*(a+(-1)/x)/(a^2*c*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^2),x,3,(a+(-1)/x)/(a^2*c^2*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^3),x,3,1/(a*c^3*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^4),x,5,2/3/(a*c^4*sqrt(1+(-1)/(a^2*x^2)))+(-1/3)/(a^2*c^4*(a+(-1)/x)*x^2*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^5),x,6,-4/5*(a+1/x)/(a^2*c^5*(1+(-1)/(a^2*x^2))^(3/2))+1/5*(a+1/x)^2/(a^3*c^5*(1+(-1)/(a^2*x^2))^(5/2))+1/5*(5*a+2/x)/(a^2*c^5*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^6),x,7,-46/35*(a+1/x)/(a^2*c^6*(1+(-1)/(a^2*x^2))^(3/2))+24/35*(a+1/x)^2/(a^3*c^6*(1+(-1)/(a^2*x^2))^(5/2))-1/7*(a+1/x)^3/(a^4*c^6*(1+(-1)/(a^2*x^2))^(7/2))+1/35*(35*a+13/x)/(a^2*c^6*sqrt(1+(-1)/(a^2*x^2)))],

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

# n>0
[exp(arccoth(a*x))*(c-a*c*x)^(9/2),x,7,-32/99*(a+(-1)/x)^3*(1+1/(a*x))^(3/2)*(c-a*c*x)^(9/2)/(a^4*(1+(-1)/(a*x))^(9/2))+9088/3465*(1+1/(a*x))^(3/2)*(c-a*c*x)^(9/2)/(a^4*(1+(-1)/(a*x))^(9/2)*x^3)-768/385*(1+1/(a*x))^(3/2)*(c-a*c*x)^(9/2)/(a^3*(1+(-1)/(a*x))^(9/2)*x^2)+128/231*(1+1/(a*x))^(3/2)*(c-a*c*x)^(9/2)/(a^2*(1+(-1)/(a*x))^(9/2)*x)+2/11*(a+(-1)/x)^4*(1+1/(a*x))^(3/2)*x*(c-a*c*x)^(9/2)/(a^4*(1+(-1)/(a*x))^(9/2))],
[exp(arccoth(a*x))*(c-a*c*x)^(7/2),x,6,-8/21*(1+1/(a*x))^(3/2)*(c-a*c*x)^(7/2)/(a*(1+(-1)/(a*x))^(7/2))-568/315*(1+1/(a*x))^(3/2)*(c-a*c*x)^(7/2)/(a^3*(1+(-1)/(a*x))^(7/2)*x^2)+48/35*(1+1/(a*x))^(3/2)*(c-a*c*x)^(7/2)/(a^2*(1+(-1)/(a*x))^(7/2)*x)+2/9*(a+(-1)/x)^3*(1+1/(a*x))^(3/2)*x*(c-a*c*x)^(7/2)/(a^3*(1+(-1)/(a*x))^(7/2))],
[exp(arccoth(a*x))*(c-a*c*x)^(5/2),x,5,64/105*a^2*c^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3/(c-a*c*x)^(3/2)+16/35*a^2*c^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3/sqrt(c-a*c*x)+2/7*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3*sqrt(c-a*c*x),-36/35*(1+1/(a*x))^(3/2)*(c-a*c*x)^(5/2)/(a*(1+(-1)/(a*x))^(5/2))+142/105*(1+1/(a*x))^(3/2)*(c-a*c*x)^(5/2)/(a^2*(1+(-1)/(a*x))^(5/2)*x)+2/7*(1+1/(a*x))^(3/2)*x*(c-a*c*x)^(5/2)/(1+(-1)/(a*x))^(5/2)],
[exp(arccoth(a*x))*(c-a*c*x)^(3/2),x,4,8/15*a^2*c^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3/(c-a*c*x)^(3/2)+2/5*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/sqrt(c-a*c*x),-14/15*(1+1/(a*x))^(3/2)*(c-a*c*x)^(3/2)/(a*(1+(-1)/(a*x))^(3/2))+2/5*(1+1/(a*x))^(3/2)*x*(c-a*c*x)^(3/2)/(1+(-1)/(a*x))^(3/2)],
[exp(arccoth(a*x))*(c-a*c*x)^(1/2),x,1,2/3*exp(arccoth(a*x))*(1+a*x)*sqrt(c-a*c*x)/a],
[exp(arccoth(a*x))/(c-a*c*x)^(1/2),x,5,2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/sqrt(c-a*c*x)-2*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1+(-1)/(a*x))/(sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x))],
[exp(arccoth(a*x))/(c-a*c*x)^(3/2),x,5,-(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(a)/((1/x)^(3/2)*(c-a*c*x)^(3/2)*sqrt(2))-a*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(3/2))],
[exp(arccoth(a*x))/(c-a*c*x)^(5/2),x,6,-1/4*a^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2)*x^2/((a+(-1)/x)^2*(c-a*c*x)^(5/2))+1/8*a^(3/2)*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(5/2)*(c-a*c*x)^(5/2)*sqrt(2))+1/8*a^2*(1+(-1)/(a*x))^(5/2)*x^2*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(5/2))],
[exp(arccoth(a*x))/(c-a*c*x)^(7/2),x,7,-1/6*a^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(3/2)*x^2/((a+(-1)/x)^3*(c-a*c*x)^(7/2))+1/16*a^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(3/2)*x^3/((a+(-1)/x)^2*(c-a*c*x)^(7/2))-1/32*a^(5/2)*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(7/2)*(c-a*c*x)^(7/2)*sqrt(2))-1/32*a^3*(1+(-1)/(a*x))^(7/2)*x^3*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(7/2))],
[exp(2*arccoth(a*x))*(c-a*c*x)^(7/2),x,5,4/7*(c-a*c*x)^(7/2)/a-2/9*(c-a*c*x)^(9/2)/(a*c)],
[exp(2*arccoth(a*x))*(c-a*c*x)^(5/2),x,5,4/5*(c-a*c*x)^(5/2)/a-2/7*(c-a*c*x)^(7/2)/(a*c)],
[exp(2*arccoth(a*x))*(c-a*c*x)^(3/2),x,5,4/3*(c-a*c*x)^(3/2)/a-2/5*(c-a*c*x)^(5/2)/(a*c)],
[exp(2*arccoth(a*x))*(c-a*c*x)^(1/2),x,5,-2/3*(c-a*c*x)^(3/2)/(a*c)+4*sqrt(c-a*c*x)/a],
[exp(2*arccoth(a*x))/(c-a*c*x)^(1/2),x,5,(-4)/(a*sqrt(c-a*c*x))-2*sqrt(c-a*c*x)/(a*c)],
[exp(2*arccoth(a*x))/(c-a*c*x)^(3/2),x,5,(-4/3)/(a*(c-a*c*x)^(3/2))+2/(a*c*sqrt(c-a*c*x))],
[exp(2*arccoth(a*x))/(c-a*c*x)^(5/2),x,5,(-4/5)/(a*(c-a*c*x)^(5/2))+2/3/(a*c*(c-a*c*x)^(3/2))],
[exp(2*arccoth(a*x))/(c-a*c*x)^(7/2),x,5,(-4/7)/(a*(c-a*c*x)^(7/2))+2/5/(a*c*(c-a*c*x)^(5/2))],
[exp(3*arccoth(a*x))*(c-a*c*x)^(9/2),x,6,-8/33*(1+1/(a*x))^(5/2)*(c-a*c*x)^(9/2)/(a*(1+(-1)/(a*x))^(9/2))-856/1155*(1+1/(a*x))^(5/2)*(c-a*c*x)^(9/2)/(a^3*(1+(-1)/(a*x))^(9/2)*x^2)+16/21*(1+1/(a*x))^(5/2)*(c-a*c*x)^(9/2)/(a^2*(1+(-1)/(a*x))^(9/2)*x)+2/11*(a+(-1)/x)^3*(1+1/(a*x))^(5/2)*x*(c-a*c*x)^(9/2)/(a^3*(1+(-1)/(a*x))^(9/2))],
[exp(3*arccoth(a*x))*(c-a*c*x)^(7/2),x,5,-44/63*(1+1/(a*x))^(5/2)*(c-a*c*x)^(7/2)/(a*(1+(-1)/(a*x))^(7/2))+214/315*(1+1/(a*x))^(5/2)*(c-a*c*x)^(7/2)/(a^2*(1+(-1)/(a*x))^(7/2)*x)+2/9*(1+1/(a*x))^(5/2)*x*(c-a*c*x)^(7/2)/(1+(-1)/(a*x))^(7/2)],
[exp(3*arccoth(a*x))*(c-a*c*x)^(5/2),x,4,-18/35*(1+1/(a*x))^(5/2)*(c-a*c*x)^(5/2)/(a*(1+(-1)/(a*x))^(5/2))+2/7*(1+1/(a*x))^(5/2)*x*(c-a*c*x)^(5/2)/(1+(-1)/(a*x))^(5/2)],
[exp(3*arccoth(a*x))*(c-a*c*x)^(3/2),x,1,2/5*exp(3*arccoth(a*x))*(1+a*x)*(c-a*c*x)^(3/2)/a],
[exp(3*arccoth(a*x))*(c-a*c*x)^(1/2),x,6,2/3*(1+1/(a*x))^(3/2)*x*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))+4*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/(a^(3/2)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))/(c-a*c*x)^(1/2),x,6,2*a*(1+1/(a*x))^(3/2)*x*sqrt(1+(-1)/(a*x))/((a+(-1)/x)*sqrt(c-a*c*x))-6*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/((a+(-1)/x)*sqrt(c-a*c*x))-3*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1+(-1)/(a*x))/(sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x))],
[exp(3*arccoth(a*x))/(c-a*c*x)^(3/2),x,6,-1/2*a^2*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2)*x/((a+(-1)/x)^2*(c-a*c*x)^(3/2))-3/4*(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(a)/((1/x)^(3/2)*(c-a*c*x)^(3/2)*sqrt(2))-3/4*a*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(3/2))],
[exp(3*arccoth(a*x))/(c-a*c*x)^(5/2),x,7,1/24*a^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2)*x^2/((a+(-1)/x)^2*(c-a*c*x)^(5/2))-1/6*a^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(5/2)*x^2/((a+(-1)/x)^3*(c-a*c*x)^(5/2))+1/16*a^(3/2)*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(5/2)*(c-a*c*x)^(5/2)*sqrt(2))+1/16*a^2*(1+(-1)/(a*x))^(5/2)*x^2*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(5/2))],
[exp(3*arccoth(a*x))/(c-a*c*x)^(7/2),x,8,-1/8*a^5*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2)*x^2/((a+(-1)/x)^4*(c-a*c*x)^(7/2))-1/128*a^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(3/2)*x^3/((a+(-1)/x)^2*(c-a*c*x)^(7/2))+1/32*a^5*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2)*x^3/((a+(-1)/x)^3*(c-a*c*x)^(7/2))-3/256*a^(5/2)*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(7/2)*(c-a*c*x)^(7/2)*sqrt(2))-3/256*a^3*(1+(-1)/(a*x))^(7/2)*x^3*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(7/2))],

# n<0
[(c-a*c*x)^(9/2)/exp(arccoth(a*x)),x,8,512/231*c^3*x*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))+640/693*c^2*x*(c-a*c*x)^(5/2)*sqrt(1+(-1)/(a^2*x^2))+40/99*c*x*(c-a*c*x)^(7/2)*sqrt(1+(-1)/(a^2*x^2))+2/11*x*(c-a*c*x)^(9/2)*sqrt(1+(-1)/(a^2*x^2))+16384/693*c^5*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a*c*x)+4096/693*c^4*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x),-40/99*(a+(-1)/x)^4*(c-a*c*x)^(9/2)*sqrt(1+1/(a*x))/(a^5*(1+(-1)/(a*x))^(9/2))-22016/693*(c-a*c*x)^(9/2)*sqrt(1+1/(a*x))/(a^5*(1+(-1)/(a*x))^(9/2)*x^4)+1024/99*(c-a*c*x)^(9/2)*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(9/2)*x^3)-512/231*(c-a*c*x)^(9/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(9/2)*x^2)+640/693*(a+(-1)/x)^3*(c-a*c*x)^(9/2)*sqrt(1+1/(a*x))/(a^5*(1+(-1)/(a*x))^(9/2)*x)+2/11*(a+(-1)/x)^5*x*(c-a*c*x)^(9/2)*sqrt(1+1/(a*x))/(a^5*(1+(-1)/(a*x))^(9/2))],
[(c-a*c*x)^(7/2)/exp(arccoth(a*x)),x,7,128/105*c^2*x*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))+32/63*c*x*(c-a*c*x)^(5/2)*sqrt(1+(-1)/(a^2*x^2))+2/9*x*(c-a*c*x)^(7/2)*sqrt(1+(-1)/(a^2*x^2))+4096/315*c^4*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a*c*x)+1024/315*c^3*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x),-32/63*(a+(-1)/x)^3*(c-a*c*x)^(7/2)*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(7/2))+5504/315*(c-a*c*x)^(7/2)*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(7/2)*x^3)-256/45*(c-a*c*x)^(7/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(7/2)*x^2)+128/105*(c-a*c*x)^(7/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(7/2)*x)+2/9*(a+(-1)/x)^4*x*(c-a*c*x)^(7/2)*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(7/2))],
[(c-a*c*x)^(5/2)/exp(arccoth(a*x)),x,6,24/35*c*x*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))+2/7*x*(c-a*c*x)^(5/2)*sqrt(1+(-1)/(a^2*x^2))+256/35*c^3*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a*c*x)+64/35*c^2*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x),-24/35*(c-a*c*x)^(5/2)*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^(5/2))-344/35*(c-a*c*x)^(5/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(5/2)*x^2)+16/5*(c-a*c*x)^(5/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(5/2)*x)+2/7*(a+(-1)/x)^3*x*(c-a*c*x)^(5/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(5/2))],
[(c-a*c*x)^(3/2)/exp(arccoth(a*x)),x,5,2/5*x*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))+64/15*c^2*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a*c*x)+16/15*c*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x),-28/15*(c-a*c*x)^(3/2)*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^(3/2))+86/15*(c-a*c*x)^(3/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(3/2)*x)+2/5*x*(c-a*c*x)^(3/2)*sqrt(1+1/(a*x))/(1+(-1)/(a*x))^(3/2)],
[(c-a*c*x)^(1/2)/exp(arccoth(a*x)),x,4,8/3*c*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a*c*x)+2/3*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x),-10/3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/3*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^(1/2)),x,1,2*(1+a*x)/(a*exp(arccoth(a*x))*sqrt(c-a*c*x))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^(3/2)),x,4,-(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(a)/((1/x)^(3/2)*(c-a*c*x)^(3/2))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^(5/2)),x,5,1/2*a^(3/2)*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(5/2)*(c-a*c*x)^(5/2)*sqrt(2))-1/2*a^2*(1+(-1)/(a*x))^(5/2)*x^2*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(5/2))],
[1/(exp(arccoth(a*x))*(c-a*c*x)^(7/2)),x,6,-3/16*a^(5/2)*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(7/2)*(c-a*c*x)^(7/2)*sqrt(2))-1/4*a^3*(1+(-1)/(a*x))^(7/2)*x^2*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-a*c*x)^(7/2))+3/16*a^3*(1+(-1)/(a*x))^(7/2)*x^3*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-a*c*x)^(7/2))],
[(c-a*c*x)^(7/2)/exp(2*arccoth(a*x)),x,10,-16/3*c^2*(c-a*c*x)^(3/2)/a-8/5*c*(c-a*c*x)^(5/2)/a-4/7*(c-a*c*x)^(7/2)/a-2/9*(c-a*c*x)^(9/2)/(a*c)+32*c^(7/2)*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/a-32*c^3*sqrt(c-a*c*x)/a],
[(c-a*c*x)^(5/2)/exp(2*arccoth(a*x)),x,9,-8/3*c*(c-a*c*x)^(3/2)/a-4/5*(c-a*c*x)^(5/2)/a-2/7*(c-a*c*x)^(7/2)/(a*c)+16*c^(5/2)*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/a-16*c^2*sqrt(c-a*c*x)/a],
[(c-a*c*x)^(3/2)/exp(2*arccoth(a*x)),x,8,-4/3*(c-a*c*x)^(3/2)/a-2/5*(c-a*c*x)^(5/2)/(a*c)+8*c^(3/2)*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/a-8*c*sqrt(c-a*c*x)/a],
[(c-a*c*x)^(1/2)/exp(2*arccoth(a*x)),x,7,-2/3*(c-a*c*x)^(3/2)/(a*c)+4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a-4*sqrt(c-a*c*x)/a],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^(1/2)),x,6,2*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*sqrt(c))-2*sqrt(c-a*c*x)/(a*c)],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^(3/2)),x,5,arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*c^(3/2))],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^(5/2)),x,6,arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))/(a*c^(5/2)*sqrt(2))+(-1)/(a*c^2*sqrt(c-a*c*x))],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^(7/2)),x,7,(-1/3)/(a*c^2*(c-a*c*x)^(3/2))+1/2*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))/(a*c^(7/2)*sqrt(2))+(-1/2)/(a*c^3*sqrt(c-a*c*x))],
[1/(exp(2*arccoth(a*x))*(c-a*c*x)^(9/2)),x,8,(-1/5)/(a*c^2*(c-a*c*x)^(5/2))+(-1/6)/(a*c^3*(c-a*c*x)^(3/2))+1/4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))/(a*c^(9/2)*sqrt(2))+(-1/4)/(a*c^4*sqrt(c-a*c*x))],
[(c-a*c*x)^(9/2)/exp(3*arccoth(a*x)),x,9,-16/33*(a+(-1)/x)^5*(c-a*c*x)^(9/2)/(a^6*(1+(-1)/(a*x))^(9/2)*sqrt(1+1/(a*x)))-94208/231*(c-a*c*x)^(9/2)/(a^6*(1+(-1)/(a*x))^(9/2)*x^5*sqrt(1+1/(a*x)))-40960/231*(c-a*c*x)^(9/2)/(a^5*(1+(-1)/(a*x))^(9/2)*x^4*sqrt(1+1/(a*x)))+4096/231*(c-a*c*x)^(9/2)/(a^4*(1+(-1)/(a*x))^(9/2)*x^3*sqrt(1+1/(a*x)))-1024/231*(a+(-1)/x)^3*(c-a*c*x)^(9/2)/(a^6*(1+(-1)/(a*x))^(9/2)*x^2*sqrt(1+1/(a*x)))+320/231*(a+(-1)/x)^4*(c-a*c*x)^(9/2)/(a^6*(1+(-1)/(a*x))^(9/2)*x*sqrt(1+1/(a*x)))+2/11*(a+(-1)/x)^6*x*(c-a*c*x)^(9/2)/(a^6*(1+(-1)/(a*x))^(9/2)*sqrt(1+1/(a*x)))],
[(c-a*c*x)^(7/2)/exp(3*arccoth(a*x)),x,8,-40/63*(a+(-1)/x)^4*(c-a*c*x)^(7/2)/(a^5*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x)))+11776/63*(c-a*c*x)^(7/2)/(a^5*(1+(-1)/(a*x))^(7/2)*x^4*sqrt(1+1/(a*x)))+5120/63*(c-a*c*x)^(7/2)/(a^4*(1+(-1)/(a*x))^(7/2)*x^3*sqrt(1+1/(a*x)))-512/63*(c-a*c*x)^(7/2)/(a^3*(1+(-1)/(a*x))^(7/2)*x^2*sqrt(1+1/(a*x)))+128/63*(a+(-1)/x)^3*(c-a*c*x)^(7/2)/(a^5*(1+(-1)/(a*x))^(7/2)*x*sqrt(1+1/(a*x)))+2/9*(a+(-1)/x)^5*x*(c-a*c*x)^(7/2)/(a^5*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x)))],
[(c-a*c*x)^(5/2)/exp(3*arccoth(a*x)),x,7,-32/35*(a+(-1)/x)^3*(c-a*c*x)^(5/2)/(a^4*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x)))-2944/35*(c-a*c*x)^(5/2)/(a^4*(1+(-1)/(a*x))^(5/2)*x^3*sqrt(1+1/(a*x)))-256/7*(c-a*c*x)^(5/2)/(a^3*(1+(-1)/(a*x))^(5/2)*x^2*sqrt(1+1/(a*x)))+128/35*(c-a*c*x)^(5/2)/(a^2*(1+(-1)/(a*x))^(5/2)*x*sqrt(1+1/(a*x)))+2/7*(a+(-1)/x)^4*x*(c-a*c*x)^(5/2)/(a^4*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x)))],
[(c-a*c*x)^(3/2)/exp(3*arccoth(a*x)),x,6,-8/5*(c-a*c*x)^(3/2)/(a*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))+184/5*(c-a*c*x)^(3/2)/(a^3*(1+(-1)/(a*x))^(3/2)*x^2*sqrt(1+1/(a*x)))+16*(c-a*c*x)^(3/2)/(a^2*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x)))+2/5*(a+(-1)/x)^3*x*(c-a*c*x)^(3/2)/(a^3*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))],
[(c-a*c*x)^(1/2)/exp(3*arccoth(a*x)),x,5,-20/3*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-46/3*sqrt(c-a*c*x)/(a^2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+2/3*x*sqrt(c-a*c*x)/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^(1/2)),x,4,6*sqrt(1+(-1)/(a*x))/(a*sqrt(1+1/(a*x))*sqrt(c-a*c*x))+2*x*sqrt(1+(-1)/(a*x))/(sqrt(1+1/(a*x))*sqrt(c-a*c*x))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^(3/2)),x,1,-2*(1+a*x)/(a*exp(3*arccoth(a*x))*(c-a*c*x)^(3/2))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^(5/2)),x,5,-a^(3/2)*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(5/2)*(c-a*c*x)^(5/2)*sqrt(2))+a*(1+(-1)/(a*x))^(5/2)*x^2/((c-a*c*x)^(5/2)*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-a*c*x)^(7/2)),x,6,3/4*a^(5/2)*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))/((1/x)^(7/2)*(c-a*c*x)^(7/2)*sqrt(2))-1/2*a^2*(1+(-1)/(a*x))^(7/2)*x^2/((a+(-1)/x)*(c-a*c*x)^(7/2)*sqrt(1+1/(a*x)))-3/4*a^2*(1+(-1)/(a*x))^(7/2)*x^3/((c-a*c*x)^(7/2)*sqrt(1+1/(a*x)))],

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

# p>0
[exp(arccoth(x))*x*(1+x),x,7,arctanh(sqrt(1+1/x)*sqrt((-1+x)/x))+1/3*(1+1/x)^(3/2)*x^2*sqrt((-1+x)/x)+1/3*(1+1/x)^(5/2)*x^3*sqrt((-1+x)/x)+x*sqrt(1+1/x)*sqrt((-1+x)/x)],
[exp(arccoth(x))*(1+x),x,6,3/2*arctanh(sqrt(1+1/x)*sqrt((-1+x)/x))+1/2*(1+1/x)^(3/2)*x^2*sqrt((-1+x)/x)+3/2*x*sqrt(1+1/x)*sqrt((-1+x)/x)],
[exp(arccoth(x))*(1-x)*x,x,3,-1/3*(1+(-1)/x^2)^(3/2)*x^3],
[exp(arccoth(x))*(1-x),x,6,1/2*arctanh(sqrt(1+(-1)/x^2))-1/2*x^2*sqrt(1+(-1)/x^2)],
[exp(arccoth(x))*x*(1+x)^2,x,8,15/8*arctanh(sqrt(1+1/x)*sqrt((-1+x)/x))+5/8*(1+1/x)^(3/2)*x^2*sqrt((-1+x)/x)+1/4*(1+1/x)^(5/2)*x^3*sqrt((-1+x)/x)+1/4*(1+1/x)^(7/2)*x^4*sqrt((-1+x)/x)+15/8*x*sqrt(1+1/x)*sqrt((-1+x)/x)],
[exp(arccoth(x))*(1+x)^2,x,7,5/2*arctanh(sqrt(1+1/x)*sqrt((-1+x)/x))+5/6*(1+1/x)^(3/2)*x^2*sqrt((-1+x)/x)+1/3*(1+1/x)^(5/2)*x^3*sqrt((-1+x)/x)+5/2*x*sqrt(1+1/x)*sqrt((-1+x)/x)],
[exp(arccoth(x))*(1-x)^2*x,x,8,-1/3*(1+(-1)/x^2)^(3/2)*x^3+1/4*(1+(-1)/x^2)^(3/2)*x^4-1/8*arctanh(sqrt(1+(-1)/x^2))+1/8*x^2*sqrt(1+(-1)/x^2)],
[exp(arccoth(x))*(1-x)^2,x,7,1/3*(1+(-1)/x^2)^(3/2)*x^3+1/2*arctanh(sqrt(1+(-1)/x^2))-1/2*x^2*sqrt(1+(-1)/x^2)],

# p<0
[exp(arccoth(x))*x/(1+x),x,3,x*sqrt(1+1/x)*sqrt((-1+x)/x)],
[exp(arccoth(x))/(1+x),x,4,arctanh(sqrt(1+1/x)*sqrt((-1+x)/x))],
[exp(arccoth(x))*x/(1-x),x,8,-2*arctanh(sqrt(1+(-1)/x^2))+2*(1+1/x)/sqrt(1+(-1)/x^2)-x*sqrt(1+(-1)/x^2)],
[exp(arccoth(x))/(1-x),x,7,-arctanh(sqrt(1+(-1)/x^2))+2*(1+1/x)/sqrt(1+(-1)/x^2)],
[exp(arccoth(x))*x/(1+x)^2,x,5,arctanh(sqrt(1+1/x)*sqrt((-1+x)/x))-sqrt((-1+x)/x)/sqrt(1+1/x)],
[exp(arccoth(x))/(1+x)^2,x,3,sqrt((-1+x)/x)/sqrt(1+1/x)],
[exp(arccoth(x))*x/(1-x)^2,x,9,-4/3*(1+1/x)/(1+(-1)/x^2)^(3/2)+arctanh(sqrt(1+(-1)/x^2))+1/3*(-3+(-5)/x)/sqrt(1+(-1)/x^2)],
[exp(arccoth(x))/(1-x)^2,x,3,-1/3*(1+(-1)/x^2)^(3/2)/(1+(-1)/x)^3],

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

# n>0
[exp(arccoth(a*x))*x^m*sqrt(c-a*c*x),x,3,2*x^(1+m)*hypergeom([-1/2,-3/2-m],[-1/2-m],(-1)/(a*x))*sqrt(c-a*c*x)/((3+2*m)*sqrt(1+(-1)/(a*x)))],
[exp(arccoth(a*x))*x^2*sqrt(c-a*c*x),x,5,16/105*(1+1/(a*x))^(3/2)*x*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))-8/35*(1+1/(a*x))^(3/2)*x^2*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/7*(1+1/(a*x))^(3/2)*x^3*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(arccoth(a*x))*x*sqrt(c-a*c*x),x,4,-4/15*(1+1/(a*x))^(3/2)*x*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/5*(1+1/(a*x))^(3/2)*x^2*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(arccoth(a*x))*sqrt(c-a*c*x),x,1,2/3*exp(arccoth(a*x))*(1+a*x)*sqrt(c-a*c*x)/a],
[exp(arccoth(a*x))*sqrt(c-a*c*x)/x,x,5,2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-2*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/(sqrt(a)*sqrt(1+(-1)/(a*x)))],
[exp(arccoth(a*x))*sqrt(c-a*c*x)/x^2,x,5,-sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))-arcsinh(sqrt(1/x)/sqrt(a))*sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(2*arccoth(a*x))*x^3*sqrt(c-a*c*x),x,5,-14/3*(c-a*c*x)^(3/2)/(a^4*c)+18/5*(c-a*c*x)^(5/2)/(a^4*c^2)-10/7*(c-a*c*x)^(7/2)/(a^4*c^3)+2/9*(c-a*c*x)^(9/2)/(a^4*c^4)+4*sqrt(c-a*c*x)/a^4],
[exp(2*arccoth(a*x))*x^2*sqrt(c-a*c*x),x,5,-10/3*(c-a*c*x)^(3/2)/(a^3*c)+8/5*(c-a*c*x)^(5/2)/(a^3*c^2)-2/7*(c-a*c*x)^(7/2)/(a^3*c^3)+4*sqrt(c-a*c*x)/a^3],
[exp(2*arccoth(a*x))*x*sqrt(c-a*c*x),x,5,-2*(c-a*c*x)^(3/2)/(a^2*c)+2/5*(c-a*c*x)^(5/2)/(a^2*c^2)+4*sqrt(c-a*c*x)/a^2],
[exp(2*arccoth(a*x))*sqrt(c-a*c*x),x,5,-2/3*(c-a*c*x)^(3/2)/(a*c)+4*sqrt(c-a*c*x)/a],
[exp(2*arccoth(a*x))*sqrt(c-a*c*x)/x,x,6,2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+2*sqrt(c-a*c*x)],
[exp(2*arccoth(a*x))*sqrt(c-a*c*x)/x^2,x,6,3*a*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+sqrt(c-a*c*x)/x],
[exp(2*arccoth(a*x))*sqrt(c-a*c*x)/x^3,x,7,7/4*a^2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+1/2*sqrt(c-a*c*x)/x^2+7/4*a*sqrt(c-a*c*x)/x],
[exp(2*arccoth(a*x))*sqrt(c-a*c*x)/x^4,x,8,11/8*a^3*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+1/3*sqrt(c-a*c*x)/x^3+11/12*a*sqrt(c-a*c*x)/x^2+11/8*a^2*sqrt(c-a*c*x)/x],
[exp(2*arccoth(a*x))*sqrt(c-a*c*x)/x^5,x,9,75/64*a^4*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+1/4*sqrt(c-a*c*x)/x^4+5/8*a*sqrt(c-a*c*x)/x^3+25/32*a^2*sqrt(c-a*c*x)/x^2+75/64*a^3*sqrt(c-a*c*x)/x],
[exp(3*arccoth(a*x))*x^3*sqrt(c-a*c*x),x,10,1576/315*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^4*sqrt(1+(-1)/(a*x)))+472/315*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^3*sqrt(1+(-1)/(a*x)))+92/105*x^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))+38/63*x^3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/9*x^4*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/(a^(9/2)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))*x^2*sqrt(c-a*c*x),x,9,104/21*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^3*sqrt(1+(-1)/(a*x)))+32/21*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))+6/7*x^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/7*x^3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/(a^(7/2)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))*x*sqrt(c-a*c*x),x,7,2/3*(1+1/(a*x))^(3/2)*x*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/5*(1+1/(a*x))^(5/2)*x^2*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))+4*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/(a^(5/2)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))*sqrt(c-a*c*x),x,6,2/3*(1+1/(a*x))^(3/2)*x*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))+4*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/(a^(3/2)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))*sqrt(c-a*c*x)/x,x,8,2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))+2*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/(sqrt(a)*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/(sqrt(a)*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))*sqrt(c-a*c*x)/x^2,x,8,sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+5*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-4*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-a*c*x)/x^3,x,9,1/2*a*(1+1/(a*x))^(3/2)*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+7/4*a*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+23/4*a^(3/2)*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-4*a^(3/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-a*c*x)/x^4,x,10,1/3*a*(1+1/(a*x))^(3/2)*sqrt(c-a*c*x)/(x^2*sqrt(1+(-1)/(a*x)))+3/4*a^2*(1+1/(a*x))^(3/2)*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+13/8*a^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+45/8*a^(5/2)*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-4*a^(5/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-a*c*x)/x^5,x,11,1/4*a*(1+1/(a*x))^(3/2)*sqrt(c-a*c*x)/(x^3*sqrt(1+(-1)/(a*x)))+11/24*a^2*(1+1/(a*x))^(3/2)*sqrt(c-a*c*x)/(x^2*sqrt(1+(-1)/(a*x)))+21/32*a^3*(1+1/(a*x))^(3/2)*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+107/64*a^3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+363/64*a^(7/2)*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))-4*a^(7/2)*arctanh(sqrt(2)*sqrt(1/x)/(sqrt(a)*sqrt(1+1/(a*x))))*sqrt(2)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[exp(arccoth(x))*x*(1+x)^(3/2),x,6,46/21*(1+x)^(3/2)*sqrt((-1+x)/x)/(1+1/x)^(3/2)+92/21*(1+x)^(3/2)*sqrt((-1+x)/x)/((1+1/x)^(3/2)*x)+8/7*x*(1+x)^(3/2)*sqrt((-1+x)/x)/(1+1/x)^(3/2)+2/7*x^2*(1+x)^(3/2)*sqrt((-1+x)/x)/(1+1/x)^(3/2)],
[exp(arccoth(x))*(1+x)^(3/2),x,5,28/15*(1+x)^(3/2)*sqrt((-1+x)/x)/(1+1/x)^(3/2)+86/15*(1+x)^(3/2)*sqrt((-1+x)/x)/((1+1/x)^(3/2)*x)+2/5*x*(1+x)^(3/2)*sqrt((-1+x)/x)/(1+1/x)^(3/2)],
[exp(arccoth(x))*(1-x)^(3/2)*x,x,5,44/105*(1+1/x)^(3/2)*(1-x)^(3/2)/(1+(-1)/x)^(3/2)-22/35*(1+1/x)^(3/2)*(1-x)^(3/2)*x/(1+(-1)/x)^(3/2)+2/7*(1+1/x)^(3/2)*(1-x)^(3/2)*x^2/(1+(-1)/x)^(3/2)],
[exp(arccoth(x))*(1-x)^(3/2),x,4,-14/15*(1+1/x)^(3/2)*(1-x)^(3/2)/(1+(-1)/x)^(3/2)+2/5*(1+1/x)^(3/2)*(1-x)^(3/2)*x/(1+(-1)/x)^(3/2)],
[exp(arccoth(x))*x*sqrt(1+x),x,5,12/5*sqrt((-1+x)/x)*sqrt(1+x)/sqrt(1+1/x)+6/5*x*sqrt((-1+x)/x)*sqrt(1+x)/sqrt(1+1/x)+2/5*x^2*sqrt((-1+x)/x)*sqrt(1+x)/sqrt(1+1/x)],
[exp(arccoth(x))*sqrt(1+x),x,4,10/3*sqrt((-1+x)/x)*sqrt(1+x)/sqrt(1+1/x)+2/3*x*sqrt((-1+x)/x)*sqrt(1+x)/sqrt(1+1/x)],
[exp(arccoth(x))*x*sqrt(1-x),x,4,-4/15*(1+1/x)^(3/2)*x*sqrt(1-x)/sqrt(1+(-1)/x)+2/5*(1+1/x)^(3/2)*x^2*sqrt(1-x)/sqrt(1+(-1)/x)],
[exp(arccoth(x))*sqrt(1-x),x,1,2/3*exp(arccoth(x))*(1+x)*sqrt(1-x)],
[exp(arccoth(x))*x/sqrt(1+x),x,4,4/3*x*sqrt(1+1/x)*sqrt((-1+x)/x)/sqrt(1+x)+2/3*x^2*sqrt(1+1/x)*sqrt((-1+x)/x)/sqrt(1+x)],
[exp(arccoth(x))/sqrt(1+x),x,3,2*x*sqrt(1+1/x)*sqrt((-1+x)/x)/sqrt(1+x)],
[exp(arccoth(x))*x/sqrt(1-x),x,6,2/3*(1+1/x)^(3/2)*x^2*sqrt(1+(-1)/x)/sqrt(1-x)+2*x*sqrt(1+(-1)/x)*sqrt(1+1/x)/sqrt(1-x)-2*arctanh(sqrt(2)*sqrt(1/x)/sqrt(1+1/x))*sqrt(2)*sqrt(1+(-1)/x)/(sqrt(1-x)*sqrt(1/x))],
[exp(arccoth(x))/sqrt(1-x),x,5,2*x*sqrt(1+(-1)/x)*sqrt(1+1/x)/sqrt(1-x)-2*arctanh(sqrt(2)*sqrt(1/x)/sqrt(1+1/x))*sqrt(2)*sqrt(1+(-1)/x)/(sqrt(1-x)*sqrt(1/x))],
[exp(arccoth(x))*x/(1+x)^(3/2),x,5,(1+1/x)^(3/2)*arctan(sqrt(2)*sqrt(1/x)/sqrt((-1+x)/x))*sqrt(2)/((1/x)^(3/2)*(1+x)^(3/2))+2*(1+1/x)^(3/2)*x^2*sqrt((-1+x)/x)/(1+x)^(3/2)],
[exp(arccoth(x))/(1+x)^(3/2),x,4,-(1+1/x)^(3/2)*arctan(sqrt(2)*sqrt(1/x)/sqrt((-1+x)/x))*sqrt(2)/((1/x)^(3/2)*(1+x)^(3/2))],
[exp(arccoth(x))*x/(1-x)^(3/2),x,6,-5*(1+(-1)/x)^(3/2)*arctanh(sqrt(2)*sqrt(1/x)/sqrt(1+1/x))/((1-x)^(3/2)*(1/x)^(3/2)*sqrt(2))-1/2*(1+1/x)^(3/2)*x^2*sqrt(1+(-1)/x)/(1-x)^(3/2)+5/2*(1+(-1)/x)^(3/2)*x^2*sqrt(1+1/x)/(1-x)^(3/2)],
[exp(arccoth(x))/(1-x)^(3/2),x,5,-(1+(-1)/x)^(3/2)*arctanh(sqrt(2)*sqrt(1/x)/sqrt(1+1/x))/((1-x)^(3/2)*(1/x)^(3/2)*sqrt(2))-x*sqrt(1+(-1)/x)*sqrt(1+1/x)/(1-x)^(3/2)],

# n<0
[x^m*sqrt(c-a*c*x)/exp(arccoth(a*x)),x,4,-2*(5+4*m)*x^m*hypergeom([1/2,-1/2-m],[1/2-m],(-1)/(a*x))*sqrt(c-a*c*x)/(a*(3+8*m+4*m^2)*sqrt(1+(-1)/(a*x)))+2*x^(1+m)*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/((3+2*m)*sqrt(1+(-1)/(a*x)))],
[x^2*sqrt(c-a*c*x)/exp(arccoth(a*x)),x,6,6/35*x*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))/(a^2*c)-2/7*x^2*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))/(a*c)+152/105*c*x*sqrt(1+(-1)/(a^2*x^2))/(a^2*sqrt(c-a*c*x))+38/105*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x)/a^2,-208/105*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^3*sqrt(1+(-1)/(a*x)))+104/105*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))-26/35*x^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/7*x^3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[x*sqrt(c-a*c*x)/exp(arccoth(a*x)),x,5,-2/5*x*(c-a*c*x)^(3/2)*sqrt(1+(-1)/(a^2*x^2))/(a*c)-8/5*c*x*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-a*c*x))-2/5*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x)/a,12/5*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))-6/5*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/5*x^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[sqrt(c-a*c*x)/exp(arccoth(a*x)),x,4,8/3*c*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a*c*x)+2/3*x*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-a*c*x),-10/3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x)))+2/3*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[sqrt(c-a*c*x)/(exp(arccoth(a*x))*x),x,5,2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))+2*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/(sqrt(a)*sqrt(1+(-1)/(a*x)))],
[sqrt(c-a*c*x)/(exp(arccoth(a*x))*x^2),x,5,sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))-3*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[x^3*sqrt(c-a*c*x)/exp(2*arccoth(a*x)),x,9,2/3*(c-a*c*x)^(3/2)/(a^4*c)+2/5*(c-a*c*x)^(5/2)/(a^4*c^2)-2/7*(c-a*c*x)^(7/2)/(a^4*c^3)+2/9*(c-a*c*x)^(9/2)/(a^4*c^4)-4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^4+4*sqrt(c-a*c*x)/a^4],
[x^2*sqrt(c-a*c*x)/exp(2*arccoth(a*x)),x,9,-2/3*(c-a*c*x)^(3/2)/(a^3*c)-2/7*(c-a*c*x)^(7/2)/(a^3*c^3)+4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^3-4*sqrt(c-a*c*x)/a^3],
[x*sqrt(c-a*c*x)/exp(2*arccoth(a*x)),x,8,2/3*(c-a*c*x)^(3/2)/(a^2*c)+2/5*(c-a*c*x)^(5/2)/(a^2*c^2)-4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^2+4*sqrt(c-a*c*x)/a^2],
[sqrt(c-a*c*x)/exp(2*arccoth(a*x)),x,7,-2/3*(c-a*c*x)^(3/2)/(a*c)+4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a-4*sqrt(c-a*c*x)/a],
[sqrt(c-a*c*x)/(exp(2*arccoth(a*x))*x),x,9,2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+2*sqrt(c-a*c*x)],
[sqrt(c-a*c*x)/(exp(2*arccoth(a*x))*x^2),x,9,-5*a*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+4*a*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+sqrt(c-a*c*x)/x],
[sqrt(c-a*c*x)/(exp(2*arccoth(a*x))*x^3),x,10,23/4*a^2*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-4*a^2*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+1/2*sqrt(c-a*c*x)/x^2-9/4*a*sqrt(c-a*c*x)/x],
[sqrt(c-a*c*x)/(exp(2*arccoth(a*x))*x^4),x,11,-45/8*a^3*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)+4*a^3*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+1/3*sqrt(c-a*c*x)/x^3-13/12*a*sqrt(c-a*c*x)/x^2+19/8*a^2*sqrt(c-a*c*x)/x],
[sqrt(c-a*c*x)/(exp(2*arccoth(a*x))*x^5),x,12,363/64*a^4*arctanh(sqrt(c-a*c*x)/sqrt(c))*sqrt(c)-4*a^4*arctanh(sqrt(c-a*c*x)/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+1/4*sqrt(c-a*c*x)/x^4-17/24*a*sqrt(c-a*c*x)/x^3+107/96*a^2*sqrt(c-a*c*x)/x^2-149/64*a^3*sqrt(c-a*c*x)/x],
[x^3*sqrt(c-a*c*x)/exp(3*arccoth(a*x)),x,8,-82/9*x^2*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-8/9*x^3*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+2/9*x^4*sqrt(c-a*c*x)/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+1312/45*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^4*sqrt(1+(-1)/(a*x)))-656/45*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^3*sqrt(1+(-1)/(a*x)))+164/15*x^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))],
[x^2*sqrt(c-a*c*x)/exp(3*arccoth(a*x)),x,7,-334/35*x*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-44/35*x^2*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+2/7*x^3*sqrt(c-a*c*x)/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-2672/105*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^3*sqrt(1+(-1)/(a*x)))+1336/105*x*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))],
[x*sqrt(c-a*c*x)/exp(3*arccoth(a*x)),x,6,-158/15*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-32/15*x*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+2/5*x^2*sqrt(c-a*c*x)/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+316/15*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(a^2*sqrt(1+(-1)/(a*x)))],
[sqrt(c-a*c*x)/exp(3*arccoth(a*x)),x,5,-20/3*sqrt(c-a*c*x)/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-46/3*sqrt(c-a*c*x)/(a^2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+2/3*x*sqrt(c-a*c*x)/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[sqrt(c-a*c*x)/(exp(3*arccoth(a*x))*x),x,6,2*sqrt(c-a*c*x)/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+10*sqrt(c-a*c*x)/(a*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-2*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/(sqrt(a)*sqrt(1+(-1)/(a*x)))],
[sqrt(c-a*c*x)/(exp(3*arccoth(a*x))*x^2),x,6,-8*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+7*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(a)*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[sqrt(c-a*c*x)/(exp(3*arccoth(a*x))*x^3),x,7,-8*sqrt(c-a*c*x)/(x^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-1/2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x^2*sqrt(1+(-1)/(a*x)))+47/4*a*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))-47/4*a^(3/2)*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[sqrt(c-a*c*x)/(exp(3*arccoth(a*x))*x^4),x,8,-8*sqrt(c-a*c*x)/(x^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-1/3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x^3*sqrt(1+(-1)/(a*x)))+119/12*a*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x^2*sqrt(1+(-1)/(a*x)))-119/8*a^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))+119/8*a^(5/2)*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],
[sqrt(c-a*c*x)/(exp(3*arccoth(a*x))*x^5),x,9,-8*sqrt(c-a*c*x)/(x^4*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-1/4*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x^4*sqrt(1+(-1)/(a*x)))+223/24*a*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x^3*sqrt(1+(-1)/(a*x)))-1115/96*a^2*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x^2*sqrt(1+(-1)/(a*x)))+1115/64*a^3*sqrt(1+1/(a*x))*sqrt(c-a*c*x)/(x*sqrt(1+(-1)/(a*x)))-1115/64*a^(7/2)*arcsinh(sqrt(1/x)/sqrt(a))*sqrt(1/x)*sqrt(c-a*c*x)/sqrt(1+(-1)/(a*x))],

# Integrands of the form E^(n ArcCoth[a x])  (c-c a x)^p with n symbolic
[exp(n*arccoth(a*x))*(c-a*c*x)^(2+1/2*n),x,6,-(56+14*n+n^2)*(1+(-1)/(a*x))^(-2-1/2*n)*(1+1/(a*x))^(1/2*(2+n))*(c-a*c*x)^(1/2*(4+n))/(a*(4+n)*(6+n))+2*(56+14*n+n^2)*(1+(-1)/(a*x))^(-2-1/2*n)*(1+1/(a*x))^(1/2*(2+n))*(c-a*c*x)^(1/2*(4+n))/(a^2*(6+n)*(8+6*n+n^2)*x)+(8+n)*(1+(-1)/(a*x))^(-2-1/2*n)*(1+1/(a*x))^(1/2*(2+n))*x*(c-a*c*x)^(1/2*(4+n))/(6+n)-(a+(-1)/x)*(1+(-1)/(a*x))^(-2-1/2*n)*(1+1/(a*x))^(1/2*(2+n))*x*(c-a*c*x)^(1/2*(4+n))/a],
[exp(n*arccoth(a*x))*(c-a*c*x)^(1+1/2*n),x,4,-2*(6+n)*(1+(-1)/(a*x))^(-1-1/2*n)*(1+1/(a*x))^(1/2*(2+n))*(c-a*c*x)^(1/2*(2+n))/(a*(8+6*n+n^2))+2*(1+(-1)/(a*x))^(-1-1/2*n)*(1+1/(a*x))^(1/2*(2+n))*x*(c-a*c*x)^(1/2*(2+n))/(4+n)],
[exp(n*arccoth(a*x))*(c-a*c*x)^(1/2*n),x,1,2*exp(n*arccoth(a*x))*(1+a*x)*(c-a*c*x)^(1/2*n)/(a*(2+n))],
[exp(n*arccoth(a*x))*(c-a*c*x)^(-1+1/2*n),x,3,2*(1+(-1)/(a*x))^(1-1/2*n)*(1+1/(a*x))^(1/2*n)*x*(c-a*c*x)^(1/2*(-2+n))*hypergeom([1,-1/2*n],[1-1/2*n],2/((a+1/x)*x))/n],
[exp(n*arccoth(a*x))*(c-a*c*x)^(-2+1/2*n),x,3,-2*(1+(-1)/(a*x))^(2-1/2*n)*(1+1/(a*x))^(1/2*(-2+n))*x*(c-a*c*x)^(1/2*(-4+n))*hypergeom([2,1-1/2*n],[2-1/2*n],2/((a+1/x)*x))/(2-n)],
[exp(n*arccoth(a*x))*(c-a*c*x)^p,x,3,((a+(-1)/x)/(a+1/x))^(1/2*(n-2*p))*(1+1/(a*x))^(1/2*(2+n))*x*(c-a*c*x)^p*hypergeom([1/2*(n-2*p),-1-p],[-p],2/((a+1/x)*x))/((1+p)*(1+(-1)/(a*x))^(1/2*n))],
[exp(n*arccoth(a*x))*(c-a*c*x)^3,x,3,-32*c^3*(1+(-1)/(a*x))^(4-1/2*n)*(1+1/(a*x))^(1/2*(-8+n))*hypergeom([5,4-1/2*n],[5-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(8-n))],
[exp(n*arccoth(a*x))*(c-a*c*x)^2,x,3,16*c^2*(1+(-1)/(a*x))^(3-1/2*n)*(1+1/(a*x))^(1/2*(-6+n))*hypergeom([4,3-1/2*n],[4-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(6-n))],
[exp(n*arccoth(a*x))*(c-a*c*x),x,3,-8*c*(1+(-1)/(a*x))^(2-1/2*n)*(1+1/(a*x))^(1/2*(-4+n))*hypergeom([3,2-1/2*n],[3-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(4-n))],
[exp(n*arccoth(a*x))/(c-a*c*x),x,3,2*(1+1/(a*x))^(1/2*n)*hypergeom([1,-1/2*n],[1-1/2*n],(a+(-1)/x)/(a+1/x))/(a*c*n*(1+(-1)/(a*x))^(1/2*n))],
[exp(n*arccoth(a*x))/(c-a*c*x)^2,x,3,-(1+(-1)/(a*x))^(-1-1/2*n)*(1+1/(a*x))^(1/2*(2+n))/(a*c^2*(2+n))],
[exp(n*arccoth(a*x))/(c-a*c*x)^3,x,4,-(3+n)*(1+(-1)/(a*x))^(1/2*(-2-n))*(1+1/(a*x))^(1/2*(2+n))/(a*c^3*(8+6*n+n^2))+(1+(-1)/(a*x))^(-2-1/2*n)*(1+1/(a*x))^(1/2*(2+n))/(a*c^3*(4+n))],
[exp(n*arccoth(a*x))/(c-a*c*x)^4,x,6,-(14+8*n+n^2)*(1+(-1)/(a*x))^(1/2*(-4-n))*(1+1/(a*x))^(1/2*(2+n))/(a*c^4*(24+10*n+n^2))-(14+8*n+n^2)*(1+(-1)/(a*x))^(1/2*(-2-n))*(1+1/(a*x))^(1/2*(2+n))/(a*c^4*(48+44*n+12*n^2+n^3))+(5+n)*(1+(-1)/(a*x))^(-3-1/2*n)*(1+1/(a*x))^(1/2*(2+n))/(a*c^4*(6+n))-(1+(-1)/(a*x))^(-3-1/2*n)*(1+1/(a*x))^(1/2*(2+n))/(a^2*c^4*x)],
[exp(n*arccoth(a*x))*(c-a*c*x)^(5/2),x,3,2/7*((a+(-1)/x)/(a+1/x))^(1/2*(-5+n))*(1+1/(a*x))^(1/2*(2+n))*x*(c-a*c*x)^(5/2)*hypergeom([-7/2,1/2*(-5+n)],[-5/2],2/((a+1/x)*x))/(1+(-1)/(a*x))^(1/2*n)],
[exp(n*arccoth(a*x))*(c-a*c*x)^(3/2),x,3,2/5*((a+(-1)/x)/(a+1/x))^(1/2*(-3+n))*(1+1/(a*x))^(1/2*(2+n))*x*(c-a*c*x)^(3/2)*hypergeom([-5/2,1/2*(-3+n)],[-3/2],2/((a+1/x)*x))/(1+(-1)/(a*x))^(1/2*n)],
[exp(n*arccoth(a*x))*(c-a*c*x)^(1/2),x,3,2/3*((a+(-1)/x)/(a+1/x))^(1/2*(-1+n))*(1+1/(a*x))^(1/2*(2+n))*x*hypergeom([-3/2,1/2*(-1+n)],[-1/2],2/((a+1/x)*x))*sqrt(c-a*c*x)/(1+(-1)/(a*x))^(1/2*n)],
[exp(n*arccoth(a*x))/(c-a*c*x)^(1/2),x,3,2*((a+(-1)/x)/(a+1/x))^(1/2*(1+n))*(1+1/(a*x))^(1/2*(2+n))*x*hypergeom([-1/2,1/2*(1+n)],[1/2],2/((a+1/x)*x))/((1+(-1)/(a*x))^(1/2*n)*sqrt(c-a*c*x))],
[exp(n*arccoth(a*x))/(c-a*c*x)^(3/2),x,3,-2*((a+(-1)/x)/(a+1/x))^(1/2*(3+n))*(1+1/(a*x))^(1/2*(2+n))*x*hypergeom([1/2,1/2*(3+n)],[3/2],2/((a+1/x)*x))/((1+(-1)/(a*x))^(1/2*n)*(c-a*c*x)^(3/2))],
[exp(n*arccoth(a*x))/(c-a*c*x)^(5/2),x,4,-a*(1+(-1)/(a*x))^(1/2*(2-n))*(1+1/(a*x))^(1/2*(2+n))*x^2/((3+n)*(c-a*c*x)^(5/2))+a*((a+(-1)/x)/(a+1/x))^(1/2*(3+n))*(1+(-1)/(a*x))^(1/2*(2-n))*(1+1/(a*x))^(1/2*(2+n))*x^2*hypergeom([1/2,1/2*(3+n)],[3/2],2/((a+1/x)*x))/((3+n)*(c-a*c*x)^(5/2))],
[exp(n*arccoth(a*x))/(c-a*c*x)^(7/2),x,5,-a*(1+(-1)/(a*x))^(1/2*(2-n))*(1+1/(a*x))^(1/2*(2+n))*x^2/((5+n)*(c-a*c*x)^(7/2))+3/2*a^2*(1+(-1)/(a*x))^(1/2*(4-n))*(1+1/(a*x))^(1/2*(2+n))*x^3/((15+8*n+n^2)*(c-a*c*x)^(7/2))-3/2*a^2*((a+(-1)/x)/(a+1/x))^(1/2*(3+n))*(1+(-1)/(a*x))^(1/2*(4-n))*(1+1/(a*x))^(1/2*(2+n))*x^3*hypergeom([1/2,1/2*(3+n)],[3/2],2/((a+1/x)*x))/((15+8*n+n^2)*(c-a*c*x)^(7/2))],

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

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

# n>0
[exp(arccoth(a*x))*(c-c/(a*x))^4,x,9,-1/3*c^4*(1+(-1)/(a^2*x^2))^(3/2)/a+c^4*(1+(-1)/(a^2*x^2))^(3/2)*x-1/2*c^4*arccsc(a*x)/a-3*c^4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+1/2*c^4*(6*a+(-1)/x)*sqrt(1+(-1)/(a^2*x^2))/a^2],
[exp(arccoth(a*x))*(c-c/(a*x))^3,x,8,c^3*(1+(-1)/(a^2*x^2))^(3/2)*x+1/2*c^3*arccsc(a*x)/a-2*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+1/2*c^3*(4*a+1/x)*sqrt(1+(-1)/(a^2*x^2))/a^2],
[exp(arccoth(a*x))*(c-c/(a*x))^2,x,7,c^2*arccsc(a*x)/a-c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+c^2*(a+1/x)*x*sqrt(1+(-1)/(a^2*x^2))/a],
[exp(arccoth(a*x))*(c-c/(a*x)),x,3,c*arccsc(a*x)/a+c*x*sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))/(c-c/(a*x)),x,7,2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)-2*(a+1/x)/(a^2*c*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c],
[exp(arccoth(a*x))/(c-c/(a*x))^2,x,8,-4/3*(a+1/x)/(a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2))+3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^2)+1/3*(-9*a+(-11)/x)/(a^2*c^2*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^2],
[exp(arccoth(a*x))/(c-c/(a*x))^3,x,9,-8/5*(a+1/x)/(a^2*c^3*(1+(-1)/(a^2*x^2))^(5/2))-4/15*(5*a+8/x)/(a^2*c^3*(1+(-1)/(a^2*x^2))^(3/2))+4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^3)+1/15*(-60*a+(-79)/x)/(a^2*c^3*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^3],
[exp(arccoth(a*x))/(c-c/(a*x))^4,x,10,-16/7*(a+1/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(7/2))-4/35*(7*a+17/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(5/2))+1/105*(-175*a+(-307)/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(3/2))+5*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^4)+1/105*(-525*a+(-719)/x)/(a^2*c^4*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^4],
[exp(2*arccoth(a*x))*(c-c/(a*x))^5,x,5,-1/4*c^5/(a^5*x^4)+c^5/(a^4*x^3)-c^5/(a^3*x^2)-2*c^5/(a^2*x)+c^5*x-3*c^5*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a*x))^4,x,5,1/3*c^4/(a^4*x^3)-c^4/(a^3*x^2)+c^4*x-2*c^4*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a*x))^3,x,5,-1/2*c^3/(a^3*x^2)+c^3/(a^2*x)+c^3*x-c^3*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a*x))^2,x,6,c^2/(a^2*x)+c^2*x],
[exp(2*arccoth(a*x))*(c-c/(a*x)),x,5,c*x+c*log(x)/a],
[exp(2*arccoth(a*x))/(c-c/(a*x)),x,5,x/c+2/(a*c*(1-a*x))+3*log(1-a*x)/(a*c)],
[exp(2*arccoth(a*x))/(c-c/(a*x))^2,x,5,x/c^2+(-1)/(a*c^2*(1-a*x)^2)+5/(a*c^2*(1-a*x))+4*log(1-a*x)/(a*c^2)],
[exp(2*arccoth(a*x))/(c-c/(a*x))^3,x,5,x/c^3+2/3/(a*c^3*(1-a*x)^3)+(-7/2)/(a*c^3*(1-a*x)^2)+9/(a*c^3*(1-a*x))+5*log(1-a*x)/(a*c^3)],
[exp(2*arccoth(a*x))/(c-c/(a*x))^4,x,5,x/c^4+(-1/2)/(a*c^4*(1-a*x)^4)+3/(a*c^4*(1-a*x)^3)+(-8)/(a*c^4*(1-a*x)^2)+14/(a*c^4*(1-a*x))+6*log(1-a*x)/(a*c^4)],
[exp(3*arccoth(a*x))*(c-c/(a*x))^4,x,8,1/3*c^4*(1+(-1)/(a^2*x^2))^(3/2)*(3*a+1/x)*x/a+3/2*c^4*arccsc(a*x)/a-c^4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+1/2*c^4*(2*a+3/x)*sqrt(1+(-1)/(a^2*x^2))/a^2],
[exp(3*arccoth(a*x))*(c-c/(a*x))^3,x,4,c^3*(1+(-1)/(a^2*x^2))^(3/2)*x+3/2*c^3*arccsc(a*x)/a+3/2*c^3*sqrt(1+(-1)/(a^2*x^2))/(a^2*x)],
[exp(3*arccoth(a*x))*(c-c/(a*x))^2,x,8,c^2*arccsc(a*x)/a+c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+c^2*(a+(-1)/x)*x*sqrt(1+(-1)/(a^2*x^2))/a],
[exp(3*arccoth(a*x))*(c-c/(a*x)),x,8,-c*arccsc(a*x)/a+2*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+c*x*sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))/(c-c/(a*x)),x,8,-8/3*(a+1/x)/(a^2*c*(1+(-1)/(a^2*x^2))^(3/2))+4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)-4/3*(3*a+4/x)/(a^2*c*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c],
[exp(3*arccoth(a*x))/(c-c/(a*x))^2,x,9,-16/5*(a+1/x)/(a^2*c^2*(1+(-1)/(a^2*x^2))^(5/2))-4/15*(5*a+11/x)/(a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2))+5*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^2)+1/15*(-75*a+(-103)/x)/(a^2*c^2*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^2],
[exp(3*arccoth(a*x))/(c-c/(a*x))^3,x,10,-32/7*(a+1/x)/(a^2*c^3*(1+(-1)/(a^2*x^2))^(7/2))-2/7*(7*a+13/x)/(a^2*c^3*(1+(-1)/(a^2*x^2))^(3/2))+(-16/7)/(a^2*c^3*(1+(-1)/(a^2*x^2))^(5/2)*x)+6*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^3)+1/7*(-42*a+(-59)/x)/(a^2*c^3*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^3],
[exp(3*arccoth(a*x))/(c-c/(a*x))^4,x,11,16/63*(9*a+(-5)/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(7/2))-64/9*(a+1/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(9/2))-8/105*(21*a+41/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(5/2))+1/315*(-735*a+(-1417)/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(3/2))+7*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^4)+1/315*(-2205*a+(-3149)/x)/(a^2*c^4*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^4],
[exp(4*arccoth(a*x))*(c-c/(a*x))^5,x,5,1/4*c^5/(a^5*x^4)-1/3*c^5/(a^4*x^3)-c^5/(a^3*x^2)+2*c^5/(a^2*x)+c^5*x-c^5*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a*x))^4,x,6,-1/3*c^4/(a^4*x^3)+2*c^4/(a^2*x)+c^4*x],
[exp(4*arccoth(a*x))*(c-c/(a*x))^3,x,5,1/2*c^3/(a^3*x^2)+c^3/(a^2*x)+c^3*x+c^3*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a*x))^2,x,5,-c^2/(a^2*x)+c^2*x+2*c^2*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a*x)),x,5,c*x-c*log(x)/a+4*c*log(1-a*x)/a],
[exp(4*arccoth(a*x))/(c-c/(a*x)),x,5,x/c+(-2)/(a*c*(1-a*x)^2)+8/(a*c*(1-a*x))+5*log(1-a*x)/(a*c)],
[exp(4*arccoth(a*x))/(c-c/(a*x))^2,x,5,x/c^2+4/3/(a*c^2*(1-a*x)^3)+(-6)/(a*c^2*(1-a*x)^2)+13/(a*c^2*(1-a*x))+6*log(1-a*x)/(a*c^2)],
[exp(4*arccoth(a*x))/(c-c/(a*x))^3,x,5,x/c^3+(-1)/(a*c^3*(1-a*x)^4)+16/3/(a*c^3*(1-a*x)^3)+(-25/2)/(a*c^3*(1-a*x)^2)+19/(a*c^3*(1-a*x))+7*log(1-a*x)/(a*c^3)],
[exp(4*arccoth(a*x))/(c-c/(a*x))^4,x,5,x/c^4+4/5/(a*c^4*(1-a*x)^5)+(-5)/(a*c^4*(1-a*x)^4)+41/3/(a*c^4*(1-a*x)^3)+(-22)/(a*c^4*(1-a*x)^2)+26/(a*c^4*(1-a*x))+8*log(1-a*x)/(a*c^4)],

# n<0
[(c-c/(a*x))^4/exp(arccoth(a*x)),x,10,-25/2*c^4*arccsc(a*x)/a-5*c^4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-32/3*c^4*sqrt(1+(-1)/(a^2*x^2))/a-1/3*c^4*sqrt(1+(-1)/(a^2*x^2))/(a^3*x^2)+5/2*c^4*sqrt(1+(-1)/(a^2*x^2))/(a^2*x)+c^4*x*sqrt(1+(-1)/(a^2*x^2))],
[(c-c/(a*x))^3/exp(arccoth(a*x)),x,9,-13/2*c^3*arccsc(a*x)/a-4*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-4*c^3*sqrt(1+(-1)/(a^2*x^2))/a+1/2*c^3*sqrt(1+(-1)/(a^2*x^2))/(a^2*x)+c^3*x*sqrt(1+(-1)/(a^2*x^2))],
[(c-c/(a*x))^2/exp(arccoth(a*x)),x,8,-3*c^2*arccsc(a*x)/a-3*c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a-c^2*sqrt(1+(-1)/(a^2*x^2))/a+c^2*x*sqrt(1+(-1)/(a^2*x^2))],
[(c-c/(a*x))/exp(arccoth(a*x)),x,7,-c*arccsc(a*x)/a-2*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+c*x*sqrt(1+(-1)/(a^2*x^2))],
[1/(exp(arccoth(a*x))*(c-c/(a*x))),x,2,x*sqrt(1+(-1)/(a^2*x^2))/c],
[1/(exp(arccoth(a*x))*(c-c/(a*x))^2),x,6,arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^2)+2*x*sqrt(1+(-1)/(a^2*x^2))/c^2-a*x*sqrt(1+(-1)/(a^2*x^2))/(c^2*(a+(-1)/x))],
[1/(exp(arccoth(a*x))*(c-c/(a*x))^3),x,8,-2/3*(a+1/x)/(a^2*c^3*(1+(-1)/(a^2*x^2))^(3/2))+2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^3)+1/3*(-6*a+(-7)/x)/(a^2*c^3*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^3],
[1/(exp(arccoth(a*x))*(c-c/(a*x))^4),x,9,-4/5*(a+1/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(5/2))+1/5*(-5*a+(-7)/x)/(a^2*c^4*(1+(-1)/(a^2*x^2))^(3/2))+3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^4)+1/5*(-15*a+(-19)/x)/(a^2*c^4*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^4],
[(c-c/(a*x))^4/exp(2*arccoth(a*x)),x,5,1/3*c^4/(a^4*x^3)-3*c^4/(a^3*x^2)+16*c^4/(a^2*x)+c^4*x+26*c^4*log(x)/a-32*c^4*log(1+a*x)/a],
[(c-c/(a*x))^3/exp(2*arccoth(a*x)),x,5,-1/2*c^3/(a^3*x^2)+5*c^3/(a^2*x)+c^3*x+11*c^3*log(x)/a-16*c^3*log(1+a*x)/a],
[(c-c/(a*x))^2/exp(2*arccoth(a*x)),x,5,c^2/(a^2*x)+c^2*x+4*c^2*log(x)/a-8*c^2*log(1+a*x)/a],
[(c-c/(a*x))/exp(2*arccoth(a*x)),x,5,c*x+c*log(x)/a-4*c*log(1+a*x)/a],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))),x,5,x/c-log(1+a*x)/(a*c)],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^2),x,6,x/c^2-arctanh(a*x)/(a*c^2)],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^3),x,5,x/c^3+1/2/(a*c^3*(1-a*x))+5/4*log(1-a*x)/(a*c^3)-1/4*log(1+a*x)/(a*c^3)],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^4),x,5,x/c^4+(-1/4)/(a*c^4*(1-a*x)^2)+7/4/(a*c^4*(1-a*x))+17/8*log(1-a*x)/(a*c^4)-1/8*log(1+a*x)/(a*c^4)],
[(c-c/(a*x))^4/exp(3*arccoth(a*x)),x,11,91/2*c^4*arccsc(a*x)/a-7*c^4*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+64*c^4*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+68/3*c^4*sqrt(1+(-1)/(a^2*x^2))/a+1/3*c^4*sqrt(1+(-1)/(a^2*x^2))/(a^3*x^2)-7/2*c^4*sqrt(1+(-1)/(a^2*x^2))/(a^2*x)+c^4*x*sqrt(1+(-1)/(a^2*x^2))],
[(c-c/(a*x))^3/exp(3*arccoth(a*x)),x,10,33/2*c^3*arccsc(a*x)/a-6*c^3*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+32*c^3*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+6*c^3*sqrt(1+(-1)/(a^2*x^2))/a-1/2*c^3*sqrt(1+(-1)/(a^2*x^2))/(a^2*x)+c^3*x*sqrt(1+(-1)/(a^2*x^2))],
[(c-c/(a*x))^2/exp(3*arccoth(a*x)),x,9,5*c^2*arccsc(a*x)/a-5*c^2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+16*c^2*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+c^2*sqrt(1+(-1)/(a^2*x^2))/a+c^2*x*sqrt(1+(-1)/(a^2*x^2))],
[(c-c/(a*x))/exp(3*arccoth(a*x)),x,8,c*arccsc(a*x)/a-4*c*arctanh(sqrt(1+(-1)/(a^2*x^2)))/a+8*c*(a+(-1)/x)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+c*x*sqrt(1+(-1)/(a^2*x^2))],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))),x,6,-2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c)+2*(a+(-1)/x)/(a^2*c*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^2),x,6,-arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^2)-(a+(-1)/x)*x/(a*c^2*sqrt(1+(-1)/(a^2*x^2)))+2*x*sqrt(1+(-1)/(a^2*x^2))/c^2],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^3),x,3,(-2)/(a^2*c^3*x*sqrt(1+(-1)/(a^2*x^2)))+x/(c^3*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^4),x,7,arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^4)-1/3*a*x/(c^4*(a+(-1)/x)*sqrt(1+(-1)/(a^2*x^2)))-1/3*(4*a+3/x)*x/(a*c^4*sqrt(1+(-1)/(a^2*x^2)))+8/3*x*sqrt(1+(-1)/(a^2*x^2))/c^4],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^5),x,9,-2/5*(a+1/x)/(a^2*c^5*(1+(-1)/(a^2*x^2))^(5/2))+1/15*(-10*a+(-13)/x)/(a^2*c^5*(1+(-1)/(a^2*x^2))^(3/2))+2*arctanh(sqrt(1+(-1)/(a^2*x^2)))/(a*c^5)+1/15*(-30*a+(-41)/x)/(a^2*c^5*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(1+(-1)/(a^2*x^2))/c^5],

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

# n>0
[exp(arccoth(a*x))*(c-c/(a*x))^(9/2),x,8,-7*c^(9/2)*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/a+59/35*c^3*(c-c/(a*x))^(3/2)*sqrt(1+(-1)/(a^2*x^2))/a+9/7*c^2*(c-c/(a*x))^(5/2)*sqrt(1+(-1)/(a^2*x^2))/a+c*(c-c/(a*x))^(7/2)*x*sqrt(1+(-1)/(a^2*x^2))+173/105*c^5*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))+227/105*c^4*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))/a,-7*(c-c/(a*x))^(9/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(9/2))+1/105*(400*a+(-227)/x)*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(9/2))+59/35*(a+(-1)/x)^2*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(9/2))+9/7*(a+(-1)/x)^3*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(9/2))+(a+(-1)/x)^4*(c-c/(a*x))^(9/2)*x*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(9/2))],
[exp(arccoth(a*x))*(c-c/(a*x))^(7/2),x,7,-5*c^(7/2)*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/a+7/5*c^2*(c-c/(a*x))^(3/2)*sqrt(1+(-1)/(a^2*x^2))/a+c*(c-c/(a*x))^(5/2)*x*sqrt(1+(-1)/(a^2*x^2))+49/15*c^4*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))+31/15*c^3*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))/a,-5*(c-c/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(7/2))+1/15*(80*a+(-31)/x)*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(7/2))+7/5*(a+(-1)/x)^2*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(7/2))+(a+(-1)/x)^3*(c-c/(a*x))^(7/2)*x*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(7/2))],
[exp(arccoth(a*x))*(c-c/(a*x))^(5/2),x,7,-2/3*c^4*(1+(-1)/(a^2*x^2))^(3/2)/(a*(c-c/(a*x))^(3/2))+c^4*(1+(-1)/(a^2*x^2))^(3/2)*x/(c-c/(a*x))^(3/2)-3*c^(5/2)*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/a+3*c^3*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x))),-2/3*(1+1/(a*x))^(3/2)*(c-c/(a*x))^(5/2)/(a*(1+(-1)/(a*x))^(5/2))+(1+1/(a*x))^(3/2)*(c-c/(a*x))^(5/2)*x/(1+(-1)/(a*x))^(5/2)-3*(c-c/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(5/2))+3*(c-c/(a*x))^(5/2)*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^(5/2))],
[exp(arccoth(a*x))*(c-c/(a*x))^(3/2),x,5,c^3*(1+(-1)/(a^2*x^2))^(3/2)*x/(c-c/(a*x))^(3/2)-c^(3/2)*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/a+c^2*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))],
[exp(arccoth(a*x))*sqrt(c-c/(a*x)),x,4,arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a+c*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))/sqrt(c-c/(a*x)),x,8,3*arctanh(sqrt(1+1/(a*x)))*sqrt(1+(-1)/(a*x))/(a*sqrt(c-c/(a*x)))-2*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(1+(-1)/(a*x))/(a*sqrt(c-c/(a*x)))+x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))/(c-c/(a*x))^(3/2),x,9,5*(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(3/2))-7*(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(3/2)*sqrt(2))-2*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(3/2))+a*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(3/2))],
[exp(arccoth(a*x))/(c-c/(a*x))^(5/2),x,10,7*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(5/2))-79/8*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(5/2)*sqrt(2))-3/2*a*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(5/2))-23/8*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(5/2))+a^2*(1+(-1)/(a*x))^(5/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(5/2))],
[exp(2*arccoth(a*x))*(c-c/(a*x))^(9/2),x,12,5/3*c^3*(c-c/(a*x))^(3/2)/a+c^2*(c-c/(a*x))^(5/2)/a+5/7*c*(c-c/(a*x))^(7/2)/a+(c-c/(a*x))^(9/2)*x-5*c^(9/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+5*c^4*sqrt(c-c/(a*x))/a],
[exp(2*arccoth(a*x))*(c-c/(a*x))^(7/2),x,11,c^2*(c-c/(a*x))^(3/2)/a+3/5*c*(c-c/(a*x))^(5/2)/a+(c-c/(a*x))^(7/2)*x-3*c^(7/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+3*c^3*sqrt(c-c/(a*x))/a],
[exp(2*arccoth(a*x))*(c-c/(a*x))^(5/2),x,10,1/3*c*(c-c/(a*x))^(3/2)/a+(c-c/(a*x))^(5/2)*x-c^(5/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+c^2*sqrt(c-c/(a*x))/a],
[exp(2*arccoth(a*x))*(c-c/(a*x))^(3/2),x,9,(c-c/(a*x))^(3/2)*x+c^(3/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a-c*sqrt(c-c/(a*x))/a],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x)),x,8,3*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a+x*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))/sqrt(c-c/(a*x)),x,9,5*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*sqrt(c))+(-5)/(a*sqrt(c-c/(a*x)))+x/sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))/(c-c/(a*x))^(3/2),x,10,(-7/3)/(a*(c-c/(a*x))^(3/2))+x/(c-c/(a*x))^(3/2)+7*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(3/2))+(-7)/(a*c*sqrt(c-c/(a*x)))],
[exp(2*arccoth(a*x))/(c-c/(a*x))^(5/2),x,11,(-9/5)/(a*(c-c/(a*x))^(5/2))+(-3)/(a*c*(c-c/(a*x))^(3/2))+x/(c-c/(a*x))^(5/2)+9*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(5/2))+(-9)/(a*c^2*sqrt(c-c/(a*x)))],
[exp(2*arccoth(a*x))/(c-c/(a*x))^(7/2),x,12,(-11/7)/(a*(c-c/(a*x))^(7/2))+(-11/5)/(a*c*(c-c/(a*x))^(5/2))+(-11/3)/(a*c^2*(c-c/(a*x))^(3/2))+x/(c-c/(a*x))^(7/2)+11*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(7/2))+(-11)/(a*c^3*sqrt(c-c/(a*x)))],
[exp(3*arccoth(a*x))*(c-c/(a*x))^(9/2),x,8,3/35*(28*a+(-17)/x)*(1+1/(a*x))^(3/2)*(c-c/(a*x))^(9/2)/(a^2*(1+(-1)/(a*x))^(9/2))+9/7*(a+(-1)/x)^2*(1+1/(a*x))^(3/2)*(c-c/(a*x))^(9/2)/(a^3*(1+(-1)/(a*x))^(9/2))+(a+(-1)/x)^3*(1+1/(a*x))^(3/2)*(c-c/(a*x))^(9/2)*x/(a^3*(1+(-1)/(a*x))^(9/2))-3*(c-c/(a*x))^(9/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(9/2))+3*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^(9/2))],
[exp(3*arccoth(a*x))*(c-c/(a*x))^(7/2),x,8,1/3*(1+1/(a*x))^(3/2)*(c-c/(a*x))^(7/2)/(a*(1+(-1)/(a*x))^(7/2))-2/5*(1+1/(a*x))^(5/2)*(c-c/(a*x))^(7/2)/(a*(1+(-1)/(a*x))^(7/2))+(1+1/(a*x))^(5/2)*(c-c/(a*x))^(7/2)*x/(1+(-1)/(a*x))^(7/2)-(c-c/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(7/2))+(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^(7/2))],
[exp(3*arccoth(a*x))*(c-c/(a*x))^(5/2),x,6,-1/3*c^4*(1+(-1)/(a^2*x^2))^(3/2)/(a*(c-c/(a*x))^(3/2))+c^5*(1+(-1)/(a^2*x^2))^(5/2)*x/(c-c/(a*x))^(5/2)+c^(5/2)*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/a-c^3*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))],
[exp(3*arccoth(a*x))*(c-c/(a*x))^(3/2),x,5,c^3*(1+(-1)/(a^2*x^2))^(3/2)*x/(c-c/(a*x))^(3/2)+3*c^(3/2)*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/a-3*c^2*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x)),x,8,5*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+x*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))/sqrt(c-c/(a*x)),x,9,7*arctanh(sqrt(1+1/(a*x)))*sqrt(1+(-1)/(a*x))/(a*sqrt(c-c/(a*x)))-5*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(1+(-1)/(a*x))/(a*sqrt(c-c/(a*x)))-3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/((a+(-1)/x)*sqrt(c-c/(a*x)))+a*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/((a+(-1)/x)*sqrt(c-c/(a*x)))],
[exp(3*arccoth(a*x))/(c-c/(a*x))^(3/2),x,10,9*(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(3/2))-51/4*(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(3/2)*sqrt(2))-2*a*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(3/2))-15/4*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(3/2))+a^2*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(3/2))],
[exp(3*arccoth(a*x))/(c-c/(a*x))^(5/2),x,11,11*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(5/2))-249/16*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(5/2)*sqrt(2))-5/3*a^2*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/((a+(-1)/x)^3*(c-c/(a*x))^(5/2))-29/12*a*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(5/2))-73/16*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(5/2))+a^3*(1+(-1)/(a*x))^(5/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)^3*(c-c/(a*x))^(5/2))],

# n<0
[(c-c/(a*x))^(7/2)/exp(arccoth(a*x)),x,7,-9*(c-c/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(7/2))-1/5*(80*a+(-7)/x)*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(7/2))+3/5*(a+(-1)/x)^2*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(7/2))+(a+(-1)/x)^3*(c-c/(a*x))^(7/2)*x*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(7/2))],
[(c-c/(a*x))^(5/2)/exp(arccoth(a*x)),x,6,-7*(c-c/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(5/2))-1/3*(16*a+1/x)*(c-c/(a*x))^(5/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(5/2))+(a+(-1)/x)^2*(c-c/(a*x))^(5/2)*x*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(5/2))],
[(c-c/(a*x))^(3/2)/exp(arccoth(a*x)),x,6,-5*(c-c/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(3/2))-2*(c-c/(a*x))^(3/2)*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^(3/2))+(c-c/(a*x))^(3/2)*x*sqrt(1+1/(a*x))/(1+(-1)/(a*x))^(3/2)],
[sqrt(c-c/(a*x))/exp(arccoth(a*x)),x,4,-3*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a+c*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[1/(exp(arccoth(a*x))*sqrt(c-c/(a*x))),x,4,-arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/(a*sqrt(c))+x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[1/(exp(arccoth(a*x))*(c-c/(a*x))^(3/2)),x,9,(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(3/2))-(1+(-1)/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)/(a*(c-c/(a*x))^(3/2))+(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))/(c-c/(a*x))^(3/2)],
[1/(exp(arccoth(a*x))*(c-c/(a*x))^(5/2)),x,10,3*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(5/2))-9/2*(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(5/2)*sqrt(2))-3/2*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(5/2))+a*(1+(-1)/(a*x))^(5/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(5/2))],
[1/(exp(arccoth(a*x))*(c-c/(a*x))^(7/2)),x,11,5*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(7/2))-115/16*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(7/2)*sqrt(2))-5/4*a*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(7/2))-35/16*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x))/((a+(-1)/x)*(c-c/(a*x))^(7/2))+a^2*(1+(-1)/(a*x))^(7/2)*x*sqrt(1+1/(a*x))/((a+(-1)/x)^2*(c-c/(a*x))^(7/2))],
[(c-c/(a*x))^(7/2)/exp(2*arccoth(a*x)),x,14,-5/3*c^2*(c-c/(a*x))^(3/2)/a+3/5*c*(c-c/(a*x))^(5/2)/a+(c-c/(a*x))^(7/2)*x-11*c^(7/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+32*c^(7/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a-21*c^3*sqrt(c-c/(a*x))/a],
[(c-c/(a*x))^(5/2)/exp(2*arccoth(a*x)),x,13,1/3*c*(c-c/(a*x))^(3/2)/a+(c-c/(a*x))^(5/2)*x-9*c^(5/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+16*c^(5/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a-7*c^2*sqrt(c-c/(a*x))/a],
[(c-c/(a*x))^(3/2)/exp(2*arccoth(a*x)),x,12,(c-c/(a*x))^(3/2)*x-7*c^(3/2)*arctanh(sqrt(c-c/(a*x))/sqrt(c))/a+8*c^(3/2)*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/a-c*sqrt(c-c/(a*x))/a],
[sqrt(c-c/(a*x))/exp(2*arccoth(a*x)),x,11,-5*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a+4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a+x*sqrt(c-c/(a*x))],
[1/(exp(2*arccoth(a*x))*sqrt(c-c/(a*x))),x,11,-3*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*sqrt(c))+2*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*sqrt(c))+x*sqrt(c-c/(a*x))/c],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^(3/2)),x,12,-arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(3/2))+arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)/(a*c^(3/2))+x*sqrt(c-c/(a*x))/c^2],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^(5/2)),x,12,arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(5/2))+arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))/(a*c^(5/2)*sqrt(2))+(-2)/(a*c^2*sqrt(c-c/(a*x)))+x/(c^2*sqrt(c-c/(a*x)))],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^(7/2)),x,13,(-4/3)/(a*c^2*(c-c/(a*x))^(3/2))+x/(c^2*(c-c/(a*x))^(3/2))+3*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(7/2))+1/2*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))/(a*c^(7/2)*sqrt(2))+(-7/2)/(a*c^3*sqrt(c-c/(a*x)))],
[1/(exp(2*arccoth(a*x))*(c-c/(a*x))^(9/2)),x,14,(-6/5)/(a*c^2*(c-c/(a*x))^(5/2))+(-11/6)/(a*c^3*(c-c/(a*x))^(3/2))+x/(c^2*(c-c/(a*x))^(5/2))+5*arctanh(sqrt(c-c/(a*x))/sqrt(c))/(a*c^(9/2))+1/4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))/(a*c^(9/2)*sqrt(2))+(-21/4)/(a*c^4*sqrt(c-c/(a*x)))],
[(c-c/(a*x))^(9/2)/exp(3*arccoth(a*x)),x,9,-15*(c-c/(a*x))^(9/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(9/2))+10*(a+(-1)/x)^4*(c-c/(a*x))^(9/2)/(a^5*(1+(-1)/(a*x))^(9/2)*sqrt(1+1/(a*x)))+(a+(-1)/x)^5*(c-c/(a*x))^(9/2)*x/(a^5*(1+(-1)/(a*x))^(9/2)*sqrt(1+1/(a*x)))+5/7*(304*a+(-65)/x)*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(9/2))+135/7*(a+(-1)/x)^2*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(9/2))+65/7*(a+(-1)/x)^3*(c-c/(a*x))^(9/2)*sqrt(1+1/(a*x))/(a^4*(1+(-1)/(a*x))^(9/2))],
[(c-c/(a*x))^(7/2)/exp(3*arccoth(a*x)),x,8,-13*(c-c/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(7/2))+10*(a+(-1)/x)^3*(c-c/(a*x))^(7/2)/(a^4*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x)))+(a+(-1)/x)^4*(c-c/(a*x))^(7/2)*x/(a^4*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x)))+1/15*(1360*a+(-311)/x)*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(7/2))+47/5*(a+(-1)/x)^2*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x))/(a^3*(1+(-1)/(a*x))^(7/2))],
[(c-c/(a*x))^(5/2)/exp(3*arccoth(a*x)),x,7,-11*(c-c/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(5/2))+10*(a+(-1)/x)^2*(c-c/(a*x))^(5/2)/(a^3*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x)))+(a+(-1)/x)^3*(c-c/(a*x))^(5/2)*x/(a^3*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x)))+1/3*(112*a+(-29)/x)*(c-c/(a*x))^(5/2)*sqrt(1+1/(a*x))/(a^2*(1+(-1)/(a*x))^(5/2))],
[(c-c/(a*x))^(3/2)/exp(3*arccoth(a*x)),x,6,-9*(c-c/(a*x))^(3/2)*arctanh(sqrt(1+1/(a*x)))/(a*(1+(-1)/(a*x))^(3/2))+(21*a+1/x)*(c-c/(a*x))^(3/2)/(a^2*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))+(a+(-1)/x)^2*(c-c/(a*x))^(3/2)*x/(a^2*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))],
[sqrt(c-c/(a*x))/exp(3*arccoth(a*x)),x,6,-7*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+9*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+x*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*sqrt(c-c/(a*x))),x,5,-5*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/(a*sqrt(c))+5*sqrt(c-c/(a*x))/(a*c*sqrt(1+(-1)/(a^2*x^2)))+x*sqrt(c-c/(a*x))/(c*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^(3/2)),x,5,-3*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))/(a*c^(3/2))+3*x*sqrt(1+(-1)/(a^2*x^2))/(c*sqrt(c-c/(a*x)))-2*x*sqrt(c-c/(a*x))/(c^2*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^(5/2)),x,9,-(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(5/2))-(1+(-1)/(a*x))^(5/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(5/2)*sqrt(2))+2*(1+(-1)/(a*x))^(5/2)/(a*(c-c/(a*x))^(5/2)*sqrt(1+1/(a*x)))+(1+(-1)/(a*x))^(5/2)*x/((c-c/(a*x))^(5/2)*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a*x))^(7/2)),x,10,(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x)))/(a*(c-c/(a*x))^(7/2))-11/4*(1+(-1)/(a*x))^(7/2)*arctanh(sqrt(1+1/(a*x))/sqrt(2))/(a*(c-c/(a*x))^(7/2)*sqrt(2))+7/4*(1+(-1)/(a*x))^(7/2)/(a*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x)))-3/2*(1+(-1)/(a*x))^(7/2)/((a+(-1)/x)*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x)))+a*(1+(-1)/(a*x))^(7/2)*x/((a+(-1)/x)*(c-c/(a*x))^(7/2)*sqrt(1+1/(a*x)))],

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

# n>0
[exp(arccoth(a*x))*x^m*sqrt(c-c/(a*x)),x,3,x^(1+m)*hypergeom([-1/2,-1-m],[-m],(-1)/(a*x))*sqrt(c-c/(a*x))/((1+m)*sqrt(1+(-1)/(a*x)))],
[exp(arccoth(a*x))*x^2*sqrt(c-c/(a*x)),x,6,1/8*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a^3-1/8*c*x*sqrt(1+(-1)/(a^2*x^2))/(a^2*sqrt(c-c/(a*x)))+1/12*c*x^2*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))+1/3*c*x^3*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))*x*sqrt(c-c/(a*x)),x,5,-1/4*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a^2+1/4*c*x*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))+1/2*c*x^2*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))*sqrt(c-c/(a*x)),x,4,arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a+c*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))*sqrt(c-c/(a*x))/x,x,4,2*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)-2*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))*sqrt(c-c/(a*x))/x^2,x,2,-2/3*a*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)],
[exp(arccoth(a*x))*sqrt(c-c/(a*x))/x^3,x,3,-2/15*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)+2/5*a^2*c*(1+(-1)/(a^2*x^2))^(3/2)/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))*sqrt(c-c/(a*x))/x^4,x,4,8/105*a^3*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)-2/7*a*c^2*(1+(-1)/(a^2*x^2))^(3/2)/((c-c/(a*x))^(3/2)*x^2)-8/35*a^3*c*(1+(-1)/(a^2*x^2))^(3/2)/sqrt(c-c/(a*x))],
[exp(arccoth(a*x))*sqrt(c-c/(a*x))/x^5,x,5,-16/315*a^4*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)-2/9*a*c^2*(1+(-1)/(a^2*x^2))^(3/2)/((c-c/(a*x))^(3/2)*x^3)+4/21*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)/((c-c/(a*x))^(3/2)*x^2)+16/105*a^4*c*(1+(-1)/(a^2*x^2))^(3/2)/sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*x^3*sqrt(c-c/(a*x)),x,11,75/64*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^4+75/64*x*sqrt(c-c/(a*x))/a^3+25/32*x^2*sqrt(c-c/(a*x))/a^2+5/8*x^3*sqrt(c-c/(a*x))/a+1/4*x^4*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*x^2*sqrt(c-c/(a*x)),x,10,11/8*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^3+11/8*x*sqrt(c-c/(a*x))/a^2+11/12*x^2*sqrt(c-c/(a*x))/a+1/3*x^3*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*x*sqrt(c-c/(a*x)),x,9,7/4*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^2+7/4*x*sqrt(c-c/(a*x))/a+1/2*x^2*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x)),x,8,3*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a+x*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x))/x,x,8,2*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)+2*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x))/x^2,x,7,-2/3*a*(c-c/(a*x))^(3/2)/c+4*a*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x))/x^3,x,7,-2*a^2*(c-c/(a*x))^(3/2)/c+2/5*a^2*(c-c/(a*x))^(5/2)/c^2+4*a^2*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x))/x^4,x,7,-10/3*a^3*(c-c/(a*x))^(3/2)/c+8/5*a^3*(c-c/(a*x))^(5/2)/c^2-2/7*a^3*(c-c/(a*x))^(7/2)/c^3+4*a^3*sqrt(c-c/(a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a*x))/x^5,x,7,-14/3*a^4*(c-c/(a*x))^(3/2)/c+18/5*a^4*(c-c/(a*x))^(5/2)/c^2-10/7*a^4*(c-c/(a*x))^(7/2)/c^3+2/9*a^4*(c-c/(a*x))^(9/2)/c^4+4*a^4*sqrt(c-c/(a*x))],
[exp(3*arccoth(a*x))*x^3*sqrt(c-c/(a*x)),x,11,363/64*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a^4*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/(a^4*sqrt(1+(-1)/(a*x)))+149/64*x*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/(a^3*sqrt(1+(-1)/(a*x)))+107/96*x^2*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x)))+17/24*x^3*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+1/4*x^4*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*x^2*sqrt(c-c/(a*x)),x,10,45/8*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a^3*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/(a^3*sqrt(1+(-1)/(a*x)))+19/8*x*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x)))+13/12*x^2*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+1/3*x^3*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*x*sqrt(c-c/(a*x)),x,9,23/4*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x)))+9/4*x*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+1/2*x^2*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x)),x,8,5*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))-4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+x*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x))/x,x,8,2*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))+2*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x))/x^2,x,5,2/3*a*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)-4*a*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/(sqrt(2)*sqrt(c-c/(a*x))))*sqrt(2)*sqrt(c)+4*a*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x))/x^3,x,6,2/5*a^2*c^3*(1+(-1)/(a^2*x^2))^(5/2)/(c-c/(a*x))^(5/2)+2/3*a^2*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)-4*a^2*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/(sqrt(2)*sqrt(c-c/(a*x))))*sqrt(2)*sqrt(c)+4*a^2*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x))/x^4,x,7,4/7*a^3*c^3*(1+(-1)/(a^2*x^2))^(5/2)/(c-c/(a*x))^(5/2)+2/3*a^3*c^2*(1+(-1)/(a^2*x^2))^(3/2)/(c-c/(a*x))^(3/2)-2/7*a^3*c^2*(1+(-1)/(a^2*x^2))^(5/2)/(c-c/(a*x))^(3/2)-4*a^3*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/(sqrt(2)*sqrt(c-c/(a*x))))*sqrt(2)*sqrt(c)+4*a^3*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a*x))/x^5,x,8,2/3*a^4*(1+1/(a*x))^(3/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))+2/5*a^4*(1+1/(a*x))^(5/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-2/7*a^4*(1+1/(a*x))^(7/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))+2/9*a^4*(1+1/(a*x))^(9/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-4*a^4*arctanh(sqrt(1+1/(a*x))/sqrt(2))*sqrt(2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))+4*a^4*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],

# n<0
[x^m*sqrt(c-c/(a*x))/exp(arccoth(a*x)),x,4,-1/2*(3+4*m)*x^m*hypergeom([1/2,-m],[1-m],(-1)/(a*x))*sqrt(c-c/(a*x))/(a*m*(1+m)*sqrt(1+(-1)/(a*x)))+x^(1+m)*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/((1+m)*sqrt(1+(-1)/(a*x)))],
[x^2*sqrt(c-c/(a*x))/exp(arccoth(a*x)),x,6,-11/8*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a^3+11/8*c*x*sqrt(1+(-1)/(a^2*x^2))/(a^2*sqrt(c-c/(a*x)))-11/12*c*x^2*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))+1/3*c*x^3*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[x*sqrt(c-c/(a*x))/exp(arccoth(a*x)),x,5,7/4*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a^2-7/4*c*x*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a*x)))+1/2*c*x^2*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/exp(arccoth(a*x)),x,4,-3*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)/a+c*x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(arccoth(a*x))*x),x,4,2*arctanh(sqrt(c)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x)))*sqrt(c)+2*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(arccoth(a*x))*x^2),x,3,-8/3*a*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))-2/3*a*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(arccoth(a*x))*x^3),x,4,2/5*a^2*(c-c/(a*x))^(3/2)*sqrt(1+(-1)/(a^2*x^2))/c+8/5*a^2*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))+2/5*a^2*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(arccoth(a*x))*x^4),x,5,-104/105*a^3*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))+2/7*c*sqrt(1+(-1)/(a^2*x^2))/(x^3*sqrt(c-c/(a*x)))-26/35*a*c*sqrt(1+(-1)/(a^2*x^2))/(x^2*sqrt(c-c/(a*x)))-104/105*a^3*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))],
[x^3*sqrt(c-c/(a*x))/exp(2*arccoth(a*x)),x,14,363/64*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^4-4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^4-149/64*x*sqrt(c-c/(a*x))/a^3+107/96*x^2*sqrt(c-c/(a*x))/a^2-17/24*x^3*sqrt(c-c/(a*x))/a+1/4*x^4*sqrt(c-c/(a*x))],
[x^2*sqrt(c-c/(a*x))/exp(2*arccoth(a*x)),x,13,-45/8*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^3+4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^3+19/8*x*sqrt(c-c/(a*x))/a^2-13/12*x^2*sqrt(c-c/(a*x))/a+1/3*x^3*sqrt(c-c/(a*x))],
[x*sqrt(c-c/(a*x))/exp(2*arccoth(a*x)),x,12,23/4*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a^2-4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a^2-9/4*x*sqrt(c-c/(a*x))/a+1/2*x^2*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/exp(2*arccoth(a*x)),x,11,-5*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)/a+4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)/a+x*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arccoth(a*x))*x),x,11,2*arctanh(sqrt(c-c/(a*x))/sqrt(c))*sqrt(c)-4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+2*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arccoth(a*x))*x^2),x,9,-2/3*a*(c-c/(a*x))^(3/2)/c+4*a*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-4*a*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arccoth(a*x))*x^3),x,10,2/3*a^2*(c-c/(a*x))^(3/2)/c+2/5*a^2*(c-c/(a*x))^(5/2)/c^2-4*a^2*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+4*a^2*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arccoth(a*x))*x^4),x,11,-2/3*a^3*(c-c/(a*x))^(3/2)/c-2/7*a^3*(c-c/(a*x))^(7/2)/c^3+4*a^3*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)-4*a^3*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(2*arccoth(a*x))*x^5),x,11,2/3*a^4*(c-c/(a*x))^(3/2)/c+2/5*a^4*(c-c/(a*x))^(5/2)/c^2-2/7*a^4*(c-c/(a*x))^(7/2)/c^3+2/9*a^4*(c-c/(a*x))^(9/2)/c^4-4*a^4*arctanh(sqrt(c-c/(a*x))/(sqrt(2)*sqrt(c)))*sqrt(2)*sqrt(c)+4*a^4*sqrt(c-c/(a*x))],
[x^3*sqrt(c-c/(a*x))/exp(3*arccoth(a*x)),x,9,1115/64*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a^4*sqrt(1+(-1)/(a*x)))-1115/64*sqrt(c-c/(a*x))/(a^4*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-1115/192*x*sqrt(c-c/(a*x))/(a^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+223/96*x^2*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-25/24*x^3*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+1/4*x^4*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[x^2*sqrt(c-c/(a*x))/exp(3*arccoth(a*x)),x,8,-119/8*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a^3*sqrt(1+(-1)/(a*x)))+119/8*sqrt(c-c/(a*x))/(a^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+119/24*x*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-19/12*x^2*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+1/3*x^3*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[x*sqrt(c-c/(a*x))/exp(3*arccoth(a*x)),x,7,47/4*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x)))-47/4*sqrt(c-c/(a*x))/(a^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-13/4*x*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+1/2*x^2*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[sqrt(c-c/(a*x))/exp(3*arccoth(a*x)),x,6,-7*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x)))+9*sqrt(c-c/(a*x))/(a*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+x*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))],
[sqrt(c-c/(a*x))/(exp(3*arccoth(a*x))*x),x,6,2*arctanh(sqrt(1+1/(a*x)))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-8*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-2*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],
[sqrt(c-c/(a*x))/(exp(3*arccoth(a*x))*x^2),x,4,-16/3*a*(c-c/(a*x))^(3/2)/(c*sqrt(1+(-1)/(a^2*x^2)))-2/3*a*(c-c/(a*x))^(5/2)/(c^2*sqrt(1+(-1)/(a^2*x^2)))+64/3*a*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a*x))/(exp(3*arccoth(a*x))*x^3),x,5,-a^2*(c-c/(a*x))^(7/2)/(c^3*sqrt(1+(-1)/(a^2*x^2)))-7/5*a^2*(c-c/(a*x))^(3/2)*sqrt(1+(-1)/(a^2*x^2))/c-224/15*a^2*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))-56/15*a^2*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(3*arccoth(a*x))*x^4),x,6,a^3*(c-c/(a*x))^(7/2)/(c^3*sqrt(1+(-1)/(a^2*x^2)))+59/35*a^3*(c-c/(a*x))^(3/2)*sqrt(1+(-1)/(a^2*x^2))/c+2/7*a^3*(c-c/(a*x))^(5/2)*sqrt(1+(-1)/(a^2*x^2))/c^2+1888/105*a^3*c*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a*x))+472/105*a^3*sqrt(1+(-1)/(a^2*x^2))*sqrt(c-c/(a*x))],
[sqrt(c-c/(a*x))/(exp(3*arccoth(a*x))*x^5),x,4,50/3*a^4*(1+1/(a*x))^(3/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-38/5*a^4*(1+1/(a*x))^(5/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))+2*a^4*(1+1/(a*x))^(7/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-2/9*a^4*(1+1/(a*x))^(9/2)*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))-8*a^4*sqrt(c-c/(a*x))/(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))-32*a^4*sqrt(1+1/(a*x))*sqrt(c-c/(a*x))/sqrt(1+(-1)/(a*x))],

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

# Integrands of the form E^(n ArcCoth[a x]) (c-c/(a x))^p with p symbolic
[exp(n*arccoth(a*x))*(c-c/(a*x))^p,x,3,-2^(1-1/2*n+p)*(1+1/(a*x))^(1/2*(2+n))*(c-c/(a*x))^p*AppellF1(1/2*(2+n),1/2*(n-2*p),2,1/2*(4+n),1/2*(a+1/x)/a,1+1/(a*x))/(a*(2+n)*(1+(-1)/(a*x))^p)],
[exp(2*p*arccoth(a*x))*(c-c/(a*x))^p,x,3,-(1+1/(a*x))^(1+p)*(c-c/(a*x))^p*hypergeom([2,1+p],[2+p],1+1/(a*x))/(a*(1+p)*(1+(-1)/(a*x))^p)],
[(c-c/(a*x))^p/exp(2*p*arccoth(a*x)),x,3,-4^p*(1+1/(a*x))^(1-p)*(c-c/(a*x))^p*AppellF1(1-p,-2*p,2,2-p,1/2*(a+1/x)/a,1+1/(a*x))/(a*(1-p)*(1+(-1)/(a*x))^p)],
[exp(2*arccoth(a*x))*(c-c/(a*x))^p,x,7,(c-c/(a*x))^p*x+(2-p)*(c-c/(a*x))^p*hypergeom([1,p],[1+p],1+(-1)/(a*x))/(a*p)],
[exp(arccoth(a*x))*(c-c/(a*x))^p,x,3,-1/3*2^(1/2+p)*(1+1/(a*x))^(3/2)*(c-c/(a*x))^p*AppellF1(3/2,1/2-p,2,5/2,1/2*(a+1/x)/a,1+1/(a*x))/(a*(1+(-1)/(a*x))^p)],
[(c-c/(a*x))^p/exp(arccoth(a*x)),x,3,-2^(3/2+p)*(c-c/(a*x))^p*AppellF1(1/2,-1/2-p,2,3/2,1/2*(a+1/x)/a,1+1/(a*x))*sqrt(1+1/(a*x))/(a*(1+(-1)/(a*x))^p)],
[(c-c/(a*x))^p/exp(2*arccoth(a*x)),x,9,(c-c/(a*x))^(2+p)*x/c^2+1/2*(c-c/(a*x))^(2+p)*hypergeom([1,2+p],[3+p],1/2*(a+(-1)/x)/a)/(a*c^2*(2+p))-(c-c/(a*x))^(2+p)*hypergeom([1,2+p],[3+p],1+(-1)/(a*x))/(a*c^2)],

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

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

# n>0
[exp(arccoth(a*x))*(c-a^2*c*x^2)^4,x,13,5/72*a^6*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(11/2)*x^7-7/72*a^7*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(11/2)*x^8+1/9*a^8*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(11/2)*x^9+35/128*c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+35/384*a*c^4*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))+7/192*a^2*c^4*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))+1/64*a^3*c^4*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))+1/144*a^4*c^4*(1+1/(a*x))^(9/2)*x^5*sqrt(1+(-1)/(a*x))-5/144*a^5*c^4*(1+1/(a*x))^(11/2)*x^6*sqrt(1+(-1)/(a*x))+35/128*c^4*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(arccoth(a*x))*(c-a^2*c*x^2)^3,x,11,5/42*a^5*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(9/2)*x^6-1/7*a^6*c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(9/2)*x^7+5/16*c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+5/48*a*c^3*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))+1/24*a^2*c^3*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))+1/56*a^3*c^3*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))-1/14*a^4*c^3*(1+1/(a*x))^(9/2)*x^5*sqrt(1+(-1)/(a*x))+5/16*c^3*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(arccoth(a*x))*(c-a^2*c*x^2)^2,x,9,1/5*a^4*c^2*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(7/2)*x^5+3/8*c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+1/8*a*c^2*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))+1/20*a^2*c^2*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-3/20*a^3*c^2*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))+3/8*c^2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(arccoth(a*x))*(c-a^2*c*x^2),x,7,1/2*c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+1/6*a*c*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-1/3*a^2*c*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))+1/2*c*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(arccoth(a*x))/(c-a^2*c*x^2),x,1,exp(arccoth(a*x))/(a*c)],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^2,x,2,2/3*exp(arccoth(a*x))/(a*c^2)-1/3*exp(arccoth(a*x))*(1-2*a*x)/(a*c^2*(1-a^2*x^2))],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^3,x,3,8/15*exp(arccoth(a*x))/(a*c^3)-1/15*exp(arccoth(a*x))*(1-4*a*x)/(a*c^3*(1-a^2*x^2)^2)-4/15*exp(arccoth(a*x))*(1-2*a*x)/(a*c^3*(1-a^2*x^2))],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^4,x,4,16/35*exp(arccoth(a*x))/(a*c^4)-1/35*exp(arccoth(a*x))*(1-6*a*x)/(a*c^4*(1-a^2*x^2)^3)-2/35*exp(arccoth(a*x))*(1-4*a*x)/(a*c^4*(1-a^2*x^2)^2)-8/35*exp(arccoth(a*x))*(1-2*a*x)/(a*c^4*(1-a^2*x^2))],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^5,x,4,-16/7*c^5*(1+a*x)^7/a+4*c^5*(1+a*x)^8/a-8/3*c^5*(1+a*x)^9/a+4/5*c^5*(1+a*x)^10/a-1/11*c^5*(1+a*x)^11/a],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^4,x,4,-4/3*c^4*(1+a*x)^6/a+12/7*c^4*(1+a*x)^7/a-3/4*c^4*(1+a*x)^8/a+1/9*c^4*(1+a*x)^9/a],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^3,x,4,-4/5*c^3*(1+a*x)^5/a+2/3*c^3*(1+a*x)^6/a-1/7*c^3*(1+a*x)^7/a],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^2,x,4,-1/2*c^2*(1+a*x)^4/a+1/5*c^2*(1+a*x)^5/a],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2),x,3,-1/3*c*(1+a*x)^3/a],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2),x,3,(-1)/(a*c*(1-a*x))],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^2,x,5,(-1/4)/(a*c^2*(1-a*x)^2)+(-1/4)/(a*c^2*(1-a*x))-1/4*arctanh(a*x)/(a*c^2)],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^3,x,5,(-1/12)/(a*c^3*(1-a*x)^3)+(-1/8)/(a*c^3*(1-a*x)^2)+(-3/16)/(a*c^3*(1-a*x))+1/16/(a*c^3*(1+a*x))-1/4*arctanh(a*x)/(a*c^3)],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^4,x,5,(-1/32)/(a*c^4*(1-a*x)^4)+(-1/16)/(a*c^4*(1-a*x)^3)+(-3/32)/(a*c^4*(1-a*x)^2)+(-5/32)/(a*c^4*(1-a*x))+1/64/(a*c^4*(1+a*x)^2)+5/64/(a*c^4*(1+a*x))-15/64*arctanh(a*x)/(a*c^4)],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^4,x,13,-5/72*a^7*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(13/2)*x^8+1/9*a^8*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(13/2)*x^9-55/128*c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-55/384*a*c^4*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-11/192*a^2*c^4*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-11/448*a^3*c^4*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))-11/1008*a^4*c^4*(1+1/(a*x))^(9/2)*x^5*sqrt(1+(-1)/(a*x))-5/1008*a^5*c^4*(1+1/(a*x))^(11/2)*x^6*sqrt(1+(-1)/(a*x))+5/168*a^6*c^4*(1+1/(a*x))^(13/2)*x^7*sqrt(1+(-1)/(a*x))-55/128*c^4*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^3,x,11,-1/7*a^6*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(11/2)*x^7-9/16*c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-3/16*a*c^3*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-3/40*a^2*c^3*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-9/280*a^3*c^3*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))-1/70*a^4*c^3*(1+1/(a*x))^(9/2)*x^5*sqrt(1+(-1)/(a*x))+1/14*a^5*c^3*(1+1/(a*x))^(11/2)*x^6*sqrt(1+(-1)/(a*x))-9/16*c^3*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^2,x,9,-7/8*c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-7/24*a*c^2*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-7/60*a^2*c^2*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-1/20*a^3*c^2*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))+1/5*a^4*c^2*(1+1/(a*x))^(9/2)*x^5*sqrt(1+(-1)/(a*x))-7/8*c^2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2),x,7,-5/2*c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-5/6*a*c*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-1/3*a^2*c*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-5/2*c*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2),x,1,1/3*exp(3*arccoth(a*x))/(a*c)],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2)^2,x,2,-2/15*exp(3*arccoth(a*x))/(a*c^2)+1/5*exp(3*arccoth(a*x))*(3-2*a*x)/(a*c^2*(1-a^2*x^2))],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2)^3,x,3,-8/35*exp(3*arccoth(a*x))/(a*c^3)-1/7*exp(3*arccoth(a*x))*(3-4*a*x)/(a*c^3*(1-a^2*x^2)^2)+12/35*exp(3*arccoth(a*x))*(3-2*a*x)/(a*c^3*(1-a^2*x^2))],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2)^4,x,4,-16/63*exp(3*arccoth(a*x))/(a*c^4)-1/9*exp(3*arccoth(a*x))*(1-2*a*x)/(a*c^4*(1-a^2*x^2)^3)-10/63*exp(3*arccoth(a*x))*(3-4*a*x)/(a*c^4*(1-a^2*x^2)^2)+8/21*exp(3*arccoth(a*x))*(3-2*a*x)/(a*c^4*(1-a^2*x^2))],
[exp(4*arccoth(a*x))*(c-a^2*c*x^2)^5,x,4,c^5*(1+a*x)^8/a-4/3*c^5*(1+a*x)^9/a+3/5*c^5*(1+a*x)^10/a-1/11*c^5*(1+a*x)^11/a],
[exp(4*arccoth(a*x))*(c-a^2*c*x^2)^4,x,4,4/7*c^4*(1+a*x)^7/a-1/2*c^4*(1+a*x)^8/a+1/9*c^4*(1+a*x)^9/a],
[exp(4*arccoth(a*x))*(c-a^2*c*x^2)^3,x,4,1/3*c^3*(1+a*x)^6/a-1/7*c^3*(1+a*x)^7/a],
[exp(4*arccoth(a*x))*(c-a^2*c*x^2)^2,x,3,1/5*c^2*(1+a*x)^5/a],
[exp(4*arccoth(a*x))*(c-a^2*c*x^2),x,4,-4*c*x-c*(1+a*x)^2/a-1/3*c*(1+a*x)^3/a-8*c*log(1-a*x)/a],
[exp(4*arccoth(a*x))/(c-a^2*c*x^2),x,3,x/(c*(1-a*x)^2)],
[exp(4*arccoth(a*x))/(c-a^2*c*x^2)^2,x,3,1/3/(a*c^2*(1-a*x)^3)],
[exp(4*arccoth(a*x))/(c-a^2*c*x^2)^3,x,5,1/8/(a*c^3*(1-a*x)^4)+1/12/(a*c^3*(1-a*x)^3)+1/16/(a*c^3*(1-a*x)^2)+1/16/(a*c^3*(1-a*x))+1/16*arctanh(a*x)/(a*c^3)],
[exp(4*arccoth(a*x))/(c-a^2*c*x^2)^4,x,5,1/20/(a*c^4*(1-a*x)^5)+1/16/(a*c^4*(1-a*x)^4)+1/16/(a*c^4*(1-a*x)^3)+1/16/(a*c^4*(1-a*x)^2)+5/64/(a*c^4*(1-a*x))+(-1/64)/(a*c^4*(1+a*x))+3/32*arctanh(a*x)/(a*c^4)],

# n<0
[(c-a^2*c*x^2)^4/exp(arccoth(a*x)),x,13,-5/48*a^5*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(9/2)*x^6+1/8*a^6*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(9/2)*x^7-1/8*a^7*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(9/2)*x^8+1/9*a^8*c^4*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(9/2)*x^9-35/128*c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-35/384*a*c^4*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-7/192*a^2*c^4*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-1/64*a^3*c^4*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))+1/16*a^4*c^4*(1+1/(a*x))^(9/2)*x^5*sqrt(1+(-1)/(a*x))-35/128*c^4*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[(c-a^2*c*x^2)^3/exp(arccoth(a*x)),x,11,-1/6*a^4*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(7/2)*x^5+1/6*a^5*c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(7/2)*x^6-1/7*a^6*c^3*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(7/2)*x^7-5/16*c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-5/48*a*c^3*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-1/24*a^2*c^3*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))+1/8*a^3*c^3*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))-5/16*c^3*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[(c-a^2*c*x^2)^2/exp(arccoth(a*x)),x,9,-1/4*a^3*c^2*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2)*x^4+1/5*a^4*c^2*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(5/2)*x^5-3/8*c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-1/8*a*c^2*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))+1/4*a^2*c^2*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-3/8*c^2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[(c-a^2*c*x^2)/exp(arccoth(a*x)),x,7,-1/3*a^2*c*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2)*x^3-1/2*c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+1/2*a*c*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-1/2*c*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)),x,1,(-1)/(a*c*exp(arccoth(a*x)))],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)^2),x,2,(-2/3)/(a*c^2*exp(arccoth(a*x)))+1/3*(1+2*a*x)/(a*c^2*exp(arccoth(a*x))*(1-a^2*x^2))],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)^3),x,3,(-8/15)/(a*c^3*exp(arccoth(a*x)))+1/15*(1+4*a*x)/(a*c^3*exp(arccoth(a*x))*(1-a^2*x^2)^2)+4/15*(1+2*a*x)/(a*c^3*exp(arccoth(a*x))*(1-a^2*x^2))],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)^4),x,4,(-16/35)/(a*c^4*exp(arccoth(a*x)))+1/35*(1+6*a*x)/(a*c^4*exp(arccoth(a*x))*(1-a^2*x^2)^3)+2/35*(1+4*a*x)/(a*c^4*exp(arccoth(a*x))*(1-a^2*x^2)^2)+8/35*(1+2*a*x)/(a*c^4*exp(arccoth(a*x))*(1-a^2*x^2))],
[(c-a^2*c*x^2)^4/exp(2*arccoth(a*x)),x,4,4/3*c^4*(1-a*x)^6/a-12/7*c^4*(1-a*x)^7/a+3/4*c^4*(1-a*x)^8/a-1/9*c^4*(1-a*x)^9/a],
[(c-a^2*c*x^2)^3/exp(2*arccoth(a*x)),x,4,4/5*c^3*(1-a*x)^5/a-2/3*c^3*(1-a*x)^6/a+1/7*c^3*(1-a*x)^7/a],
[(c-a^2*c*x^2)^2/exp(2*arccoth(a*x)),x,4,1/2*c^2*(1-a*x)^4/a-1/5*c^2*(1-a*x)^5/a],
[(c-a^2*c*x^2)/exp(2*arccoth(a*x)),x,3,1/3*c*(1-a*x)^3/a],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)),x,3,1/(a*c*(1+a*x))],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^2),x,5,1/4/(a*c^2*(1+a*x)^2)+1/4/(a*c^2*(1+a*x))-1/4*arctanh(a*x)/(a*c^2)],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^3),x,5,(-1/16)/(a*c^3*(1-a*x))+1/12/(a*c^3*(1+a*x)^3)+1/8/(a*c^3*(1+a*x)^2)+3/16/(a*c^3*(1+a*x))-1/4*arctanh(a*x)/(a*c^3)],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^4),x,5,(-1/64)/(a*c^4*(1-a*x)^2)+(-5/64)/(a*c^4*(1-a*x))+1/32/(a*c^4*(1+a*x)^4)+1/16/(a*c^4*(1+a*x)^3)+3/32/(a*c^4*(1+a*x)^2)+5/32/(a*c^4*(1+a*x))-15/64*arctanh(a*x)/(a*c^4)],
[(c-a^2*c*x^2)^4/exp(3*arccoth(a*x)),x,13,11/48*a^4*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(7/2)*x^5-11/48*a^5*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(7/2)*x^6+11/56*a^6*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(7/2)*x^7-11/72*a^7*c^4*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(7/2)*x^8+1/9*a^8*c^4*(1+(-1)/(a*x))^(11/2)*(1+1/(a*x))^(7/2)*x^9+55/128*c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+55/384*a*c^4*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))+11/192*a^2*c^4*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))-11/64*a^3*c^4*(1+1/(a*x))^(7/2)*x^4*sqrt(1+(-1)/(a*x))+55/128*c^4*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[(c-a^2*c*x^2)^3/exp(3*arccoth(a*x)),x,11,3/8*a^3*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2)*x^4-3/10*a^4*c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(5/2)*x^5+3/14*a^5*c^3*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2)*x^6-1/7*a^6*c^3*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(5/2)*x^7+9/16*c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+3/16*a*c^3*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))-3/8*a^2*c^3*(1+1/(a*x))^(5/2)*x^3*sqrt(1+(-1)/(a*x))+9/16*c^3*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[(c-a^2*c*x^2)^2/exp(3*arccoth(a*x)),x,9,7/12*a^2*c^2*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2)*x^3-7/20*a^3*c^2*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2)*x^4+1/5*a^4*c^2*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(3/2)*x^5+7/8*c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-7/8*a*c^2*(1+1/(a*x))^(3/2)*x^2*sqrt(1+(-1)/(a*x))+7/8*c^2*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[(c-a^2*c*x^2)/exp(3*arccoth(a*x)),x,7,5/2*c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+5/6*a*c*(1+(-1)/(a*x))^(3/2)*x^2*sqrt(1+1/(a*x))-1/3*a^2*c*(1+(-1)/(a*x))^(5/2)*x^3*sqrt(1+1/(a*x))-5/2*c*x*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)),x,1,(-1/3)/(a*c*exp(3*arccoth(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)^2),x,2,2/15/(a*c^2*exp(3*arccoth(a*x)))+1/5*(-3-2*a*x)/(a*c^2*exp(3*arccoth(a*x))*(1-a^2*x^2))],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)^3),x,3,8/35/(a*c^3*exp(3*arccoth(a*x)))+1/7*(3+4*a*x)/(a*c^3*exp(3*arccoth(a*x))*(1-a^2*x^2)^2)-12/35*(3+2*a*x)/(a*c^3*exp(3*arccoth(a*x))*(1-a^2*x^2))],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)^4),x,4,16/63/(a*c^4*exp(3*arccoth(a*x)))+1/9*(1+2*a*x)/(a*c^4*exp(3*arccoth(a*x))*(1-a^2*x^2)^3)+10/63*(3+4*a*x)/(a*c^4*exp(3*arccoth(a*x))*(1-a^2*x^2)^2)-8/21*(3+2*a*x)/(a*c^4*exp(3*arccoth(a*x))*(1-a^2*x^2))],

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

# n>0
[exp(arccoth(a*x))*(c-a^2*c*x^2)^(9/2),x,4,8/3*(1+a*x)^6*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)-32/7*(1+a*x)^7*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+3*(1+a*x)^8*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)-8/9*(1+a*x)^9*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+1/10*(1+a*x)^10*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)],
[exp(arccoth(a*x))*(c-a^2*c*x^2)^(7/2),x,4,-8/5*(1+a*x)^5*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)+2*(1+a*x)^6*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)-6/7*(1+a*x)^7*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)+1/8*(1+a*x)^8*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)],
[exp(arccoth(a*x))*(c-a^2*c*x^2)^(5/2),x,4,(1+a*x)^4*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)-4/5*(1+a*x)^5*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)+1/6*(1+a*x)^6*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)],
[exp(arccoth(a*x))*(c-a^2*c*x^2)^(3/2),x,4,-2/3*(1+a*x)^3*(c-a^2*c*x^2)^(3/2)/(a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3)+1/4*(1+a*x)^4*(c-a^2*c*x^2)^(3/2)/(a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3)],
[exp(arccoth(a*x))*sqrt(c-a^2*c*x^2),x,3,sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))/sqrt(c-a^2*c*x^2),x,3,x*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a^2*c*x^2)],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^(3/2),x,5,1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))+1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3*arctanh(a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^(5/2),x,5,-1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))-1/4*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))-3/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^(7/2),x,5,1/24*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)^3*(c-a^2*c*x^2)^(7/2))+3/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)^2*(c-a^2*c*x^2)^(7/2))+3/16*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)*(c-a^2*c*x^2)^(7/2))-1/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)^2*(c-a^2*c*x^2)^(7/2))-1/8*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)*(c-a^2*c*x^2)^(7/2))+5/16*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7*arctanh(a*x)/(c-a^2*c*x^2)^(7/2)],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(9/2),x,10,-77/384*c^3*x*(c-a^2*c*x^2)^(3/2)-77/480*c^2*x*(c-a^2*c*x^2)^(5/2)-11/80*c*x*(c-a^2*c*x^2)^(7/2)+11/90*(c-a^2*c*x^2)^(9/2)/a+1/10*(1+a*x)*(c-a^2*c*x^2)^(9/2)/a-77/256*c^(9/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a-77/256*c^4*x*sqrt(c-a^2*c*x^2)],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(7/2),x,9,-15/64*c^2*x*(c-a^2*c*x^2)^(3/2)-3/16*c*x*(c-a^2*c*x^2)^(5/2)+9/56*(c-a^2*c*x^2)^(7/2)/a+1/8*(1+a*x)*(c-a^2*c*x^2)^(7/2)/a-45/128*c^(7/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a-45/128*c^3*x*sqrt(c-a^2*c*x^2)],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(5/2),x,8,-7/24*c*x*(c-a^2*c*x^2)^(3/2)+7/30*(c-a^2*c*x^2)^(5/2)/a+1/6*(1+a*x)*(c-a^2*c*x^2)^(5/2)/a-7/16*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a-7/16*c^2*x*sqrt(c-a^2*c*x^2)],
[exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(3/2),x,7,5/12*(c-a^2*c*x^2)^(3/2)/a+1/4*(1+a*x)*(c-a^2*c*x^2)^(3/2)/a-5/8*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a-5/8*c*x*sqrt(c-a^2*c*x^2)],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2),x,6,-3/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a+3/2*sqrt(c-a^2*c*x^2)/a+1/2*(1+a*x)*sqrt(c-a^2*c*x^2)/a],
[exp(2*arccoth(a*x))/sqrt(c-a^2*c*x^2),x,5,arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a*sqrt(c))-2*(1+a*x)/(a*sqrt(c-a^2*c*x^2))],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^(3/2),x,4,-2/3*(1+a*x)/(a*(c-a^2*c*x^2)^(3/2))-1/3*x/(c*sqrt(c-a^2*c*x^2))],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^(5/2),x,5,-2/5*(1+a*x)/(a*(c-a^2*c*x^2)^(5/2))-1/5*x/(c*(c-a^2*c*x^2)^(3/2))-2/5*x/(c^2*sqrt(c-a^2*c*x^2))],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^(7/2),x,6,-2/7*(1+a*x)/(a*(c-a^2*c*x^2)^(7/2))-1/7*x/(c*(c-a^2*c*x^2)^(5/2))-4/21*x/(c^2*(c-a^2*c*x^2)^(3/2))-8/21*x/(c^3*sqrt(c-a^2*c*x^2))],
[exp(2*arccoth(a*x))/(c-a^2*c*x^2)^(9/2),x,7,-2/9*(1+a*x)/(a*(c-a^2*c*x^2)^(9/2))-1/9*x/(c*(c-a^2*c*x^2)^(7/2))-2/15*x/(c^2*(c-a^2*c*x^2)^(5/2))-8/45*x/(c^3*(c-a^2*c*x^2)^(3/2))-16/45*x/(c^4*sqrt(c-a^2*c*x^2))],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(9/2),x,4,-8/7*(1+a*x)^7*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+3/2*(1+a*x)^8*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)-2/3*(1+a*x)^9*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+1/10*(1+a*x)^10*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(7/2),x,4,2/3*(1+a*x)^6*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)-4/7*(1+a*x)^7*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)+1/8*(1+a*x)^8*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(5/2),x,4,-2/5*(1+a*x)^5*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)+1/6*(1+a*x)^6*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)],
[exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(3/2),x,3,1/4*(1+a*x)^4*(c-a^2*c*x^2)^(3/2)/(a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3)],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2),x,4,3*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))/sqrt(c-a^2*c*x^2),x,4,2*x*sqrt(1+(-1)/(a^2*x^2))/((1-a*x)*sqrt(c-a^2*c*x^2))+x*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a^2*c*x^2)],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2)^(3/2),x,3,-1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)^2*(c-a^2*c*x^2)^(3/2))],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2)^(5/2),x,5,1/6*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^3*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(3*arccoth(a*x))/(c-a^2*c*x^2)^(7/2),x,5,-1/16*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)^4*(c-a^2*c*x^2)^(7/2))-1/12*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)^3*(c-a^2*c*x^2)^(7/2))-3/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)^2*(c-a^2*c*x^2)^(7/2))-1/8*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)*(c-a^2*c*x^2)^(7/2))+1/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)*(c-a^2*c*x^2)^(7/2))-5/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7*arctanh(a*x)/(c-a^2*c*x^2)^(7/2)],

# n<0
[(c-a^2*c*x^2)^(9/2)/exp(arccoth(a*x)),x,4,8/3*(1-a*x)^6*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)-32/7*(1-a*x)^7*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+3*(1-a*x)^8*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)-8/9*(1-a*x)^9*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+1/10*(1-a*x)^10*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)],
[(c-a^2*c*x^2)^(7/2)/exp(arccoth(a*x)),x,4,-8/5*(1-a*x)^5*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)+2*(1-a*x)^6*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)-6/7*(1-a*x)^7*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)+1/8*(1-a*x)^8*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)],
[(c-a^2*c*x^2)^(5/2)/exp(arccoth(a*x)),x,4,(1-a*x)^4*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)-4/5*(1-a*x)^5*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)+1/6*(1-a*x)^6*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)],
[(c-a^2*c*x^2)^(3/2)/exp(arccoth(a*x)),x,4,-2/3*(1-a*x)^3*(c-a^2*c*x^2)^(3/2)/(a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3)+1/4*(1-a*x)^4*(c-a^2*c*x^2)^(3/2)/(a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3)],
[sqrt(c-a^2*c*x^2)/exp(arccoth(a*x)),x,3,-sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[1/(exp(arccoth(a*x))*sqrt(c-a^2*c*x^2)),x,3,x*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a^2*c*x^2)],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)^(3/2)),x,5,1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1+a*x)*(c-a^2*c*x^2)^(3/2))-1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3*arctanh(a*x)/(c-a^2*c*x^2)^(3/2)],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)^(5/2)),x,5,1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))-1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)^2*(c-a^2*c*x^2)^(5/2))-1/4*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))+3/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[1/(exp(arccoth(a*x))*(c-a^2*c*x^2)^(7/2)),x,5,-1/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)^2*(c-a^2*c*x^2)^(7/2))-1/8*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)*(c-a^2*c*x^2)^(7/2))+1/24*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)^3*(c-a^2*c*x^2)^(7/2))+3/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)^2*(c-a^2*c*x^2)^(7/2))+3/16*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)*(c-a^2*c*x^2)^(7/2))-5/16*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7*arctanh(a*x)/(c-a^2*c*x^2)^(7/2)],
[(c-a^2*c*x^2)^(5/2)/exp(2*arccoth(a*x)),x,8,-7/24*c*x*(c-a^2*c*x^2)^(3/2)-7/30*(c-a^2*c*x^2)^(5/2)/a-1/6*(1-a*x)*(c-a^2*c*x^2)^(5/2)/a-7/16*c^(5/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a-7/16*c^2*x*sqrt(c-a^2*c*x^2)],
[(c-a^2*c*x^2)^(3/2)/exp(2*arccoth(a*x)),x,7,-5/12*(c-a^2*c*x^2)^(3/2)/a-1/4*(1-a*x)*(c-a^2*c*x^2)^(3/2)/a-5/8*c^(3/2)*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/a-5/8*c*x*sqrt(c-a^2*c*x^2)],
[sqrt(c-a^2*c*x^2)/exp(2*arccoth(a*x)),x,6,-3/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a-3/2*sqrt(c-a^2*c*x^2)/a-1/2*(1-a*x)*sqrt(c-a^2*c*x^2)/a],
[1/(exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2)),x,5,arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))/(a*sqrt(c))+2*(1-a*x)/(a*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(3/2)),x,4,2/3*(1-a*x)/(a*(c-a^2*c*x^2)^(3/2))-1/3*x/(c*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(5/2)),x,5,2/5*(1-a*x)/(a*(c-a^2*c*x^2)^(5/2))-1/5*x/(c*(c-a^2*c*x^2)^(3/2))-2/5*x/(c^2*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(7/2)),x,6,2/7*(1-a*x)/(a*(c-a^2*c*x^2)^(7/2))-1/7*x/(c*(c-a^2*c*x^2)^(5/2))-4/21*x/(c^2*(c-a^2*c*x^2)^(3/2))-8/21*x/(c^3*sqrt(c-a^2*c*x^2))],
[1/(exp(2*arccoth(a*x))*(c-a^2*c*x^2)^(9/2)),x,7,2/9*(1-a*x)/(a*(c-a^2*c*x^2)^(9/2))-1/9*x/(c*(c-a^2*c*x^2)^(7/2))-2/15*x/(c^2*(c-a^2*c*x^2)^(5/2))-8/45*x/(c^3*(c-a^2*c*x^2)^(3/2))-16/45*x/(c^4*sqrt(c-a^2*c*x^2))],
[(c-a^2*c*x^2)^(9/2)/exp(3*arccoth(a*x)),x,4,-8/7*(1-a*x)^7*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+3/2*(1-a*x)^8*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)-2/3*(1-a*x)^9*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)+1/10*(1-a*x)^10*(c-a^2*c*x^2)^(9/2)/(a^10*(1+(-1)/(a^2*x^2))^(9/2)*x^9)],
[(c-a^2*c*x^2)^(7/2)/exp(3*arccoth(a*x)),x,4,2/3*(1-a*x)^6*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)-4/7*(1-a*x)^7*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)+1/8*(1-a*x)^8*(c-a^2*c*x^2)^(7/2)/(a^8*(1+(-1)/(a^2*x^2))^(7/2)*x^7)],
[(c-a^2*c*x^2)^(5/2)/exp(3*arccoth(a*x)),x,4,-2/5*(1-a*x)^5*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)+1/6*(1-a*x)^6*(c-a^2*c*x^2)^(5/2)/(a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5)],
[(c-a^2*c*x^2)^(3/2)/exp(3*arccoth(a*x)),x,3,1/4*(1-a*x)^4*(c-a^2*c*x^2)^(3/2)/(a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3)],
[sqrt(c-a^2*c*x^2)/exp(3*arccoth(a*x)),x,4,-3*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))],
[1/(exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2)),x,4,2*x*sqrt(1+(-1)/(a^2*x^2))/((1+a*x)*sqrt(c-a^2*c*x^2))+x*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-a^2*c*x^2)],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(3/2)),x,3,-1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1+a*x)^2*(c-a^2*c*x^2)^(3/2))],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(5/2)),x,5,1/6*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)^3*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)^2*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))-1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[1/(exp(3*arccoth(a*x))*(c-a^2*c*x^2)^(7/2)),x,5,1/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1-a*x)*(c-a^2*c*x^2)^(7/2))-1/16*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)^4*(c-a^2*c*x^2)^(7/2))-1/12*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)^3*(c-a^2*c*x^2)^(7/2))-3/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)^2*(c-a^2*c*x^2)^(7/2))-1/8*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7/((1+a*x)*(c-a^2*c*x^2)^(7/2))+5/32*a^6*(1+(-1)/(a^2*x^2))^(7/2)*x^7*arctanh(a*x)/(c-a^2*c*x^2)^(7/2)],

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

# n>0
[exp(arccoth(a*x))*x^2*sqrt(c-a^2*c*x^2),x,4,1/3*x^2*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/4*x^3*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*x*sqrt(c-a^2*c*x^2),x,4,1/2*x*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^2*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*sqrt(c-a^2*c*x^2),x,3,sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*sqrt(c-a^2*c*x^2)/x,x,4,sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+log(x)*sqrt(c-a^2*c*x^2)/(a*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^2,x,4,-sqrt(c-a^2*c*x^2)/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))+log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(2*arccoth(a*x))*x^3*sqrt(c-a^2*c*x^2),x,8,-3/4*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^4+3/5*x^2*sqrt(c-a^2*c*x^2)/a^2+1/2*x^3*sqrt(c-a^2*c*x^2)/a+1/5*x^4*sqrt(c-a^2*c*x^2)+3/20*(8+5*a*x)*sqrt(c-a^2*c*x^2)/a^4],
[exp(2*arccoth(a*x))*x^2*sqrt(c-a^2*c*x^2),x,7,-7/8*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^3+2/3*x^2*sqrt(c-a^2*c*x^2)/a+1/4*x^3*sqrt(c-a^2*c*x^2)+1/24*(32+21*a*x)*sqrt(c-a^2*c*x^2)/a^3],
[exp(2*arccoth(a*x))*x*sqrt(c-a^2*c*x^2),x,6,-arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^2+1/3*x^2*sqrt(c-a^2*c*x^2)+1/3*(5+3*a*x)*sqrt(c-a^2*c*x^2)/a^2],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2),x,6,-3/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a+3/2*sqrt(c-a^2*c*x^2)/a+1/2*(1+a*x)*sqrt(c-a^2*c*x^2)/a],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x,x,9,-2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)+arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+sqrt(c-a^2*c*x^2)],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^2,x,9,-a*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)+2*a*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+sqrt(c-a^2*c*x^2)/x],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^3,x,7,3/2*a^2*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+1/2*sqrt(c-a^2*c*x^2)/x^2+2*a*sqrt(c-a^2*c*x^2)/x],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^4,x,8,a^3*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+1/3*sqrt(c-a^2*c*x^2)/x^3+a*sqrt(c-a^2*c*x^2)/x^2+5/3*a^2*sqrt(c-a^2*c*x^2)/x],
[exp(2*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^5,x,9,7/8*a^4*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+1/4*sqrt(c-a^2*c*x^2)/x^4+2/3*a*sqrt(c-a^2*c*x^2)/x^3+7/8*a^2*sqrt(c-a^2*c*x^2)/x^2+4/3*a^3*sqrt(c-a^2*c*x^2)/x],
[exp(3*arccoth(a*x))*x^3*sqrt(c-a^2*c*x^2),x,4,4*sqrt(c-a^2*c*x^2)/(a^4*sqrt(1+(-1)/(a^2*x^2)))+2*x*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1+(-1)/(a^2*x^2)))+4/3*x^2*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+3/4*x^3*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/5*x^4*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^5*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*x^2*sqrt(c-a^2*c*x^2),x,4,4*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1+(-1)/(a^2*x^2)))+2*x*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+x^2*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/4*x^3*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^4*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*x*sqrt(c-a^2*c*x^2),x,4,4*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1+(-1)/(a^2*x^2)))+3/2*x*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^2*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^3*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2),x,4,3*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x,x,4,sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))-log(x)*sqrt(c-a^2*c*x^2)/(a*x*sqrt(1+(-1)/(a^2*x^2)))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(a*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^2,x,4,sqrt(c-a^2*c*x^2)/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))-3*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))+4*log(1-a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^3,x,4,1/2*sqrt(c-a^2*c*x^2)/(a*x^3*sqrt(1+(-1)/(a^2*x^2)))+3*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1+(-1)/(a^2*x^2)))-4*a*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))+4*a*log(1-a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^4,x,4,1/3*sqrt(c-a^2*c*x^2)/(a*x^4*sqrt(1+(-1)/(a^2*x^2)))+3/2*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1+(-1)/(a^2*x^2)))+4*a*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1+(-1)/(a^2*x^2)))-4*a^2*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))+4*a^2*log(1-a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-a^2*c*x^2)/x^5,x,4,1/4*sqrt(c-a^2*c*x^2)/(a*x^5*sqrt(1+(-1)/(a^2*x^2)))+sqrt(c-a^2*c*x^2)/(x^4*sqrt(1+(-1)/(a^2*x^2)))+2*a*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1+(-1)/(a^2*x^2)))+4*a^2*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1+(-1)/(a^2*x^2)))-4*a^3*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))+4*a^3*log(1-a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*x^4/(c-a^2*c*x^2)^(3/2),x,4,(1+(-1)/(a^2*x^2))^(3/2)*x^4/(a*(c-a^2*c*x^2)^(3/2))+1/2*(1+(-1)/(a^2*x^2))^(3/2)*x^5/(c-a^2*c*x^2)^(3/2)+1/2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/(a^2*(1-a*x)*(c-a^2*c*x^2)^(3/2))+7/4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1-a*x)/(a^2*(c-a^2*c*x^2)^(3/2))+1/4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1+a*x)/(a^2*(c-a^2*c*x^2)^(3/2))],
[exp(arccoth(a*x))*x^3/(c-a^2*c*x^2)^(3/2),x,4,(1+(-1)/(a^2*x^2))^(3/2)*x^4/(c-a^2*c*x^2)^(3/2)+1/2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/(a*(1-a*x)*(c-a^2*c*x^2)^(3/2))+5/4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1-a*x)/(a*(c-a^2*c*x^2)^(3/2))-1/4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1+a*x)/(a*(c-a^2*c*x^2)^(3/2))],
[exp(arccoth(a*x))*x^2/(c-a^2*c*x^2)^(3/2),x,4,1/2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))+3/4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1-a*x)/(c-a^2*c*x^2)^(3/2)+1/4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1+a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))*x/(c-a^2*c*x^2)^(3/2),x,5,1/2*a*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))-1/2*a*(1+(-1)/(a^2*x^2))^(3/2)*x^3*arctanh(a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^(3/2),x,5,1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))+1/2*a^2*(1+(-1)/(a^2*x^2))^(3/2)*x^3*arctanh(a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))/(x*(c-a^2*c*x^2)^(3/2)),x,4,1/2*a^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))+a^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(x)/(c-a^2*c*x^2)^(3/2)-3/4*a^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1-a*x)/(c-a^2*c*x^2)^(3/2)-1/4*a^3*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1+a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))/(x^2*(c-a^2*c*x^2)^(3/2)),x,4,-a^3*(1+(-1)/(a^2*x^2))^(3/2)*x^2/(c-a^2*c*x^2)^(3/2)+1/2*a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))+a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(x)/(c-a^2*c*x^2)^(3/2)-5/4*a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1-a*x)/(c-a^2*c*x^2)^(3/2)+1/4*a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1+a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))/(x^3*(c-a^2*c*x^2)^(3/2)),x,4,-1/2*a^3*(1+(-1)/(a^2*x^2))^(3/2)*x/(c-a^2*c*x^2)^(3/2)-a^4*(1+(-1)/(a^2*x^2))^(3/2)*x^2/(c-a^2*c*x^2)^(3/2)+1/2*a^5*(1+(-1)/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(c-a^2*c*x^2)^(3/2))+2*a^5*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(x)/(c-a^2*c*x^2)^(3/2)-7/4*a^5*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1-a*x)/(c-a^2*c*x^2)^(3/2)-1/4*a^5*(1+(-1)/(a^2*x^2))^(3/2)*x^3*log(1+a*x)/(c-a^2*c*x^2)^(3/2)],
[exp(arccoth(a*x))*x^5/(c-a^2*c*x^2)^(5/2),x,4,(1+(-1)/(a^2*x^2))^(5/2)*x^6/(c-a^2*c*x^2)^(5/2)-1/8*(1+(-1)/(a^2*x^2))^(5/2)*x^5/(a*(1-a*x)^2*(c-a^2*c*x^2)^(5/2))+(1+(-1)/(a^2*x^2))^(5/2)*x^5/(a*(1-a*x)*(c-a^2*c*x^2)^(5/2))-1/8*(1+(-1)/(a^2*x^2))^(5/2)*x^5/(a*(1+a*x)*(c-a^2*c*x^2)^(5/2))+23/16*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1-a*x)/(a*(c-a^2*c*x^2)^(5/2))-7/16*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1+a*x)/(a*(c-a^2*c*x^2)^(5/2))],
[exp(arccoth(a*x))*x^4/(c-a^2*c*x^2)^(5/2),x,4,-1/8*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))+3/4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))+1/8*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))+11/16*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1-a*x)/(c-a^2*c*x^2)^(5/2)+5/16*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1+a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))*x^3/(c-a^2*c*x^2)^(5/2),x,5,-1/8*a*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))+1/2*a*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))-1/8*a*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))-3/8*a*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))*x^2/(c-a^2*c*x^2)^(5/2),x,5,-1/8*a^2*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))+1/4*a^2*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^2*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^2*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))*x/(c-a^2*c*x^2)^(5/2),x,5,-1/8*a^3*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))-1/8*a^3*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^3*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))/(c-a^2*c*x^2)^(5/2),x,5,-1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))-1/4*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))-3/8*a^4*(1+(-1)/(a^2*x^2))^(5/2)*x^5*arctanh(a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))/(x*(c-a^2*c*x^2)^(5/2)),x,4,-1/8*a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))-1/2*a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))-1/8*a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))-a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(x)/(c-a^2*c*x^2)^(5/2)+11/16*a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1-a*x)/(c-a^2*c*x^2)^(5/2)+5/16*a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1+a*x)/(c-a^2*c*x^2)^(5/2)],
[exp(arccoth(a*x))/(x^2*(c-a^2*c*x^2)^(5/2)),x,4,a^5*(1+(-1)/(a^2*x^2))^(5/2)*x^4/(c-a^2*c*x^2)^(5/2)-1/8*a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(c-a^2*c*x^2)^(5/2))-3/4*a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1-a*x)*(c-a^2*c*x^2)^(5/2))+1/8*a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5/((1+a*x)*(c-a^2*c*x^2)^(5/2))-a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(x)/(c-a^2*c*x^2)^(5/2)+23/16*a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1-a*x)/(c-a^2*c*x^2)^(5/2)-7/16*a^6*(1+(-1)/(a^2*x^2))^(5/2)*x^5*log(1+a*x)/(c-a^2*c*x^2)^(5/2)],

# n<0
[x^2*sqrt(c-a^2*c*x^2)/exp(arccoth(a*x)),x,4,-1/3*x^2*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/4*x^3*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[x*sqrt(c-a^2*c*x^2)/exp(arccoth(a*x)),x,4,-1/2*x*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^2*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-a^2*c*x^2)/exp(arccoth(a*x)),x,3,-sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-a^2*c*x^2)/(exp(arccoth(a*x))*x),x,4,sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))-log(x)*sqrt(c-a^2*c*x^2)/(a*x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/(exp(arccoth(a*x))*x^2),x,4,sqrt(c-a^2*c*x^2)/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))+log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[x^3*sqrt(c-a^2*c*x^2)/exp(2*arccoth(a*x)),x,8,3/4*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^4+3/5*x^2*sqrt(c-a^2*c*x^2)/a^2-1/2*x^3*sqrt(c-a^2*c*x^2)/a+1/5*x^4*sqrt(c-a^2*c*x^2)+3/20*(8-5*a*x)*sqrt(c-a^2*c*x^2)/a^4],
[x^2*sqrt(c-a^2*c*x^2)/exp(2*arccoth(a*x)),x,7,-7/8*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^3-2/3*x^2*sqrt(c-a^2*c*x^2)/a+1/4*x^3*sqrt(c-a^2*c*x^2)-1/24*(32-21*a*x)*sqrt(c-a^2*c*x^2)/a^3],
[x*sqrt(c-a^2*c*x^2)/exp(2*arccoth(a*x)),x,6,arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a^2+1/3*x^2*sqrt(c-a^2*c*x^2)+1/3*(5-3*a*x)*sqrt(c-a^2*c*x^2)/a^2],
[sqrt(c-a^2*c*x^2)/exp(2*arccoth(a*x)),x,6,-3/2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)/a-3/2*sqrt(c-a^2*c*x^2)/a-1/2*(1-a*x)*sqrt(c-a^2*c*x^2)/a],
[sqrt(c-a^2*c*x^2)/(exp(2*arccoth(a*x))*x),x,9,2*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)+arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+sqrt(c-a^2*c*x^2)],
[sqrt(c-a^2*c*x^2)/(exp(2*arccoth(a*x))*x^2),x,9,-a*arctan(a*x*sqrt(c)/sqrt(c-a^2*c*x^2))*sqrt(c)-2*a*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+sqrt(c-a^2*c*x^2)/x],
[sqrt(c-a^2*c*x^2)/(exp(2*arccoth(a*x))*x^3),x,7,3/2*a^2*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+1/2*sqrt(c-a^2*c*x^2)/x^2-2*a*sqrt(c-a^2*c*x^2)/x],
[sqrt(c-a^2*c*x^2)/(exp(2*arccoth(a*x))*x^4),x,8,-a^3*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+1/3*sqrt(c-a^2*c*x^2)/x^3-a*sqrt(c-a^2*c*x^2)/x^2+5/3*a^2*sqrt(c-a^2*c*x^2)/x],
[sqrt(c-a^2*c*x^2)/(exp(2*arccoth(a*x))*x^5),x,9,7/8*a^4*arctanh(sqrt(c-a^2*c*x^2)/sqrt(c))*sqrt(c)+1/4*sqrt(c-a^2*c*x^2)/x^4-2/3*a*sqrt(c-a^2*c*x^2)/x^3+7/8*a^2*sqrt(c-a^2*c*x^2)/x^2-4/3*a^3*sqrt(c-a^2*c*x^2)/x],
[x^3*sqrt(c-a^2*c*x^2)/exp(3*arccoth(a*x)),x,4,4*sqrt(c-a^2*c*x^2)/(a^4*sqrt(1+(-1)/(a^2*x^2)))-2*x*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1+(-1)/(a^2*x^2)))+4/3*x^2*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1+(-1)/(a^2*x^2)))-3/4*x^3*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/5*x^4*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))-4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^5*x*sqrt(1+(-1)/(a^2*x^2)))],
[x^2*sqrt(c-a^2*c*x^2)/exp(3*arccoth(a*x)),x,4,-4*sqrt(c-a^2*c*x^2)/(a^3*sqrt(1+(-1)/(a^2*x^2)))+2*x*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1+(-1)/(a^2*x^2)))-x^2*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/4*x^3*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^4*x*sqrt(1+(-1)/(a^2*x^2)))],
[x*sqrt(c-a^2*c*x^2)/exp(3*arccoth(a*x)),x,4,4*sqrt(c-a^2*c*x^2)/(a^2*sqrt(1+(-1)/(a^2*x^2)))-3/2*x*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^2*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))-4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^3*x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/exp(3*arccoth(a*x)),x,4,-3*sqrt(c-a^2*c*x^2)/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x*sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/(exp(3*arccoth(a*x))*x),x,4,sqrt(c-a^2*c*x^2)/sqrt(1+(-1)/(a^2*x^2))+log(x)*sqrt(c-a^2*c*x^2)/(a*x*sqrt(1+(-1)/(a^2*x^2)))-4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(a*x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/(exp(3*arccoth(a*x))*x^2),x,4,-sqrt(c-a^2*c*x^2)/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))-3*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))+4*log(1+a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/(exp(3*arccoth(a*x))*x^3),x,4,-1/2*sqrt(c-a^2*c*x^2)/(a*x^3*sqrt(1+(-1)/(a^2*x^2)))+3*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1+(-1)/(a^2*x^2)))+4*a*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a*log(1+a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/(exp(3*arccoth(a*x))*x^4),x,4,-1/3*sqrt(c-a^2*c*x^2)/(a*x^4*sqrt(1+(-1)/(a^2*x^2)))+3/2*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1+(-1)/(a^2*x^2)))-4*a*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1+(-1)/(a^2*x^2)))-4*a^2*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))+4*a^2*log(1+a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-a^2*c*x^2)/(exp(3*arccoth(a*x))*x^5),x,4,-1/4*sqrt(c-a^2*c*x^2)/(a*x^5*sqrt(1+(-1)/(a^2*x^2)))+sqrt(c-a^2*c*x^2)/(x^4*sqrt(1+(-1)/(a^2*x^2)))-2*a*sqrt(c-a^2*c*x^2)/(x^3*sqrt(1+(-1)/(a^2*x^2)))+4*a^2*sqrt(c-a^2*c*x^2)/(x^2*sqrt(1+(-1)/(a^2*x^2)))+4*a^3*log(x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a^3*log(1+a*x)*sqrt(c-a^2*c*x^2)/(x*sqrt(1+(-1)/(a^2*x^2)))],

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

# Integrands of the form x^m E^(n ArcCoth[a x]) (c-c a^2 x^2)^p with n symbolic
[exp(n*arccoth(a*x))*(c-a^2*c*x^2)^3,x,3,-256*c^3*(1+(-1)/(a*x))^(4-1/2*n)*(1+1/(a*x))^(1/2*(-8+n))*hypergeom([8,4-1/2*n],[5-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(8-n))],
[exp(n*arccoth(a*x))*(c-a^2*c*x^2)^2,x,3,64*c^2*(1+(-1)/(a*x))^(3-1/2*n)*(1+1/(a*x))^(1/2*(-6+n))*hypergeom([6,3-1/2*n],[4-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(6-n))],
[exp(n*arccoth(a*x))*(c-a^2*c*x^2),x,3,-16*c*(1+(-1)/(a*x))^(2-1/2*n)*(1+1/(a*x))^(1/2*(-4+n))*hypergeom([4,2-1/2*n],[3-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(4-n))],
[exp(n*arccoth(a*x)),x,2,4*(1+(-1)/(a*x))^(1-1/2*n)*(1+1/(a*x))^(1/2*(-2+n))*hypergeom([2,1-1/2*n],[2-1/2*n],(a+(-1)/x)/(a+1/x))/(a*(2-n))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2),x,1,exp(n*arccoth(a*x))/(a*c*n)],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^2,x,2,2*exp(n*arccoth(a*x))/(a*c^2*n*(4-n^2))-exp(n*arccoth(a*x))*(n-2*a*x)/(a*c^2*(4-n^2)*(1-a^2*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^3,x,3,24*exp(n*arccoth(a*x))/(a*c^3*n*(64-20*n^2+n^4))-exp(n*arccoth(a*x))*(n-4*a*x)/(a*c^3*(16-n^2)*(1-a^2*x^2)^2)-12*exp(n*arccoth(a*x))*(n-2*a*x)/(a*c^3*(64-20*n^2+n^4)*(1-a^2*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^4,x,4,720*exp(n*arccoth(a*x))/(a*c^4*n*(36-n^2)*(64-20*n^2+n^4))-exp(n*arccoth(a*x))*(n-6*a*x)/(a*c^4*(36-n^2)*(1-a^2*x^2)^3)-30*exp(n*arccoth(a*x))*(n-4*a*x)/(a*c^4*(576-52*n^2+n^4)*(1-a^2*x^2)^2)-360*exp(n*arccoth(a*x))*(n-2*a*x)/(a*c^4*(36-n^2)*(64-20*n^2+n^4)*(1-a^2*x^2))],
[exp(n*arccoth(a*x))*(c-a^2*c*x^2)^(3/2),x,3,32*(1+(-1)/(a*x))^(1/2*(5-n))*(1+1/(a*x))^(1/2*(-5+n))*(c-a^2*c*x^2)^(3/2)*hypergeom([5,1/2*(5-n)],[1/2*(7-n)],(a+(-1)/x)/(a+1/x))/(a^4*(5-n)*(1+(-1)/(a^2*x^2))^(3/2)*x^3)],
[exp(n*arccoth(a*x))*(c-a^2*c*x^2)^(1/2),x,3,8*(1+(-1)/(a*x))^(1/2*(3-n))*(1+1/(a*x))^(1/2*(-3+n))*hypergeom([3,1/2*(3-n)],[1/2*(5-n)],(a+(-1)/x)/(a+1/x))*sqrt(c-a^2*c*x^2)/(a^2*(3-n)*x*sqrt(1+(-1)/(a^2*x^2)))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(1/2),x,3,2*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x*hypergeom([1,1/2*(1-n)],[1/2*(3-n)],(a+(-1)/x)/(a+1/x))*sqrt(1+(-1)/(a^2*x^2))/((1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(3/2),x,1,-exp(n*arccoth(a*x))*(n-a*x)/(a*c*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(5/2),x,2,-exp(n*arccoth(a*x))*(n-3*a*x)/(a*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))-6*exp(n*arccoth(a*x))*(n-a*x)/(a*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(7/2),x,3,-exp(n*arccoth(a*x))*(n-5*a*x)/(a*c*(25-n^2)*(c-a^2*c*x^2)^(5/2))-20*exp(n*arccoth(a*x))*(n-3*a*x)/(a*c^2*(225-34*n^2+n^4)*(c-a^2*c*x^2)^(3/2))-120*exp(n*arccoth(a*x))*(n-a*x)/(a*c^3*(25-n^2)*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(9/2),x,4,-exp(n*arccoth(a*x))*(n-7*a*x)/(a*c*(49-n^2)*(c-a^2*c*x^2)^(7/2))-42*exp(n*arccoth(a*x))*(n-5*a*x)/(a*c^2*(1225-74*n^2+n^4)*(c-a^2*c*x^2)^(5/2))-840*exp(n*arccoth(a*x))*(n-3*a*x)/(a*c^3*(49-n^2)*(225-34*n^2+n^4)*(c-a^2*c*x^2)^(3/2))-5040*exp(n*arccoth(a*x))*(n-a*x)/(a*c^4*(1225-74*n^2+n^4)*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))*x^3/(c-a^2*c*x^2)^(3/2),x,7,-(2+n)*(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^3/(a*(1+n)*(c-a^2*c*x^2)^(3/2))+(2+2*n+n^2)*(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^3/(a*(1-n^2)*(c-a^2*c*x^2)^(3/2))+(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^4/(c-a^2*c*x^2)^(3/2)-2*n*(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^3*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(a+1/x)/(a+(-1)/x))/(a*(1-n)*(c-a^2*c*x^2)^(3/2))],
[exp(n*arccoth(a*x))*x^2/(c-a^2*c*x^2)^(3/2),x,4,-exp(n*arccoth(a*x))*(n-a*x)/(a^3*c*(1-n^2)*sqrt(c-a^2*c*x^2))-2*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x*hypergeom([1,1/2*(1-n)],[1/2*(3-n)],(a+(-1)/x)/(a+1/x))*sqrt(1+(-1)/(a^2*x^2))/(a^2*c*(1-n)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))*x/(c-a^2*c*x^2)^(3/2),x,1,exp(n*arccoth(a*x))*(1-a*n*x)/(a^2*c*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(3/2),x,1,-exp(n*arccoth(a*x))*(n-a*x)/(a*c*(1-n^2)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(x*(c-a^2*c*x^2)^(3/2)),x,5,-a^3*(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^3/((1+n)*(c-a^2*c*x^2)^(3/2))+a^3*(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^3/((1-n^2)*(c-a^2*c*x^2)^(3/2))-2^(1/2*(1+n))*a^3*(1+(-1)/(a^2*x^2))^(3/2)*(1+(-1)/(a*x))^(1/2*(1-n))*x^3*hypergeom([1/2*(1-n),1/2*(1-n)],[1/2*(3-n)],1/2*(a+(-1)/x)/a)/((1-n)*(c-a^2*c*x^2)^(3/2))],
[exp(n*arccoth(a*x))*x^4/(c-a^2*c*x^2)^(5/2),x,8,-(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+n)*(c-a^2*c*x^2)^(5/2))-(6+n)*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+4*n+n^2)*(c-a^2*c*x^2)^(5/2))+(15+6*n+n^2)*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+n-3*n^2-n^3)*(c-a^2*c*x^2)^(5/2))-(18+7*n-2*n^2-n^3)*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(3-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((9-10*n^2+n^4)*(c-a^2*c*x^2)^(5/2))-2*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^5*hypergeom([1,1/2*(-1+n)],[1/2*(1+n)],(a+1/x)/(a+(-1)/x))/((1-n)*(c-a^2*c*x^2)^(5/2))],
[exp(n*arccoth(a*x))*x^3/(c-a^2*c*x^2)^(5/2),x,6,-a*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+n)*(c-a^2*c*x^2)^(5/2))-3*a*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+4*n+n^2)*(c-a^2*c*x^2)^(5/2))+6*a*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+n-3*n^2-n^3)*(c-a^2*c*x^2)^(5/2))-6*a*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(3-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((9-10*n^2+n^4)*(c-a^2*c*x^2)^(5/2))],
[exp(n*arccoth(a*x))*x^2/(c-a^2*c*x^2)^(5/2),x,2,-exp(n*arccoth(a*x))*(n-3*a*x)/(a^3*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))+exp(n*arccoth(a*x))*(3-n^2)*(n-a*x)/(a^3*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))*x/(c-a^2*c*x^2)^(5/2),x,2,exp(n*arccoth(a*x))*(3-a*n*x)/(a^2*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))+2*exp(n*arccoth(a*x))*n*(n-a*x)/(a^2*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(c-a^2*c*x^2)^(5/2),x,2,-exp(n*arccoth(a*x))*(n-3*a*x)/(a*c*(9-n^2)*(c-a^2*c*x^2)^(3/2))-6*exp(n*arccoth(a*x))*(n-a*x)/(a*c^2*(9-10*n^2+n^4)*sqrt(c-a^2*c*x^2))],
[exp(n*arccoth(a*x))/(x*(c-a^2*c*x^2)^(5/2)),x,15,-a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+n)*(c-a^2*c*x^2)^(5/2))-3*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+4*n+n^2)*(c-a^2*c*x^2)^(5/2))+6*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((3+n-3*n^2-n^3)*(c-a^2*c*x^2)^(5/2))-6*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(3-n))*(1+1/(a*x))^(1/2*(-3+n))*x^5/((9-10*n^2+n^4)*(c-a^2*c*x^2)^(5/2))+4*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*(1+1/(a*x))^(1/2*(-1+n))*x^5/((3+n)*(c-a^2*c*x^2)^(5/2))+8*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^5/((3+4*n+n^2)*(c-a^2*c*x^2)^(5/2))-8*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(1-n))*(1+1/(a*x))^(1/2*(-1+n))*x^5/((3+n-3*n^2-n^3)*(c-a^2*c*x^2)^(5/2))-6*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*(1+1/(a*x))^(1/2*(1+n))*x^5/((3+n)*(c-a^2*c*x^2)^(5/2))-6*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-1-n))*(1+1/(a*x))^(1/2*(1+n))*x^5/((3+4*n+n^2)*(c-a^2*c*x^2)^(5/2))+4*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*(1+1/(a*x))^(1/2*(3+n))*x^5/((3+n)*(c-a^2*c*x^2)^(5/2))-2^(1/2*(5+n))*a^5*(1+(-1)/(a^2*x^2))^(5/2)*(1+(-1)/(a*x))^(1/2*(-3-n))*x^5*hypergeom([1/2*(-3-n),1/2*(-3-n)],[1/2*(-1-n)],1/2*(a+(-1)/x)/a)/((3+n)*(c-a^2*c*x^2)^(5/2))],

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

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

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

# n>0
[exp(arccoth(a*x))*(c-c/(a^2*x^2))^4,x,14,47/42*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(9/2)/a+8/7*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(9/2)/a+c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(9/2)*x+35/16*c^4*arccsc(a*x)/a+c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-67/48*c^4*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-91/120*c^4*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a-131/280*c^4*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x))/a+61/70*c^4*(1+1/(a*x))^(9/2)*sqrt(1+(-1)/(a*x))/a-51/16*c^4*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(arccoth(a*x))*(c-c/(a^2*x^2))^3,x,12,6/5*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(7/2)/a+c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(7/2)*x+15/8*c^3*arccsc(a*x)/a+c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-31/24*c^3*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-43/60*c^3*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a+23/20*c^3*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x))/a-23/8*c^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(arccoth(a*x))*(c-c/(a^2*x^2))^2,x,10,c^2*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2)*x+3/2*c^2*arccsc(a*x)/a+c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-7/6*c^2*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a+4/3*c^2*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a-5/2*c^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(arccoth(a*x))*(c-c/(a^2*x^2)),x,9,c*arccsc(a*x)/a+c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+c*(1+1/(a*x))^(3/2)*x*sqrt(1+(-1)/(a*x))-2*c*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(arccoth(a*x))/(c-c/(a^2*x^2)),x,6,arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c)-2*sqrt(1+1/(a*x))/(a*c*sqrt(1+(-1)/(a*x)))+x*sqrt(1+1/(a*x))/(c*sqrt(1+(-1)/(a*x)))],
[exp(arccoth(a*x))/(c-c/(a^2*x^2))^2,x,8,arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^2)+(-4/3)/(a*c^2*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))+x/(c^2*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))+(-11/3)/(a*c^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+8/3*sqrt(1+(-1)/(a*x))/(a*c^2*sqrt(1+1/(a*x)))],
[exp(arccoth(a*x))/(c-c/(a^2*x^2))^3,x,10,(-6/5)/(a*c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2))+(-29/15)/(a*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2))+x/(c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2))+arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^3)+(-34/5)/(a*c^3*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x)))+21/5*sqrt(1+(-1)/(a*x))/(a*c^3*(1+1/(a*x))^(3/2))+16/5*sqrt(1+(-1)/(a*x))/(a*c^3*sqrt(1+1/(a*x)))],
[exp(arccoth(a*x))/(c-c/(a^2*x^2))^4,x,12,(-8/7)/(a*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2))+(-11/7)/(a*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(5/2))+(-62/21)/(a*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2))+x/(c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2))+arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^4)+(-269/21)/(a*c^4*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x)))+262/35*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(5/2))+163/35*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(3/2))+128/35*sqrt(1+(-1)/(a*x))/(a*c^4*sqrt(1+1/(a*x)))],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^5,x,5,-1/9*c^5/(a^10*x^9)-1/4*c^5/(a^9*x^8)+3/7*c^5/(a^8*x^7)+4/3*c^5/(a^7*x^6)-2/5*c^5/(a^6*x^5)-3*c^5/(a^5*x^4)-2/3*c^5/(a^4*x^3)+4*c^5/(a^3*x^2)+3*c^5/(a^2*x)+c^5*x+2*c^5*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^4,x,5,1/7*c^4/(a^8*x^7)+1/3*c^4/(a^7*x^6)-2/5*c^4/(a^6*x^5)-3/2*c^4/(a^5*x^4)+3*c^4/(a^3*x^2)+2*c^4/(a^2*x)+c^4*x+2*c^4*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^3,x,5,-1/5*c^3/(a^6*x^5)-1/2*c^3/(a^5*x^4)+1/3*c^3/(a^4*x^3)+2*c^3/(a^3*x^2)+c^3/(a^2*x)+c^3*x+2*c^3*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^2,x,5,1/3*c^2/(a^4*x^3)+c^2/(a^3*x^2)+c^2*x+2*c^2*log(x)/a],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2)),x,5,-c/(a^2*x)+c*x+2*c*log(x)/a],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2)),x,5,x/c+1/(a*c*(1-a*x))+2*log(1-a*x)/(a*c)],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2))^2,x,5,x/c^2+(-1/4)/(a*c^2*(1-a*x)^2)+7/4/(a*c^2*(1-a*x))+17/8*log(1-a*x)/(a*c^2)-1/8*log(1+a*x)/(a*c^2)],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2))^3,x,5,x/c^3+1/12/(a*c^3*(1-a*x)^3)+(-5/8)/(a*c^3*(1-a*x)^2)+39/16/(a*c^3*(1-a*x))+(-1/16)/(a*c^3*(1+a*x))+9/4*log(1-a*x)/(a*c^3)-1/4*log(1+a*x)/(a*c^3)],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2))^4,x,5,x/c^4+(-1/32)/(a*c^4*(1-a*x)^4)+13/48/(a*c^4*(1-a*x)^3)+(-35/32)/(a*c^4*(1-a*x)^2)+99/32/(a*c^4*(1-a*x))+1/64/(a*c^4*(1+a*x)^2)+(-11/64)/(a*c^4*(1+a*x))+303/128*log(1-a*x)/(a*c^4)-47/128*log(1+a*x)/(a*c^4)],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^4,x,14,8/7*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(11/2)/a+c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(11/2)*x+15/16*c^4*arccsc(a*x)/a+3*c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-37/16*c^4*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-61/40*c^4*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a-303/280*c^4*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x))/a-57/70*c^4*(1+1/(a*x))^(9/2)*sqrt(1+(-1)/(a*x))/a+15/14*c^4*(1+1/(a*x))^(11/2)*sqrt(1+(-1)/(a*x))/a-63/16*c^4*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^3,x,12,c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(9/2)*x+3/8*c^3*arccsc(a*x)/a+3*c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-17/8*c^3*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-29/20*c^3*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a-21/20*c^3*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x))/a+6/5*c^3*(1+1/(a*x))^(9/2)*sqrt(1+(-1)/(a*x))/a-27/8*c^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^2,x,10,-1/2*c^2*arccsc(a*x)/a+3*c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-11/6*c^2*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-4/3*c^2*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a+c^2*(1+1/(a*x))^(7/2)*x*sqrt(1+(-1)/(a*x))-5/2*c^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2)),x,8,-3*c*arccsc(a*x)/a+3*c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+c*(1+1/(a*x))^(3/2)*x*sqrt(1+(-1)/(a*x))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2)),x,7,3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c)-5/3*sqrt(1+1/(a*x))/(a*c*(1+(-1)/(a*x))^(3/2))+x*sqrt(1+1/(a*x))/(c*(1+(-1)/(a*x))^(3/2))-14/3*sqrt(1+1/(a*x))/(a*c*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2))^2,x,9,3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^2)-6/5*sqrt(1+1/(a*x))/(a*c^2*(1+(-1)/(a*x))^(5/2))-9/5*sqrt(1+1/(a*x))/(a*c^2*(1+(-1)/(a*x))^(3/2))+x*sqrt(1+1/(a*x))/(c^2*(1+(-1)/(a*x))^(5/2))-24/5*sqrt(1+1/(a*x))/(a*c^2*sqrt(1+(-1)/(a*x)))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2))^3,x,10,3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^3)+(-8/7)/(a*c^3*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x)))+(-53/35)/(a*c^3*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x)))+(-88/35)/(a*c^3*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x)))+x/(c^3*(1+(-1)/(a*x))^(7/2)*sqrt(1+1/(a*x)))+(-281/35)/(a*c^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))+176/35*sqrt(1+(-1)/(a*x))/(a*c^3*sqrt(1+1/(a*x)))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2))^4,x,12,(-10/9)/(a*c^4*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(3/2))+(-29/21)/(a*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(3/2))+(-208/105)/(a*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2))+(-1147/315)/(a*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2))+x/(c^4*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(3/2))+3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^4)+(-1462/105)/(a*c^4*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x)))+2609/315*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(3/2))+1664/315*sqrt(1+(-1)/(a*x))/(a*c^4*sqrt(1+1/(a*x)))],
[exp(4*arccoth(a*x))*(c-c/(a^2*x^2))^5,x,5,1/9*c^5/(a^10*x^9)+1/2*c^5/(a^9*x^8)+3/7*c^5/(a^8*x^7)-4/3*c^5/(a^7*x^6)-14/5*c^5/(a^6*x^5)+14/3*c^5/(a^4*x^3)+4*c^5/(a^3*x^2)-3*c^5/(a^2*x)+c^5*x+4*c^5*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a^2*x^2))^4,x,5,-1/7*c^4/(a^8*x^7)-2/3*c^4/(a^7*x^6)-4/5*c^4/(a^6*x^5)+c^4/(a^5*x^4)+10/3*c^4/(a^4*x^3)+2*c^4/(a^3*x^2)-4*c^4/(a^2*x)+c^4*x+4*c^4*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a^2*x^2))^3,x,5,1/5*c^3/(a^6*x^5)+c^3/(a^5*x^4)+5/3*c^3/(a^4*x^3)-5*c^3/(a^2*x)+c^3*x+4*c^3*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a^2*x^2))^2,x,5,-1/3*c^2/(a^4*x^3)-2*c^2/(a^3*x^2)-6*c^2/(a^2*x)+c^2*x+4*c^2*log(x)/a],
[exp(4*arccoth(a*x))*(c-c/(a^2*x^2)),x,5,c/(a^2*x)+c*x-4*c*log(x)/a+8*c*log(1-a*x)/a],
[exp(4*arccoth(a*x))/(c-c/(a^2*x^2)),x,5,x/c+(-1)/(a*c*(1-a*x)^2)+5/(a*c*(1-a*x))+4*log(1-a*x)/(a*c)],
[exp(4*arccoth(a*x))/(c-c/(a^2*x^2))^2,x,5,x/c^2+1/3/(a*c^2*(1-a*x)^3)+(-2)/(a*c^2*(1-a*x)^2)+6/(a*c^2*(1-a*x))+4*log(1-a*x)/(a*c^2)],
[exp(4*arccoth(a*x))/(c-c/(a^2*x^2))^3,x,5,x/c^3+(-1/8)/(a*c^3*(1-a*x)^4)+11/12/(a*c^3*(1-a*x)^3)+(-49/16)/(a*c^3*(1-a*x)^2)+111/16/(a*c^3*(1-a*x))+129/32*log(1-a*x)/(a*c^3)-1/32*log(1+a*x)/(a*c^3)],
[exp(4*arccoth(a*x))/(c-c/(a^2*x^2))^4,x,5,x/c^4+1/20/(a*c^4*(1-a*x)^5)+(-7/16)/(a*c^4*(1-a*x)^4)+83/48/(a*c^4*(1-a*x)^3)+(-67/16)/(a*c^4*(1-a*x)^2)+501/64/(a*c^4*(1-a*x))+(-1/64)/(a*c^4*(1+a*x))+261/64*log(1-a*x)/(a*c^4)-5/64*log(1+a*x)/(a*c^4)],

# n<0
[(c-c/(a^2*x^2))^4/exp(arccoth(a*x)),x,14,29/30*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(7/2)/a+7/6*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(7/2)/a+8/7*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(7/2)/a+c^4*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(7/2)*x+35/16*c^4*arccsc(a*x)/a-c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a-1/16*c^4*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a+7/40*c^4*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a+19/40*c^4*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x))/a-19/16*c^4*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[(c-c/(a^2*x^2))^3/exp(arccoth(a*x)),x,12,5/4*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2)/a+6/5*c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(5/2)/a+c^3*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2)*x+15/8*c^3*arccsc(a*x)/a-c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+1/24*c^3*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a+11/12*c^3*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a-7/8*c^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[(c-c/(a^2*x^2))^2/exp(arccoth(a*x)),x,10,4/3*c^2*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2)/a+c^2*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2)*x+3/2*c^2*arccsc(a*x)/a-c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+3/2*c^2*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-1/2*c^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[(c-c/(a^2*x^2))/exp(arccoth(a*x)),x,9,c*arccsc(a*x)/a-c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+c*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))+2*c*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[1/(exp(arccoth(a*x))*(c-c/(a^2*x^2))),x,6,-arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c)+2*sqrt(1+(-1)/(a*x))/(a*c*sqrt(1+1/(a*x)))+x*sqrt(1+(-1)/(a*x))/(c*sqrt(1+1/(a*x)))],
[1/(exp(arccoth(a*x))*(c-c/(a^2*x^2))^2),x,8,-arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^2)+(-2)/(a*c^2*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x)))+x/(c^2*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x)))+5/3*sqrt(1+(-1)/(a*x))/(a*c^2*(1+1/(a*x))^(3/2))+8/3*sqrt(1+(-1)/(a*x))/(a*c^2*sqrt(1+1/(a*x)))],
[1/(exp(arccoth(a*x))*(c-c/(a^2*x^2))^3),x,10,(-4/3)/(a*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2))+x/(c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2))-arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^3)+(-13/3)/(a*c^3*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x)))+14/5*sqrt(1+(-1)/(a*x))/(a*c^3*(1+1/(a*x))^(5/2))+11/5*sqrt(1+(-1)/(a*x))/(a*c^3*(1+1/(a*x))^(3/2))+16/5*sqrt(1+(-1)/(a*x))/(a*c^3*sqrt(1+1/(a*x)))],
[1/(exp(arccoth(a*x))*(c-c/(a^2*x^2))^4),x,12,(-6/5)/(a*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(7/2))+(-31/15)/(a*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(7/2))+x/(c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(7/2))-arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^4)+(-28/3)/(a*c^4*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x)))+115/21*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(7/2))+122/35*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(5/2))+93/35*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(3/2))+128/35*sqrt(1+(-1)/(a*x))/(a*c^4*sqrt(1+1/(a*x)))],
[(c-c/(a^2*x^2))^4/exp(2*arccoth(a*x)),x,5,1/7*c^4/(a^8*x^7)-1/3*c^4/(a^7*x^6)-2/5*c^4/(a^6*x^5)+3/2*c^4/(a^5*x^4)-3*c^4/(a^3*x^2)+2*c^4/(a^2*x)+c^4*x-2*c^4*log(x)/a],
[(c-c/(a^2*x^2))^3/exp(2*arccoth(a*x)),x,5,-1/5*c^3/(a^6*x^5)+1/2*c^3/(a^5*x^4)+1/3*c^3/(a^4*x^3)-2*c^3/(a^3*x^2)+c^3/(a^2*x)+c^3*x-2*c^3*log(x)/a],
[(c-c/(a^2*x^2))^2/exp(2*arccoth(a*x)),x,5,1/3*c^2/(a^4*x^3)-c^2/(a^3*x^2)+c^2*x-2*c^2*log(x)/a],
[(c-c/(a^2*x^2))/exp(2*arccoth(a*x)),x,5,-c/(a^2*x)+c*x-2*c*log(x)/a],
[1/(exp(2*arccoth(a*x))*(c-c/(a^2*x^2))),x,5,x/c+(-1)/(a*c*(1+a*x))-2*log(1+a*x)/(a*c)],
[1/(exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^2),x,5,x/c^2+1/4/(a*c^2*(1+a*x)^2)+(-7/4)/(a*c^2*(1+a*x))+1/8*log(1-a*x)/(a*c^2)-17/8*log(1+a*x)/(a*c^2)],
[1/(exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^3),x,5,x/c^3+1/16/(a*c^3*(1-a*x))+(-1/12)/(a*c^3*(1+a*x)^3)+5/8/(a*c^3*(1+a*x)^2)+(-39/16)/(a*c^3*(1+a*x))+1/4*log(1-a*x)/(a*c^3)-9/4*log(1+a*x)/(a*c^3)],
[1/(exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^4),x,5,x/c^4+(-1/64)/(a*c^4*(1-a*x)^2)+11/64/(a*c^4*(1-a*x))+1/32/(a*c^4*(1+a*x)^4)+(-13/48)/(a*c^4*(1+a*x)^3)+35/32/(a*c^4*(1+a*x)^2)+(-99/32)/(a*c^4*(1+a*x))+47/128*log(1-a*x)/(a*c^4)-303/128*log(1+a*x)/(a*c^4)],
[(c-c/(a^2*x^2))^4/exp(3*arccoth(a*x)),x,14,5/8*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(5/2)/a+11/10*c^4*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(5/2)/a+17/14*c^4*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(5/2)/a+8/7*c^4*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(5/2)/a+c^4*(1+(-1)/(a*x))^(11/2)*(1+1/(a*x))^(5/2)*x+15/16*c^4*arccsc(a*x)/a-3*c^4*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+27/16*c^4*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a-3/8*c^4*(1+1/(a*x))^(5/2)*sqrt(1+(-1)/(a*x))/a+33/16*c^4*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[(c-c/(a^2*x^2))^3/exp(3*arccoth(a*x)),x,12,5/4*c^3*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(3/2)/a+27/20*c^3*(1+(-1)/(a*x))^(5/2)*(1+1/(a*x))^(3/2)/a+6/5*c^3*(1+(-1)/(a*x))^(7/2)*(1+1/(a*x))^(3/2)/a+c^3*(1+(-1)/(a*x))^(9/2)*(1+1/(a*x))^(3/2)*x+3/8*c^3*arccsc(a*x)/a-3*c^3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+3/8*c^3*(1+1/(a*x))^(3/2)*sqrt(1+(-1)/(a*x))/a+21/8*c^3*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[(c-c/(a^2*x^2))^2/exp(3*arccoth(a*x)),x,10,-1/2*c^2*arccsc(a*x)/a-3*c^2*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+11/6*c^2*(1+(-1)/(a*x))^(3/2)*sqrt(1+1/(a*x))/a+4/3*c^2*(1+(-1)/(a*x))^(5/2)*sqrt(1+1/(a*x))/a+c^2*(1+(-1)/(a*x))^(7/2)*x*sqrt(1+1/(a*x))+5/2*c^2*sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x))/a],
[(c-c/(a^2*x^2))/exp(3*arccoth(a*x)),x,8,-3*c*arccsc(a*x)/a-3*c*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/a+c*(1+(-1)/(a*x))^(3/2)*x*sqrt(1+1/(a*x))],
[1/(exp(3*arccoth(a*x))*(c-c/(a^2*x^2))),x,7,-3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c)+5/3*sqrt(1+(-1)/(a*x))/(a*c*(1+1/(a*x))^(3/2))+x*sqrt(1+(-1)/(a*x))/(c*(1+1/(a*x))^(3/2))+14/3*sqrt(1+(-1)/(a*x))/(a*c*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^2),x,9,-3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^2)+6/5*sqrt(1+(-1)/(a*x))/(a*c^2*(1+1/(a*x))^(5/2))+9/5*sqrt(1+(-1)/(a*x))/(a*c^2*(1+1/(a*x))^(3/2))+x*sqrt(1+(-1)/(a*x))/(c^2*(1+1/(a*x))^(5/2))+24/5*sqrt(1+(-1)/(a*x))/(a*c^2*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^3),x,10,-3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^3)+(-2)/(a*c^3*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x)))+x/(c^3*(1+1/(a*x))^(7/2)*sqrt(1+(-1)/(a*x)))+11/7*sqrt(1+(-1)/(a*x))/(a*c^3*(1+1/(a*x))^(7/2))+54/35*sqrt(1+(-1)/(a*x))/(a*c^3*(1+1/(a*x))^(5/2))+71/35*sqrt(1+(-1)/(a*x))/(a*c^3*(1+1/(a*x))^(3/2))+176/35*sqrt(1+(-1)/(a*x))/(a*c^3*sqrt(1+1/(a*x)))],
[1/(exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^4),x,12,(-4/3)/(a*c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(9/2))+x/(c^4*(1+(-1)/(a*x))^(3/2)*(1+1/(a*x))^(9/2))-3*arctanh(sqrt(1+(-1)/(a*x))*sqrt(1+1/(a*x)))/(a*c^4)+(-5)/(a*c^4*(1+1/(a*x))^(9/2)*sqrt(1+(-1)/(a*x)))+28/9*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(9/2))+139/63*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(7/2))+202/105*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(5/2))+719/315*sqrt(1+(-1)/(a*x))/(a*c^4*(1+1/(a*x))^(3/2))+1664/315*sqrt(1+(-1)/(a*x))/(a*c^4*sqrt(1+1/(a*x)))],

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

# n>0
[exp(arccoth(a*x))*(c-c/(a^2*x^2))^(7/2),x,4,1/6*c^3*sqrt(c-c/(a^2*x^2))/(a^7*x^6*sqrt(1+(-1)/(a^2*x^2)))+1/5*c^3*sqrt(c-c/(a^2*x^2))/(a^6*x^5*sqrt(1+(-1)/(a^2*x^2)))-3/4*c^3*sqrt(c-c/(a^2*x^2))/(a^5*x^4*sqrt(1+(-1)/(a^2*x^2)))-c^3*sqrt(c-c/(a^2*x^2))/(a^4*x^3*sqrt(1+(-1)/(a^2*x^2)))+3/2*c^3*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))+3*c^3*sqrt(c-c/(a^2*x^2))/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))+c^3*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+c^3*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*(c-c/(a^2*x^2))^(5/2),x,4,-1/4*c^2*sqrt(c-c/(a^2*x^2))/(a^5*x^4*sqrt(1+(-1)/(a^2*x^2)))-1/3*c^2*sqrt(c-c/(a^2*x^2))/(a^4*x^3*sqrt(1+(-1)/(a^2*x^2)))+c^2*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))+2*c^2*sqrt(c-c/(a^2*x^2))/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))+c^2*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+c^2*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*(c-c/(a^2*x^2))^(3/2),x,4,1/2*c*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))+c*sqrt(c-c/(a^2*x^2))/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))+c*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+c*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*sqrt(c-c/(a^2*x^2)),x,4,x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))/sqrt(c-c/(a^2*x^2)),x,4,x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a^2*x^2))+log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a^2*x^2)))],
[exp(arccoth(a*x))/(c-c/(a^2*x^2))^(3/2),x,4,x*sqrt(1+(-1)/(a^2*x^2))/(c*sqrt(c-c/(a^2*x^2)))+1/2*sqrt(1+(-1)/(a^2*x^2))/(a*c*(1-a*x)*sqrt(c-c/(a^2*x^2)))+5/4*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c*sqrt(c-c/(a^2*x^2)))-1/4*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c*sqrt(c-c/(a^2*x^2)))],
[exp(arccoth(a*x))/(c-c/(a^2*x^2))^(5/2),x,4,x*sqrt(1+(-1)/(a^2*x^2))/(c^2*sqrt(c-c/(a^2*x^2)))-1/8*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*(1-a*x)^2*sqrt(c-c/(a^2*x^2)))+sqrt(1+(-1)/(a^2*x^2))/(a*c^2*(1-a*x)*sqrt(c-c/(a^2*x^2)))-1/8*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*(1+a*x)*sqrt(c-c/(a^2*x^2)))+23/16*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*sqrt(c-c/(a^2*x^2)))-7/16*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*sqrt(c-c/(a^2*x^2)))],
[exp(arccoth(a*x))/(c-c/(a^2*x^2))^(7/2),x,4,x*sqrt(1+(-1)/(a^2*x^2))/(c^3*sqrt(c-c/(a^2*x^2)))+1/24*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)^3*sqrt(c-c/(a^2*x^2)))-11/32*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)^2*sqrt(c-c/(a^2*x^2)))+3/2*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)*sqrt(c-c/(a^2*x^2)))+1/32*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1+a*x)^2*sqrt(c-c/(a^2*x^2)))-5/16*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1+a*x)*sqrt(c-c/(a^2*x^2)))+51/32*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*sqrt(c-c/(a^2*x^2)))-19/32*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*sqrt(c-c/(a^2*x^2)))],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^(7/2),x,15,11/30*a^3*(c-c/(a^2*x^2))^(7/2)*x^4/(1-a*x)^3-57/16*a^6*(c-c/(a^2*x^2))^(7/2)*x^7/((1-a*x)^3*(1+a*x)^3)+41/24*a^5*(c-c/(a^2*x^2))^(7/2)*x^6/((1-a*x)^3*(1+a*x)^2)+57/80*a^4*(c-c/(a^2*x^2))^(7/2)*x^5/((1-a*x)^3*(1+a*x))-13/40*a^2*(c-c/(a^2*x^2))^(7/2)*x^3*(1+a*x)/(1-a*x)^3+1/15*a*(c-c/(a^2*x^2))^(7/2)*x^2*(1+a*x)/(1-a*x)^2+1/6*(c-c/(a^2*x^2))^(7/2)*x*(1+a*x)/(1-a*x)+2*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*arcsin(a*x)/((1-a*x)^(7/2)*(1+a*x)^(7/2))+25/16*a^6*(c-c/(a^2*x^2))^(7/2)*x^7*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(7/2)*(1+a*x)^(7/2))],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^(5/2),x,13,-5/8*a^2*(c-c/(a^2*x^2))^(5/2)*x^3/(1-a*x)^2+25/8*a^4*(c-c/(a^2*x^2))^(5/2)*x^5/((1-a*x)^2*(1+a*x)^2)-17/12*a^3*(c-c/(a^2*x^2))^(5/2)*x^4/((1-a*x)^2*(1+a*x))+1/6*a*(c-c/(a^2*x^2))^(5/2)*x^2*(1+a*x)/(1-a*x)^2+1/4*(c-c/(a^2*x^2))^(5/2)*x*(1+a*x)/(1-a*x)-2*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*arcsin(a*x)/((1-a*x)^(5/2)*(1+a*x)^(5/2))-9/8*a^4*(c-c/(a^2*x^2))^(5/2)*x^5*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(5/2)*(1+a*x)^(5/2))],
[exp(2*arccoth(a*x))*(c-c/(a^2*x^2))^(3/2),x,11,a*(c-c/(a^2*x^2))^(3/2)*x^2/(1-a*x)-5/2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3/((1-a*x)*(1+a*x))+1/2*(c-c/(a^2*x^2))^(3/2)*x*(1+a*x)/(1-a*x)+2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*arcsin(a*x)/((1-a*x)^(3/2)*(1+a*x)^(3/2))+1/2*a^2*(c-c/(a^2*x^2))^(3/2)*x^3*arctanh(sqrt(1-a*x)*sqrt(1+a*x))/((1-a*x)^(3/2)*(1+a*x)^(3/2))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2)),x,9,x*sqrt(c-c/(a^2*x^2))-2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))+x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))/sqrt(c-c/(a^2*x^2)),x,7,-2*(1-a*x)*(1+a*x)/(a^2*x*sqrt(c-c/(a^2*x^2)))-(1+a*x)^2/(a^2*x*sqrt(c-c/(a^2*x^2)))+2*arcsin(a*x)*sqrt(1-a*x)*sqrt(1+a*x)/(a^2*x*sqrt(c-c/(a^2*x^2)))],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2))^(3/2),x,7,-1/3*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(3/2)*x)+2/3*(5-2*a*x)*(1-a*x)*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)-2*(1-a*x)^(3/2)*(1+a*x)^(3/2)*arcsin(a*x)/(a^4*(c-c/(a^2*x^2))^(3/2)*x^3)],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2))^(5/2),x,9,-1/5*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(5/2)*x)+2/3*(1-a*x)*(1+a*x)^2/(a^3*(c-c/(a^2*x^2))^(5/2)*x^2)-58/15*(1-a*x)^2*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(5/2)*x^3)-2/15*(1-a*x)^3*(1+a*x)^2*(28+43*a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)+2*(1-a*x)^(5/2)*(1+a*x)^(5/2)*arcsin(a*x)/(a^6*(c-c/(a^2*x^2))^(5/2)*x^5)],
[exp(2*arccoth(a*x))/(c-c/(a^2*x^2))^(7/2),x,11,-1/7*(1+a*x)^2/(a^2*(c-c/(a^2*x^2))^(7/2)*x)+2/5*(1-a*x)*(1+a*x)^2/(a^3*(c-c/(a^2*x^2))^(7/2)*x^2)-124/105*(1-a*x)^2*(1+a*x)^2/(a^4*(c-c/(a^2*x^2))^(7/2)*x^3)+782/105*(1-a*x)^3*(1+a*x)^2/(a^5*(c-c/(a^2*x^2))^(7/2)*x^4)+142/35*(1-a*x)^4*(1+a*x)^2/(a^6*(c-c/(a^2*x^2))^(7/2)*x^5)+2/35*(1-a*x)^4*(1+a*x)^3*(72+107*a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)-2*(1-a*x)^(7/2)*(1+a*x)^(7/2)*arcsin(a*x)/(a^8*(c-c/(a^2*x^2))^(7/2)*x^7)],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^(9/2),x,4,1/8*c^4*sqrt(c-c/(a^2*x^2))/(a^9*x^8*sqrt(1+(-1)/(a^2*x^2)))+3/7*c^4*sqrt(c-c/(a^2*x^2))/(a^8*x^7*sqrt(1+(-1)/(a^2*x^2)))-8/5*c^4*sqrt(c-c/(a^2*x^2))/(a^6*x^5*sqrt(1+(-1)/(a^2*x^2)))-3/2*c^4*sqrt(c-c/(a^2*x^2))/(a^5*x^4*sqrt(1+(-1)/(a^2*x^2)))+2*c^4*sqrt(c-c/(a^2*x^2))/(a^4*x^3*sqrt(1+(-1)/(a^2*x^2)))+4*c^4*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))+c^4*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+3*c^4*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^(7/2),x,4,-1/6*c^3*sqrt(c-c/(a^2*x^2))/(a^7*x^6*sqrt(1+(-1)/(a^2*x^2)))-3/5*c^3*sqrt(c-c/(a^2*x^2))/(a^6*x^5*sqrt(1+(-1)/(a^2*x^2)))-1/4*c^3*sqrt(c-c/(a^2*x^2))/(a^5*x^4*sqrt(1+(-1)/(a^2*x^2)))+5/3*c^3*sqrt(c-c/(a^2*x^2))/(a^4*x^3*sqrt(1+(-1)/(a^2*x^2)))+5/2*c^3*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))-c^3*sqrt(c-c/(a^2*x^2))/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))+c^3*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+3*c^3*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^(5/2),x,4,1/4*c^2*sqrt(c-c/(a^2*x^2))/(a^5*x^4*sqrt(1+(-1)/(a^2*x^2)))+c^2*sqrt(c-c/(a^2*x^2))/(a^4*x^3*sqrt(1+(-1)/(a^2*x^2)))+c^2*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))-2*c^2*sqrt(c-c/(a^2*x^2))/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))+c^2*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+3*c^2*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*(c-c/(a^2*x^2))^(3/2),x,4,-1/2*c*sqrt(c-c/(a^2*x^2))/(a^3*x^2*sqrt(1+(-1)/(a^2*x^2)))-3*c*sqrt(c-c/(a^2*x^2))/(a^2*x*sqrt(1+(-1)/(a^2*x^2)))+c*x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+3*c*log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2)),x,4,x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))-log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))/sqrt(c-c/(a^2*x^2)),x,4,x*sqrt(1+(-1)/(a^2*x^2))/sqrt(c-c/(a^2*x^2))+2*sqrt(1+(-1)/(a^2*x^2))/(a*(1-a*x)*sqrt(c-c/(a^2*x^2)))+3*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*sqrt(c-c/(a^2*x^2)))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2))^(3/2),x,4,x*sqrt(1+(-1)/(a^2*x^2))/(c*sqrt(c-c/(a^2*x^2)))-1/2*sqrt(1+(-1)/(a^2*x^2))/(a*c*(1-a*x)^2*sqrt(c-c/(a^2*x^2)))+3*sqrt(1+(-1)/(a^2*x^2))/(a*c*(1-a*x)*sqrt(c-c/(a^2*x^2)))+3*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c*sqrt(c-c/(a^2*x^2)))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2))^(5/2),x,4,x*sqrt(1+(-1)/(a^2*x^2))/(c^2*sqrt(c-c/(a^2*x^2)))+1/6*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*(1-a*x)^3*sqrt(c-c/(a^2*x^2)))-9/8*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*(1-a*x)^2*sqrt(c-c/(a^2*x^2)))+31/8*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*(1-a*x)*sqrt(c-c/(a^2*x^2)))+49/16*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*sqrt(c-c/(a^2*x^2)))-1/16*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^2*sqrt(c-c/(a^2*x^2)))],
[exp(3*arccoth(a*x))/(c-c/(a^2*x^2))^(7/2),x,4,x*sqrt(1+(-1)/(a^2*x^2))/(c^3*sqrt(c-c/(a^2*x^2)))-1/16*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)^4*sqrt(c-c/(a^2*x^2)))+1/2*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)^3*sqrt(c-c/(a^2*x^2)))-59/32*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)^2*sqrt(c-c/(a^2*x^2)))+75/16*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1-a*x)*sqrt(c-c/(a^2*x^2)))-1/32*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*(1+a*x)*sqrt(c-c/(a^2*x^2)))+201/64*log(1-a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*sqrt(c-c/(a^2*x^2)))-9/64*log(1+a*x)*sqrt(1+(-1)/(a^2*x^2))/(a*c^3*sqrt(c-c/(a^2*x^2)))],

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

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

# n>0
[exp(arccoth(a*x))*x^m*sqrt(c-c/(a^2*x^2)),x,4,x^m*sqrt(c-c/(a^2*x^2))/(a*m*sqrt(1+(-1)/(a^2*x^2)))+x^(1+m)*sqrt(c-c/(a^2*x^2))/((1+m)*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*x^2*sqrt(c-c/(a^2*x^2)),x,4,1/2*x^2*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*x*sqrt(c-c/(a^2*x^2)),x,3,x*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*sqrt(c-c/(a^2*x^2)),x,4,x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x,x,4,-sqrt(c-c/(a^2*x^2))/(a*x*sqrt(1+(-1)/(a^2*x^2)))+log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^2,x,3,-1/2*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))],
[exp(2*arccoth(a*x))*x^3*sqrt(c-c/(a^2*x^2)),x,9,7/8*x*sqrt(c-c/(a^2*x^2))/a^3+7/24*x*(1+a*x)*sqrt(c-c/(a^2*x^2))/a^3+1/6*x*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/a^3+1/4*x^2*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/a^2-7/8*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*x^2*sqrt(c-c/(a^2*x^2)),x,8,x*sqrt(c-c/(a^2*x^2))/a^2+1/3*x*(1+a*x)*sqrt(c-c/(a^2*x^2))/a^2+1/3*x*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/a^2-x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*x*sqrt(c-c/(a^2*x^2)),x,7,3/2*x*sqrt(c-c/(a^2*x^2))/a+1/2*x*(1+a*x)*sqrt(c-c/(a^2*x^2))/a-3/2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2)),x,9,x*sqrt(c-c/(a^2*x^2))-2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))+x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x,x,9,sqrt(c-c/(a^2*x^2))-a*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))+2*a*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^2,x,7,3/2*a*sqrt(c-c/(a^2*x^2))+1/2*(1+a*x)*sqrt(c-c/(a^2*x^2))/x+3/2*a^2*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^3,x,8,a^2*sqrt(c-c/(a^2*x^2))+1/3*a*(1+a*x)*sqrt(c-c/(a^2*x^2))/x+1/3*(1+a*x)^2*sqrt(c-c/(a^2*x^2))/x^2+a^3*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^4,x,10,4/3*a^3*sqrt(c-c/(a^2*x^2))+1/4*sqrt(c-c/(a^2*x^2))/x^3+2/3*a*sqrt(c-c/(a^2*x^2))/x^2+7/8*a^2*sqrt(c-c/(a^2*x^2))/x+7/8*a^4*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(2*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^5,x,11,6/5*a^4*sqrt(c-c/(a^2*x^2))+1/5*sqrt(c-c/(a^2*x^2))/x^4+1/2*a*sqrt(c-c/(a^2*x^2))/x^3+3/5*a^2*sqrt(c-c/(a^2*x^2))/x^2+3/4*a^3*sqrt(c-c/(a^2*x^2))/x+3/4*a^5*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[exp(3*arccoth(a*x))*x^3*sqrt(c-c/(a^2*x^2)),x,4,4*x*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1+(-1)/(a^2*x^2)))+2*x^2*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1+(-1)/(a^2*x^2)))+x^3*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/4*x^4*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a^4*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*x^2*sqrt(c-c/(a^2*x^2)),x,4,4*x*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1+(-1)/(a^2*x^2)))+3/2*x^2*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*x*sqrt(c-c/(a^2*x^2)),x,4,3*x*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2)),x,4,x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))-log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x,x,4,sqrt(c-c/(a^2*x^2))/(a*x*sqrt(1+(-1)/(a^2*x^2)))-3*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^2,x,4,1/2*sqrt(c-c/(a^2*x^2))/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))+3*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*a*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^3,x,4,1/3*sqrt(c-c/(a^2*x^2))/(a*x^3*sqrt(1+(-1)/(a^2*x^2)))+3/2*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1+(-1)/(a^2*x^2)))+4*a*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a^2*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*a^2*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^4,x,4,1/4*sqrt(c-c/(a^2*x^2))/(a*x^4*sqrt(1+(-1)/(a^2*x^2)))+sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1+(-1)/(a^2*x^2)))+2*a*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1+(-1)/(a^2*x^2)))+4*a^2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a^3*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*a^3*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[exp(3*arccoth(a*x))*sqrt(c-c/(a^2*x^2))/x^5,x,4,1/5*sqrt(c-c/(a^2*x^2))/(a*x^5*sqrt(1+(-1)/(a^2*x^2)))+3/4*sqrt(c-c/(a^2*x^2))/(x^4*sqrt(1+(-1)/(a^2*x^2)))+4/3*a*sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1+(-1)/(a^2*x^2)))+2*a^2*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1+(-1)/(a^2*x^2)))+4*a^3*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a^4*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*a^4*log(1-a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],

# n<0
[x^m*sqrt(c-c/(a^2*x^2))/exp(arccoth(a*x)),x,4,-x^m*sqrt(c-c/(a^2*x^2))/(a*m*sqrt(1+(-1)/(a^2*x^2)))+x^(1+m)*sqrt(c-c/(a^2*x^2))/((1+m)*sqrt(1+(-1)/(a^2*x^2)))],
[x^2*sqrt(c-c/(a^2*x^2))/exp(arccoth(a*x)),x,4,-1/2*x^2*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[x*sqrt(c-c/(a^2*x^2))/exp(arccoth(a*x)),x,3,-x*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/exp(arccoth(a*x)),x,4,x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))-log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-c/(a^2*x^2))/(exp(arccoth(a*x))*x),x,4,sqrt(c-c/(a^2*x^2))/(a*x*sqrt(1+(-1)/(a^2*x^2)))+log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/(exp(arccoth(a*x))*x^2),x,3,1/2*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))],
[x^3*sqrt(c-c/(a^2*x^2))/exp(2*arccoth(a*x)),x,9,-7/8*x*sqrt(c-c/(a^2*x^2))/a^3-7/24*x*(1-a*x)*sqrt(c-c/(a^2*x^2))/a^3-1/6*x*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/a^3+1/4*x^2*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/a^2-7/8*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1-a*x)*sqrt(1+a*x))],
[x^2*sqrt(c-c/(a^2*x^2))/exp(2*arccoth(a*x)),x,8,x*sqrt(c-c/(a^2*x^2))/a^2+1/3*x*(1-a*x)*sqrt(c-c/(a^2*x^2))/a^2+1/3*x*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/a^2+x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1-a*x)*sqrt(1+a*x))],
[x*sqrt(c-c/(a^2*x^2))/exp(2*arccoth(a*x)),x,7,-3/2*x*sqrt(c-c/(a^2*x^2))/a-1/2*x*(1-a*x)*sqrt(c-c/(a^2*x^2))/a-3/2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/exp(2*arccoth(a*x)),x,9,x*sqrt(c-c/(a^2*x^2))+2*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))+x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arccoth(a*x))*x),x,9,sqrt(c-c/(a^2*x^2))-a*x*arcsin(a*x)*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))-2*a*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arccoth(a*x))*x^2),x,7,-3/2*a*sqrt(c-c/(a^2*x^2))+1/2*(1-a*x)*sqrt(c-c/(a^2*x^2))/x+3/2*a^2*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arccoth(a*x))*x^3),x,8,a^2*sqrt(c-c/(a^2*x^2))-1/3*a*(1-a*x)*sqrt(c-c/(a^2*x^2))/x+1/3*(1-a*x)^2*sqrt(c-c/(a^2*x^2))/x^2-a^3*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arccoth(a*x))*x^4),x,10,-4/3*a^3*sqrt(c-c/(a^2*x^2))+1/4*sqrt(c-c/(a^2*x^2))/x^3-2/3*a*sqrt(c-c/(a^2*x^2))/x^2+7/8*a^2*sqrt(c-c/(a^2*x^2))/x+7/8*a^4*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[sqrt(c-c/(a^2*x^2))/(exp(2*arccoth(a*x))*x^5),x,11,6/5*a^4*sqrt(c-c/(a^2*x^2))+1/5*sqrt(c-c/(a^2*x^2))/x^4-1/2*a*sqrt(c-c/(a^2*x^2))/x^3+3/5*a^2*sqrt(c-c/(a^2*x^2))/x^2-3/4*a^3*sqrt(c-c/(a^2*x^2))/x-3/4*a^5*x*arctanh(sqrt(1-a*x)*sqrt(1+a*x))*sqrt(c-c/(a^2*x^2))/(sqrt(1-a*x)*sqrt(1+a*x))],
[x^3*sqrt(c-c/(a^2*x^2))/exp(3*arccoth(a*x)),x,4,-4*x*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1+(-1)/(a^2*x^2)))+2*x^2*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1+(-1)/(a^2*x^2)))-x^3*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/4*x^4*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a^4*sqrt(1+(-1)/(a^2*x^2)))],
[x^2*sqrt(c-c/(a^2*x^2))/exp(3*arccoth(a*x)),x,4,4*x*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1+(-1)/(a^2*x^2)))-3/2*x^2*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/3*x^3*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))-4*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a^3*sqrt(1+(-1)/(a^2*x^2)))],
[x*sqrt(c-c/(a^2*x^2))/exp(3*arccoth(a*x)),x,4,-3*x*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))+1/2*x^2*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a^2*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-c/(a^2*x^2))/exp(3*arccoth(a*x)),x,4,x*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+log(x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))-4*log(1+a*x)*sqrt(c-c/(a^2*x^2))/(a*sqrt(1+(-1)/(a^2*x^2)))],
[sqrt(c-c/(a^2*x^2))/(exp(3*arccoth(a*x))*x),x,4,-sqrt(c-c/(a^2*x^2))/(a*x*sqrt(1+(-1)/(a^2*x^2)))-3*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/(exp(3*arccoth(a*x))*x^2),x,4,-1/2*sqrt(c-c/(a^2*x^2))/(a*x^2*sqrt(1+(-1)/(a^2*x^2)))+3*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))+4*a*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))-4*a*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/(exp(3*arccoth(a*x))*x^3),x,4,-1/3*sqrt(c-c/(a^2*x^2))/(a*x^3*sqrt(1+(-1)/(a^2*x^2)))+3/2*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1+(-1)/(a^2*x^2)))-4*a*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a^2*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*a^2*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/(exp(3*arccoth(a*x))*x^4),x,4,-1/4*sqrt(c-c/(a^2*x^2))/(a*x^4*sqrt(1+(-1)/(a^2*x^2)))+sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1+(-1)/(a^2*x^2)))-2*a*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1+(-1)/(a^2*x^2)))+4*a^2*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))+4*a^3*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))-4*a^3*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],
[sqrt(c-c/(a^2*x^2))/(exp(3*arccoth(a*x))*x^5),x,4,-1/5*sqrt(c-c/(a^2*x^2))/(a*x^5*sqrt(1+(-1)/(a^2*x^2)))+3/4*sqrt(c-c/(a^2*x^2))/(x^4*sqrt(1+(-1)/(a^2*x^2)))-4/3*a*sqrt(c-c/(a^2*x^2))/(x^3*sqrt(1+(-1)/(a^2*x^2)))+2*a^2*sqrt(c-c/(a^2*x^2))/(x^2*sqrt(1+(-1)/(a^2*x^2)))-4*a^3*sqrt(c-c/(a^2*x^2))/(x*sqrt(1+(-1)/(a^2*x^2)))-4*a^4*log(x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))+4*a^4*log(1+a*x)*sqrt(c-c/(a^2*x^2))/sqrt(1+(-1)/(a^2*x^2))],

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

# Integrands of the form E^(n ArcCoth[a x]) (c-c/(a^2 x^2))^p with p symbolic
[exp(n*arccoth(a*x))*(c-c/(a^2*x^2))^p,x,3,-2^(1-1/2*n+p)*(c-c/(a^2*x^2))^p*(1+1/(a*x))^(1+1/2*n+p)*AppellF1(1+1/2*n+p,1/2*(n-2*p),2,2+1/2*n+p,1/2*(a+1/x)/a,1+1/(a*x))/(a*(2+n+2*p)*(1+(-1)/(a^2*x^2))^p)],
[(c-c/(a^2*x^2))^p/exp(2*p*arccoth(a*x)),x,3,(c-c/(a^2*x^2))^p*(1+(-1)/(a*x))^(1+2*p)*hypergeom([2,1+2*p],[2*(1+p)],1+(-1)/(a*x))/(a*(1+2*p)*(1+(-1)/(a^2*x^2))^p)],
[exp(2*p*arccoth(a*x))*(c-c/(a^2*x^2))^p,x,3,-(c-c/(a^2*x^2))^p*(1+1/(a*x))^(1+2*p)*hypergeom([2,1+2*p],[2*(1+p)],1+1/(a*x))/(a*(1+2*p)*(1+(-1)/(a^2*x^2))^p)]]:
