# Maple integration test file: "4 Trig functions\4.4 Cotangent\4.4.9 trig^m (a+b cot^n+c cot^(2 n))^p.txt"

lst:=[

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

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

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

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

# Integrands of the form Cot[d+e x]^m (a+b Cot[d+e x]+c Cot[d+e x]^2)^(p/2)
[cot(d+e*x)^5/sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,15,-1/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(3/2)*e)+1/16*b*(5*b^2-12*a*c)*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(7/2)*e)+sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c*e)-1/3*cot(d+e*x)^2*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c*e)-1/24*(15*b^2-16*a*c-10*b*c*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^3*e)-arctanh((a-c+b*cot(d+e*x)-sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))+arctanh((a-c+b*cot(d+e*x)+sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))],
[cot(d+e*x)^3/sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,11,1/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(3/2)*e)-sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c*e)+arctanh((a-c+b*cot(d+e*x)-sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))-arctanh((a-c+b*cot(d+e*x)+sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))],
[cot(d+e*x)/sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,6,-arctanh((a-c+b*cot(d+e*x)-sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))+arctanh((a-c+b*cot(d+e*x)+sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))],
[tan(d+e*x)/sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,10,arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(e*sqrt(a))+arctanh((a-c+b*cot(d+e*x)-sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))-arctanh((a-c+b*cot(d+e*x)+sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))],
[tan(d+e*x)^3/sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,14,1/8*(3*b^2-4*a*c)*arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(a^(5/2)*e)-arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(e*sqrt(a))-arctanh((a-c+b*cot(d+e*x)-sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c-sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))+arctanh((a-c+b*cot(d+e*x)+sqrt(a^2+b^2-2*a*c+c^2))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(a-c+sqrt(a^2+b^2-2*a*c+c^2))/(e*sqrt(2)*sqrt(a^2+b^2-2*a*c+c^2))-3/4*b*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x)/(a^2*e)+1/2*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x)^2/(a*e)],
[cot(d+e*x)^5*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,21,1/16*b*(b^2-4*a*c)*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(5/2)*e)-1/256*b*(7*b^2-12*a*c)*(b^2-4*a*c)*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(9/2)*e)+1/3*(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2)/(c*e)-1/5*cot(d+e*x)^2*(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2)/(c*e)-1/240*(35*b^2-32*a*c-42*b*c*cot(d+e*x))*(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2)/(c^3*e)-1/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(e*sqrt(c))-sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/e-1/8*b*(b+2*c*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^2*e)+1/128*b*(7*b^2-12*a*c)*(b+2*c*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^4*e)+arctanh((b^2+b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2)+(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))-arctan((b^2+(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))],
[cot(d+e*x)^3*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,16,-1/16*b*(b^2-4*a*c)*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(5/2)*e)-1/3*(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2)/(c*e)+1/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(e*sqrt(c))+sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/e+1/8*b*(b+2*c*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^2*e)-arctanh((b^2+b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2)+(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))+arctan((b^2+(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))],
[cot(d+e*x)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2),x,10,-1/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(e*sqrt(c))-sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/e+arctanh((b^2+b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2)+(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))-arctan((b^2+(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))],
[sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x),x,18,arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))*sqrt(a)/e-arctanh((b^2+b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2)+(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))+arctan((b^2+(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))],
[sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x)^3,x,21,-1/8*(b^2-4*a*c)*arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(a^(3/2)*e)-arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))*sqrt(a)/e+arctanh((b^2+b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2)+(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c-sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c-sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))-arctan((b^2+(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(1/4)*sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))))*sqrt(a^2+b^2+c*(c+sqrt(a^2+b^2-2*a*c+c^2))-a*(2*c+sqrt(a^2+b^2-2*a*c+c^2)))/((a^2+b^2-2*a*c+c^2)^(1/4)*e*sqrt(2))+1/4*(2*a+b*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x)^2/(a*e)],
[cot(d+e*x)^7/(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2),x,20,-3/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(5/2)*e)+5/16*b*(7*b^2-12*a*c)*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(9/2)*e)-2*(2*a+b*cot(d+e*x))/((b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+2*cot(d+e*x)^2*(2*a+b*cot(d+e*x))/((b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-2*cot(d+e*x)^4*(2*a+b*cot(d+e*x))/((b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(d+e*x))/((b^2+(a-c)^2)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-1/3*(7*b^2-16*a*c)*cot(d+e*x)^2*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^2*(b^2-4*a*c)*e)+2*b*cot(d+e*x)^3*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c*(b^2-4*a*c)*e)+(3*b^2-8*a*c-2*b*c*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^2*(b^2-4*a*c)*e)-1/24*(105*b^4-460*a*b^2*c+256*a^2*c^2-2*b*c*(35*b^2-116*a*c)*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^4*(b^2-4*a*c)*e)-arctanh((b^2-(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))+arctanh((b^2-b*cot(d+e*x)*(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))-(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))],
[cot(d+e*x)^5/(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2),x,14,3/2*b*arctanh(1/2*(b+2*c*cot(d+e*x))/(sqrt(c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(c^(5/2)*e)+2*(2*a+b*cot(d+e*x))/((b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-2*cot(d+e*x)^2*(2*a+b*cot(d+e*x))/((b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(d+e*x))/((b^2+(a-c)^2)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-(3*b^2-8*a*c-2*b*c*cot(d+e*x))*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)/(c^2*(b^2-4*a*c)*e)+arctanh((b^2-(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))-arctanh((b^2-b*cot(d+e*x)*(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))-(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))],
[cot(d+e*x)^3/(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2),x,10,-2*(2*a+b*cot(d+e*x))/((b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(d+e*x))/((b^2+(a-c)^2)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-arctanh((b^2-(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))+arctanh((b^2-b*cot(d+e*x)*(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))-(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))],
[cot(d+e*x)/(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2),x,7,-2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(d+e*x))/((b^2+(a-c)^2)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+arctanh((b^2-(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))-arctanh((b^2-b*cot(d+e*x)*(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))-(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))],
[tan(d+e*x)/(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2),x,13,arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(a^(3/2)*e)-2*(b^2-2*a*c+b*c*cot(d+e*x))/(a*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(d+e*x))/((b^2+(a-c)^2)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-arctanh((b^2-(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))+arctanh((b^2-b*cot(d+e*x)*(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))-(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))],
[tan(d+e*x)^3/(a+b*cot(d+e*x)+c*cot(d+e*x)^2)^(3/2),x,18,-arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(a^(3/2)*e)+3/8*(5*b^2-4*a*c)*arctanh(1/2*(2*a+b*cot(d+e*x))/(sqrt(a)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)))/(a^(7/2)*e)+2*(b^2-2*a*c+b*c*cot(d+e*x))/(a*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))-2*(a*(b^2-2*(a-c)*c)+b*c*(a+c)*cot(d+e*x))/((b^2+(a-c)^2)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+arctanh((b^2-(a-c)*(a-c-sqrt(a^2+b^2-2*a*c+c^2))-b*cot(d+e*x)*(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c+sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2-(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))-arctanh((b^2-b*cot(d+e*x)*(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))-(a-c)*(a-c+sqrt(a^2+b^2-2*a*c+c^2)))/(sqrt(2)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))))*sqrt(2*a-2*c-sqrt(a^2+b^2-2*a*c+c^2))*sqrt(a^2-b^2-2*a*c+c^2+(a-c)*sqrt(a^2+b^2-2*a*c+c^2))/((a^2+b^2-2*a*c+c^2)^(3/2)*e*sqrt(2))-1/4*b*(15*b^2-52*a*c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x)/(a^3*(b^2-4*a*c)*e)-2*(b^2-2*a*c+b*c*cot(d+e*x))*tan(d+e*x)^2/(a*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2))+1/2*(5*b^2-12*a*c)*sqrt(a+b*cot(d+e*x)+c*cot(d+e*x)^2)*tan(d+e*x)^2/(a^2*(b^2-4*a*c)*e)],

# Integrands of the form Cot[d+e x]^m (a+b Cot[d+e x]^2+c Cot[d+e x]^4)^(p/2)
[cot(d+e*x)^5/sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,8,1/4*(b+2*c)*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(c^(3/2)*e)+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a-b+c))-1/2*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)/(c*e)],
[cot(d+e*x)^3/sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,7,-1/2*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(c))-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a-b+c))],
[cot(d+e*x)/sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,4,1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a-b+c))],
[tan(d+e*x)/sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,8,1/2*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a))-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a-b+c))],
[tan(d+e*x)^3/sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,11,-1/4*b*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(a^(3/2)*e)-1/2*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a))+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a-b+c))+1/2*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)*tan(d+e*x)^2/(a*e)],
[cot(d+e*x)^5*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,9,-1/32*(b^3+2*b^2*c-4*b*(a-2*c)*c-8*c^2*(a+2*c))*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(c^(5/2)*e)-1/6*(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2)/(c*e)+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a-b+c)/e+1/16*((b-2*c)*(b+4*c)+2*c*(b+2*c)*cot(d+e*x)^2)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)/(c^2*e)],
[cot(d+e*x)^3*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,8,1/16*(b^2+4*b*c-4*c*(a+2*c))*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(c^(3/2)*e)-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a-b+c)/e-1/8*(b-4*c+2*c*cot(d+e*x)^2)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)/(c*e)],
[cot(d+e*x)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4),x,8,-1/4*(b-2*c)*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(c))+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a-b+c)/e-1/2*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)/e],
[sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)*tan(d+e*x),x,10,1/2*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a)/e-1/2*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(c)/e-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a-b+c)/e],
[sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)*tan(d+e*x)^3,x,22,1/4*b*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(a))-1/2*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a)/e+1/4*b*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(c))-1/4*(b-2*c)*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(e*sqrt(c))-1/2*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(c)/e+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))*sqrt(a-b+c)/e+1/2*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)*tan(d+e*x)^2/e],
[cot(d+e*x)^7/(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2),x,8,-1/2*arctanh(1/2*(b+2*c*cot(d+e*x)^2)/(sqrt(c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(c^(3/2)*e)-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/((a-b+c)^(3/2)*e)+(-a*(b^2-a*(b+2*c))-(b^3+2*a^2*c-a*b*(b+3*c))*cot(d+e*x)^2)/(c*(a-b+c)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))],
[cot(d+e*x)^5/(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2),x,6,1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/((a-b+c)^(3/2)*e)+(-a*(2*a-b)-((a-b)*b+2*a*c)*cot(d+e*x)^2)/((a-b+c)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))],
[cot(d+e*x)^3/(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2),x,6,-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/((a-b+c)^(3/2)*e)+(a*(b-2*c)+(2*a-b)*c*cot(d+e*x)^2)/((a-b+c)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))],
[cot(d+e*x)/(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2),x,6,1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/((a-b+c)^(3/2)*e)+(-b^2+2*a*c+b*c-(b-2*c)*c*cot(d+e*x)^2)/((a-b+c)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))],
[tan(d+e*x)/(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2),x,12,1/2*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(a^(3/2)*e)-1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/((a-b+c)^(3/2)*e)+(-b^2+2*a*c-b*c*cot(d+e*x)^2)/(a*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))+(b^2-2*a*c-b*c+(b-2*c)*c*cot(d+e*x)^2)/((a-b+c)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))],
[tan(d+e*x)^3/(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)^(3/2),x,16,-1/2*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(a^(3/2)*e)-3/4*b*arctanh(1/2*(2*a+b*cot(d+e*x)^2)/(sqrt(a)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/(a^(5/2)*e)+1/2*arctanh(1/2*(2*a-b+(b-2*c)*cot(d+e*x)^2)/(sqrt(a-b+c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)))/((a-b+c)^(3/2)*e)+(b^2-2*a*c+b*c*cot(d+e*x)^2)/(a*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))+(-b^2+2*a*c+b*c-(b-2*c)*c*cot(d+e*x)^2)/((a-b+c)*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))-(b^2-2*a*c+b*c*cot(d+e*x)^2)*tan(d+e*x)^2/(a*(b^2-4*a*c)*e*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4))+1/2*(3*b^2-8*a*c)*sqrt(a+b*cot(d+e*x)^2+c*cot(d+e*x)^4)*tan(d+e*x)^2/(a^2*(b^2-4*a*c)*e)]]:
