# Maple integration test file: "4 Trig functions\4.2 Cosine\4.2.9 trig^m (a+b cos^n+c cos^(2 n))^p.txt"

lst:=[

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

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

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

# p>0

# p<0
[sin(x)^5/(a+b*cos(x)+c*cos(x)^2),x,7,-(b^2-c*(a+2*c))*cos(x)/c^3+1/2*b*cos(x)^2/c^2-1/3*cos(x)^3/c+1/2*b*(b^2-2*c*(a+c))*log(a+b*cos(x)+c*cos(x)^2)/c^4+(b^4+2*c^2*(a+c)^2-2*b^2*c*(2*a+c))*arctanh((b+2*c*cos(x))/sqrt(b^2-4*a*c))/(c^4*sqrt(b^2-4*a*c))],
[sin(x)^3/(a+b*cos(x)+c*cos(x)^2),x,7,cos(x)/c-1/2*b*log(a+b*cos(x)+c*cos(x)^2)/c^2-(b^2-2*c*(a+c))*arctanh((b+2*c*cos(x))/sqrt(b^2-4*a*c))/(c^2*sqrt(b^2-4*a*c))],
[sin(x)/(a+b*cos(x)+c*cos(x)^2),x,3,2*arctanh((b+2*c*cos(x))/sqrt(b^2-4*a*c))/sqrt(b^2-4*a*c)],
[csc(x)/(a+b*cos(x)+c*cos(x)^2),x,9,1/2*log(1-cos(x))/(a+b+c)-1/2*log(1+cos(x))/(a-b+c)+1/2*b*log(a+b*cos(x)+c*cos(x)^2)/((a-b+c)*(a+b+c))-(b^2-2*a*c-2*c^2)*arctanh((b+2*c*cos(x))/sqrt(b^2-4*a*c))/((a-b+c)*(a+b+c)*sqrt(b^2-4*a*c))],
[csc(x)^3/(a+b*cos(x)+c*cos(x)^2),x,10,1/2*(b-(a+c)*cos(x))*csc(x)^2/((a-b+c)*(a+b+c))+1/4*(a+2*b+3*c)*log(1-cos(x))/(a+b+c)^2-1/4*(a-2*b+3*c)*log(1+cos(x))/(a-b+c)^2-1/2*b*(b^2-2*c*(a+c))*log(a+b*cos(x)+c*cos(x)^2)/(a^2-b^2+2*a*c+c^2)^2+(b^4+2*c^2*(a+c)^2-2*b^2*c*(2*a+c))*arctanh((b+2*c*cos(x))/sqrt(b^2-4*a*c))/((a^2-b^2+2*a*c+c^2)^2*sqrt(b^2-4*a*c))],
[sin(x)^4/(a+b*cos(x)+c*cos(x)^2),x,10,1/2*x/c+(b^2-c*(a+2*c))*x/c^3-b*sin(x)/c^2+1/2*cos(x)*sin(x)/c-2*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b*(b^2-2*c*(a+c))+(-b^4-2*c^2*(a+c)^2+2*b^2*c*(2*a+c))/sqrt(b^2-4*a*c))/(c^3*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-2*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b^4+2*c^2*(a+c)^2-2*b^2*c*(2*a+c)+b^3*sqrt(b^2-4*a*c)-2*b*c*(a+c)*sqrt(b^2-4*a*c))/(c^3*sqrt(b^2-4*a*c)*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[sin(x)^2/(a+b*cos(x)+c*cos(x)^2),x,7,-x/c+2*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b+(-b^2+2*c*(a+c))/sqrt(b^2-4*a*c))/(c*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))+2*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b+(b^2-2*c*(a+c))/sqrt(b^2-4*a*c))/(c*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[csc(x)^2/(a+b*cos(x)+c*cos(x)^2),x,9,-1/2*sin(x)/((a+b+c)*(1-cos(x)))+1/2*sin(x)/((a-b+c)*(1+cos(x)))-2*b*c*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(1+(b^2-2*c*(a+c))/(b*sqrt(b^2-4*a*c)))/((a-b+c)*(a+b+c)*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-2*b*c*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(1+(-b^2+2*c*(a+c))/(b*sqrt(b^2-4*a*c)))/((a-b+c)*(a+b+c)*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[sin(x)/(-2+cos(x)+cos(x)^2),x,4,-1/3*log(1-cos(x))+1/3*log(2+cos(x))],
[sin(x)/(4-5*cos(x)+cos(x)^2),x,4,1/3*log(1-cos(x))-1/3*log(4-cos(x))],
[sin(x)/(3-2*cos(x)+cos(x)^2),x,3,arctan((1-cos(x))/sqrt(2))/sqrt(2)],
[sin(x)/(13-4*cos(x)+cos(x)^2)^2,x,4,-1/54*arctan(1/3*(-2+cos(x)))+1/18*(2-cos(x))/(13-4*cos(x)+cos(x)^2)],

# Integrands of the form Sin[d+e x]^m (a+b Cos[d+e x]^2+c Cos[d+e x]^4)^p

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

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

# p>0

# p<0
[cos(x)^4/(a+b*cos(x)+c*cos(x)^2),x,10,1/2*x/c+(b^2-a*c)*x/c^3-b*sin(x)/c^2+1/2*cos(x)*sin(x)/c-2*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b^3-2*a*b*c+(-b^4+4*a*b^2*c-2*a^2*c^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-2*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b^3-2*a*b*c+(b^4-4*a*b^2*c+2*a^2*c^2)/sqrt(b^2-4*a*c))/(c^3*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[cos(x)^3/(a+b*cos(x)+c*cos(x)^2),x,8,-b*x/c^2+sin(x)/c+2*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b^2-a*c-b^3/sqrt(b^2-4*a*c)+3*a*b*c/sqrt(b^2-4*a*c))/(c^2*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))+2*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b^2-a*c+b^3/sqrt(b^2-4*a*c)-3*a*b*c/sqrt(b^2-4*a*c))/(c^2*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[cos(x)^2/(a+b*cos(x)+c*cos(x)^2),x,7,x/c-2*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b+(-b^2+2*a*c)/sqrt(b^2-4*a*c))/(c*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-2*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b+(b^2-2*a*c)/sqrt(b^2-4*a*c))/(c*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[cos(x)/(a+b*cos(x)+c*cos(x)^2),x,6,2*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(1-b/sqrt(b^2-4*a*c))/(sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))+2*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(1+b/sqrt(b^2-4*a*c))/(sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[1/(a+b*cos(x)+c*cos(x)^2),x,5,4*c*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))/(sqrt(b^2-4*a*c)*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-4*c*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))/(sqrt(b^2-4*a*c)*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[sec(x)/(a+b*cos(x)+c*cos(x)^2),x,8,arctanh(sin(x))/a-2*c*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(1+b/sqrt(b^2-4*a*c))/(a*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))-2*c*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(1-b/sqrt(b^2-4*a*c))/(a*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))],
[sec(x)^2/(a+b*cos(x)+c*cos(x)^2),x,10,-b*arctanh(sin(x))/a^2+2*b*c*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(1+(b^2-2*a*c)/(b*sqrt(b^2-4*a*c)))/(a^2*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))+2*b*c*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(1+(-b^2+2*a*c)/(b*sqrt(b^2-4*a*c)))/(a^2*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))+tan(x)/a],
[sec(x)^3/(a+b*cos(x)+c*cos(x)^2),x,12,1/2*arctanh(sin(x))/a+(b^2-a*c)*arctanh(sin(x))/a^3-2*c*arctan(sqrt(b-2*c-sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c-sqrt(b^2-4*a*c)))*(b^3-3*a*b*c+(b^2-a*c)*sqrt(b^2-4*a*c))/(a^3*sqrt(b^2-4*a*c)*sqrt(b-2*c-sqrt(b^2-4*a*c))*sqrt(b+2*c-sqrt(b^2-4*a*c)))+2*c*arctan(sqrt(b-2*c+sqrt(b^2-4*a*c))*tan(1/2*x)/sqrt(b+2*c+sqrt(b^2-4*a*c)))*(b^3-3*a*b*c-(b^2-a*c)*sqrt(b^2-4*a*c))/(a^3*sqrt(b^2-4*a*c)*sqrt(b-2*c+sqrt(b^2-4*a*c))*sqrt(b+2*c+sqrt(b^2-4*a*c)))-b*tan(x)/a^2+1/2*sec(x)*tan(x)/a]]:

# Integrands of the form Cos[d+e x]^m (a+b Cos[d+e x]^2+c Cos[d+e x]^4)^p

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