# Maple integration test file: "4 Trig functions\4.2 Cosine\4.2.8 (a+b cos)^m (c+d trig)^n.txt"

lst:=[

# Integrands of the form (a+b Cos[c+d x])^m (A+B Trig[c+d x])

# Integrands of the form (a+b Cos[c+d x])^m (A+B Sin[c+d x])
[(A+B*sin(x))/(a+b*cos(x)),x,6,-B*log(a+b*cos(x))/b+2*A*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],
[(A+B*sin(x))/(1+cos(x)),x,5,-B*log(1+cos(x))+A*sin(x)/(1+cos(x))],
[(A+B*sin(x))/(1-cos(x)),x,5,B*log(1-cos(x))-A*sin(x)/(1-cos(x))],
[(b+c+sin(x))/(a+b*cos(x)),x,6,-log(a+b*cos(x))/b+2*(b+c)*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],
[(b+c+sin(x))/(a-b*cos(x)),x,6,log(a-b*cos(x))/b+2*(b+c)*arctan(sqrt(a+b)*tan(1/2*x)/sqrt(a-b))/(sqrt(a-b)*sqrt(a+b))],

# Integrands of the form (a+b Cos[c+d x])^m (A+B Tan[c+d x])
[(A+B*tan(x))/(a+b*cos(x)),x,8,-B*log(cos(x))/a+B*log(a+b*cos(x))/a+2*A*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],

# Integrands of the form (a+b Cos[c+d x])^m (A+B Cot[c+d x])
[(A+B*cot(x))/(a+b*cos(x)),x,7,1/2*B*log(1-cos(x))/(a+b)+1/2*B*log(1+cos(x))/(a-b)-a*B*log(a+b*cos(x))/(a^2-b^2)+2*A*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],

# Integrands of the form (a+b Cos[c+d x])^m (A+B Sec[c+d x])

# Integrands of the form (a+b Cos[c+d x])^m (A+B Csc[c+d x])
[(A+B*csc(x))/(a+b*cos(x)),x,11,1/2*B*log(1-cos(x))/(a+b)-1/2*B*log(1+cos(x))/(a-b)+b*B*log(a+b*cos(x))/(a^2-b^2)+2*A*arctan(sqrt(a-b)*tan(1/2*x)/sqrt(a+b))/(sqrt(a-b)*sqrt(a+b))],

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

# Integrands of the form (a+b Cos[e+f x])^m (c+d Sec[e+f x])^n
[(c+d*sec(e+f*x))^4/(a+b*cos(e+f*x)),x,12,1/2*d^3*(4*a*c-b*d)*arctanh(sin(e+f*x))/(a^2*f)+d*(2*a*c-b*d)*(2*a^2*c^2-2*a*b*c*d+b^2*d^2)*arctanh(sin(e+f*x))/(a^4*f)+2*(a*c-b*d)^4*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a^4*f*sqrt(a-b)*sqrt(a+b))+d^4*tan(e+f*x)/(a*f)+d^2*(6*a^2*c^2-4*a*b*c*d+b^2*d^2)*tan(e+f*x)/(a^3*f)+1/2*d^3*(4*a*c-b*d)*sec(e+f*x)*tan(e+f*x)/(a^2*f)+1/3*d^4*tan(e+f*x)^3/(a*f)],
[(c+d*sec(e+f*x))^3/(a+b*cos(e+f*x)),x,10,1/2*d^3*arctanh(sin(e+f*x))/(a*f)+d*(3*a^2*c^2-3*a*b*c*d+b^2*d^2)*arctanh(sin(e+f*x))/(a^3*f)+2*(a*c-b*d)^3*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a^3*f*sqrt(a-b)*sqrt(a+b))+d^2*(3*a*c-b*d)*tan(e+f*x)/(a^2*f)+1/2*d^3*sec(e+f*x)*tan(e+f*x)/(a*f)],
[(c+d*sec(e+f*x))^2/(a+b*cos(e+f*x)),x,8,d*(2*a*c-b*d)*arctanh(sin(e+f*x))/(a^2*f)+2*(a*c-b*d)^2*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a^2*f*sqrt(a-b)*sqrt(a+b))+d^2*tan(e+f*x)/(a*f)],
[(c+d*sec(e+f*x))/(a+b*cos(e+f*x)),x,5,d*arctanh(sin(e+f*x))/(a*f)+2*(a*c-b*d)*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/(a*f*sqrt(a-b)*sqrt(a+b))],
[1/((a+b*cos(e+f*x))*(c+d*sec(e+f*x))),x,6,2*a*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((a*c-b*d)*f*sqrt(a-b)*sqrt(a+b))-2*d*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((a*c-b*d)*f*sqrt(c-d)*sqrt(c+d))],
[1/((a+b*cos(e+f*x))*(c+d*sec(e+f*x))^2),x,7,-2*d*(2*a*c^2-b*c*d-a*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/((c-d)^(3/2)*(c+d)^(3/2)*(a*c-b*d)^2*f)+d^2*sin(e+f*x)/((a*c-b*d)*(c^2-d^2)*f*(d+c*cos(e+f*x)))+2*a^2*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((a*c-b*d)^2*f*sqrt(a-b)*sqrt(a+b))],
[1/((a+b*cos(e+f*x))*(c+d*sec(e+f*x))^3),x,16,-2*d^3*(3*a*c-2*b*d)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(c^2*(c-d)^(3/2)*(c+d)^(3/2)*(a*c-b*d)^2*f)-d^3*(c^2+2*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(c^2*(c-d)^(5/2)*(c+d)^(5/2)*(a*c-b*d)*f)-1/2*d^3*sin(e+f*x)/(c*(a*c-b*d)*(c^2-d^2)*f*(d+c*cos(e+f*x))^2)+3/2*d^4*sin(e+f*x)/(c*(a*c-b*d)*(c^2-d^2)^2*f*(d+c*cos(e+f*x)))+d^2*(3*a*c-2*b*d)*sin(e+f*x)/(c*(a*c-b*d)^2*(c^2-d^2)*f*(d+c*cos(e+f*x)))+2*a^3*arctan(sqrt(a-b)*tan(1/2*(e+f*x))/sqrt(a+b))/((a*c-b*d)^3*f*sqrt(a-b)*sqrt(a+b))-2*d*(3*a^2*c^2-3*a*b*c*d+b^2*d^2)*arctanh(sqrt(c-d)*tan(1/2*(e+f*x))/sqrt(c+d))/(c^2*(a*c-b*d)^3*f*sqrt(c-d)*sqrt(c+d))],

#  {(c + d*Sec[e + f*x])^(5/2)/(a + b*Cos[e + f*x]), x, 0, 0} 

#  {(c + d*Sec[e + f*x])^(3/2)/(a + b*Cos[e + f*x]), x, 0, 0} 
[(c+d*sec(e+f*x))^(1/2)/(a+b*cos(e+f*x)),x,4,2*cot(e+f*x)*EllipticF(sqrt(c+d*sec(e+f*x))/sqrt(c+d),sqrt((c+d)/(c-d)))*sqrt(c+d)*sqrt(d*(1-sec(e+f*x))/(c+d))*sqrt(-d*(1+sec(e+f*x))/(c-d))/(a*f)+2*(a*c-b*d)*EllipticPi(sqrt(1-sec(e+f*x))/sqrt(2),2*a/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sec(e+f*x))/(c+d))*tan(e+f*x)/(a*(a+b)*f*sqrt(c+d*sec(e+f*x))*sqrt(-tan(e+f*x)^2))],
[1/((a+b*cos(e+f*x))*(c+d*sec(e+f*x))^(1/2)),x,2,2*EllipticPi(sqrt(1-sec(e+f*x))/sqrt(2),2*a/(a+b),sqrt(2*d/(c+d)))*sqrt((c+d*sec(e+f*x))/(c+d))*tan(e+f*x)/((a+b)*f*sqrt(c+d*sec(e+f*x))*sqrt(-tan(e+f*x)^2))],

#  {1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^(3/2)), x, 0, 0} 

#  {1/((a + b*Cos[e + f*x])*(c + d*Sec[e + f*x])^(5/2)), x, 0, 0} 

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

# Integrands of the form (a+b Cos[c+d x])^m (A+B Cos[c+d x]+C Sin[c+d x]^2)
[(A+B*cos(d+e*x)+C*sin(d+e*x))/(a+b*cos(d+e*x)),x,6,B*x/b-C*log(a+b*cos(d+e*x))/(b*e)+2*(A*b-a*B)*arctan(sqrt(a-b)*tan(1/2*(d+e*x))/sqrt(a+b))/(b*e*sqrt(a-b)*sqrt(a+b))],
[(A+B*cos(d+e*x)+C*sin(d+e*x))/(a+b*cos(d+e*x))^2,x,7,2*(a*A-b*B)*arctan(sqrt(a-b)*tan(1/2*(d+e*x))/sqrt(a+b))/((a-b)^(3/2)*(a+b)^(3/2)*e)+C/(b*e*(a+b*cos(d+e*x)))-(A*b-a*B)*sin(d+e*x)/((a^2-b^2)*e*(a+b*cos(d+e*x)))],
[(A+B*cos(d+e*x)+C*sin(d+e*x))/(a+b*cos(d+e*x))^3,x,8,(2*a^2*A+A*b^2-3*a*b*B)*arctan(sqrt(a-b)*tan(1/2*(d+e*x))/sqrt(a+b))/((a-b)^(5/2)*(a+b)^(5/2)*e)+1/2*C/(b*e*(a+b*cos(d+e*x))^2)-1/2*(A*b-a*B)*sin(d+e*x)/((a^2-b^2)*e*(a+b*cos(d+e*x))^2)-1/2*(3*a*A*b-a^2*B-2*b^2*B)*sin(d+e*x)/((a^2-b^2)^2*e*(a+b*cos(d+e*x)))],
[(A+B*cos(d+e*x)+C*sin(d+e*x))/(a+b*cos(d+e*x))^4,x,9,(2*a^3*A+3*a*A*b^2-4*a^2*b*B-b^3*B)*arctan(sqrt(a-b)*tan(1/2*(d+e*x))/sqrt(a+b))/((a-b)^(7/2)*(a+b)^(7/2)*e)+1/3*C/(b*e*(a+b*cos(d+e*x))^3)-1/3*(A*b-a*B)*sin(d+e*x)/((a^2-b^2)*e*(a+b*cos(d+e*x))^3)-1/6*(5*a*A*b-2*a^2*B-3*b^2*B)*sin(d+e*x)/((a^2-b^2)^2*e*(a+b*cos(d+e*x))^2)-1/6*(11*a^2*A*b+4*A*b^3-2*a^3*B-13*a*b^2*B)*sin(d+e*x)/((a^2-b^2)^3*e*(a+b*cos(d+e*x)))]]:
