Integrand size = 32, antiderivative size = 379 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {i (e+f x)^2}{2 a f}-\frac {i \left (a^2-b^2\right ) (e+f x)^2}{2 a b^2 f}-\frac {f \cos (c+d x)}{b d^2}+\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d}+\frac {(e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a d}-\frac {i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^2 d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a d^2}-\frac {(e+f x) \sin (c+d x)}{b d} \]
-1/2*I*(f*x+e)^2/a/f-1/2*I*(a^2-b^2)*(f*x+e)^2/a/b^2/f-f*cos(d*x+c)/b/d^2+ (f*x+e)*ln(1-exp(2*I*(d*x+c)))/a/d+(a^2-b^2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c ))/(a-(a^2-b^2)^(1/2)))/a/b^2/d+(a^2-b^2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c))/ (a+(a^2-b^2)^(1/2)))/a/b^2/d-1/2*I*f*polylog(2,exp(2*I*(d*x+c)))/a/d^2-I*( a^2-b^2)*f*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a/b^2/d^2-I*( a^2-b^2)*f*polylog(2,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a/b^2/d^2-(f* x+e)*sin(d*x+c)/b/d
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(884\) vs. \(2(379)=758\).
Time = 3.18 (sec) , antiderivative size = 884, normalized size of antiderivative = 2.33 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-a b f \cos (c+d x)+a^2 d e \log \left (1+\frac {b \sin (c+d x)}{a}\right )-b^2 d e \log \left (1+\frac {b \sin (c+d x)}{a}\right )-a^2 c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )+b^2 c f \log \left (1+\frac {b \sin (c+d x)}{a}\right )+b^2 d e (\log (\cos (c+d x))+\log (\tan (c+d x)))-b^2 c f (\log (\cos (c+d x))+\log (\tan (c+d x)))+\frac {1}{8} a^2 f \left (i (-2 c+\pi -2 d x)^2-32 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \cot \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}{\sqrt {a^2-b^2}}\right )-4 \left (-2 c+\pi -2 d x+4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1-\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )-4 \left (-2 c+\pi -2 d x-4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+4 (-2 c+\pi -2 d x) \log (a+b \sin (c+d x))+8 (c+d x) \log (a+b \sin (c+d x))+8 i \left (\operatorname {PolyLog}\left (2,\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )\right )\right )-\frac {1}{8} b^2 f \left (i (-2 c+\pi -2 d x)^2-32 i \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right ) \arctan \left (\frac {(a-b) \cot \left (\frac {1}{4} (2 c+\pi +2 d x)\right )}{\sqrt {a^2-b^2}}\right )-4 \left (-2 c+\pi -2 d x+4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1-\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )-4 \left (-2 c+\pi -2 d x-4 \arcsin \left (\frac {\sqrt {\frac {a+b}{b}}}{\sqrt {2}}\right )\right ) \log \left (1+\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+4 (-2 c+\pi -2 d x) \log (a+b \sin (c+d x))+8 (c+d x) \log (a+b \sin (c+d x))+8 i \left (\operatorname {PolyLog}\left (2,\frac {i \left (-a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )+\operatorname {PolyLog}\left (2,-\frac {i \left (a+\sqrt {a^2-b^2}\right ) e^{-i (c+d x)}}{b}\right )\right )\right )+b^2 f \left ((c+d x) \log \left (1-e^{2 i (c+d x)}\right )-\frac {1}{2} i \left ((c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )\right )\right )-a b d (e+f x) \sin (c+d x)}{a b^2 d^2} \]
(-(a*b*f*Cos[c + d*x]) + a^2*d*e*Log[1 + (b*Sin[c + d*x])/a] - b^2*d*e*Log [1 + (b*Sin[c + d*x])/a] - a^2*c*f*Log[1 + (b*Sin[c + d*x])/a] + b^2*c*f*L og[1 + (b*Sin[c + d*x])/a] + b^2*d*e*(Log[Cos[c + d*x]] + Log[Tan[c + d*x] ]) - b^2*c*f*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]) + (a^2*f*(I*(-2*c + P i - 2*d*x)^2 - (32*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Cot[ (2*c + Pi + 2*d*x)/4])/Sqrt[a^2 - b^2]] - 4*(-2*c + Pi - 2*d*x + 4*ArcSin[ Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 - (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] - 4*(-2*c + Pi - 2*d*x - 4*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + (I*(a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + 4*(-2*c + Pi - 2*d*x)*L og[a + b*Sin[c + d*x]] + 8*(c + d*x)*Log[a + b*Sin[c + d*x]] + (8*I)*(Poly Log[2, (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + PolyLog[2, ((-I)* (a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))])))/8 - (b^2*f*(I*(-2*c + Pi - 2*d*x)^2 - (32*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[((a - b)*Cot[(2*c + Pi + 2*d*x)/4])/Sqrt[a^2 - b^2]] - 4*(-2*c + Pi - 2*d*x + 4*ArcSin[Sqrt [(a + b)/b]/Sqrt[2]])*Log[1 - (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x) ))] - 4*(-2*c + Pi - 2*d*x - 4*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + (I *(a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + 4*(-2*c + Pi - 2*d*x)*Log[a + b*Sin[c + d*x]] + 8*(c + d*x)*Log[a + b*Sin[c + d*x]] + (8*I)*(PolyLog[ 2, (I*(-a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))] + PolyLog[2, ((-I)*(a + Sqrt[a^2 - b^2]))/(b*E^(I*(c + d*x)))])))/8 + b^2*f*((c + d*x)*Log[1 -...
Time = 2.67 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.22, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {5054, 4908, 3042, 25, 4202, 2620, 2715, 2838, 4904, 3042, 3115, 24, 5036, 3042, 3777, 25, 3042, 3118, 4904, 3042, 3115, 24, 5030, 2620, 2715, 2838}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 5054 |
| \(\displaystyle \frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4908 |
| \(\displaystyle \frac {\int (e+f x) \cot (c+d x)dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\left ((e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\int (e+f x) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\) |
| \(\Big \downarrow \) 4202 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)}{1+e^{i (2 c+2 d x+\pi )}}dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {i f \int \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\int (e+f x) \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {f \int e^{-i (2 c+2 d x+\pi )} \log \left (1+e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\int (e+f x) \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\int (e+f x) \cos (c+d x) \sin (c+d x)dx+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {f \int \sin ^2(c+d x)dx}{2 d}+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {f \int \sin (c+d x)^2dx}{2 d}+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 5036 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x) \cos (c+d x)dx}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 4904 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \int \sin ^2(c+d x)dx}{2 d}}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \int \sin (c+d x)^2dx}{2 d}}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\) |
| \(\Big \downarrow \) 5030 |
| \(\displaystyle \frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}-\frac {b \left (-\frac {\left (a^2-b^2\right ) \left (\int \frac {e^{i (c+d x)} (e+f x)}{a-i b e^{i (c+d x)}-\sqrt {a^2-b^2}}dx+\int \frac {e^{i (c+d x)} (e+f x)}{a-i b e^{i (c+d x)}+\sqrt {a^2-b^2}}dx-\frac {i (e+f x)^2}{2 b f}\right )}{b^2}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle \frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}-\frac {b \left (-\frac {\left (a^2-b^2\right ) \left (-\frac {f \int \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )dx}{b d}-\frac {f \int \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )dx}{b d}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i (e+f x)^2}{2 b f}\right )}{b^2}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}-\frac {b \left (-\frac {\left (a^2-b^2\right ) \left (\frac {i f \int e^{-i (c+d x)} \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (c+d x)}}{b d^2}+\frac {i f \int e^{-i (c+d x)} \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (c+d x)}}{b d^2}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i (e+f x)^2}{2 b f}\right )}{b^2}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}-\frac {b \left (-\frac {\left (a^2-b^2\right ) \left (-\frac {i f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}-\frac {i f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}+\frac {(e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {i (e+f x)^2}{2 b f}\right )}{b^2}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}}{b}\right )}{a}\) |
(((-1/2*I)*(e + f*x)^2)/f + (2*I)*(((-1/2*I)*(e + f*x)*Log[1 + E^(I*(2*c + Pi + 2*d*x))])/d - (f*PolyLog[2, -E^(I*(2*c + Pi + 2*d*x))])/(4*d^2)) - ( (e + f*x)*Sin[c + d*x]^2)/(2*d) + (f*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2 *d)))/(2*d))/a - (b*(-(((a^2 - b^2)*(((-1/2*I)*(e + f*x)^2)/(b*f) + ((e + f*x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d) + ((e + f *x)*Log[1 - (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d) - (I*f*Pol yLog[2, (I*b*E^(I*(c + d*x)))/(a - Sqrt[a^2 - b^2])])/(b*d^2) - (I*f*PolyL og[2, (I*b*E^(I*(c + d*x)))/(a + Sqrt[a^2 - b^2])])/(b*d^2)))/b^2) + (a*(( f*Cos[c + d*x])/d^2 + ((e + f*x)*Sin[c + d*x])/d))/b^2 - (((e + f*x)*Sin[c + d*x]^2)/(2*d) - (f*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/(2*d))/b) )/a
3.4.31.3.1 Defintions of rubi rules used
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I *((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I Int[(c + d*x)^m*(E^(2*I*( e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt Q[m, 0]
Int[Cos[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x _)]^(n_.), x_Symbol] :> Simp[(c + d*x)^m*(Sin[a + b*x]^(n + 1)/(b*(n + 1))) , x] - Simp[d*(m/(b*(n + 1))) Int[(c + d*x)^(m - 1)*Sin[a + b*x]^(n + 1), x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d _.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^ (p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x] /; Fr eeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[ (c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-I)*((e + f*x)^(m + 1)/(b*f*(m + 1 ))), x] + (Int[(e + f*x)^m*(E^(I*(c + d*x))/(a - Rt[a^2 - b^2, 2] - I*b*E^( I*(c + d*x)))), x] + Int[(e + f*x)^m*(E^(I*(c + d*x))/(a + Rt[a^2 - b^2, 2] - I*b*E^(I*(c + d*x)))), x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && PosQ[a^2 - b^2]
Int[(Cos[(c_.) + (d_.)*(x_)]^(n_)*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.) *Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[a/b^2 Int[(e + f*x)^m*Cos[c + d*x]^(n - 2), x], x] + (-Simp[1/b Int[(e + f*x)^m*Cos[c + d*x]^(n - 2)* Sin[c + d*x], x], x] - Simp[(a^2 - b^2)/b^2 Int[(e + f*x)^m*(Cos[c + d*x] ^(n - 2)/(a + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[n, 1] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(Cos[(c_.) + (d_.)*(x_)]^(p_.)*Cot[(c_.) + (d_.)*(x_)]^(n_.)*((e_.) + ( f_.)*(x_))^(m_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp [1/a Int[(e + f*x)^m*Cos[c + d*x]^p*Cot[c + d*x]^n, x], x] - Simp[b/a I nt[(e + f*x)^m*Cos[c + d*x]^(p + 1)*(Cot[c + d*x]^(n - 1)/(a + b*Sin[c + d* x])), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1720 vs. \(2 (344 ) = 688\).
Time = 2.51 (sec) , antiderivative size = 1721, normalized size of antiderivative = 4.54
-1/d*b^2*f/a/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(- a^2+b^2)^(1/2)))*x-1/d*b^2*f/a/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b ^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*x-1/2*I*(d*x*f-I*f+d*e)/b/d^2*exp(-I*(d *x+c))+I/d^2*b^2*f/a/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/ 2))/(I*a-(-a^2+b^2)^(1/2)))+I/d^2*b^2*f/a/(-a^2+b^2)*dilog((I*a+b*exp(I*(d *x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))-1/d^2/b^2*a^3*f/(-a^2+b^2 )*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c+1/d *f/a*ln(exp(I*(d*x+c))+1)*x+I/d^2*f/a*dilog(exp(I*(d*x+c)))-1/d^2*f*c/a*ln (exp(I*(d*x+c))-1)+I/d^2/b^2*f*a^3/(-a^2+b^2)*dilog((I*a+b*exp(I*(d*x+c))+ (-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+I/d^2/b^2*f*a^3/(-a^2+b^2)*dilog ((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))-1/d^2*b^2 *f/a/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2) ^(1/2)))*c+1/d*e/a*ln(exp(I*(d*x+c))-1)+1/d*e/a*ln(exp(I*(d*x+c))+1)-1/d^2 /b^2*a^3*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^ 2+b^2)^(1/2)))*c+1/d/b^2*a*e*ln(I*b*exp(2*I*(d*x+c))-I*b-2*a*exp(I*(d*x+c) ))-2/d/b^2*a*e*ln(exp(I*(d*x+c)))+1/d^2*f*c/a*ln(I*b*exp(2*I*(d*x+c))-I*b- 2*a*exp(I*(d*x+c)))-I/d^2*f/a*dilog(exp(I*(d*x+c))+1)-1/d/b^2*a^3*f/(-a^2+ b^2)*ln((I*a+b*exp(I*(d*x+c))-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*x- 1/d/b^2*a^3*f/(-a^2+b^2)*ln((I*a+b*exp(I*(d*x+c))+(-a^2+b^2)^(1/2))/(I*a+( -a^2+b^2)^(1/2)))*x+1/2*I*(d*x*f+I*f+d*e)/b/d^2*exp(I*(d*x+c))-1/d*e/a*...
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1293 vs. \(2 (338) = 676\).
Time = 0.50 (sec) , antiderivative size = 1293, normalized size of antiderivative = 3.41 \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]
-1/2*(2*a*b*f*cos(d*x + c) + I*b^2*f*dilog(cos(d*x + c) + I*sin(d*x + c)) - I*b^2*f*dilog(cos(d*x + c) - I*sin(d*x + c)) - I*b^2*f*dilog(-cos(d*x + c) + I*sin(d*x + c)) + I*b^2*f*dilog(-cos(d*x + c) - I*sin(d*x + c)) + I*( a^2 - b^2)*f*dilog((I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) + I*(a^2 - b^2)*f*dil og((I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin(d*x + c) )*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - I*(a^2 - b^2)*f*dilog((-I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b + 1) - I*(a^2 - b^2)*f*dilog((-I*a*cos(d*x + c) - a*sin(d *x + c) - (b*cos(d*x + c) - I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/ b + 1) - ((a^2 - b^2)*d*e - (a^2 - b^2)*c*f)*log(2*b*cos(d*x + c) + 2*I*b* sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - ((a^2 - b^2)*d*e - (a ^2 - b^2)*c*f)*log(2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) - ((a^2 - b^2)*d*e - (a^2 - b^2)*c*f)*log(-2*b*cos(d* x + c) + 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) + 2*I*a) - ((a^2 - b^2)*d*e - (a^2 - b^2)*c*f)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) + 2*b*sqrt(-(a^2 - b^2)/b^2) - 2*I*a) - ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c* f)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) + (b*cos(d*x + c) + I*b*sin(d*x + c))*sqrt(-(a^2 - b^2)/b^2) - b)/b) - ((a^2 - b^2)*d*f*x + (a^2 - b^2)*c *f)*log(-(I*a*cos(d*x + c) - a*sin(d*x + c) - (b*cos(d*x + c) + I*b*sin...
\[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos ^{2}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Exception generated. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
\[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )}{b \sin \left (d x + c\right ) + a} \,d x } \]
Timed out. \[ \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \]