\(\int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [339]

Optimal result

Integrand size = 32, antiderivative size = 386 \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i \left (a^2-b^2\right ) f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2} \] Output:

1/2*I*b*(f*x+e)^2/a^2/f+1/2*I*(a^2-b^2)*(f*x+e)^2/a^2/b/f-f*arctanh(cos(d* 
x+c))/a/d^2-(f*x+e)*csc(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^2/b/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^2/b/d-b*(f*x+e)*ln(1-exp(2*I*(d*x+c)))/a^2/d+I*(a^2 
-b^2)*f*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b/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^2/b/d^2+1/2*I* 
b*f*polylog(2,exp(2*I*(d*x+c)))/a^2/d^2
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1111\) vs. \(2(386)=772\).

Time = 9.01 (sec) , antiderivative size = 1111, normalized size of antiderivative = 2.88 \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[((e + f*x)*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
 

Output:

((-(d*e*Cos[(c + d*x)/2]) + c*f*Cos[(c + d*x)/2] - f*(c + d*x)*Cos[(c + d* 
x)/2])*Csc[(c + d*x)/2])/(2*a*d^2) - (b*e*Log[Sin[c + d*x]])/(a^2*d) + (b* 
c*f*Log[Sin[c + d*x]])/(a^2*d^2) - (e*Log[1 + (b*Sin[c + d*x])/a])/(b*d) + 
 (b*e*Log[1 + (b*Sin[c + d*x])/a])/(a^2*d) + (c*f*Log[1 + (b*Sin[c + d*x]) 
/a])/(b*d^2) - (b*c*f*Log[1 + (b*Sin[c + d*x])/a])/(a^2*d^2) + (f*Log[Tan[ 
(c + d*x)/2]])/(a*d^2) - (f*(((c + d*x)*Log[a + b*Sin[c + d*x]])/b - ((-1/ 
2*I)*(-c + Pi/2 - d*x)^2 + (4*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTan[(( 
a - b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[a^2 - b^2]] + (-c + Pi/2 - d*x + 2*A 
rcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a - Sqrt[a^2 - b^2])*E^(I*(-c + 
Pi/2 - d*x)))/b] + (-c + Pi/2 - d*x - 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*L 
og[1 + ((a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] - (-c + Pi/2 - d 
*x)*Log[a + b*Sin[c + d*x]] - I*(PolyLog[2, ((-a - Sqrt[a^2 - b^2])*E^(I*( 
-c + Pi/2 - d*x)))/b] + PolyLog[2, ((-a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 
 - d*x)))/b]))/b))/d^2 + (b^2*f*(((c + d*x)*Log[a + b*Sin[c + d*x]])/b - ( 
(-1/2*I)*(-c + Pi/2 - d*x)^2 + (4*I)*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]]*ArcTa 
n[((a - b)*Tan[(-c + Pi/2 - d*x)/2])/Sqrt[a^2 - b^2]] + (-c + Pi/2 - d*x + 
 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2]])*Log[1 + ((a - Sqrt[a^2 - b^2])*E^(I*(- 
c + Pi/2 - d*x)))/b] + (-c + Pi/2 - d*x - 2*ArcSin[Sqrt[(a + b)/b]/Sqrt[2] 
])*Log[1 + ((a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] - (-c + Pi/2 
 - d*x)*Log[a + b*Sin[c + d*x]] - I*(PolyLog[2, ((-a - Sqrt[a^2 - b^2])...
 

Rubi [F]

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.

\(\displaystyle \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 5054

\(\displaystyle \frac {\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\int (e+f x) \cos (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\frac {f \int -\sin (c+d x)dx}{d}-\frac {(e+f x) \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx+\frac {f \int \sin (c+d x)dx}{d}-\frac {(e+f x) \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx+\frac {f \int \sin (c+d x)dx}{d}-\frac {(e+f x) \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4910

\(\displaystyle \frac {\frac {f \int \csc (c+d x)dx}{d}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {f \int \csc (c+d x)dx}{d}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5054

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (\frac {\int (e+f x) \cot (c+d x)dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (\frac {\int -\left ((e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (\frac {-\int (e+f x) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)}{1+e^{i (2 c+2 d x+\pi )}}dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {i f \int \log \left (1+e^{i (2 c+2 d x+\pi )}\right )dx}{2 d}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\int (e+f x) \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (\frac {f \int e^{-i (2 c+2 d x+\pi )} \log \left (1+e^{i (2 c+2 d x+\pi )}\right )de^{i (2 c+2 d x+\pi )}}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\int (e+f x) \cos (c+d x) \sin (c+d x)dx-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {-\int (e+f x) \cos (c+d x) \sin (c+d x)dx+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 4904

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {f \int \sin ^2(c+d x)dx}{2 d}+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {f \int \sin (c+d x)^2dx}{2 d}+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {\frac {f \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}+2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 5036

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x) \cos (c+d x)dx}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}+\frac {a \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\int (e+f x) \cos (c+d x) \sin (c+d x)dx}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 4904

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \int \sin ^2(c+d x)dx}{2 d}}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}}{a}-\frac {b \left (-\frac {b \left (-\frac {\left (a^2-b^2\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)}dx}{b^2}-\frac {\frac {(e+f x) \sin ^2(c+d x)}{2 d}-\frac {f \int \sin (c+d x)^2dx}{2 d}}{b}+\frac {a \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{b^2}\right )}{a}+\frac {2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{i (2 c+2 d x+\pi )}\right )}{4 d^2}-\frac {i (e+f x) \log \left (1+e^{i (2 c+2 d x+\pi )}\right )}{2 d}\right )-\frac {(e+f x) \sin ^2(c+d x)}{2 d}+\frac {f \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

Input:

Int[((e + f*x)*Cos[c + d*x]*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2620

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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]
 

rule 3118

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ 
[{c, d}, x]
 

rule 3777

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]
 

rule 4202

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]
 

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

rule 4904

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]
 

rule 4908

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]
 

rule 4910

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d 
_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Csc[a + b*x]^n/(b*n)), x 
] + Simp[d*(m/(b*n))   Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Free 
Q[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]
 

rule 5036

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]
 

rule 5054

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1731 vs. \(2 (351 ) = 702\).

Time = 2.03 (sec) , antiderivative size = 1732, normalized size of antiderivative = 4.49

method	result	size
risch	\(\text {Expression too large to display}\)	\(1732\)

Input:

int((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x,method=_RETURNVERBO 
SE)
 

Output:

1/d^2/a*f*ln(exp(I*(d*x+c))-1)-1/d^2/a*f*ln(exp(I*(d*x+c))+1)-2/b/d^2*f*c* 
ln(exp(I*(d*x+c)))+I/b/d^2*f*c^2+b/d/a^2*e*ln(I*b*exp(2*I*(d*x+c))-2*a*exp 
(I*(d*x+c))-I*b)-b/d/a^2*e*ln(exp(I*(d*x+c))-1)-b/d/a^2*e*ln(exp(I*(d*x+c) 
)+1)-I/b*e*x-1/b/d*e*ln(I*b*exp(2*I*(d*x+c))-2*a*exp(I*(d*x+c))-I*b)+2/b/d 
*e*ln(exp(I*(d*x+c)))+1/2*I/b*f*x^2+b^3/d/a^2*f/(-a^2+b^2)*ln((I*a+exp(I*( 
d*x+c))*b-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*x+b^3/d/a^2*f/(-a^2+b^ 
2)*ln((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x+1/ 
b/d*a^2*f/(-a^2+b^2)*ln((I*a+exp(I*(d*x+c))*b-(-a^2+b^2)^(1/2))/(I*a-(-a^2 
+b^2)^(1/2)))*x+1/b/d*a^2*f/(-a^2+b^2)*ln((I*a+exp(I*(d*x+c))*b+(-a^2+b^2) 
^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x+1/b/d^2*a^2*f/(-a^2+b^2)*ln((I*a+exp(I*( 
d*x+c))*b-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*c+1/b/d^2*a^2*f/(-a^2+ 
b^2)*ln((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*c+ 
b^3/d^2/a^2*f/(-a^2+b^2)*ln((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+( 
-a^2+b^2)^(1/2)))*c+b^3/d^2/a^2*f/(-a^2+b^2)*ln((I*a+exp(I*(d*x+c))*b-(-a^ 
2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))*c-I*b^3/d^2/a^2*f/(-a^2+b^2)*dilog(( 
I*a+exp(I*(d*x+c))*b-(-a^2+b^2)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))-I*b^3/d^2/a 
^2*f/(-a^2+b^2)*dilog((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b 
^2)^(1/2)))-I/b/d^2*a^2*f/(-a^2+b^2)*dilog((I*a+exp(I*(d*x+c))*b-(-a^2+b^2 
)^(1/2))/(I*a-(-a^2+b^2)^(1/2)))-I/b/d^2*a^2*f/(-a^2+b^2)*dilog((I*a+exp(I 
*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))-2*I*(f*x+e)*exp(I...
 

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1419 vs. \(2 (343) = 686\).

Time = 0.30 (sec) , antiderivative size = 1419, normalized size of antiderivative = 3.68 \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="f 
ricas")
 

Output:

-1/2*(2*a*b*d*f*x - I*b^2*f*dilog(cos(d*x + c) + I*sin(d*x + c))*sin(d*x + 
 c) + I*b^2*f*dilog(cos(d*x + c) - I*sin(d*x + c))*sin(d*x + c) + I*b^2*f* 
dilog(-cos(d*x + c) + I*sin(d*x + c))*sin(d*x + c) - I*b^2*f*dilog(-cos(d* 
x + c) - I*sin(d*x + c))*sin(d*x + c) + 2*a*b*d*e - 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))*s 
qrt(-(a^2 - b^2)/b^2) - b)/b + 1)*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)*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)*sin(d*x + c) + I*(a^2 - b^2)*f*dilog((-I*a*co 
s(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)*sin(d*x + c) + ((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)*sin(d*x + c) + ((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)*sin(d*x 
 + c) + ((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)*sin(d*x + c) + ((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)*sin(d*x + c) + ((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)...
 

Sympy [F]

\[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)
 

Output:

Integral((e + f*x)*cos(c + d*x)*cot(c + d*x)**2/(a + b*sin(c + d*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="m 
axima")
 

Output:

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
 

Giac [F]

\[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cos \left (d x + c\right ) \cot \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm="g 
iac")
 

Output:

integrate((f*x + e)*cos(d*x + c)*cot(d*x + c)^2/(b*sin(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \] Input:

int((cos(c + d*x)*cot(c + d*x)^2*(e + f*x))/(a + b*sin(c + d*x)),x)
                                                                                    
                                                                                    
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (f x +e \right ) \cos \left (d x +c \right ) \cot \left (d x +c \right )^{2}}{\sin \left (d x +c \right ) b +a}d x \] Input:

int((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
 

Output:

int((f*x+e)*cos(d*x+c)*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)