\(\int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\) [335]

Optimal result

Integrand size = 32, antiderivative size = 512 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {(e+f x)^2}{4 b f}+\frac {\left (a^2-b^2\right ) (e+f x)^2}{2 b^3 f}-\frac {2 (e+f x) \text {arctanh}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a b^2 d}-\frac {f \cos ^2(c+d x)}{4 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {i f \operatorname {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \operatorname {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {\left (a^2-b^2\right ) f \sin (c+d x)}{a b^2 d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d} \] Output:

-1/4*(f*x+e)^2/b/f+1/2*(a^2-b^2)*(f*x+e)^2/b^3/f-2*(f*x+e)*arctanh(exp(I*( 
d*x+c)))/a/d+(f*x+e)*cos(d*x+c)/a/d+(a^2-b^2)*(f*x+e)*cos(d*x+c)/a/b^2/d-1 
/4*f*cos(d*x+c)^2/b/d^2+I*(a^2-b^2)^(3/2)*(f*x+e)*ln(1-I*b*exp(I*(d*x+c))/ 
(a-(a^2-b^2)^(1/2)))/a/b^3/d-I*(a^2-b^2)^(3/2)*(f*x+e)*ln(1-I*b*exp(I*(d*x 
+c))/(a+(a^2-b^2)^(1/2)))/a/b^3/d+I*f*polylog(2,-exp(I*(d*x+c)))/a/d^2-I*f 
*polylog(2,exp(I*(d*x+c)))/a/d^2+(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I*(d* 
x+c))/(a-(a^2-b^2)^(1/2)))/a/b^3/d^2-(a^2-b^2)^(3/2)*f*polylog(2,I*b*exp(I 
*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a/b^3/d^2-f*sin(d*x+c)/a/d^2-(a^2-b^2)*f*si 
n(d*x+c)/a/b^2/d^2-1/2*(f*x+e)*cos(d*x+c)*sin(d*x+c)/b/d
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2194\) vs. \(2(512)=1024\).

Time = 17.19 (sec) , antiderivative size = 2194, normalized size of antiderivative = 4.29 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Result too large to show} \] Input:

Integrate[((e + f*x)*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]
 

Output:

-1/4*((-2*a^2 + 3*b^2)*(c + d*x)*(2*d*e - 2*c*f + f*(c + d*x)))/(b^3*d^2) 
+ (a*(d*e - c*f + f*(c + d*x))*Cos[c + d*x])/(b^2*d^2) - (f*Cos[2*(c + d*x 
)])/(8*b*d^2) + (e*Log[Tan[(c + d*x)/2]])/(a*d) - (c*f*Log[Tan[(c + d*x)/2 
]])/(a*d^2) + (f*((c + d*x)*(Log[1 - E^(I*(c + d*x))] - Log[1 + E^(I*(c + 
d*x))]) + I*(PolyLog[2, -E^(I*(c + d*x))] - PolyLog[2, E^(I*(c + d*x))]))) 
/(a*d^2) - (a*f*Sin[c + d*x])/(b^2*d^2) - ((d*e - c*f + f*(c + d*x))*Sin[2 
*(c + d*x)])/(4*b*d^2) + (((2*(d*e - c*f)*ArcTan[(b + a*Tan[(c + d*x)/2])/ 
Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (I*f*Log[1 + I*Tan[(c + d*x)/2]]*Log[( 
b - Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I*a + b - Sqrt[-a^2 + b^2])])/ 
Sqrt[-a^2 + b^2] + (I*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[-((b - Sqrt[-a^2 + 
 b^2] + a*Tan[(c + d*x)/2])/(I*a - b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^ 
2] - (I*f*Log[1 - I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c 
 + d*x)/2])/((-I)*a + b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] + (I*f*Log[ 
1 + I*Tan[(c + d*x)/2]]*Log[(b + Sqrt[-a^2 + b^2] + a*Tan[(c + d*x)/2])/(I 
*a + b + Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] - (I*f*PolyLog[2, (a*(1 - I* 
Tan[(c + d*x)/2]))/(a + I*(b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2] + (I* 
f*PolyLog[2, (a*(1 + I*Tan[(c + d*x)/2]))/(a - I*(b + Sqrt[-a^2 + b^2]))]) 
/Sqrt[-a^2 + b^2] + (I*f*PolyLog[2, (a*(I + Tan[(c + d*x)/2]))/(I*a - b + 
Sqrt[-a^2 + b^2])])/Sqrt[-a^2 + b^2] - (I*f*PolyLog[2, (a + I*a*Tan[(c + d 
*x)/2])/(a + I*(-b + Sqrt[-a^2 + b^2]))])/Sqrt[-a^2 + b^2])*(-(((a^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.

\(\displaystyle \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 5054

\(\displaystyle \frac {\int (e+f x) \cos ^3(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx-\int (e+f x) \cos ^2(c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4905

\(\displaystyle \frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx-\frac {f \int \cos ^3(c+d x)dx}{3 d}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx-\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx}{3 d}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3113

\(\displaystyle \frac {\frac {f \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{3 d^2}+\int (e+f x) \cos (c+d x) \cot (c+d x)dx+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\int (e+f x) \cos (c+d x) \cot (c+d x)dx+\frac {f \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{3 d^2}+\frac {(e+f x) \cos ^3(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 5036

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 4905

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3113

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5036

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

\(\Big \downarrow \) 17

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3777

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3117

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

\(\Big \downarrow \) 3804

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

\(\Big \downarrow \) 2694

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

Input:

Int[((e + f*x)*Cos[c + d*x]^3*Cot[c + d*x])/(a + b*Sin[c + d*x]),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 17

Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 
)/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && 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 3113

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] 
 && IGtQ[(n - 1)/2, 0]
 

rule 3117

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[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 3791

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> 
 Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x 
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n)   Int[(c + d* 
x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 
 1]
 

rule 3804

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Sy 
mbol] :> Simp[2   Int[(c + d*x)^m*(E^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x 
)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ 
[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 4905

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[a + b*x]^(n + 1)/(b*(n + 1 
))), x] + Simp[d*(m/(b*(n + 1)))   Int[(c + d*x)^(m - 1)*Cos[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 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. 1873 vs. \(2 (466 ) = 932\).

Time = 8.29 (sec) , antiderivative size = 1874, normalized size of antiderivative = 3.66

method	result	size
risch	\(\text {Expression too large to display}\)	\(1874\)

Input:

int((f*x+e)*cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x,method=_RETURNVERBO 
SE)
 

Output:

2*I*b/d^2*f*c/a/(-a^2+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a 
^2+b^2)^(1/2))-1/d/a*e*ln(exp(I*(d*x+c))+1)+1/d/a*e*ln(exp(I*(d*x+c))-1)-4 
*I/d^2/b*a*f*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^ 
2+b^2)^(1/2))+2*I/d^2/b^3*a^3*f*c/(-a^2+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I 
*(d*x+c))-2*a)/(-a^2+b^2)^(1/2))-1/d/a*f*ln(exp(I*(d*x+c))+1)*x-1/d^2/a*c* 
f*ln(exp(I*(d*x+c))-1)-1/16*I*(2*d*x*f-I*f+2*d*e)/d^2/b*exp(-2*I*(d*x+c))+ 
1/2*a*(d*x*f-I*f+d*e)/b^2/d^2*exp(-I*(d*x+c))+1/d^2/b^3*f*a^3/(-a^2+b^2)^( 
1/2)*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/d^2/b^3*f*a^3/(-a^2+b^2)^(1/2)*dilog((-I*a-exp(I*(d*x+c))*b+(-a^2+b^2)^( 
1/2))/(-I*a+(-a^2+b^2)^(1/2)))-I/d^2/b^3*f*a^3/(-a^2+b^2)^(1/2)*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/d/b^3*a^3* 
e/(-a^2+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+b^2)^(1/2)) 
+4*I/d/b*a*e/(-a^2+b^2)^(1/2)*arctan(1/2*(2*I*b*exp(I*(d*x+c))-2*a)/(-a^2+ 
b^2)^(1/2))-2*I/d^2/b*f*a/(-a^2+b^2)^(1/2)*dilog((-I*a-exp(I*(d*x+c))*b+(- 
a^2+b^2)^(1/2))/(-I*a+(-a^2+b^2)^(1/2)))-3/4/b*f*x^2-3/2/b*e*x+I/d^2*f/a*d 
ilog(exp(I*(d*x+c)))+I/d^2*f/a*dilog(exp(I*(d*x+c))+1)+1/2*a*(d*x*f+I*f+d* 
e)/b^2/d^2*exp(I*(d*x+c))+b/d*f/a/(-a^2+b^2)^(1/2)*ln((I*a+exp(I*(d*x+c))* 
b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))*x-I*b/d^2*f/a/(-a^2+b^2)^(1/2) 
*dilog((I*a+exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(I*a+(-a^2+b^2)^(1/2)))+2/b 
/d^2*f*a/(-a^2+b^2)^(1/2)*ln((-I*a-exp(I*(d*x+c))*b+(-a^2+b^2)^(1/2))/(...
 

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. 1611 vs. \(2 (454) = 908\).

Time = 0.49 (sec) , antiderivative size = 1611, normalized size of antiderivative = 3.15 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="f 
ricas")
 

Output:

1/4*((2*a^3 - 3*a*b^2)*d^2*f*x^2 - a*b^2*f*cos(d*x + c)^2 + 2*(2*a^3 - 3*a 
*b^2)*d^2*e*x - 2*I*b^3*f*dilog(cos(d*x + c) + I*sin(d*x + c)) + 2*I*b^3*f 
*dilog(cos(d*x + c) - I*sin(d*x + c)) - 2*I*b^3*f*dilog(-cos(d*x + c) + I* 
sin(d*x + c)) + 2*I*b^3*f*dilog(-cos(d*x + c) - I*sin(d*x + c)) - 2*I*(a^2 
*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*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) 
 + 2*I*(a^2*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*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) + 2*I*(a^2*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*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) - 2*I*(a^2*b - b^3)*f*sqrt(-(a^2 - b^2)/b^2)*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) - 2*((a^2*b - b^3)*d*e - (a^2*b - b^3) 
*c*f)*sqrt(-(a^2 - b^2)/b^2)*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) - 2*((a^2*b - b^3)*d*e - (a^2*b - b^3)* 
c*f)*sqrt(-(a^2 - b^2)/b^2)*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) + 2*((a^2*b - b^3)*d*e - (a^2*b - b^3)*c 
*f)*sqrt(-(a^2 - b^2)/b^2)*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) + 2*((a^2*b - b^3)*d*e - (a^2*b - b^3)*c 
*f)*sqrt(-(a^2 - b^2)/b^2)*log(-2*b*cos(d*x + c) - 2*I*b*sin(d*x + c) +...
 

Sympy [F]

\[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos ^{3}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*cos(d*x+c)**3*cot(d*x+c)/(a+b*sin(d*x+c)),x)
 

Output:

Integral((e + f*x)*cos(c + d*x)**3*cot(c + d*x)/(a + b*sin(c + d*x)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((f*x+e)*cos(d*x+c)^3*cot(d*x+c)/(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(-1)]

Timed out. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:

integrate((f*x+e)*cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x, algorithm="g 
iac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \] Input:

int((cos(c + d*x)^3*cot(c + d*x)*(e + f*x))/(a + b*sin(c + d*x)),x)
                                                                                    
                                                                                    
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (f x +e \right ) \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )}{\sin \left (d x +c \right ) b +a}d x \] Input:

int((f*x+e)*cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x)
 

Output:

int((f*x+e)*cos(d*x+c)^3*cot(d*x+c)/(a+b*sin(d*x+c)),x)