\(\int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\) [347]

Optimal result

Integrand size = 34, antiderivative size = 641 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {b f x}{4 a^2 d}-\frac {\left (a^2-b^2\right ) f x}{4 a^2 b d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {i \left (a^2-b^2\right )^2 (e+f x)^2}{2 a^2 b^3 f}-\frac {f \text {arctanh}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {\left (a^2-b^2\right ) f \cos (c+d x)}{a b^2 d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}+\frac {\left (a^2-b^2\right )^2 (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {i \left (a^2-b^2\right )^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}-\frac {i \left (a^2-b^2\right )^2 f \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b^3 d^2}+\frac {i b f \operatorname {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {(e+f x) \sin (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x) \sin (c+d x)}{a b^2 d}+\frac {b f \cos (c+d x) \sin (c+d x)}{4 a^2 d^2}+\frac {\left (a^2-b^2\right ) f \cos (c+d x) \sin (c+d x)}{4 a^2 b d^2}+\frac {b (e+f x) \sin ^2(c+d x)}{2 a^2 d}+\frac {\left (a^2-b^2\right ) (e+f x) \sin ^2(c+d x)}{2 a^2 b d} \] Output:

-1/4*b*f*x/a^2/d-1/4*(a^2-b^2)*f*x/a^2/b/d-1/2*I*(a^2-b^2)^2*(f*x+e)^2/a^2 
/b^3/f-I*(a^2-b^2)^2*f*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a 
^2/b^3/d^2-f*arctanh(cos(d*x+c))/a/d^2-f*cos(d*x+c)/a/d^2-(a^2-b^2)*f*cos( 
d*x+c)/a/b^2/d^2-(f*x+e)*csc(d*x+c)/a/d+(a^2-b^2)^2*(f*x+e)*ln(1-I*b*exp(I 
*(d*x+c))/(a-(a^2-b^2)^(1/2)))/a^2/b^3/d+(a^2-b^2)^2*(f*x+e)*ln(1-I*b*exp( 
I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d-b*(f*x+e)*ln(1-exp(2*I*(d*x+c))) 
/a^2/d+1/2*I*b*(f*x+e)^2/a^2/f-I*(a^2-b^2)^2*f*polylog(2,I*b*exp(I*(d*x+c) 
)/(a+(a^2-b^2)^(1/2)))/a^2/b^3/d^2+1/2*I*b*f*polylog(2,exp(2*I*(d*x+c)))/a 
^2/d^2-(f*x+e)*sin(d*x+c)/a/d-(a^2-b^2)*(f*x+e)*sin(d*x+c)/a/b^2/d+1/4*b*f 
*cos(d*x+c)*sin(d*x+c)/a^2/d^2+1/4*(a^2-b^2)*f*cos(d*x+c)*sin(d*x+c)/a^2/b 
/d^2+1/2*b*(f*x+e)*sin(d*x+c)^2/a^2/d+1/2*(a^2-b^2)*(f*x+e)*sin(d*x+c)^2/a 
^2/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. \(1650\) vs. \(2(641)=1282\).

Time = 10.32 (sec) , antiderivative size = 1650, normalized size of antiderivative = 2.57 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

Integrate[((e + f*x)*Cos[c + d*x]^3*Cot[c + d*x]^2)/(a + b*Sin[c + d*x]),x 
]
 

Output:

-((a*f*Cos[c + d*x])/(b^2*d^2)) - ((d*e - c*f + f*(c + d*x))*Cos[2*(c + d* 
x)])/(4*b*d^2) + ((-(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) + (a^2*e*Log[1 + (b*Sin[ 
c + d*x])/a])/(b^3*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) - (a^2*c*f*Log[1 + (b*Sin[c + d*x])/a])/ 
(b^3*d^2) + (2*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) - (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*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])*E^(I*(-c + Pi/2 - d*x)))/b] + PolyL 
og[2, ((-a + Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b]))/b))/d^2 + (a^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*ArcSin[Sqrt[(a + b)/b]/Sqr 
t[2]])*Log[1 + ((a - Sqrt[a^2 - b^2])*E^(I*(-c + Pi/2 - d*x)))/b] + (-c...
 

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.

\(\displaystyle \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx\)

\(\Big \downarrow \) 5054

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

\(\Big \downarrow \) 4908

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle \frac {-\frac {2}{3} \int (e+f x) \cos (c+d x)dx+\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2}{3} \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {2}{3} \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )+\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {2}{3} \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )+\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2}{3} \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )+\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {\int (e+f x) \cos (c+d x) \cot ^2(c+d x)dx-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {-\int (e+f x) \cos (c+d x)dx+\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\frac {f \int -\sin (c+d x)dx}{d}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx+\frac {f \int \sin (c+d x)dx}{d}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx+\frac {f \int \sin (c+d x)dx}{d}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \frac {\int (e+f x) \cot (c+d x) \csc (c+d x)dx-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4910

\(\displaystyle \frac {\frac {f \int \csc (c+d x)dx}{d}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {f \int \csc (c+d x)dx}{d}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x) \cot (c+d x)}{a+b \sin (c+d x)}dx}{a}\)

\(\Big \downarrow \) 5054

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^4(c+d x) \cot (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx-\int (e+f x) \cos ^3(c+d x) \sin (c+d x)dx}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 4905

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx-\frac {f \int \cos ^4(c+d x)dx}{4 d}+\frac {(e+f x) \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx-\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx}{4 d}+\frac {(e+f x) \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx-\frac {f \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{4 d}+\frac {(e+f x) \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx-\frac {f \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{4 d}+\frac {(e+f x) \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx-\frac {f \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )}{4 d}+\frac {(e+f x) \cos ^4(c+d x)}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cos ^2(c+d x) \cot (c+d x)dx+\frac {(e+f x) \cos ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 4908

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int (e+f x) \cot (c+d x)dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx+\frac {(e+f x) \cos ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {\int -\left ((e+f x) \tan \left (c+d x+\frac {\pi }{2}\right )\right )dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx+\frac {(e+f x) \cos ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (\frac {-\int (e+f x) \tan \left (\frac {1}{2} (2 c+\pi )+d x\right )dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx+\frac {(e+f x) \cos ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}}{a}-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}\right )}{a}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {-\frac {f \text {arctanh}(\cos (c+d x))}{d^2}-\frac {2}{3} \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )-\frac {f \cos ^3(c+d x)}{9 d^2}-\frac {f \cos (c+d x)}{d^2}-\frac {(e+f x) \sin (c+d x)}{d}-\frac {(e+f x) \csc (c+d x)}{d}-\frac {(e+f x) \sin (c+d x) \cos ^2(c+d x)}{3 d}}{a}-\frac {b \left (-\frac {b \int \frac {(e+f x) \cos ^5(c+d x)}{a+b \sin (c+d x)}dx}{a}+\frac {2 i \int \frac {e^{i (2 c+2 d x+\pi )} (e+f x)}{1+e^{i (2 c+2 d x+\pi )}}dx-\int (e+f x) \cos (c+d x) \sin (c+d x)dx+\frac {(e+f x) \cos ^4(c+d x)}{4 d}-\frac {f \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )}{4 d}-\frac {i (e+f x)^2}{2 f}}{a}\right )}{a}\)

Input:

Int[((e + f*x)*Cos[c + d*x]^3*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 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 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 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 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 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 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. 5209 vs. \(2 (594 ) = 1188\).

Time = 17.61 (sec) , antiderivative size = 5210, normalized size of antiderivative = 8.13

method	result	size
risch	\(\text {Expression too large to display}\)	\(5210\)

Input:

int((f*x+e)*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x,method=_RETURNVER 
BOSE)
 

Output:

result too large to display
 

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. 1707 vs. \(2 (582) = 1164\).

Time = 0.34 (sec) , antiderivative size = 1707, normalized size of antiderivative = 2.66 \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((f*x+e)*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm= 
"fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (e + f x\right ) \cos ^{3}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \] Input:

integrate((f*x+e)*cos(d*x+c)**3*cot(d*x+c)**2/(a+b*sin(d*x+c)),x)
 

Output:

Integral((e + f*x)*cos(c + d*x)**3*cot(c + d*x)**2/(a + b*sin(c + d*x)), x 
)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Exception raised: RuntimeError} \] Input:

integrate((f*x+e)*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm= 
"maxima")
 

Output:

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un 
defined.
 

Giac [F]

\[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )} \cos \left (d x + c\right )^{3} \cot \left (d x + c\right )^{2}}{b \sin \left (d x + c\right ) + a} \,d x } \] Input:

integrate((f*x+e)*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x, algorithm= 
"giac")
 

Output:

integrate((f*x + e)*cos(d*x + c)^3*cot(d*x + c)^2/(b*sin(d*x + c) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Hanged} \] Input:

int((cos(c + d*x)^3*cot(c + d*x)^2*(e + f*x))/(a + b*sin(c + d*x)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {(e+f x) \cos ^3(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx=\int \frac {\left (f x +e \right ) \cos \left (d x +c \right )^{3} \cot \left (d x +c \right )^{2}}{\sin \left (d x +c \right ) b +a}d x \] Input:

int((f*x+e)*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)
 

Output:

int((f*x+e)*cos(d*x+c)^3*cot(d*x+c)^2/(a+b*sin(d*x+c)),x)