3.3.28 \(\int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx\) [228]

3.3.28.1 Optimal result

Integrand size = 28, antiderivative size = 802 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {3 e f^2 x}{4 b d^2}-\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cos (c+d x)}{b^2 d^3}+\frac {a (e+f x)^3 \cos (c+d x)}{b^2 d}+\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}-\frac {i a^3 (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d}+\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}-\frac {3 a^3 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^2}+\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 i a^3 f^2 (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^3}-\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a^3 f^3 \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b^3 \sqrt {a^2-b^2} d^4}+\frac {6 a f^3 \sin (c+d x)}{b^2 d^4}-\frac {3 a f (e+f x)^2 \sin (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cos (c+d x) \sin (c+d x)}{4 b d^3}-\frac {(e+f x)^3 \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {3 f^3 \sin ^2(c+d x)}{8 b d^4}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 b d^2} \]

output

-3/4*e*f^2*x/b/d^2-3/8*f^3*x^2/b/d^2+1/4*a^2*(f*x+e)^4/b^3/f+1/8*(f*x+e)^4 
/b/f-6*a*f^2*(f*x+e)*cos(d*x+c)/b^2/d^3+a*(f*x+e)^3*cos(d*x+c)/b^2/d+6*a*f 
^3*sin(d*x+c)/b^2/d^4-3*a*f*(f*x+e)^2*sin(d*x+c)/b^2/d^2+3/4*f^2*(f*x+e)*c 
os(d*x+c)*sin(d*x+c)/b/d^3-1/2*(f*x+e)^3*cos(d*x+c)*sin(d*x+c)/b/d-3/8*f^3 
*sin(d*x+c)^2/b/d^4+3/4*f*(f*x+e)^2*sin(d*x+c)^2/b/d^2+6*I*a^3*f^2*(f*x+e) 
*polylog(3,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^3/d^3/(a^2-b^2)^(1/2) 
-6*I*a^3*f^2*(f*x+e)*polylog(3,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3 
/d^3/(a^2-b^2)^(1/2)+3*a^3*f*(f*x+e)^2*polylog(2,I*b*exp(I*(d*x+c))/(a-(a^ 
2-b^2)^(1/2)))/b^3/d^2/(a^2-b^2)^(1/2)-3*a^3*f*(f*x+e)^2*polylog(2,I*b*exp 
(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3/d^2/(a^2-b^2)^(1/2)+I*a^3*(f*x+e)^3*l 
n(1-I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^3/d/(a^2-b^2)^(1/2)-I*a^3*(f 
*x+e)^3*ln(1-I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3/d/(a^2-b^2)^(1/2) 
-6*a^3*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a-(a^2-b^2)^(1/2)))/b^3/d^4/(a^2- 
b^2)^(1/2)+6*a^3*f^3*polylog(4,I*b*exp(I*(d*x+c))/(a+(a^2-b^2)^(1/2)))/b^3 
/d^4/(a^2-b^2)^(1/2)
 

3.3.28.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1923\) vs. \(2(802)=1604\).

Time = 4.44 (sec) , antiderivative size = 1923, normalized size of antiderivative = 2.40 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx =\text {Too large to display} \]

input

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

output

(16*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e^3*x + 8*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*e 
^3*x + 24*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e^2*f*x^2 + 12*b^2*Sqrt[-(-a^2 + b^ 
2)^2]*d^4*e^2*f*x^2 + 16*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*e*f^2*x^3 + 8*b^2*Sq 
rt[-(-a^2 + b^2)^2]*d^4*e*f^2*x^3 + 4*a^2*Sqrt[-(a^2 - b^2)^2]*d^4*f^3*x^4 
 + 2*b^2*Sqrt[-(-a^2 + b^2)^2]*d^4*f^3*x^4 - 32*a^3*Sqrt[-a^2 + b^2]*d^3*e 
^3*ArcTan[(I*a + b*E^(I*(c + d*x)))/Sqrt[a^2 - b^2]] + 16*a*b*Sqrt[-(a^2 - 
 b^2)^2]*d^3*e^3*Cos[c + d*x] - 96*a*b*Sqrt[-(a^2 - b^2)^2]*d*e*f^2*Cos[c 
+ d*x] + 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*e^2*f*x*Cos[c + d*x] - 96*a*b*Sqr 
t[-(a^2 - b^2)^2]*d*f^3*x*Cos[c + d*x] + 48*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*e 
*f^2*x^2*Cos[c + d*x] + 16*a*b*Sqrt[-(a^2 - b^2)^2]*d^3*f^3*x^3*Cos[c + d* 
x] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e^2*f*Cos[2*(c + d*x)] + 3*b^2*Sqrt[-( 
a^2 - b^2)^2]*f^3*Cos[2*(c + d*x)] - 12*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*e*f^2 
*x*Cos[2*(c + d*x)] - 6*b^2*Sqrt[-(a^2 - b^2)^2]*d^2*f^3*x^2*Cos[2*(c + d* 
x)] - 48*a^3*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 - (b*E^(I*(c + d*x)))/((-I) 
*a + Sqrt[-a^2 + b^2])] - 48*a^3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 - (b* 
E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] - 16*a^3*Sqrt[a^2 - b^2]*d^3 
*f^3*x^3*Log[1 - (b*E^(I*(c + d*x)))/((-I)*a + Sqrt[-a^2 + b^2])] + 48*a^3 
*Sqrt[a^2 - b^2]*d^3*e^2*f*x*Log[1 + (b*E^(I*(c + d*x)))/(I*a + Sqrt[-a^2 
+ b^2])] + 48*a^3*Sqrt[a^2 - b^2]*d^3*e*f^2*x^2*Log[1 + (b*E^(I*(c + d*x)) 
)/(I*a + Sqrt[-a^2 + b^2])] + 16*a^3*Sqrt[a^2 - b^2]*d^3*f^3*x^3*Log[1 ...
 

3.3.28.3 Rubi [A] (verified)

Time = 4.06 (sec) , antiderivative size = 713, normalized size of antiderivative = 0.89, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5026, 3042, 3792, 17, 3042, 3791, 17, 5026, 3042, 3777, 3042, 3777, 25, 3042, 3777, 3042, 3117, 5026, 17, 3042, 3804, 2694, 27, 2620, 3011, 7163, 2720, 7143}

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.

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

\(\Big \downarrow \) 5026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \sin ^2(c+d x)dx}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^2 \int (e+f x) \sin (c+d x)^2dx}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)}dx}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {1}{2} \int (e+f x)dx+\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)}dx}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \int \frac {(e+f x)^3 \sin ^2(c+d x)}{a+b \sin (c+d x)}dx}{b}\)

\(\Big \downarrow \) 5026

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \sin (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\int (e+f x)^3 \sin (c+d x)dx}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \int (e+f x)^2 \cos (c+d x)dx}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \int (e+f x)^2 \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {2 f \int -((e+f x) \sin (c+d x))dx}{d}+\frac {(e+f x)^2 \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \int (e+f x) \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3777

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \int \cos (c+d x)dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \int \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 3117

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \int \frac {(e+f x)^3 \sin (c+d x)}{a+b \sin (c+d x)}dx}{b}\right )}{b}\)

\(\Big \downarrow \) 5026

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {\int (e+f x)^3dx}{b}-\frac {a \int \frac {(e+f x)^3}{a+b \sin (c+d x)}dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {a \int \frac {(e+f x)^3}{a+b \sin (c+d x)}dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {a \int \frac {(e+f x)^3}{a+b \sin (c+d x)}dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3804

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \int \frac {e^{i (c+d x)} (e+f x)^3}{2 e^{i (c+d x)} a-i b e^{2 i (c+d x)}+i b}dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \int \frac {e^{i (c+d x)} (e+f x)^3}{2 \left (a-i b e^{i (c+d x)}+\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i (c+d x)} (e+f x)^3}{2 \left (a-i b e^{i (c+d x)}-\sqrt {a^2-b^2}\right )}dx}{\sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \int \frac {e^{i (c+d x)} (e+f x)^3}{a-i b e^{i (c+d x)}+\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}-\frac {i b \int \frac {e^{i (c+d x)} (e+f x)^3}{a-i b e^{i (c+d x)}-\sqrt {a^2-b^2}}dx}{2 \sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {3 f \int (e+f x)^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )dx}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )dx}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \int (e+f x) \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )dx}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7163

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \left (\frac {i f \int \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )dx}{d}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \left (\frac {f \int e^{-i (c+d x)} \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )de^{i (c+d x)}}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {-\frac {3 f^2 \left (\frac {f \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^2}{4 f}\right )}{2 d^2}+\frac {3 f (e+f x)^2 \sin ^2(c+d x)}{4 d^2}-\frac {(e+f x)^3 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^4}{8 f}}{b}-\frac {a \left (\frac {\frac {3 f \left (\frac {(e+f x)^2 \sin (c+d x)}{d}-\frac {2 f \left (\frac {f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x)}{d}\right )}{d}\right )}{d}-\frac {(e+f x)^3 \cos (c+d x)}{d}}{b}-\frac {a \left (\frac {(e+f x)^4}{4 b f}-\frac {2 a \left (\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{d}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}-\frac {i b \left (\frac {(e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {3 f \left (\frac {i (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}-\frac {2 i f \left (\frac {f \operatorname {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d^2}-\frac {i (e+f x) \operatorname {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{d}\right )}{d}\right )}{b d}\right )}{2 \sqrt {a^2-b^2}}\right )}{b}\right )}{b}\right )}{b}\)

input

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

output

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

3.3.28.3.1 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 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

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 3792

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo 
l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim 
p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 
2*((n - 1)/n)   Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 
*m*((m - 1)/(f^2*n^2))   Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) 
/; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 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 5026

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_. 
)*Sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[1/b   Int[(e + f*x)^m*Sin[c + 
 d*x]^(n - 1), x], x] - Simp[a/b   Int[(e + f*x)^m*(Sin[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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.3.28.4 Maple [F]

\[\int \frac {\left (f x +e \right )^{3} \left (\sin ^{3}\left (d x +c \right )\right )}{a +b \sin \left (d x +c \right )}d x\]

input

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

output

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

3.3.28.5 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. 3008 vs. \(2 (712) = 1424\).

Time = 0.59 (sec) , antiderivative size = 3008, normalized size of antiderivative = 3.75 \[ \int \frac {(e+f x)^3 \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Too large to display} \]

input

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

output

1/8*((2*a^4 - a^2*b^2 - b^4)*d^4*f^3*x^4 + 4*(2*a^4 - a^2*b^2 - b^4)*d^4*e 
*f^2*x^3 + 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(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) - 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(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) - 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, -(-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) + 24*I*a^3*b*f^3*sqrt(-(a^2 - b^2)/b^2)*polylog(4, - 
(-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) + 3*(2*(2*a^4 - a^2*b^2 - b^4)*d^4*e^2*f + (a^2 
*b^2 - b^4)*d^2*f^3)*x^2 - 3*(2*(a^2*b^2 - b^4)*d^2*f^3*x^2 + 4*(a^2*b^2 - 
 b^4)*d^2*e*f^2*x + 2*(a^2*b^2 - b^4)*d^2*e^2*f - (a^2*b^2 - b^4)*f^3)*cos 
(d*x + c)^2 + 12*(-I*a^3*b*d^2*f^3*x^2 - 2*I*a^3*b*d^2*e*f^2*x - I*a^3*b*d 
^2*e^2*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) + 
 12*(I*a^3*b*d^2*f^3*x^2 + 2*I*a^3*b*d^2*e*f^2*x + I*a^3*b*d^2*e^2*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) + 12*(I*a^3*b*d 
^2*f^3*x^2 + 2*I*a^3*b*d^2*e*f^2*x + I*a^3*b*d^2*e^2*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*...
 

3.3.28.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.3.28.7 Maxima [F(-2)]

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

input

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

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
 

3.3.28.8 Giac [F]

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

input

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

output

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

3.3.28.9 Mupad [F(-1)]

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

input

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

output

\text{Hanged}