\(\int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx\) [192]

Optimal result

Integrand size = 28, antiderivative size = 278 \[ \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=-\frac {f^2 x}{4 a d^2}+\frac {i (e+f x)^2}{a d}+\frac {(e+f x)^3}{2 a f}-\frac {2 f^2 \cos (c+d x)}{a d^3}+\frac {(e+f x)^2 \cos (c+d x)}{a d}+\frac {(e+f x)^2 \cot \left (\frac {c}{2}+\frac {\pi }{4}+\frac {d x}{2}\right )}{a d}-\frac {4 f (e+f x) \log \left (1-i e^{i (c+d x)}\right )}{a d^2}+\frac {4 i f^2 \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right )}{a d^3}-\frac {2 f (e+f x) \sin (c+d x)}{a d^2}+\frac {f^2 \cos (c+d x) \sin (c+d x)}{4 a d^3}-\frac {(e+f x)^2 \cos (c+d x) \sin (c+d x)}{2 a d}+\frac {f (e+f x) \sin ^2(c+d x)}{2 a d^2} \] Output:

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

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(830\) vs. \(2(278)=556\).

Time = 5.23 (sec) , antiderivative size = 830, normalized size of antiderivative = 2.99 \[ \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/16*(-6*d^2*e^2*Cos[(3*(c + d*x))/2] - 14*d*e*f*Cos[(3*(c + d*x))/2] + 1 
5*f^2*Cos[(3*(c + d*x))/2] - 12*d^2*e*f*x*Cos[(3*(c + d*x))/2] - 14*d*f^2* 
x*Cos[(3*(c + d*x))/2] - 6*d^2*f^2*x^2*Cos[(3*(c + d*x))/2] - 2*d^2*e^2*Co 
s[(5*(c + d*x))/2] + 2*d*e*f*Cos[(5*(c + d*x))/2] + f^2*Cos[(5*(c + d*x))/ 
2] - 4*d^2*e*f*x*Cos[(5*(c + d*x))/2] + 2*d*f^2*x*Cos[(5*(c + d*x))/2] - 2 
*d^2*f^2*x^2*Cos[(5*(c + d*x))/2] - 8*Cos[(c + d*x)/2]*(-2*f^2 - 2*d*f*(e 
+ f*x) + (3 - 2*I)*d^2*(e + f*x)^2 + d^3*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 8 
*d*f*(e + f*x)*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]) + (24 + 16*I)*d^2*e 
^2*Sin[(c + d*x)/2] + 16*d*e*f*Sin[(c + d*x)/2] - 16*f^2*Sin[(c + d*x)/2] 
- 24*d^3*e^2*x*Sin[(c + d*x)/2] + (48 + 32*I)*d^2*e*f*x*Sin[(c + d*x)/2] + 
 16*d*f^2*x*Sin[(c + d*x)/2] - 24*d^3*e*f*x^2*Sin[(c + d*x)/2] + (24 + 16* 
I)*d^2*f^2*x^2*Sin[(c + d*x)/2] - 8*d^3*f^2*x^3*Sin[(c + d*x)/2] + 64*d*e* 
f*Log[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + 64*d*f^2*x*Log 
[1 + I*Cos[c + d*x] + Sin[c + d*x]]*Sin[(c + d*x)/2] + (64*I)*f^2*PolyLog[ 
2, (-I)*Cos[c + d*x] - Sin[c + d*x]]*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) 
 - 6*d^2*e^2*Sin[(3*(c + d*x))/2] + 14*d*e*f*Sin[(3*(c + d*x))/2] + 15*f^2 
*Sin[(3*(c + d*x))/2] - 12*d^2*e*f*x*Sin[(3*(c + d*x))/2] + 14*d*f^2*x*Sin 
[(3*(c + d*x))/2] - 6*d^2*f^2*x^2*Sin[(3*(c + d*x))/2] + 2*d^2*e^2*Sin[(5* 
(c + d*x))/2] + 2*d*e*f*Sin[(5*(c + d*x))/2] - f^2*Sin[(5*(c + d*x))/2] + 
4*d^2*e*f*x*Sin[(5*(c + d*x))/2] + 2*d*f^2*x*Sin[(5*(c + d*x))/2] + 2*d...
 

Rubi [A] (verified)

Time = 2.41 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.10, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.964, Rules used = {5026, 3042, 3792, 17, 3042, 3115, 24, 5026, 3042, 3777, 3042, 3777, 25, 3042, 3118, 5026, 17, 3042, 3799, 3042, 4672, 3042, 25, 4202, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5026

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3792

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

\(\Big \downarrow \) 17

\(\displaystyle \frac {-\frac {f^2 \int \sin ^2(c+d x)dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\int \frac {(e+f x)^2 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {f^2 \int \sin (c+d x)^2dx}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\int \frac {(e+f x)^2 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {-\frac {f^2 \left (\frac {\int 1dx}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}+\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\int \frac {(e+f x)^2 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\int \frac {(e+f x)^2 \sin ^2(c+d x)}{\sin (c+d x) a+a}dx\)

\(\Big \downarrow \) 5026

\(\displaystyle -\frac {\int (e+f x)^2 \sin (c+d x)dx}{a}+\int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int (e+f x)^2 \sin (c+d x)dx}{a}+\int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {2 f \int (e+f x) \cos (c+d x)dx}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {2 f \int (e+f x) \sin \left (c+d x+\frac {\pi }{2}\right )dx}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 3777

\(\displaystyle \int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {2 f \left (\frac {f \int -\sin (c+d x)dx}{d}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx-\frac {\frac {2 f \left (\frac {(e+f x) \sin (c+d x)}{d}-\frac {f \int \sin (c+d x)dx}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}\)

\(\Big \downarrow \) 3118

\(\displaystyle \int \frac {(e+f x)^2 \sin (c+d x)}{\sin (c+d x) a+a}dx+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 5026

\(\displaystyle -\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx+\frac {\int (e+f x)^2dx}{a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}\)

\(\Big \downarrow \) 17

\(\displaystyle -\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \frac {(e+f x)^2}{\sin (c+d x) a+a}dx+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 3799

\(\displaystyle -\frac {\int (e+f x)^2 \csc ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\int (e+f x)^2 \csc \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )^2dx}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 4672

\(\displaystyle -\frac {\frac {4 f \int (e+f x) \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\frac {4 f \int -\left ((e+f x) \tan \left (\frac {c}{2}+\frac {d x}{2}+\frac {3 \pi }{4}\right )\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {4 f \int (e+f x) \tan \left (\frac {1}{4} (2 c+3 \pi )+\frac {d x}{2}\right )dx}{d}-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 4202

\(\displaystyle -\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \int \frac {e^{\frac {1}{2} i (2 c+2 d x+3 \pi )} (e+f x)}{1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}dx\right )}{d}}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {i f \int \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )dx}{d}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (\frac {f \int e^{-\frac {1}{2} i (2 c+2 d x+3 \pi )} \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )de^{\frac {1}{2} i (2 c+2 d x+3 \pi )}}{d^2}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}+\frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {\frac {f (e+f x) \sin ^2(c+d x)}{2 d^2}-\frac {f^2 \left (\frac {x}{2}-\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )}{2 d^2}-\frac {(e+f x)^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {(e+f x)^3}{6 f}}{a}-\frac {-\frac {2 (e+f x)^2 \cot \left (\frac {c}{2}+\frac {d x}{2}+\frac {\pi }{4}\right )}{d}-\frac {4 f \left (\frac {i (e+f x)^2}{2 f}-2 i \left (-\frac {f \operatorname {PolyLog}\left (2,-e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d^2}-\frac {i (e+f x) \log \left (1+e^{\frac {1}{2} i (2 c+2 d x+3 \pi )}\right )}{d}\right )\right )}{d}}{2 a}-\frac {\frac {2 f \left (\frac {f \cos (c+d x)}{d^2}+\frac {(e+f x) \sin (c+d x)}{d}\right )}{d}-\frac {(e+f x)^2 \cos (c+d x)}{d}}{a}+\frac {(e+f x)^3}{3 a f}\)

Input:

Int[((e + f*x)^2*Sin[c + d*x]^3)/(a + a*Sin[c + d*x]),x]
 

Output:

(e + f*x)^3/(3*a*f) - ((-2*(e + f*x)^2*Cot[c/2 + Pi/4 + (d*x)/2])/d - (4*f 
*(((I/2)*(e + f*x)^2)/f - (2*I)*(((-I)*(e + f*x)*Log[1 + E^((I/2)*(2*c + 3 
*Pi + 2*d*x))])/d - (f*PolyLog[2, -E^((I/2)*(2*c + 3*Pi + 2*d*x))])/d^2))) 
/d)/(2*a) - (-(((e + f*x)^2*Cos[c + d*x])/d) + (2*f*((f*Cos[c + d*x])/d^2 
+ ((e + f*x)*Sin[c + d*x])/d))/d)/a + ((e + f*x)^3/(6*f) - ((e + f*x)^2*Co 
s[c + d*x]*Sin[c + d*x])/(2*d) + (f*(e + f*x)*Sin[c + d*x]^2)/(2*d^2) - (f 
^2*(x/2 - (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/(2*d^2))/a
 

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 24

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

rule 25

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

rule 2620

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

rule 2715

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

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 3115

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

rule 3118

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

rule 3777

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( 
-(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*C 
os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
 

rule 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 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Simp[(2*a)^n   Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) 
+ f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 
2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
 

rule 4202

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

rule 4672

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

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (254 ) = 508\).

Time = 2.72 (sec) , antiderivative size = 591, normalized size of antiderivative = 2.13

method	result	size
risch	\(\frac {f^{2} x^{3}}{2 a}+\frac {3 f e \,x^{2}}{2 a}+\frac {3 e^{2} x}{2 a}+\frac {e^{3}}{2 a f}+\frac {i \left (2 d^{2} x^{2} f^{2}+4 e f x \,d^{2}+2 i d \,f^{2} x +2 d^{2} e^{2}+2 i d e f -f^{2}\right ) {\mathrm e}^{2 i \left (d x +c \right )}}{16 a \,d^{3}}+\frac {\left (d^{2} x^{2} f^{2}+2 e f x \,d^{2}+2 i d \,f^{2} x +d^{2} e^{2}+2 i d e f -2 f^{2}\right ) {\mathrm e}^{i \left (d x +c \right )}}{2 a \,d^{3}}+\frac {\left (d^{2} x^{2} f^{2}+2 e f x \,d^{2}-2 i d \,f^{2} x +d^{2} e^{2}-2 i d e f -2 f^{2}\right ) {\mathrm e}^{-i \left (d x +c \right )}}{2 a \,d^{3}}+\frac {2 i f^{2} x^{2}}{d a}+\frac {2 x^{2} f^{2}+4 e f x +2 e^{2}}{d a \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}+\frac {4 f e \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {2 f e \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a \,d^{2}}-\frac {i \left (2 d^{2} x^{2} f^{2}+4 e f x \,d^{2}-2 i d \,f^{2} x +2 d^{2} e^{2}-2 i d e f -f^{2}\right ) {\mathrm e}^{-2 i \left (d x +c \right )}}{16 a \,d^{3}}+\frac {2 i f^{2} c^{2}}{a \,d^{3}}-\frac {4 i f^{2} c \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {4 i f^{2} c x}{a \,d^{2}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}-\frac {4 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}+\frac {4 i f e \arctan \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{2}}-\frac {4 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}+\frac {2 f^{2} c \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a \,d^{3}}+\frac {4 i f^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}\)	\(591\)

Input:

int((f*x+e)^2*sin(d*x+c)^3/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/2/a*f^2*x^3+3/2/a*f*e*x^2+3/2/a*e^2*x+1/2/a/f*e^3+1/16*I*(2*d^2*x^2*f^2+ 
2*I*d*f^2*x+4*e*f*x*d^2+2*I*d*e*f+2*d^2*e^2-f^2)/a/d^3*exp(2*I*(d*x+c))+1/ 
2*(d^2*x^2*f^2+2*I*d*f^2*x+2*e*f*x*d^2+2*I*d*e*f+d^2*e^2-2*f^2)/a/d^3*exp( 
I*(d*x+c))+1/2*(d^2*x^2*f^2-2*I*d*f^2*x+2*e*f*x*d^2-2*I*d*e*f+d^2*e^2-2*f^ 
2)/a/d^3*exp(-I*(d*x+c))+2*I/a/d*f^2*x^2+2*(f^2*x^2+2*e*f*x+e^2)/d/a/(exp( 
I*(d*x+c))+I)+4/a/d^2*f*e*ln(exp(I*(d*x+c)))-2/a/d^2*f*e*ln(exp(2*I*(d*x+c 
))+1)-1/16*I*(2*d^2*x^2*f^2-2*I*d*f^2*x+4*e*f*x*d^2-2*I*d*e*f+2*d^2*e^2-f^ 
2)/a/d^3*exp(-2*I*(d*x+c))+2*I/a/d^3*f^2*c^2-4*I/a/d^3*f^2*c*arctan(exp(I* 
(d*x+c)))+4*I/a/d^2*f^2*c*x-4/a/d^2*f^2*ln(1-I*exp(I*(d*x+c)))*x-4/a/d^3*f 
^2*ln(1-I*exp(I*(d*x+c)))*c+4*I/a/d^2*f*e*arctan(exp(I*(d*x+c)))-4/a/d^3*f 
^2*c*ln(exp(I*(d*x+c)))+2/a/d^3*f^2*c*ln(exp(2*I*(d*x+c))+1)+4*I*f^2*polyl 
og(2,I*exp(I*(d*x+c)))/a/d^3
 

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. 846 vs. \(2 (249) = 498\).

Time = 0.11 (sec) , antiderivative size = 846, normalized size of antiderivative = 3.04 \[ \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(2*d^3*f^2*x^3 + 4*d^2*e^2 + (2*d^2*f^2*x^2 + 2*d^2*e^2 - 2*d*e*f - f^ 
2 + 2*(2*d^2*e*f - d*f^2)*x)*cos(d*x + c)^3 - 7*d*e*f + 2*(3*d^3*e*f + 2*d 
^2*f^2)*x^2 + 2*(2*d^2*f^2*x^2 + 2*d^2*e^2 + 3*d*e*f - 4*f^2 + (4*d^2*e*f 
+ 3*d*f^2)*x)*cos(d*x + c)^2 + (6*d^3*e^2 + 8*d^2*e*f - 7*d*f^2)*x + (2*d^ 
3*f^2*x^3 + 6*d^2*e^2 + d*e*f + 6*(d^3*e*f + d^2*f^2)*x^2 - 7*f^2 + (6*d^3 
*e^2 + 12*d^2*e*f + d*f^2)*x)*cos(d*x + c) - 8*(-I*f^2*cos(d*x + c) - I*f^ 
2*sin(d*x + c) - I*f^2)*dilog(I*cos(d*x + c) - sin(d*x + c)) - 8*(I*f^2*co 
s(d*x + c) + I*f^2*sin(d*x + c) + I*f^2)*dilog(-I*cos(d*x + c) - sin(d*x + 
 c)) - 8*(d*e*f - c*f^2 + (d*e*f - c*f^2)*cos(d*x + c) + (d*e*f - c*f^2)*s 
in(d*x + c))*log(cos(d*x + c) + I*sin(d*x + c) + I) - 8*(d*f^2*x + c*f^2 + 
 (d*f^2*x + c*f^2)*cos(d*x + c) + (d*f^2*x + c*f^2)*sin(d*x + c))*log(I*co 
s(d*x + c) + sin(d*x + c) + 1) - 8*(d*f^2*x + c*f^2 + (d*f^2*x + c*f^2)*co 
s(d*x + c) + (d*f^2*x + c*f^2)*sin(d*x + c))*log(-I*cos(d*x + c) + sin(d*x 
 + c) + 1) - 8*(d*e*f - c*f^2 + (d*e*f - c*f^2)*cos(d*x + c) + (d*e*f - c* 
f^2)*sin(d*x + c))*log(-cos(d*x + c) + I*sin(d*x + c) + I) + (2*d^3*f^2*x^ 
3 - 4*d^2*e^2 - 7*d*e*f + 2*(3*d^3*e*f - 2*d^2*f^2)*x^2 - (2*d^2*f^2*x^2 + 
 2*d^2*e^2 + 2*d*e*f - f^2 + 2*(2*d^2*e*f + d*f^2)*x)*cos(d*x + c)^2 + (6* 
d^3*e^2 - 8*d^2*e*f - 7*d*f^2)*x + (2*d^2*f^2*x^2 + 2*d^2*e^2 - 8*d*e*f - 
7*f^2 + 4*(d^2*e*f - 2*d*f^2)*x)*cos(d*x + c))*sin(d*x + c))/(a*d^3*cos(d* 
x + c) + a*d^3*sin(d*x + c) + a*d^3)
 

Sympy [F]

\[ \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {e^{2} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \sin ^{3}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \] Input:

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

Output:

(Integral(e**2*sin(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(f**2*x**2 
*sin(c + d*x)**3/(sin(c + d*x) + 1), x) + Integral(2*e*f*x*sin(c + d*x)**3 
/(sin(c + d*x) + 1), x))/a
 

Maxima [F(-2)]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e+f x)^2 \sin ^3(c+d x)}{a+a \sin (c+d x)} \, dx=\text {too large to display} \] Input:

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

Output:

( - 2*cos(c + d*x)**3*tan((c + d*x)/2)*d*f**2*x - 2*cos(c + d*x)**3*d*f**2 
*x + 2*cos(c + d*x)**2*sin(c + d*x)*tan((c + d*x)/2)*d*f**2*x + 2*cos(c + 
d*x)**2*sin(c + d*x)*tan((c + d*x)/2)*f**2 + 2*cos(c + d*x)**2*sin(c + d*x 
)*d*f**2*x + 2*cos(c + d*x)**2*sin(c + d*x)*f**2 - 22*cos(c + d*x)**2*tan( 
(c + d*x)/2)*d*f**2*x - 8*cos(c + d*x)**2*tan((c + d*x)/2)*f**2 - 22*cos(c 
 + d*x)**2*d*f**2*x - 8*cos(c + d*x)**2*f**2 + 48*cos(c + d*x)*int((tan((c 
 + d*x)/2)*x)/(tan((c + d*x)/2) + 1),x)*tan((c + d*x)/2)*d**2*f**2 + 48*co 
s(c + d*x)*int((tan((c + d*x)/2)*x)/(tan((c + d*x)/2) + 1),x)*d**2*f**2 + 
24*cos(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*tan((c + d*x)/2)*d*e*f + 24*c 
os(c + d*x)*log(tan((c + d*x)/2)**2 + 1)*d*e*f - 48*cos(c + d*x)*log(tan(( 
c + d*x)/2) + 1)*tan((c + d*x)/2)*d*e*f - 48*cos(c + d*x)*log(tan((c + d*x 
)/2) + 1)*d*e*f + 6*cos(c + d*x)*sin(c + d*x)**2*tan((c + d*x)/2)*d**2*e** 
2 + 12*cos(c + d*x)*sin(c + d*x)**2*tan((c + d*x)/2)*d**2*e*f*x + 6*cos(c 
+ d*x)*sin(c + d*x)**2*tan((c + d*x)/2)*d**2*f**2*x**2 + 6*cos(c + d*x)*si 
n(c + d*x)**2*tan((c + d*x)/2)*d*e*f + 4*cos(c + d*x)*sin(c + d*x)**2*tan( 
(c + d*x)/2)*d*f**2*x - 3*cos(c + d*x)*sin(c + d*x)**2*tan((c + d*x)/2)*f* 
*2 + 6*cos(c + d*x)*sin(c + d*x)**2*d**2*e**2 + 12*cos(c + d*x)*sin(c + d* 
x)**2*d**2*e*f*x + 6*cos(c + d*x)*sin(c + d*x)**2*d**2*f**2*x**2 + 6*cos(c 
 + d*x)*sin(c + d*x)**2*d*e*f + 4*cos(c + d*x)*sin(c + d*x)**2*d*f**2*x - 
3*cos(c + d*x)*sin(c + d*x)**2*f**2 - 6*cos(c + d*x)*sin(c + d*x)*tan((...