\(\int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx\) [42]

Optimal result

Integrand size = 20, antiderivative size = 295 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}-\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}-\frac {b^2 (c+d x)^3 \cot (e+f x)}{f}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}+\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{2 f^4} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1657\) vs. \(2(295)=590\).

Time = 6.80 (sec) , antiderivative size = 1657, normalized size of antiderivative = 5.62 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx =\text {Too large to display} \] Input:

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

Output:

-1/2*(b^2*d^3*E^(I*e)*Csc[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2* 
I)*e))*f^2*x^2*Log[1 - E^((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2* 
x^2*Log[1 + E^((-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, -E^( 
(-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, E^((-I)*(e + f*x))] 
 + (6*I)*(1 - E^((-2*I)*e))*PolyLog[3, -E^((-I)*(e + f*x))] + (6*I)*(1 - E 
^((-2*I)*e))*PolyLog[3, E^((-I)*(e + f*x))]))/f^4 - (a*b*c*d^2*E^(I*e)*Csc 
[e]*((2*f^3*x^3)/E^((2*I)*e) + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 - E^ 
((-I)*(e + f*x))] + (3*I)*(1 - E^((-2*I)*e))*f^2*x^2*Log[1 + E^((-I)*(e + 
f*x))] - 6*(1 - E^((-2*I)*e))*f*x*PolyLog[2, -E^((-I)*(e + f*x))] - 6*(1 - 
 E^((-2*I)*e))*f*x*PolyLog[2, E^((-I)*(e + f*x))] + (6*I)*(1 - E^((-2*I)*e 
))*PolyLog[3, -E^((-I)*(e + f*x))] + (6*I)*(1 - E^((-2*I)*e))*PolyLog[3, E 
^((-I)*(e + f*x))]))/f^3 - (a*b*d^3*E^(I*e)*Csc[e]*((f^4*x^4)/E^((2*I)*e) 
+ (2*I)*(1 - E^((-2*I)*e))*f^3*x^3*Log[1 - E^((-I)*(e + f*x))] + (2*I)*(1 
- E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e)) 
*f^2*x^2*PolyLog[2, -E^((-I)*(e + f*x))] - 6*(1 - E^((-2*I)*e))*f^2*x^2*Po 
lyLog[2, E^((-I)*(e + f*x))] + (12*I)*(1 - E^((-2*I)*e))*f*x*PolyLog[3, -E 
^((-I)*(e + f*x))] + (12*I)*(1 - E^((-2*I)*e))*f*x*PolyLog[3, E^((-I)*(e + 
 f*x))] + 12*(1 - E^((-2*I)*e))*PolyLog[4, -E^((-I)*(e + f*x))] + 12*(1 - 
E^((-2*I)*e))*PolyLog[4, E^((-I)*(e + f*x))]))/(2*f^4) + (3*b^2*c^2*d*Csc[ 
e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^...
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {3042, 4205, 2009}

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.

\(\displaystyle \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4205

\(\displaystyle \int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \cot (e+f x)+b^2 (c+d x)^3 \cot ^2(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((-I)*b^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) - ((I/2)*a*b*(c + d*x)^ 
4)/d - (b^2*(c + d*x)^4)/(4*d) - (b^2*(c + d*x)^3*Cot[e + f*x])/f + (3*b^2 
*d*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))])/f^2 + (2*a*b*(c + d*x)^3*Log[ 
1 - E^((2*I)*(e + f*x))])/f - ((3*I)*b^2*d^2*(c + d*x)*PolyLog[2, E^((2*I) 
*(e + f*x))])/f^3 - ((3*I)*a*b*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x) 
)])/f^2 + (3*b^2*d^3*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^4) + (3*a*b*d^2 
*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/f^3 + (((3*I)/2)*a*b*d^3*PolyL 
og[4, E^((2*I)*(e + f*x))])/f^4
 

Defintions of rubi rules used

rule 2009

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

rule 3042

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

rule 4205

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, m}, 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. 1603 vs. \(2 (266 ) = 532\).

Time = 1.15 (sec) , antiderivative size = 1604, normalized size of antiderivative = 5.44

method	result	size
risch	\(\text {Expression too large to display}\)	\(1604\)

Input:

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

Output:

-12*I/f*b*d*c^2*a*e*x-12*I/f^2*b*a*c*d^2*polylog(2,exp(I*(f*x+e)))*x-12*I/ 
f^2*b*a*c*d^2*polylog(2,-exp(I*(f*x+e)))*x+12*I/f^2*b*a*c*d^2*e^2*x+d^2*a^ 
2*c*x^3+3/2*d*a^2*c^2*x^2+a^2*c^3*x-d^2*b^2*c*x^3-3/2*d*b^2*c^2*x^2+2/f*b* 
a*c^3*ln(exp(I*(f*x+e))-1)-4/f*b*a*c^3*ln(exp(I*(f*x+e)))+2/f*b*a*c^3*ln(e 
xp(I*(f*x+e))+1)+3/f^4*b^2*e^2*d^3*ln(exp(I*(f*x+e))-1)-6/f^4*b^2*e^2*d^3* 
ln(exp(I*(f*x+e)))+3/f^2*b^2*c^2*d*ln(exp(I*(f*x+e))-1)-6/f^2*b^2*c^2*d*ln 
(exp(I*(f*x+e)))+3/f^2*b^2*c^2*d*ln(exp(I*(f*x+e))+1)+3/f^2*b^2*d^3*ln(1-e 
xp(I*(f*x+e)))*x^2-2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/f/(exp(2*I* 
(f*x+e))-1)+6/f^4*b^2*d^3*polylog(3,exp(I*(f*x+e)))+6/f^4*b^2*d^3*polylog( 
3,-exp(I*(f*x+e)))-2/f^4*b*e^3*a*d^3*ln(exp(I*(f*x+e))-1)+4/f^4*b*e^3*a*d^ 
3*ln(exp(I*(f*x+e)))+12/f^3*b*a*c*d^2*polylog(3,exp(I*(f*x+e)))+12/f^3*b*a 
*c*d^2*polylog(3,-exp(I*(f*x+e)))+12/f^3*b*d^3*a*polylog(3,exp(I*(f*x+e))) 
*x+12/f^3*b*d^3*a*polylog(3,-exp(I*(f*x+e)))*x+6/f^2*b^2*d^2*c*ln(1-exp(I* 
(f*x+e)))*x+6/f^2*b^2*d^2*c*ln(exp(I*(f*x+e))+1)*x+2/f^4*b*e^3*a*d^3*ln(1- 
exp(I*(f*x+e)))+2/f*b*d^3*a*ln(1-exp(I*(f*x+e)))*x^3+2/f*b*d^3*a*ln(exp(I* 
(f*x+e))+1)*x^3-6/f^3*b^2*e*c*d^2*ln(exp(I*(f*x+e))-1)+2*I*a*b*c^3*x+1/2*I 
/d*a*b*c^4-1/2*I*d^3*a*b*x^4+3/f^2*b^2*d^3*ln(exp(I*(f*x+e))+1)*x^2-3/f^4* 
b^2*e^2*d^3*ln(1-exp(I*(f*x+e)))-2*I/f*b^2*d^3*x^3+4*I/f^4*b^2*e^3*d^3+6*I 
/f^3*b^2*e^2*d^3*x+12*I/f^4*b*d^3*a*polylog(4,exp(I*(f*x+e)))+12*I/f^4*b*d 
^3*a*polylog(4,-exp(I*(f*x+e)))+1/4*d^3*a^2*x^4+1/4/d*a^2*c^4-1/4*d^3*b...
 

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. 1154 vs. \(2 (259) = 518\).

Time = 0.18 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.91 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

-1/4*(4*b^2*d^3*f^3*x^3 + 12*b^2*c*d^2*f^3*x^2 + 12*b^2*c^2*d*f^3*x + 4*b^ 
2*c^3*f^3 - 3*I*a*b*d^3*polylog(4, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e))* 
sin(2*f*x + 2*e) + 3*I*a*b*d^3*polylog(4, cos(2*f*x + 2*e) - I*sin(2*f*x + 
 2*e))*sin(2*f*x + 2*e) + 6*(I*a*b*d^3*f^2*x^2 + I*a*b*c^2*d*f^2 + I*b^2*c 
*d^2*f + I*(2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*dilog(cos(2*f*x + 2*e) + I*sin 
(2*f*x + 2*e))*sin(2*f*x + 2*e) + 6*(-I*a*b*d^3*f^2*x^2 - I*a*b*c^2*d*f^2 
- I*b^2*c*d^2*f - I*(2*a*b*c*d^2*f^2 + b^2*d^3*f)*x)*dilog(cos(2*f*x + 2*e 
) - I*sin(2*f*x + 2*e))*sin(2*f*x + 2*e) + 2*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^ 
3 - 3*b^2*d^3*e^2 + 3*(2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - 
 b^2*c*d^2*e)*f)*log(-1/2*cos(2*f*x + 2*e) + 1/2*I*sin(2*f*x + 2*e) + 1/2) 
*sin(2*f*x + 2*e) + 2*(2*a*b*d^3*e^3 - 2*a*b*c^3*f^3 - 3*b^2*d^3*e^2 + 3*( 
2*a*b*c^2*d*e - b^2*c^2*d)*f^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f)*log(-1 
/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2)*sin(2*f*x + 2*e) - 2*( 
2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3* 
(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*(a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 
6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 
 2*e) + 1)*sin(2*f*x + 2*e) - 2*(2*a*b*d^3*f^3*x^3 + 2*a*b*d^3*e^3 + 6*a*b 
*c^2*d*e*f^2 - 3*b^2*d^3*e^2 + 3*(2*a*b*c*d^2*f^3 + b^2*d^3*f^2)*x^2 - 6*( 
a*b*c*d^2*e^2 - b^2*c*d^2*e)*f + 6*(a*b*c^2*d*f^3 + b^2*c*d^2*f^2)*x)*log( 
-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1)*sin(2*f*x + 2*e) - 3*(2*a*b...
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4024 vs. \(2 (259) = 518\).

Time = 1.13 (sec) , antiderivative size = 4024, normalized size of antiderivative = 13.64 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(4*(f*x + e)*a^2*c^3 + (f*x + e)^4*a^2*d^3/f^3 - 4*(f*x + e)^3*a^2*d^3 
*e/f^3 + 6*(f*x + e)^2*a^2*d^3*e^2/f^3 - 4*(f*x + e)*a^2*d^3*e^3/f^3 + 4*( 
f*x + e)^3*a^2*c*d^2/f^2 - 12*(f*x + e)^2*a^2*c*d^2*e/f^2 + 12*(f*x + e)*a 
^2*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^2*c^2*d/f - 12*(f*x + e)*a^2*c^2*d*e/f 
+ 8*a*b*c^3*log(sin(f*x + e)) - 8*a*b*d^3*e^3*log(sin(f*x + e))/f^3 + 24*a 
*b*c*d^2*e^2*log(sin(f*x + e))/f^2 - 24*a*b*c^2*d*e*log(sin(f*x + e))/f + 
4*((2*a*b - I*b^2)*(f*x + e)^4*d^3 + 8*b^2*d^3*e^3 - 24*b^2*c*d^2*e^2*f + 
24*b^2*c^2*d*e*f^2 - 8*b^2*c^3*f^3 - 4*((2*a*b - I*b^2)*d^3*e - (2*a*b - I 
*b^2)*c*d^2*f)*(f*x + e)^3 + 6*((2*a*b - I*b^2)*d^3*e^2 - 2*(2*a*b - I*b^2 
)*c*d^2*e*f + (2*a*b - I*b^2)*c^2*d*f^2)*(f*x + e)^2 - 4*(-I*b^2*d^3*e^3 + 
 3*I*b^2*c*d^2*e^2*f - 3*I*b^2*c^2*d*e*f^2 + I*b^2*c^3*f^3)*(f*x + e) - 4* 
(2*(f*x + e)^3*a*b*d^3 + 3*b^2*d^3*e^2 - 6*b^2*c*d^2*e*f + 3*b^2*c^2*d*f^2 
 - 3*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 6*(a*b*d^3*e^2 
+ a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e) - ( 
2*(f*x + e)^3*a*b*d^3 + 3*b^2*d^3*e^2 - 6*b^2*c*d^2*e*f + 3*b^2*c^2*d*f^2 
- 3*(2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e)^2 + 6*(a*b*d^3*e^2 + 
 a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*(f*x + e))*cos 
(2*f*x + 2*e) + (-2*I*(f*x + e)^3*a*b*d^3 - 3*I*b^2*d^3*e^2 + 6*I*b^2*c*d^ 
2*e*f - 3*I*b^2*c^2*d*f^2 + 3*(2*I*a*b*d^3*e - 2*I*a*b*c*d^2*f - I*b^2*d^3 
)*(f*x + e)^2 + 6*(-I*a*b*d^3*e^2 - I*a*b*c^2*d*f^2 + I*b^2*d^3*e + (2*...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=\frac {-12 \cos \left (f x +e \right ) b^{2} c^{2} d f x -4 \cot \left (f x +e \right ) \sin \left (f x +e \right ) b^{2} c^{3} f +8 \left (\int \cot \left (f x +e \right ) x^{3}d x \right ) \sin \left (f x +e \right ) a b \,d^{3} f^{2}+24 \left (\int \cot \left (f x +e \right ) x^{2}d x \right ) \sin \left (f x +e \right ) a b c \,d^{2} f^{2}+24 \left (\int \cot \left (f x +e \right ) x d x \right ) \sin \left (f x +e \right ) a b \,c^{2} d \,f^{2}+4 \left (\int \cot \left (f x +e \right )^{2} x^{3}d x \right ) \sin \left (f x +e \right ) b^{2} d^{3} f^{2}+12 \left (\int \cot \left (f x +e \right )^{2} x^{2}d x \right ) \sin \left (f x +e \right ) b^{2} c \,d^{2} f^{2}-8 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) \sin \left (f x +e \right ) a b \,c^{3} f -12 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}+1\right ) \sin \left (f x +e \right ) b^{2} c^{2} d +8 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right ) a b \,c^{3} f +12 \,\mathrm {log}\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \sin \left (f x +e \right ) b^{2} c^{2} d +4 \sin \left (f x +e \right ) a^{2} c^{3} f^{2} x +6 \sin \left (f x +e \right ) a^{2} c^{2} d \,f^{2} x^{2}+4 \sin \left (f x +e \right ) a^{2} c \,d^{2} f^{2} x^{3}+\sin \left (f x +e \right ) a^{2} d^{3} f^{2} x^{4}-4 \sin \left (f x +e \right ) b^{2} c^{3} f^{2} x -6 \sin \left (f x +e \right ) b^{2} c^{2} d \,f^{2} x^{2}}{4 \sin \left (f x +e \right ) f^{2}} \] Input:

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

Output:

( - 12*cos(e + f*x)*b**2*c**2*d*f*x - 4*cot(e + f*x)*sin(e + f*x)*b**2*c** 
3*f + 8*int(cot(e + f*x)*x**3,x)*sin(e + f*x)*a*b*d**3*f**2 + 24*int(cot(e 
 + f*x)*x**2,x)*sin(e + f*x)*a*b*c*d**2*f**2 + 24*int(cot(e + f*x)*x,x)*si 
n(e + f*x)*a*b*c**2*d*f**2 + 4*int(cot(e + f*x)**2*x**3,x)*sin(e + f*x)*b* 
*2*d**3*f**2 + 12*int(cot(e + f*x)**2*x**2,x)*sin(e + f*x)*b**2*c*d**2*f** 
2 - 8*log(tan((e + f*x)/2)**2 + 1)*sin(e + f*x)*a*b*c**3*f - 12*log(tan((e 
 + f*x)/2)**2 + 1)*sin(e + f*x)*b**2*c**2*d + 8*log(tan((e + f*x)/2))*sin( 
e + f*x)*a*b*c**3*f + 12*log(tan((e + f*x)/2))*sin(e + f*x)*b**2*c**2*d + 
4*sin(e + f*x)*a**2*c**3*f**2*x + 6*sin(e + f*x)*a**2*c**2*d*f**2*x**2 + 4 
*sin(e + f*x)*a**2*c*d**2*f**2*x**3 + sin(e + f*x)*a**2*d**3*f**2*x**4 - 4 
*sin(e + f*x)*b**2*c**3*f**2*x - 6*sin(e + f*x)*b**2*c**2*d*f**2*x**2)/(4* 
sin(e + f*x)*f**2)