\(\int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx\) [47]

Optimal result

Integrand size = 20, antiderivative size = 603 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=-\frac {3 i b^3 d (c+d x)^2}{2 f^2}-\frac {3 i a b^2 (c+d x)^3}{f}-\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 i a^2 b (c+d x)^4}{4 d}-\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {i b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac {b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}-\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}+\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4} \] Output:

-3/2*I*b^3*d*(d*x+c)^2/f^2-3*I*a*b^2*(d*x+c)^3/f-1/2*b^3*(d*x+c)^3/f+1/4*a 
^3*(d*x+c)^4/d-9/2*I*a^2*b*d*(d*x+c)^2*polylog(2,exp(2*I*(f*x+e)))/f^2-3/4 
*a*b^2*(d*x+c)^4/d-3/4*I*b^3*d^3*polylog(4,exp(2*I*(f*x+e)))/f^4-3/2*b^3*d 
*(d*x+c)^2*cot(f*x+e)/f^2-3*a*b^2*(d*x+c)^3*cot(f*x+e)/f-1/2*b^3*(d*x+c)^3 
*cot(f*x+e)^2/f+3*b^3*d^2*(d*x+c)*ln(1-exp(2*I*(f*x+e)))/f^3+9*a*b^2*d*(d* 
x+c)^2*ln(1-exp(2*I*(f*x+e)))/f^2+3*a^2*b*(d*x+c)^3*ln(1-exp(2*I*(f*x+e))) 
/f-b^3*(d*x+c)^3*ln(1-exp(2*I*(f*x+e)))/f+1/4*I*b^3*(d*x+c)^4/d+3/2*I*b^3* 
d*(d*x+c)^2*polylog(2,exp(2*I*(f*x+e)))/f^2-9*I*a*b^2*d^2*(d*x+c)*polylog( 
2,exp(2*I*(f*x+e)))/f^3+9/4*I*a^2*b*d^3*polylog(4,exp(2*I*(f*x+e)))/f^4+9/ 
2*a*b^2*d^3*polylog(3,exp(2*I*(f*x+e)))/f^4+9/2*a^2*b*d^2*(d*x+c)*polylog( 
3,exp(2*I*(f*x+e)))/f^3-3/2*b^3*d^2*(d*x+c)*polylog(3,exp(2*I*(f*x+e)))/f^ 
3-3/2*I*b^3*d^3*polylog(2,exp(2*I*(f*x+e)))/f^4-3/4*I*a^2*b*(d*x+c)^4/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. \(3129\) vs. \(2(603)=1206\).

Time = 7.42 (sec) , antiderivative size = 3129, normalized size of antiderivative = 5.19 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\text {Result too large to show} \] Input:

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

Output:

((-(b^3*c^3) - 3*b^3*c^2*d*x - 3*b^3*c*d^2*x^2 - b^3*d^3*x^3)*Csc[e + f*x] 
^2)/(2*f) - (3*a*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*Poly 
Log[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))]))/(2*f^4) - (3*a^2* 
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*Lo 
g[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))]))/(2*f^3) + (b^3*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))]))/(2*f^3) - (3*a^2*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))] + ...
 

Rubi [A] (verified)

Time = 1.38 (sec) , antiderivative size = 603, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4205

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^4}{4 d}+\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i a^2 b (c+d x)^4}{4 d}+\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^3}{f}-\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac {b^3 (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}-\frac {3 i b^3 d (c+d x)^2}{2 f^2}-\frac {b^3 (c+d x)^3}{2 f}+\frac {i b^3 (c+d x)^4}{4 d}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}\)

Input:

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

Output:

(((-3*I)/2)*b^3*d*(c + d*x)^2)/f^2 - ((3*I)*a*b^2*(c + d*x)^3)/f - (b^3*(c 
 + d*x)^3)/(2*f) + (a^3*(c + d*x)^4)/(4*d) - (((3*I)/4)*a^2*b*(c + d*x)^4) 
/d - (3*a*b^2*(c + d*x)^4)/(4*d) + ((I/4)*b^3*(c + d*x)^4)/d - (3*b^3*d*(c 
 + d*x)^2*Cot[e + f*x])/(2*f^2) - (3*a*b^2*(c + d*x)^3*Cot[e + f*x])/f - ( 
b^3*(c + d*x)^3*Cot[e + f*x]^2)/(2*f) + (3*b^3*d^2*(c + d*x)*Log[1 - E^((2 
*I)*(e + f*x))])/f^3 + (9*a*b^2*d*(c + d*x)^2*Log[1 - E^((2*I)*(e + f*x))] 
)/f^2 + (3*a^2*b*(c + d*x)^3*Log[1 - E^((2*I)*(e + f*x))])/f - (b^3*(c + d 
*x)^3*Log[1 - E^((2*I)*(e + f*x))])/f - (((3*I)/2)*b^3*d^3*PolyLog[2, E^(( 
2*I)*(e + f*x))])/f^4 - ((9*I)*a*b^2*d^2*(c + d*x)*PolyLog[2, E^((2*I)*(e 
+ f*x))])/f^3 - (((9*I)/2)*a^2*b*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f* 
x))])/f^2 + (((3*I)/2)*b^3*d*(c + d*x)^2*PolyLog[2, E^((2*I)*(e + f*x))])/ 
f^2 + (9*a*b^2*d^3*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^4) + (9*a^2*b*d^2 
*(c + d*x)*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) - (3*b^3*d^2*(c + d*x) 
*PolyLog[3, E^((2*I)*(e + f*x))])/(2*f^3) + (((9*I)/4)*a^2*b*d^3*PolyLog[4 
, E^((2*I)*(e + f*x))])/f^4 - (((3*I)/4)*b^3*d^3*PolyLog[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. 3160 vs. \(2 (535 ) = 1070\).

Time = 1.23 (sec) , antiderivative size = 3161, normalized size of antiderivative = 5.24

method	result	size
risch	\(\text {Expression too large to display}\)	\(3161\)

Input:

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

Output:

-1/4*I/d*b^3*c^4+b^2*(3*I*b*c^2*d+6*I*a*c^3*f+2*b*d^3*f*x^3*exp(2*I*(f*x+e 
))+6*I*b*c*d^2*x+3*I*b*d^3*x^2+18*I*a*c^2*d*f*x+6*b*c*d^2*f*x^2*exp(2*I*(f 
*x+e))-3*I*b*d^3*x^2*exp(2*I*(f*x+e))-18*I*a*c^2*d*f*x*exp(2*I*(f*x+e))-6* 
I*a*d^3*f*x^3*exp(2*I*(f*x+e))+6*b*c^2*d*f*x*exp(2*I*(f*x+e))-18*I*a*c*d^2 
*f*x^2*exp(2*I*(f*x+e))-3*I*b*c^2*d*exp(2*I*(f*x+e))-6*I*b*c*d^2*x*exp(2*I 
*(f*x+e))+2*b*c^3*f*exp(2*I*(f*x+e))+18*I*a*c*d^2*f*x^2-6*I*a*c^3*f*exp(2* 
I*(f*x+e))+6*I*a*d^3*f*x^3)/f^2/(exp(2*I*(f*x+e))-1)^2+1/4*I*b^3*d^3*x^4-9 
/2*d*a*b^2*c^2*x^2+3*I*a^2*b*c^3*x+3/4*I/d*a^2*b*c^4+d^2*a^3*c*x^3+3/2*d*a 
^3*c^2*x^2+a^3*c^3*x-3/4*d^3*a*b^2*x^4-I*b^3*c^3*x-3*a*b^2*c^3*x-3/4/d*a*b 
^2*c^4-1/f*b^3*c^3*ln(exp(I*(f*x+e))-1)+2/f*b^3*c^3*ln(exp(I*(f*x+e)))-1/f 
*b^3*c^3*ln(exp(I*(f*x+e))+1)-3/4*I*d^3*a^2*b*x^4+3/2*I*b^3*d*c^2*x^2-3*d^ 
2*a*b^2*c*x^3+3/f^3*b^3*c*d^2*ln(exp(I*(f*x+e))+1)-6/f^3*b^3*c*d^2*polylog 
(3,exp(I*(f*x+e)))-6/f^3*b^3*c*d^2*polylog(3,-exp(I*(f*x+e)))+3/f*b*a^2*c^ 
3*ln(exp(I*(f*x+e))-1)-6/f*b*a^2*c^3*ln(exp(I*(f*x+e)))+3/f*b*a^2*c^3*ln(e 
xp(I*(f*x+e))+1)-6/f^3*b^3*d^3*polylog(3,exp(I*(f*x+e)))*x-6/f^3*b^3*d^3*p 
olylog(3,-exp(I*(f*x+e)))*x+3/f^3*b^3*d^3*ln(1-exp(I*(f*x+e)))*x+3/f^3*b^3 
*d^3*ln(exp(I*(f*x+e))+1)*x-1/f*b^3*d^3*ln(1-exp(I*(f*x+e)))*x^3-1/f*b^3*d 
^3*ln(exp(I*(f*x+e))+1)*x^3-1/f^4*b^3*d^3*ln(1-exp(I*(f*x+e)))*e^3-3/f^4*b 
^3*e*d^3*ln(exp(I*(f*x+e))-1)+6/f^4*b^3*e*d^3*ln(exp(I*(f*x+e)))+1/f^4*b^3 
*e^3*d^3*ln(exp(I*(f*x+e))-1)-2/f^4*b^3*e^3*d^3*ln(exp(I*(f*x+e)))+3/f^...
 

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. 2749 vs. \(2 (521) = 1042\).

Time = 0.16 (sec) , antiderivative size = 2749, normalized size of antiderivative = 4.56 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

-1/8*(2*(a^3 - 3*a*b^2)*d^3*f^4*x^4 - 8*b^3*c^3*f^3 - 8*(b^3*d^3*f^3 - (a^ 
3 - 3*a*b^2)*c*d^2*f^4)*x^3 - 12*(2*b^3*c*d^2*f^3 - (a^3 - 3*a*b^2)*c^2*d* 
f^4)*x^2 - 8*(3*b^3*c^2*d*f^3 - (a^3 - 3*a*b^2)*c^3*f^4)*x - 2*((a^3 - 3*a 
*b^2)*d^3*f^4*x^4 + 4*(a^3 - 3*a*b^2)*c*d^2*f^4*x^3 + 6*(a^3 - 3*a*b^2)*c^ 
2*d*f^4*x^2 + 4*(a^3 - 3*a*b^2)*c^3*f^4*x)*cos(2*f*x + 2*e) + 6*(-I*(3*a^2 
*b - b^3)*d^3*f^2*x^2 - 6*I*a*b^2*c*d^2*f - I*b^3*d^3 - I*(3*a^2*b - b^3)* 
c^2*d*f^2 - 2*I*(3*a*b^2*d^3*f + (3*a^2*b - b^3)*c*d^2*f^2)*x + (I*(3*a^2* 
b - b^3)*d^3*f^2*x^2 + 6*I*a*b^2*c*d^2*f + I*b^3*d^3 + I*(3*a^2*b - b^3)*c 
^2*d*f^2 + 2*I*(3*a*b^2*d^3*f + (3*a^2*b - b^3)*c*d^2*f^2)*x)*cos(2*f*x + 
2*e))*dilog(cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) + 6*(I*(3*a^2*b - b^3)* 
d^3*f^2*x^2 + 6*I*a*b^2*c*d^2*f + I*b^3*d^3 + I*(3*a^2*b - b^3)*c^2*d*f^2 
+ 2*I*(3*a*b^2*d^3*f + (3*a^2*b - b^3)*c*d^2*f^2)*x + (-I*(3*a^2*b - b^3)* 
d^3*f^2*x^2 - 6*I*a*b^2*c*d^2*f - I*b^3*d^3 - I*(3*a^2*b - b^3)*c^2*d*f^2 
- 2*I*(3*a*b^2*d^3*f + (3*a^2*b - b^3)*c*d^2*f^2)*x)*cos(2*f*x + 2*e))*dil 
og(cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) + 4*(9*a*b^2*d^3*e^2 - 3*b^3*d^3 
*e - (3*a^2*b - b^3)*d^3*e^3 + (3*a^2*b - b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d 
- (3*a^2*b - b^3)*c^2*d*e)*f^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b 
 - b^3)*c*d^2*e^2)*f - (9*a*b^2*d^3*e^2 - 3*b^3*d^3*e - (3*a^2*b - b^3)*d^ 
3*e^3 + (3*a^2*b - b^3)*c^3*f^3 + 3*(3*a*b^2*c^2*d - (3*a^2*b - b^3)*c^2*d 
*e)*f^2 - 3*(6*a*b^2*c*d^2*e - b^3*c*d^2 - (3*a^2*b - b^3)*c*d^2*e^2)*f...
 

Sympy [F]

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

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

Output:

Integral((a + b*cot(e + f*x))**3*(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. 11252 vs. \(2 (521) = 1042\).

Time = 14.41 (sec) , antiderivative size = 11252, normalized size of antiderivative = 18.66 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*(4*(f*x + e)*a^3*c^3 + (f*x + e)^4*a^3*d^3/f^3 - 4*(f*x + e)^3*a^3*d^3 
*e/f^3 + 6*(f*x + e)^2*a^3*d^3*e^2/f^3 - 4*(f*x + e)*a^3*d^3*e^3/f^3 + 4*( 
f*x + e)^3*a^3*c*d^2/f^2 - 12*(f*x + e)^2*a^3*c*d^2*e/f^2 + 12*(f*x + e)*a 
^3*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^3*c^2*d/f - 12*(f*x + e)*a^3*c^2*d*e/f 
+ 12*a^2*b*c^3*log(sin(f*x + e)) - 12*a^2*b*d^3*e^3*log(sin(f*x + e))/f^3 
+ 36*a^2*b*c*d^2*e^2*log(sin(f*x + e))/f^2 - 36*a^2*b*c^2*d*e*log(sin(f*x 
+ e))/f - 4*(24*a*b^2*d^3*e^3 - 24*a*b^2*c^3*f^3 + (3*a^2*b - 3*I*a*b^2 - 
b^3)*(f*x + e)^4*d^3 - 12*b^3*d^3*e^2 - 4*((3*a^2*b - 3*I*a*b^2 - b^3)*d^3 
*e - (3*a^2*b - 3*I*a*b^2 - b^3)*c*d^2*f)*(f*x + e)^3 + 6*((3*a^2*b - 3*I* 
a*b^2 - b^3)*d^3*e^2 - 2*(3*a^2*b - 3*I*a*b^2 - b^3)*c*d^2*e*f + (3*a^2*b 
- 3*I*a*b^2 - b^3)*c^2*d*f^2)*(f*x + e)^2 + 12*(6*a*b^2*c^2*d*e - b^3*c^2* 
d)*f^2 - 4*((-3*I*a*b^2 - b^3)*d^3*e^3 + 3*(3*I*a*b^2 + b^3)*c*d^2*e^2*f + 
 3*(-3*I*a*b^2 - b^3)*c^2*d*e*f^2 + (3*I*a*b^2 + b^3)*c^3*f^3)*(f*x + e) - 
 24*(3*a*b^2*c*d^2*e^2 - b^3*c*d^2*e)*f - 4*(b^3*d^3*e^3 - b^3*c^3*f^3 + 9 
*a*b^2*d^3*e^2 + (3*a^2*b - b^3)*(f*x + e)^3*d^3 - 3*b^3*d^3*e + 3*(3*a*b^ 
2*d^3 - (3*a^2*b - b^3)*d^3*e + (3*a^2*b - b^3)*c*d^2*f)*(f*x + e)^2 + 3*( 
b^3*c^2*d*e + 3*a*b^2*c^2*d)*f^2 - 3*(6*a*b^2*d^3*e - b^3*d^3 - (3*a^2*b - 
 b^3)*d^3*e^2 - (3*a^2*b - b^3)*c^2*d*f^2 - 2*(3*a*b^2*c*d^2 - (3*a^2*b - 
b^3)*c*d^2*e)*f)*(f*x + e) - 3*(b^3*c*d^2*e^2 + 6*a*b^2*c*d^2*e - b^3*c*d^ 
2)*f + (b^3*d^3*e^3 - b^3*c^3*f^3 + 9*a*b^2*d^3*e^2 + (3*a^2*b - b^3)*(...
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx =\text {Too large to display} \] Input:

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

Output:

( - 36*cos(e + f*x)*sin(e + f*x)*a*b**2*c**2*d*f*x - 12*cot(e + f*x)*sin(e 
 + f*x)**2*a*b**2*c**3*f + 12*int(cot(e + f*x)*x**3,x)*sin(e + f*x)**2*a** 
2*b*d**3*f**2 + 36*int(cot(e + f*x)*x**2,x)*sin(e + f*x)**2*a**2*b*c*d**2* 
f**2 + 36*int(cot(e + f*x)*x,x)*sin(e + f*x)**2*a**2*b*c**2*d*f**2 + 4*int 
(cot(e + f*x)**3*x**3,x)*sin(e + f*x)**2*b**3*d**3*f**2 + 12*int(cot(e + f 
*x)**3*x**2,x)*sin(e + f*x)**2*b**3*c*d**2*f**2 + 12*int(cot(e + f*x)**3*x 
,x)*sin(e + f*x)**2*b**3*c**2*d*f**2 + 12*int(cot(e + f*x)**2*x**3,x)*sin( 
e + f*x)**2*a*b**2*d**3*f**2 + 36*int(cot(e + f*x)**2*x**2,x)*sin(e + f*x) 
**2*a*b**2*c*d**2*f**2 - 12*log(tan((e + f*x)/2)**2 + 1)*sin(e + f*x)**2*a 
**2*b*c**3*f - 36*log(tan((e + f*x)/2)**2 + 1)*sin(e + f*x)**2*a*b**2*c**2 
*d + 4*log(tan((e + f*x)/2)**2 + 1)*sin(e + f*x)**2*b**3*c**3*f + 12*log(t 
an((e + f*x)/2))*sin(e + f*x)**2*a**2*b*c**3*f + 36*log(tan((e + f*x)/2))* 
sin(e + f*x)**2*a*b**2*c**2*d - 4*log(tan((e + f*x)/2))*sin(e + f*x)**2*b* 
*3*c**3*f + 4*sin(e + f*x)**2*a**3*c**3*f**2*x + 6*sin(e + f*x)**2*a**3*c* 
*2*d*f**2*x**2 + 4*sin(e + f*x)**2*a**3*c*d**2*f**2*x**3 + sin(e + f*x)**2 
*a**3*d**3*f**2*x**4 - 12*sin(e + f*x)**2*a*b**2*c**3*f**2*x - 18*sin(e + 
f*x)**2*a*b**2*c**2*d*f**2*x**2 + sin(e + f*x)**2*b**3*c**3*f - 2*b**3*c** 
3*f)/(4*sin(e + f*x)**2*f**2)