Integrand size = 20, antiderivative size = 603 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=-\frac {3 i b^3 d (c+d x)^2}{2 f^2}-\frac {3 i a b^2 (c+d x)^3}{f}-\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}-\frac {3 i a^2 b (c+d x)^4}{4 d}-\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {i b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac {3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac {b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}-\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}+\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4} \]
-3/4*I*b^3*d^3*polylog(4,exp(2*I*(f*x+e)))/f^4-3/2*I*b^3*d*(d*x+c)^2/f^2-1 /2*b^3*(d*x+c)^3/f+1/4*a^3*(d*x+c)^4/d-3*I*a*b^2*(d*x+c)^3/f-3/4*a*b^2*(d* x+c)^4/d-3/2*I*b^3*d^3*polylog(2,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-3/4*I*a^2*b*(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+1/4*I*b^3*(d*x+c)^4/d+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^3-9/2*I*a^2*b*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
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3129\) vs. \(2(603)=1206\).
Time = 8.22 (sec) , antiderivative size = 3129, normalized size of antiderivative = 5.19 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\text {Result too large to show} \]
((-(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))] + ...
Time = 1.33 (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 |
\(\displaystyle \frac {a^3 (c+d x)^4}{4 d}+\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac {3 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {3 i a^2 b (c+d x)^4}{4 d}+\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^3}{f}-\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac {b^3 (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}-\frac {3 i b^3 d (c+d x)^2}{2 f^2}-\frac {b^3 (c+d x)^3}{2 f}+\frac {i b^3 (c+d x)^4}{4 d}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}\) |
(((-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
3.1.47.3.1 Defintions of rubi rules used
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]
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 = 0.94 (sec) , antiderivative size = 3161, normalized size of antiderivative = 5.24
-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*I*d^2*a^2*b*c*x^3-9/2*I*d*a^2*b*c^2*x^2+18/f^3*b *d^3*a^2*polylog(3,exp(I*(f*x+e)))*x+18/f^3*b*d^3*a^2*polylog(3,-exp(I*(f* x+e)))*x-6/f^2*b^3*e*d*c^2*ln(exp(I*(f*x+e)))+3/f^2*b^3*e*d*c^2*ln(exp(I*( f*x+e))-1)+9/f^2*b^2*a*c^2*d*ln(exp(I*(f*x+e))+1)-18/f^2*b^2*a*c^2*d*ln(ex p(I*(f*x+e)))+9/f^2*b^2*a*c^2*d*ln(exp(I*(f*x+e))-1)-18/f^4*b^2*e^2*a*d^3* ln(exp(I*(f*x+e)))+9/f^4*b^2*e^2*a*d^3*ln(exp(I*(f*x+e))-1)-3/f*b^3*c*d^2* ln(1-exp(I*(f*x+e)))*x^2-3/f*b^3*c*d^2*ln(exp(I*(f*x+e))+1)*x^2+3/f^3*b^3* c*d^2*ln(1-exp(I*(f*x+e)))*e^2-9/2*I/f^4*b*d^3*a^2*e^4+18*I/f^4*b*d^3*a^2* polylog(4,exp(I*(f*x+e)))+18*I/f^4*b*d^3*a^2*polylog(4,-exp(I*(f*x+e)))+3* I/f^2*b^3*d*c^2*e^2+3*I/f^2*b^3*d*c^2*polylog(2,exp(I*(f*x+e)))+3*I/f^2*b^ 3*d*c^2*polylog(2,-exp(I*(f*x+e)))+2*I/f^3*b^3*d^3*e^3*x-6*I/f^3*b^3*d^3*e *x+3*I/f^2*b^3*d^3*polylog(2,-exp(I*(f*x+e)))*x^2+3*I/f^2*b^3*d^3*polylog( 2,exp(I*(f*x+e)))*x^2-4*I/f^3*b^3*c*d^2*e^3-6*I/f*b^2*a*d^3*x^3+12*I/f^4*b ^2*a*d^3*e^3+6/f^4*b*e^3*d^3*a^2*ln(exp(I*(f*x+e)))-3/f^4*b*e^3*d^3*a^2*ln (exp(I*(f*x+e))-1)+6/f^3*b^3*e^2*c*d^2*ln(exp(I*(f*x+e)))-3/f^3*b^3*e^2*c* d^2*ln(exp(I*(f*x+e))-1)+3/f^4*b*d^3*a^2*ln(1-exp(I*(f*x+e)))*e^3+3/f*b*d^ 3*a^2*ln(1-exp(I*(f*x+e)))*x^3+3/f*b*d^3*a^2*ln(exp(I*(f*x+e))+1)*x^3-3/f* b^3*d*c^2*ln(1-exp(I*(f*x+e)))*x-3/f^2*b^3*d*c^2*ln(1-exp(I*(f*x+e)))*e-3/ f*b^3*d*c^2*ln(exp(I*(f*x+e))+1)*x+9/f^2*b^2*a*d^3*ln(1-exp(I*(f*x+e)))...
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 = 0.37 (sec) , antiderivative size = 2749, normalized size of antiderivative = 4.56 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\text {Too large to display} \]
-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...
\[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\int \left (a + b \cot {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{3}\, dx \]
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 = 15.55 (sec) , antiderivative size = 11252, normalized size of antiderivative = 18.66 \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\text {Too large to display} \]
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)*(...
\[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \cot \left (f x + e\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {cot}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \]