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

Optimal result

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

[Out]

-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)*polylo
g(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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3803, 3798, 2221, 2611, 6744, 2320, 6724, 3801, 32} \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=\frac {a^2 (c+d x)^4}{4 d}+\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {i a b (c+d x)^4}{2 d}+\frac {3 i a b d^3 \operatorname {PolyLog}\left (4,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4} \]

[In]

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

[Out]

((-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*Lo
g[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*PolyLog[4, E^((2*I)*(e + f*x))])/f^4

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

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]
- Dist[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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{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 2611

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] + Dist[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 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3803

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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> 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 6744

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = \frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \cot (e+f x) \, dx+b^2 \int (c+d x)^3 \cot ^2(e+f x) \, dx \\ & = \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)^3 \cot (e+f x)}{f}-(4 i a b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{1-e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x)^3 \, dx+\frac {\left (3 b^2 d\right ) \int (c+d x)^2 \cot (e+f x) \, dx}{f} \\ & = -\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 {2 a b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac {(6 a b d) \int (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}-\frac {\left (6 i b^2 d\right ) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{1-e^{2 i (e+f x)}} \, dx}{f} \\ & = -\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 a b d (c+d x)^2 \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^2}+\frac {\left (6 i a b d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f^2} \\ & = -\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 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}-\frac {\left (3 a b d^3\right ) \int \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right ) \, dx}{f^3}+\frac {\left (3 i b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,e^{2 i (e+f x)}\right ) \, dx}{f^3} \\ & = -\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 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,e^{2 i (e+f x)}\right )}{f^3}+\frac {\left (3 i a b d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}+\frac {\left (3 b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4} \\ & = -\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} \\ \end{align*}

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 = 7.24 (sec) , antiderivative size = 1657, normalized size of antiderivative = 5.62 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=-\frac {b^2 d^3 e^{i e} \csc (e) \left (2 e^{-2 i e} f^3 x^3+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (2,-e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (2,e^{-i (e+f x)}\right )+6 i \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+6 i \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )\right )}{2 f^4}-\frac {a b c d^2 e^{i e} \csc (e) \left (2 e^{-2 i e} f^3 x^3+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1-e^{-i (e+f x)}\right )+3 i \left (1-e^{-2 i e}\right ) f^2 x^2 \log \left (1+e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (2,-e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (2,e^{-i (e+f x)}\right )+6 i \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+6 i \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )\right )}{f^3}-\frac {a b d^3 e^{i e} \csc (e) \left (e^{-2 i e} f^4 x^4+2 i \left (1-e^{-2 i e}\right ) f^3 x^3 \log \left (1-e^{-i (e+f x)}\right )+2 i \left (1-e^{-2 i e}\right ) f^3 x^3 \log \left (1+e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f^2 x^2 \operatorname {PolyLog}\left (2,-e^{-i (e+f x)}\right )-6 \left (1-e^{-2 i e}\right ) f^2 x^2 \operatorname {PolyLog}\left (2,e^{-i (e+f x)}\right )+12 i \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (3,-e^{-i (e+f x)}\right )+12 i \left (1-e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (3,e^{-i (e+f x)}\right )+12 \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (4,-e^{-i (e+f x)}\right )+12 \left (1-e^{-2 i e}\right ) \operatorname {PolyLog}\left (4,e^{-i (e+f x)}\right )\right )}{2 f^4}+\frac {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^2 \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {2 a b c^3 \csc (e) (-f x \cos (e)+\log (\cos (f x) \sin (e)+\cos (e) \sin (f x)) \sin (e))}{f \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {\csc (e) \csc (e+f x) \left (4 a^2 c^3 f x \cos (f x)-4 b^2 c^3 f x \cos (f x)+6 a^2 c^2 d f x^2 \cos (f x)-6 b^2 c^2 d f x^2 \cos (f x)+4 a^2 c d^2 f x^3 \cos (f x)-4 b^2 c d^2 f x^3 \cos (f x)+a^2 d^3 f x^4 \cos (f x)-b^2 d^3 f x^4 \cos (f x)-4 a^2 c^3 f x \cos (2 e+f x)+4 b^2 c^3 f x \cos (2 e+f x)-6 a^2 c^2 d f x^2 \cos (2 e+f x)+6 b^2 c^2 d f x^2 \cos (2 e+f x)-4 a^2 c d^2 f x^3 \cos (2 e+f x)+4 b^2 c d^2 f x^3 \cos (2 e+f x)-a^2 d^3 f x^4 \cos (2 e+f x)+b^2 d^3 f x^4 \cos (2 e+f x)+8 b^2 c^3 \sin (f x)+24 b^2 c^2 d x \sin (f x)+8 a b c^3 f x \sin (f x)+24 b^2 c d^2 x^2 \sin (f x)+12 a b c^2 d f x^2 \sin (f x)+8 b^2 d^3 x^3 \sin (f x)+8 a b c d^2 f x^3 \sin (f x)+2 a b d^3 f x^4 \sin (f x)+8 a b c^3 f x \sin (2 e+f x)+12 a b c^2 d f x^2 \sin (2 e+f x)+8 a b c d^2 f x^3 \sin (2 e+f x)+2 a b d^3 f x^4 \sin (2 e+f x)\right )}{8 f}-\frac {3 b^2 c d^2 \csc (e) \sec (e) \left (e^{i \arctan (\tan (e))} f^2 x^2+\frac {\left (i f x (-\pi +2 \arctan (\tan (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x+\arctan (\tan (e))) \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )+\pi \log (\cos (f x))+2 \arctan (\tan (e)) \log (\sin (f x+\arctan (\tan (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x+\arctan (\tan (e)))}\right )\right ) \tan (e)}{\sqrt {1+\tan ^2(e)}}\right )}{f^3 \sqrt {\sec ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}-\frac {3 a b c^2 d \csc (e) \sec (e) \left (e^{i \arctan (\tan (e))} f^2 x^2+\frac {\left (i f x (-\pi +2 \arctan (\tan (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x+\arctan (\tan (e))) \log \left (1-e^{2 i (f x+\arctan (\tan (e)))}\right )+\pi \log (\cos (f x))+2 \arctan (\tan (e)) \log (\sin (f x+\arctan (\tan (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x+\arctan (\tan (e)))}\right )\right ) \tan (e)}{\sqrt {1+\tan ^2(e)}}\right )}{f^2 \sqrt {\sec ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}} \]

[In]

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

[Out]

-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))*PolyLo
g[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*PolyLog[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^2*(Cos[e]^2 + Sin[e]^2)) + (2*a*b*c^3*Csc[e]*(-(f*x*Cos[e]) + L
og[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (Csc[e]*Csc[e + f*x]*(4*a^2*c^3*f*x
*Cos[f*x] - 4*b^2*c^3*f*x*Cos[f*x] + 6*a^2*c^2*d*f*x^2*Cos[f*x] - 6*b^2*c^2*d*f*x^2*Cos[f*x] + 4*a^2*c*d^2*f*x
^3*Cos[f*x] - 4*b^2*c*d^2*f*x^3*Cos[f*x] + a^2*d^3*f*x^4*Cos[f*x] - b^2*d^3*f*x^4*Cos[f*x] - 4*a^2*c^3*f*x*Cos
[2*e + f*x] + 4*b^2*c^3*f*x*Cos[2*e + f*x] - 6*a^2*c^2*d*f*x^2*Cos[2*e + f*x] + 6*b^2*c^2*d*f*x^2*Cos[2*e + f*
x] - 4*a^2*c*d^2*f*x^3*Cos[2*e + f*x] + 4*b^2*c*d^2*f*x^3*Cos[2*e + f*x] - a^2*d^3*f*x^4*Cos[2*e + f*x] + b^2*
d^3*f*x^4*Cos[2*e + f*x] + 8*b^2*c^3*Sin[f*x] + 24*b^2*c^2*d*x*Sin[f*x] + 8*a*b*c^3*f*x*Sin[f*x] + 24*b^2*c*d^
2*x^2*Sin[f*x] + 12*a*b*c^2*d*f*x^2*Sin[f*x] + 8*b^2*d^3*x^3*Sin[f*x] + 8*a*b*c*d^2*f*x^3*Sin[f*x] + 2*a*b*d^3
*f*x^4*Sin[f*x] + 8*a*b*c^3*f*x*Sin[2*e + f*x] + 12*a*b*c^2*d*f*x^2*Sin[2*e + f*x] + 8*a*b*c*d^2*f*x^3*Sin[2*e
+ f*x] + 2*a*b*d^3*f*x^4*Sin[2*e + f*x]))/(8*f) - (3*b^2*c*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 +
((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x
+ ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2
*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2]))/(f^3*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a*b
*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Log[1 + E^((-2*I)*f
*x)] - 2*(f*x + ArcTan[Tan[e]])*Log[1 - E^((2*I)*(f*x + ArcTan[Tan[e]]))] + Pi*Log[Cos[f*x]] + 2*ArcTan[Tan[e]
]*Log[Sin[f*x + ArcTan[Tan[e]]]] + I*PolyLog[2, E^((2*I)*(f*x + ArcTan[Tan[e]]))])*Tan[e])/Sqrt[1 + Tan[e]^2])
)/(f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)])

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 = 0.85 (sec) , antiderivative size = 1604, normalized size of antiderivative = 5.44

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

[In]

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

[Out]

4/f^4*b*e^3*d^3*a*ln(exp(I*(f*x+e)))-2/f^4*b*e^3*d^3*a*ln(exp(I*(f*x+e))-1)+12*I/f^4*b*d^3*a*polylog(4,-exp(I*
(f*x+e)))-3*I/f^4*b*d^3*a*e^4-6*I/f^3*b^2*d^3*polylog(2,exp(I*(f*x+e)))*x-6*I/f^3*b^2*d^3*polylog(2,-exp(I*(f*
x+e)))*x+6*I/f^3*b^2*e^2*d^3*x-6*I/f*b^2*d^2*c*x^2-6*I/f^3*b^2*d^2*c*e^2-6*I/f^3*b^2*d^2*c*polylog(2,exp(I*(f*
x+e)))-6*I/f^3*b^2*d^2*c*polylog(2,-exp(I*(f*x+e)))+12*I/f^4*b*d^3*a*polylog(4,exp(I*(f*x+e)))-2*I*d^2*a*b*c*x
^3-3*I*d*a*b*c^2*x^2+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*I*a*b*c^3*x+1
/2*I/d*a*b*c^4-1/2*I*d^3*a*b*x^4+12/f^2*b*e*a*c^2*d*ln(exp(I*(f*x+e)))+12/f^3*b^2*e*c*d^2*ln(exp(I*(f*x+e)))-6
/f^3*b^2*e*c*d^2*ln(exp(I*(f*x+e))-1)+6/f^3*b^2*d^2*c*ln(1-exp(I*(f*x+e)))*e+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+12/f^3*b*d^3*a*polylog(3,-exp(I*(f*x+e)))*x+12/f^3*b*d^3*a*polylo
g(3,exp(I*(f*x+e)))*x+2/f*b*d^3*a*ln(exp(I*(f*x+e))+1)*x^3+2/f^4*b*d^3*a*ln(1-exp(I*(f*x+e)))*e^3+2/f*b*d^3*a*
ln(1-exp(I*(f*x+e)))*x^3+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))
)+3/f^2*b^2*d^3*ln(exp(I*(f*x+e))+1)*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(exp(I*(f*x+e))-1)+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)-6/f^4*b^2*e^2*d^3*ln(exp(I*(f*x+e)))+3/f^4*b^2*e^2*d^3*ln(exp(I*(f*x+e))-1)-3/f^4*b
^2*e^2*d^3*ln(1-exp(I*(f*x+e)))+3/f^2*b^2*d^3*ln(1-exp(I*(f*x+e)))*x^2-2*I/f*b^2*d^3*x^3+4*I/f^4*b^2*e^3*d^3-6
/f^2*b*e*a*c^2*d*ln(exp(I*(f*x+e))-1)+6/f*b*d*c^2*a*ln(1-exp(I*(f*x+e)))*x+6/f*b*d*c^2*a*ln(exp(I*(f*x+e))+1)*
x+6/f*b*a*c*d^2*ln(1-exp(I*(f*x+e)))*x^2-6/f^3*b*a*c*d^2*ln(1-exp(I*(f*x+e)))*e^2+6/f*b*a*c*d^2*ln(exp(I*(f*x+
e))+1)*x^2-12/f^3*b*e^2*a*c*d^2*ln(exp(I*(f*x+e)))+6/f^3*b*e^2*a*c*d^2*ln(exp(I*(f*x+e))-1)+6/f^2*b*d*c^2*a*ln
(1-exp(I*(f*x+e)))*e-12*I/f^2*b^2*d^2*c*e*x+8*I/f^3*b*a*c*d^2*e^3-6*I/f^2*b*d^3*a*polylog(2,-exp(I*(f*x+e)))*x
^2-6*I/f^2*b*d^3*a*polylog(2,exp(I*(f*x+e)))*x^2-4*I/f^3*b*d^3*a*e^3*x-6*I/f^2*b*d*c^2*a*e^2-6*I/f^2*b*d*c^2*a
*polylog(2,exp(I*(f*x+e)))-6*I/f^2*b*d*c^2*a*polylog(2,-exp(I*(f*x+e)))-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)))+
1/4*d^3*a^2*x^4+1/4/d*a^2*c^4-1/4*d^3*b^2*x^4-b^2*c^3*x-1/4/d*b^2*c^4-12*I/f*b*d*c^2*a*e*x+12*I/f^2*b*a*c*d^2*
e^2*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

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 = 0.35 (sec) , antiderivative size = 1154, normalized size of antiderivative = 3.91 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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 = 1.51 (sec) , antiderivative size = 4024, normalized size of antiderivative = 13.64 \[ \int (c+d x)^3 (a+b \cot (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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*I*a*b*c*d^2*e - I*b^2*c*d^2)*f)*(f*x + e))*sin(2*f*x + 2*e))*ar
ctan2(sin(f*x + e), cos(f*x + e) + 1) - 12*(b^2*d^3*e^2 - 2*b^2*c*d^2*e*f + b^2*c^2*d*f^2 - (b^2*d^3*e^2 - 2*b
^2*c*d^2*e*f + b^2*c^2*d*f^2)*cos(2*f*x + 2*e) + (-I*b^2*d^3*e^2 + 2*I*b^2*c*d^2*e*f - I*b^2*c^2*d*f^2)*sin(2*
f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) - 1) + 4*(2*(f*x + e)^3*a*b*d^3 - 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*(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
*(-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*I*a*b*c*d^2*e + I*b^2*c*d^2)*f)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), -cos(f*x + e) + 1) -
((2*a*b - I*b^2)*(f*x + e)^4*d^3 + 4*(2*b^2*d^3 - (2*a*b - I*b^2)*d^3*e + (2*a*b - I*b^2)*c*d^2*f)*(f*x + e)^
3 - 6*(4*b^2*d^3*e - (2*a*b - I*b^2)*d^3*e^2 - (2*a*b - I*b^2)*c^2*d*f^2 - 2*(2*b^2*c*d^2 - (2*a*b - I*b^2)*c*
d^2*e)*f)*(f*x + e)^2 + 4*(I*b^2*d^3*e^3 - I*b^2*c^3*f^3 + 6*b^2*d^3*e^2 + 3*(I*b^2*c^2*d*e + 2*b^2*c^2*d)*f^2
+ 3*(-I*b^2*c*d^2*e^2 - 4*b^2*c*d^2*e)*f)*(f*x + e))*cos(2*f*x + 2*e) + 24*((f*x + e)^2*a*b*d^3 + a*b*d^3*e^2
+ a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - (2*a*b*c*d^2*e - b^2*c*d^2)
*f - ((f*x + e)^2*a*b*d^3 + a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*
(f*x + e) - (2*a*b*c*d^2*e - b^2*c*d^2)*f)*cos(2*f*x + 2*e) - (I*(f*x + e)^2*a*b*d^3 + I*a*b*d^3*e^2 + I*a*b*c
^2*d*f^2 - I*b^2*d^3*e + (-2*I*a*b*d^3*e + 2*I*a*b*c*d^2*f + I*b^2*d^3)*(f*x + e) + (-2*I*a*b*c*d^2*e + I*b^2*
c*d^2)*f)*sin(2*f*x + 2*e))*dilog(-e^(I*f*x + I*e)) + 24*((f*x + e)^2*a*b*d^3 + a*b*d^3*e^2 + a*b*c^2*d*f^2 -
b^2*d^3*e - (2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - (2*a*b*c*d^2*e - b^2*c*d^2)*f - ((f*x + e)^2*a
*b*d^3 + a*b*d^3*e^2 + a*b*c^2*d*f^2 - b^2*d^3*e - (2*a*b*d^3*e - 2*a*b*c*d^2*f - b^2*d^3)*(f*x + e) - (2*a*b*
c*d^2*e - b^2*c*d^2)*f)*cos(2*f*x + 2*e) - (I*(f*x + e)^2*a*b*d^3 + I*a*b*d^3*e^2 + I*a*b*c^2*d*f^2 - I*b^2*d^
3*e + (-2*I*a*b*d^3*e + 2*I*a*b*c*d^2*f + I*b^2*d^3)*(f*x + e) + (-2*I*a*b*c*d^2*e + I*b^2*c*d^2)*f)*sin(2*f*x
+ 2*e))*dilog(e^(I*f*x + I*e)) - 2*(-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*I*a*b*c*d^2*e - I*b^2*c*d^2)*f)*(f*x + 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*I*a*b*c*d^2*e + I*b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*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))*si
n(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1) - 2*(-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*I*a*b*c*d^2*e - I*b^2*c*d^2)*f)*(f*x + 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*I*a*b*c*d^2*e + I*b^2*
c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*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))*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 - 2*cos(f*x + e)
+ 1) + 48*(a*b*d^3*cos(2*f*x + 2*e) + I*a*b*d^3*sin(2*f*x + 2*e) - a*b*d^3)*polylog(4, -e^(I*f*x + I*e)) + 48
*(a*b*d^3*cos(2*f*x + 2*e) + I*a*b*d^3*sin(2*f*x + 2*e) - a*b*d^3)*polylog(4, e^(I*f*x + I*e)) - 24*(-2*I*(f*x
+ e)*a*b*d^3 + 2*I*a*b*d^3*e - 2*I*a*b*c*d^2*f - I*b^2*d^3 + (2*I*(f*x + e)*a*b*d^3 - 2*I*a*b*d^3*e + 2*I*a*b
*c*d^2*f + I*b^2*d^3)*cos(2*f*x + 2*e) - (2*(f*x + e)*a*b*d^3 - 2*a*b*d^3*e + 2*a*b*c*d^2*f + b^2*d^3)*sin(2*f
*x + 2*e))*polylog(3, -e^(I*f*x + I*e)) - 24*(-2*I*(f*x + e)*a*b*d^3 + 2*I*a*b*d^3*e - 2*I*a*b*c*d^2*f - I*b^2
*d^3 + (2*I*(f*x + e)*a*b*d^3 - 2*I*a*b*d^3*e + 2*I*a*b*c*d^2*f + I*b^2*d^3)*cos(2*f*x + 2*e) - (2*(f*x + e)*a
*b*d^3 - 2*a*b*d^3*e + 2*a*b*c*d^2*f + b^2*d^3)*sin(2*f*x + 2*e))*polylog(3, e^(I*f*x + I*e)) + ((-2*I*a*b - b
^2)*(f*x + e)^4*d^3 - 4*(2*I*b^2*d^3 + (-2*I*a*b - b^2)*d^3*e + (2*I*a*b + b^2)*c*d^2*f)*(f*x + e)^3 - 6*(-4*I
*b^2*d^3*e + (2*I*a*b + b^2)*d^3*e^2 + (2*I*a*b + b^2)*c^2*d*f^2 + 2*(2*I*b^2*c*d^2 + (-2*I*a*b - b^2)*c*d^2*e
)*f)*(f*x + e)^2 + 4*(b^2*d^3*e^3 - b^2*c^3*f^3 - 6*I*b^2*d^3*e^2 + 3*(b^2*c^2*d*e - 2*I*b^2*c^2*d)*f^2 - 3*(b
^2*c*d^2*e^2 - 4*I*b^2*c*d^2*e)*f)*(f*x + e))*sin(2*f*x + 2*e))/(-4*I*f^3*cos(2*f*x + 2*e) + 4*f^3*sin(2*f*x +
2*e) + 4*I*f^3))/f

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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