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

Optimal. Leaf size=603 \[ \frac{9 a^2 b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{9 i a^2 b d^3 \text{PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}-\frac{9 i a b^2 d^2 (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac{9 a b^2 d^3 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}-\frac{3 b^3 d^2 (c+d x) \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{3 i b^3 d (c+d x)^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 i b^3 d^3 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^4}-\frac{3 i b^3 d^3 \text{PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}+\frac{3 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i a^2 b (c+d x)^4}{4 d}+\frac{a^3 (c+d x)^4}{4 d}+\frac{9 a b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^3}{f}-\frac{3 a b^2 (c+d x)^4}{4 d}+\frac{3 b^3 d^2 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^3}-\frac{3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac{b^3 (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}-\frac{3 i b^3 d (c+d x)^2}{2 f^2}-\frac{b^3 (c+d x)^3}{2 f}+\frac{i b^3 (c+d x)^4}{4 d} \]

[Out]

(((-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

________________________________________________________________________________________

Rubi [A]  time = 0.980552, antiderivative size = 603, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {3722, 3717, 2190, 2531, 6609, 2282, 6589, 3720, 32, 2279, 2391} \[ \frac{9 a^2 b d^2 (c+d x) \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}+\frac{9 i a^2 b d^3 \text{PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}-\frac{9 i a b^2 d^2 (c+d x) \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{f^3}+\frac{9 a b^2 d^3 \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^4}-\frac{3 b^3 d^2 (c+d x) \text{PolyLog}\left (3,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{3 i b^3 d (c+d x)^2 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 i b^3 d^3 \text{PolyLog}\left (2,e^{2 i (e+f x)}\right )}{2 f^4}-\frac{3 i b^3 d^3 \text{PolyLog}\left (4,e^{2 i (e+f x)}\right )}{4 f^4}+\frac{3 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{3 i a^2 b (c+d x)^4}{4 d}+\frac{a^3 (c+d x)^4}{4 d}+\frac{9 a b^2 d (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a b^2 (c+d x)^3 \cot (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^3}{f}-\frac{3 a b^2 (c+d x)^4}{4 d}+\frac{3 b^3 d^2 (c+d x) \log \left (1-e^{2 i (e+f x)}\right )}{f^3}-\frac{3 b^3 d (c+d x)^2 \cot (e+f x)}{2 f^2}-\frac{b^3 (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}-\frac{b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}-\frac{3 i b^3 d (c+d x)^2}{2 f^2}-\frac{b^3 (c+d x)^3}{2 f}+\frac{i b^3 (c+d x)^4}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-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

Rule 3722

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 3717

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 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2531

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 6609

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]

Rule 2282

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 6589

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 3720

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 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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2391

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]

Rubi steps

\begin{align*} \int (c+d x)^3 (a+b \cot (e+f x))^3 \, dx &=\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\\ &=\frac{a^3 (c+d x)^4}{4 d}+\left (3 a^2 b\right ) \int (c+d x)^3 \cot (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \cot ^2(e+f x) \, dx+b^3 \int (c+d x)^3 \cot ^3(e+f x) \, dx\\ &=\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)^3 \cot (e+f x)}{f}-\frac{b^3 (c+d x)^3 \cot ^2(e+f x)}{2 f}-\left (6 i a^2 b\right ) \int \frac{e^{2 i (e+f x)} (c+d x)^3}{1-e^{2 i (e+f x)}} \, dx-\left (3 a b^2\right ) \int (c+d x)^3 \, dx-b^3 \int (c+d x)^3 \cot (e+f x) \, dx+\frac{\left (9 a b^2 d\right ) \int (c+d x)^2 \cot (e+f x) \, dx}{f}+\frac{\left (3 b^3 d\right ) \int (c+d x)^2 \cot ^2(e+f x) \, dx}{2 f}\\ &=-\frac{3 i a b^2 (c+d x)^3}{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 a^2 b (c+d x)^3 \log \left (1-e^{2 i (e+f x)}\right )}{f}+\left (2 i b^3\right ) \int \frac{e^{2 i (e+f x)} (c+d x)^3}{1-e^{2 i (e+f x)}} \, dx+\frac{\left (3 b^3 d^2\right ) \int (c+d x) \cot (e+f x) \, dx}{f^2}-\frac{\left (9 a^2 b d\right ) \int (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}-\frac{\left (18 i a 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{\left (3 b^3 d\right ) \int (c+d x)^2 \, dx}{2 f}\\ &=-\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{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{9 i a^2 b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{\left (9 i a^2 b d^2\right ) \int (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (18 a b^2 d^2\right ) \int (c+d x) \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (6 i b^3 d^2\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1-e^{2 i (e+f x)}} \, dx}{f^2}+\frac{\left (3 b^3 d\right ) \int (c+d x)^2 \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=-\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{9 i a b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 i b^3 d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{9 a^2 b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac{\left (9 a^2 b d^3\right ) \int \text{Li}_3\left (e^{2 i (e+f x)}\right ) \, dx}{2 f^3}+\frac{\left (9 i a b^2 d^3\right ) \int \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^3}-\frac{\left (3 b^3 d^3\right ) \int \log \left (1-e^{2 i (e+f x)}\right ) \, dx}{f^3}-\frac{\left (3 i b^3 d^2\right ) \int (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\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{9 i a b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 i b^3 d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{9 a^2 b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 b^3 d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}+\frac{\left (9 i a^2 b d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 f^4}+\frac{\left (9 a b^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}+\frac{\left (3 i b^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}+\frac{\left (3 b^3 d^3\right ) \int \text{Li}_3\left (e^{2 i (e+f x)}\right ) \, dx}{2 f^3}\\ &=-\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 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^4}-\frac{9 i a b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 i b^3 d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{9 a b^2 d^3 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^4}+\frac{9 a^2 b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 b^3 d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}+\frac{9 i a^2 b d^3 \text{Li}_4\left (e^{2 i (e+f x)}\right )}{4 f^4}-\frac{\left (3 i b^3 d^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 f^4}\\ &=-\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 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^4}-\frac{9 i a b^2 d^2 (c+d x) \text{Li}_2\left (e^{2 i (e+f x)}\right )}{f^3}-\frac{9 i a^2 b d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 i b^3 d (c+d x)^2 \text{Li}_2\left (e^{2 i (e+f x)}\right )}{2 f^2}+\frac{9 a b^2 d^3 \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^4}+\frac{9 a^2 b d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 b^3 d^2 (c+d x) \text{Li}_3\left (e^{2 i (e+f x)}\right )}{2 f^3}+\frac{9 i a^2 b d^3 \text{Li}_4\left (e^{2 i (e+f x)}\right )}{4 f^4}-\frac{3 i b^3 d^3 \text{Li}_4\left (e^{2 i (e+f x)}\right )}{4 f^4}\\ \end{align*}

Mathematica [B]  time = 8.47522, size = 3045, normalized size = 5.05 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-(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*PolyLog[2, -E^((-I)*(e + f*x))] - I*
PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*
PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(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*Log[1 + E^
((-I)*(e + f*x))] - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, -E^((-I)*(e + f*x))] - I*PolyLog[3, -E^((-I)*(e + f*
x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x
))]))/E^((2*I)*e)))/(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))] - I*PolyLog[3, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E
^((2*I)*e))*(f*x*PolyLog[2, E^((-I)*(e + f*x))] - I*PolyLog[3, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(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)
)] + (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))] - (2*I)*f*x*PolyLog[3, -E^((-I)*(e + f*x))] - 2*PolyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I
)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))
] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)))/(4*f^4) + (b^3*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))] - (2*I)*f*x*PolyLog[3, -E^((-
I)*(e + f*x))] - 2*PolyLog[4, -E^((-I)*(e + f*x))]))/E^((2*I)*e) - (6*(-1 + E^((2*I)*e))*(f^2*x^2*PolyLog[2, E
^((-I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, E^((-I)*(e + f*x))] - 2*PolyLog[4, E^((-I)*(e + f*x))]))/E^((2*I)*e)
))/(4*f^4) + (3*b^3*c*d^2*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(f^3*(Cos[e]
^2 + Sin[e]^2)) + (9*a*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)) + (3*a^2*b*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/
(f*(Cos[e]^2 + Sin[e]^2)) - (b^3*c^3*Csc[e]*(-(f*x*Cos[e]) + Log[Cos[f*x]*Sin[e] + Cos[e]*Sin[f*x]]*Sin[e]))/(
f*(Cos[e]^2 + Sin[e]^2)) + (3*x^2*(-(a^3*c^2*d) + (3*I)*a^2*b*c^2*d + 3*a*b^2*c^2*d - I*b^3*c^2*d + a^3*c^2*d*
Cos[2*e] + (3*I)*a^2*b*c^2*d*Cos[2*e] - 3*a*b^2*c^2*d*Cos[2*e] - I*b^3*c^2*d*Cos[2*e] + I*a^3*c^2*d*Sin[2*e] -
3*a^2*b*c^2*d*Sin[2*e] - (3*I)*a*b^2*c^2*d*Sin[2*e] + b^3*c^2*d*Sin[2*e]))/(2*(-1 + Cos[2*e] + I*Sin[2*e])) +
(x^3*(-(a^3*c*d^2) + (3*I)*a^2*b*c*d^2 + 3*a*b^2*c*d^2 - I*b^3*c*d^2 + a^3*c*d^2*Cos[2*e] + (3*I)*a^2*b*c*d^2
*Cos[2*e] - 3*a*b^2*c*d^2*Cos[2*e] - I*b^3*c*d^2*Cos[2*e] + I*a^3*c*d^2*Sin[2*e] - 3*a^2*b*c*d^2*Sin[2*e] - (3
*I)*a*b^2*c*d^2*Sin[2*e] + b^3*c*d^2*Sin[2*e]))/(-1 + Cos[2*e] + I*Sin[2*e]) + (x^4*(-(a^3*d^3) + (3*I)*a^2*b*
d^3 + 3*a*b^2*d^3 - I*b^3*d^3 + a^3*d^3*Cos[2*e] + (3*I)*a^2*b*d^3*Cos[2*e] - 3*a*b^2*d^3*Cos[2*e] - I*b^3*d^3
*Cos[2*e] + I*a^3*d^3*Sin[2*e] - 3*a^2*b*d^3*Sin[2*e] - (3*I)*a*b^2*d^3*Sin[2*e] + b^3*d^3*Sin[2*e]))/(4*(-1 +
Cos[2*e] + I*Sin[2*e])) + x*(a^3*c^3 - 3*a*b^2*c^3 + ((3*I)*a^2*b*c^3)/(-1 + Cos[2*e] + I*Sin[2*e]) + ((3*I)*
a^2*b*c^3*Cos[2*e] - 3*a^2*b*c^3*Sin[2*e])/(-1 + Cos[2*e] + I*Sin[2*e]) + ((-2*I)*b^3*c^3*Cos[2*e] + 2*b^3*c^3
*Sin[2*e])/((-1 + Cos[2*e] + I*Sin[2*e])*(1 + Cos[2*e] + Cos[4*e] + I*Sin[2*e] + I*Sin[4*e])) + ((-2*I)*b^3*c^
3*Cos[4*e] + 2*b^3*c^3*Sin[4*e])/((-1 + Cos[2*e] + I*Sin[2*e])*(1 + Cos[2*e] + Cos[4*e] + I*Sin[2*e] + I*Sin[4
*e])) - (I*b^3*c^3)/(-1 + Cos[6*e] + I*Sin[6*e]) + ((-I)*b^3*c^3*Cos[6*e] + b^3*c^3*Sin[6*e])/(-1 + Cos[6*e] +
I*Sin[6*e])) + (3*Csc[e]*Csc[e + f*x]*(b^3*c^2*d*Sin[f*x] + 2*a*b^2*c^3*f*Sin[f*x] + 2*b^3*c*d^2*x*Sin[f*x] +
6*a*b^2*c^2*d*f*x*Sin[f*x] + b^3*d^3*x^2*Sin[f*x] + 6*a*b^2*c*d^2*f*x^2*Sin[f*x] + 2*a*b^2*d^3*f*x^3*Sin[f*x]
))/(2*f^2) - (3*b^3*d^3*Csc[e]*Sec[e]*(E^(I*ArcTan[Tan[e]])*f^2*x^2 + ((I*f*x*(-Pi + 2*ArcTan[Tan[e]]) - Pi*Lo
g[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])/Sq
rt[1 + Tan[e]^2]))/(2*f^4*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (9*a*b^2*c*d^2*Csc[e]*Sec[e]*(E^(I*ArcTan[Ta
n[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)]) - (9*a^2*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]))/(2*f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (3*b^3*c^2*d*Csc[e]*Sec[e]*(E^(I*ArcTan[Ta
n[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]))/(2*f^2*Sqrt[Sec[e]^2*(Cos[e]^2 + S
in[e]^2)])

________________________________________________________________________________________

Maple [B]  time = 0.777, size = 3113, normalized size = 5.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*b/f^4*a^2*d^3*e^3*ln(exp(I*(f*x+e)))+6*b^3/f^3*c*d^2*e^2*ln(exp(I*(f*x+e)))-3*b/f^4*a^2*d^3*e^3*ln(exp(I*(f*
x+e))-1)+3*b^3/f^2*c^2*d*e*ln(exp(I*(f*x+e))-1)-3*b^3/f*ln(1-exp(I*(f*x+e)))*c*d^2*x^2+18*b/f^3*a^2*d^3*polylo
g(3,-exp(I*(f*x+e)))*x+2*b^3/f*c^3*ln(exp(I*(f*x+e)))-b^3/f*c^3*ln(exp(I*(f*x+e))-1)-b^3/f*c^3*ln(exp(I*(f*x+e
))+1)-3/4*a*b^2*d^3*x^4+3/2*a^3*c^2*d*x^2-3*b^2*a*c^3*x-3*I*a^2*b*c*d^2*x^3-9/2*I*a^2*b*c^2*d*x^2+a^3*c*d^2*x^
3-I*b^3*c^3*x+1/4*a^3*d^3*x^4+a^3*c^3*x-6*b^3/f^3*c*d^2*ln(exp(I*(f*x+e)))-6*b/f*a^2*c^3*ln(exp(I*(f*x+e)))-6*
b^3/f^3*d^3*polylog(3,exp(I*(f*x+e)))*x-9*b^2/f^4*a*d^3*e^2*ln(1-exp(I*(f*x+e)))+2*I*b^3/f^3*d^3*e^3*x+3*I*b^3
/f^2*d^3*polylog(2,exp(I*(f*x+e)))*x^2+3*I*b^3/f^2*c^2*d*polylog(2,exp(I*(f*x+e)))+3*I*b^3/f^2*d^3*polylog(2,-
exp(I*(f*x+e)))*x^2+3*I*b^3/f^2*c^2*d*polylog(2,-exp(I*(f*x+e)))+12*I*b^2/f^4*a*d^3*e^3+18*I*b/f^4*a^2*d^3*pol
ylog(4,exp(I*(f*x+e)))+18*I*b/f^4*a^2*d^3*polylog(4,-exp(I*(f*x+e)))+3*I*b^3/f^2*c^2*d*e^2-9/2*I*b/f^4*a^2*d^3
*e^4-4*I*b^3/f^3*c*d^2*e^3-3*a*b^2*c*d^2*x^3-9/2*a*b^2*c^2*d*x^2+1/4*I*b^3*d^3*x^4+I*b^3*c*d^2*x^3-b^3/f*d^3*l
n(1-exp(I*(f*x+e)))*x^3-b^3/f^4*d^3*ln(1-exp(I*(f*x+e)))*e^3+9*b^2/f^4*a*d^3*e^2*ln(exp(I*(f*x+e))-1)+9*b^2/f^
2*a*c^2*d*ln(exp(I*(f*x+e))+1)+9*b^2/f^2*a*c^2*d*ln(exp(I*(f*x+e))-1)-b^3/f*d^3*ln(exp(I*(f*x+e))+1)*x^3-6*I*b
^3/f^3*d^3*e*x-6*I*b^2/f*a*d^3*x^3-18*b/f^3*a^2*c*d^2*e^2*ln(exp(I*(f*x+e)))+18*b/f^2*a^2*c^2*d*e*ln(exp(I*(f*
x+e)))+36*b^2/f^3*a*c*d^2*e*ln(exp(I*(f*x+e)))-9*I*b/f^2*a^2*d^3*polylog(2,exp(I*(f*x+e)))*x^2+6*I*b^3/f^2*pol
ylog(2,-exp(I*(f*x+e)))*c*d^2*x+6*I*b^3/f^2*polylog(2,exp(I*(f*x+e)))*c*d^2*x-18*I*b^2/f^3*a*c*d^2*polylog(2,e
xp(I*(f*x+e)))-9*I*b/f^2*a^2*c^2*d*e^2-9*b/f^3*ln(1-exp(I*(f*x+e)))*a^2*c*d^2*e^2+18*b^2/f^3*ln(1-exp(I*(f*x+e
)))*a*c*d^2*e+9*b/f*ln(1-exp(I*(f*x+e)))*a^2*c*d^2*x^2+9*b/f*ln(exp(I*(f*x+e))+1)*a^2*c^2*d*x+9*b/f*ln(1-exp(I
*(f*x+e)))*a^2*c^2*d*x+9*b/f^2*ln(1-exp(I*(f*x+e)))*a^2*c^2*d*e+3*b/f*a^2*d^3*ln(exp(I*(f*x+e))+1)*x^3+3*b/f*a
^2*d^3*ln(1-exp(I*(f*x+e)))*x^3+3*b/f^4*a^2*d^3*ln(1-exp(I*(f*x+e)))*e^3+3*I*a^2*b*c^3*x-3/4*I*a^2*b*d^3*x^4+3
/2*I*b^3*c^2*d*x^2+b^3/f^4*d^3*e^3*ln(exp(I*(f*x+e))-1)-3*b^3/f^4*d^3*e*ln(exp(I*(f*x+e))-1)+18*b^2/f^4*a*d^3*
polylog(3,-exp(I*(f*x+e)))+3*b/f*a^2*c^3*ln(exp(I*(f*x+e))-1)+3*b^3/f^3*c*d^2*ln(exp(I*(f*x+e))-1)+3*b^3/f^3*c
*d^2*ln(exp(I*(f*x+e))+1)-6*b^3/f^3*c*d^2*polylog(3,exp(I*(f*x+e)))+3*b/f*a^2*c^3*ln(exp(I*(f*x+e))+1)-6*b^3/f
^3*c*d^2*polylog(3,-exp(I*(f*x+e)))+18*b^2/f^4*a*d^3*polylog(3,exp(I*(f*x+e)))-2*b^3/f^4*d^3*e^3*ln(exp(I*(f*x
+e)))+6*b^3/f^4*d^3*e*ln(exp(I*(f*x+e)))-6*b^3/f^3*d^3*polylog(3,-exp(I*(f*x+e)))*x+3*b^3/f^3*d^3*ln(exp(I*(f*
x+e))+1)*x+3*b^3/f^3*d^3*ln(1-exp(I*(f*x+e)))*x+3*b^3/f^4*d^3*ln(1-exp(I*(f*x+e)))*e+3/2*I*b^3/f^4*d^3*e^4-3*I
*b^3/f^2*d^3*x^2-6*I*b^3/f^4*d^3*polylog(4,-exp(I*(f*x+e)))-6*I*b^3/f^4*d^3*polylog(4,exp(I*(f*x+e)))-3*I*b^3/
f^4*d^3*e^2-3*I*b^3/f^4*d^3*polylog(2,-exp(I*(f*x+e)))-3*I*b^3/f^4*d^3*polylog(2,exp(I*(f*x+e)))+18*I*b/f^2*a^
2*c*d^2*e^2*x-18*I*b/f^2*polylog(2,exp(I*(f*x+e)))*a^2*c*d^2*x-36*I*b^2/f^2*a*c*d^2*e*x-18*I*b/f*a^2*c^2*d*e*x
-18*I*b/f^2*polylog(2,-exp(I*(f*x+e)))*a^2*c*d^2*x+3*b^3/f^3*ln(1-exp(I*(f*x+e)))*c*d^2*e^2+18*b/f^3*a^2*c*d^2
*polylog(3,-exp(I*(f*x+e)))+18*b/f^3*a^2*c*d^2*polylog(3,exp(I*(f*x+e)))+18*b/f^3*a^2*d^3*polylog(3,exp(I*(f*x
+e)))*x-3*b^3/f*ln(exp(I*(f*x+e))+1)*c^2*d*x-3*b^3/f*ln(1-exp(I*(f*x+e)))*c^2*d*x-3*b^3/f^2*ln(1-exp(I*(f*x+e)
))*c^2*d*e+9*b^2/f^2*a*d^3*ln(1-exp(I*(f*x+e)))*x^2+9*b^2/f^2*a*d^3*ln(exp(I*(f*x+e))+1)*x^2-3*b^3/f*ln(exp(I*
(f*x+e))+1)*c*d^2*x^2-3*b^3/f^3*c*d^2*e^2*ln(exp(I*(f*x+e))-1)-6*b^3/f^2*c^2*d*e*ln(exp(I*(f*x+e)))-18*b^2/f^2
*a*c^2*d*ln(exp(I*(f*x+e)))-18*b^2/f^4*a*d^3*e^2*ln(exp(I*(f*x+e)))+b^2*(-18*I*a*c^2*d*f*x*exp(2*I*(f*x+e))-6*
I*b*c*d^2*x*exp(2*I*(f*x+e))+2*b*d^3*f*x^3*exp(2*I*(f*x+e))+6*I*a*c^3*f-6*I*a*c^3*f*exp(2*I*(f*x+e))+3*I*b*d^3
*x^2+6*b*c*d^2*f*x^2*exp(2*I*(f*x+e))+18*I*a*c*d^2*f*x^2-18*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*b*c^2*d*f*x*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+6*I*b*c*d^2*x+2*b*c
^3*f*exp(2*I*(f*x+e))-6*I*a*d^3*f*x^3*exp(2*I*(f*x+e))+3*I*b*c^2*d+6*I*a*d^3*f*x^3)/f^2/(exp(2*I*(f*x+e))-1)^2
-9*I*b/f^2*a^2*d^3*polylog(2,-exp(I*(f*x+e)))*x^2-18*I*b^2/f^3*a*c*d^2*polylog(2,-exp(I*(f*x+e)))-18*I*b^2/f*a
*c*d^2*x^2-18*I*b^2/f^3*a*d^3*polylog(2,exp(I*(f*x+e)))*x-18*I*b^2/f^3*a*d^3*polylog(2,-exp(I*(f*x+e)))*x+6*I*
b^3/f*c^2*d*e*x-6*I*b^3/f^2*c*d^2*e^2*x+12*I*b/f^3*a^2*c*d^2*e^3-6*I*b/f^3*a^2*d^3*e^3*x-9*I*b/f^2*a^2*c^2*d*p
olylog(2,-exp(I*(f*x+e)))-9*I*b/f^2*a^2*c^2*d*polylog(2,exp(I*(f*x+e)))-18*I*b^2/f^3*a*c*d^2*e^2+18*I*b^2/f^3*
a*d^3*e^2*x-9*b/f^2*a^2*c^2*d*e*ln(exp(I*(f*x+e))-1)+9*b/f*ln(exp(I*(f*x+e))+1)*a^2*c*d^2*x^2+18*b^2/f^2*ln(ex
p(I*(f*x+e))+1)*a*c*d^2*x+18*b^2/f^2*ln(1-exp(I*(f*x+e)))*a*c*d^2*x-18*b^2/f^3*a*c*d^2*e*ln(exp(I*(f*x+e))-1)+
9*b/f^3*a^2*c*d^2*e^2*ln(exp(I*(f*x+e))-1)

________________________________________________________________________________________

Maxima [B]  time = 101.748, size = 15125, normalized size = 25.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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 - ((12*a^2*b - 12*I*a*b^2 - 4*b^3)*d^3*e - (12*a^2*b - 12*I*a*b^2 - 4*b^3)*c*d^2*f)*(f*x + e)^3 + ((18*a
^2*b - 18*I*a*b^2 - 6*b^3)*d^3*e^2 - (36*a^2*b - 36*I*a*b^2 - 12*b^3)*c*d^2*e*f + (18*a^2*b - 18*I*a*b^2 - 6*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 + ((12*I*a*b^2 + 4*b^3)*d^3*e^3 + (-36*I*a*b
^2 - 12*b^3)*c*d^2*e^2*f + (36*I*a*b^2 + 12*b^3)*c^2*d*e*f^2 + (-12*I*a*b^2 - 4*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 - 4*b^3*c^3*f^3 + 36*a*b^2*d^3*e^2 + 4*(3*a^2*b - b^3)*(f*
x + e)^3*d^3 - 12*b^3*d^3*e + 12*(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 +
12*(b^3*c^2*d*e + 3*a*b^2*c^2*d)*f^2 - 12*(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) - 12*(b^3*c*d^2*e^2 + 6*a*b^2*c*d^2*e -
b^3*c*d^2)*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)*cos(4*f*x + 4*e)
- 8*(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)*cos(2*f*x + 2*e) - (-4*I*b^3*
d^3*e^3 + 4*I*b^3*c^3*f^3 - 36*I*a*b^2*d^3*e^2 + (-12*I*a^2*b + 4*I*b^3)*(f*x + e)^3*d^3 + 12*I*b^3*d^3*e + (-
36*I*a*b^2*d^3 + (36*I*a^2*b - 12*I*b^3)*d^3*e + (-36*I*a^2*b + 12*I*b^3)*c*d^2*f)*(f*x + e)^2 + (-12*I*b^3*c^
2*d*e - 36*I*a*b^2*c^2*d)*f^2 + (72*I*a*b^2*d^3*e - 12*I*b^3*d^3 + (-36*I*a^2*b + 12*I*b^3)*d^3*e^2 + (-36*I*a
^2*b + 12*I*b^3)*c^2*d*f^2 + (-72*I*a*b^2*c*d^2 + (72*I*a^2*b - 24*I*b^3)*c*d^2*e)*f)*(f*x + e) + (12*I*b^3*c*
d^2*e^2 + 72*I*a*b^2*c*d^2*e - 12*I*b^3*c*d^2)*f)*sin(4*f*x + 4*e) - (8*I*b^3*d^3*e^3 - 8*I*b^3*c^3*f^3 + 72*I
*a*b^2*d^3*e^2 + (24*I*a^2*b - 8*I*b^3)*(f*x + e)^3*d^3 - 24*I*b^3*d^3*e + (72*I*a*b^2*d^3 + (-72*I*a^2*b + 24
*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f)*(f*x + e)^2 + (24*I*b^3*c^2*d*e + 72*I*a*b^2*c^2*d)*f^2 + (-1
44*I*a*b^2*d^3*e + 24*I*b^3*d^3 + (72*I*a^2*b - 24*I*b^3)*d^3*e^2 + (72*I*a^2*b - 24*I*b^3)*c^2*d*f^2 + (144*I
*a*b^2*c*d^2 + (-144*I*a^2*b + 48*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-24*I*b^3*c*d^2*e^2 - 144*I*a*b^2*c*d^2*e +
24*I*b^3*c*d^2)*f)*sin(2*f*x + 2*e))*arctan2(sin(f*x + e), cos(f*x + e) + 1) - (4*b^3*d^3*e^3 - 4*b^3*c^3*f^3
+ 36*a*b^2*d^3*e^2 - 12*b^3*d^3*e + 12*(b^3*c^2*d*e + 3*a*b^2*c^2*d)*f^2 - 12*(b^3*c*d^2*e^2 + 6*a*b^2*c*d^2*e
- b^3*c*d^2)*f + 4*(b^3*d^3*e^3 - b^3*c^3*f^3 + 9*a*b^2*d^3*e^2 - 3*b^3*d^3*e + 3*(b^3*c^2*d*e + 3*a*b^2*c^2*
d)*f^2 - 3*(b^3*c*d^2*e^2 + 6*a*b^2*c*d^2*e - b^3*c*d^2)*f)*cos(4*f*x + 4*e) - 8*(b^3*d^3*e^3 - b^3*c^3*f^3 +
9*a*b^2*d^3*e^2 - 3*b^3*d^3*e + 3*(b^3*c^2*d*e + 3*a*b^2*c^2*d)*f^2 - 3*(b^3*c*d^2*e^2 + 6*a*b^2*c*d^2*e - b^3
*c*d^2)*f)*cos(2*f*x + 2*e) - (-4*I*b^3*d^3*e^3 + 4*I*b^3*c^3*f^3 - 36*I*a*b^2*d^3*e^2 + 12*I*b^3*d^3*e + (-12
*I*b^3*c^2*d*e - 36*I*a*b^2*c^2*d)*f^2 + (12*I*b^3*c*d^2*e^2 + 72*I*a*b^2*c*d^2*e - 12*I*b^3*c*d^2)*f)*sin(4*f
*x + 4*e) - (8*I*b^3*d^3*e^3 - 8*I*b^3*c^3*f^3 + 72*I*a*b^2*d^3*e^2 - 24*I*b^3*d^3*e + (24*I*b^3*c^2*d*e + 72*
I*a*b^2*c^2*d)*f^2 + (-24*I*b^3*c*d^2*e^2 - 144*I*a*b^2*c*d^2*e + 24*I*b^3*c*d^2)*f)*sin(2*f*x + 2*e))*arctan2
(sin(f*x + e), cos(f*x + e) - 1) + (4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 + 12*(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 - 12*(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) + 4*((3*a^2*b - b^3)*(f*x + e)^3*d^3
+ 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*(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))*cos(4*f*x + 4*e) - 8*((3*a^2*b - b^3)*(f*x + e)^3*d^3 + 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*(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))*cos(2*f*x + 2*e) + ((12*I*a^2*b - 4*I*b^3)*(f*x +
e)^3*d^3 + (36*I*a*b^2*d^3 + (-36*I*a^2*b + 12*I*b^3)*d^3*e + (36*I*a^2*b - 12*I*b^3)*c*d^2*f)*(f*x + e)^2 +
(-72*I*a*b^2*d^3*e + 12*I*b^3*d^3 + (36*I*a^2*b - 12*I*b^3)*d^3*e^2 + (36*I*a^2*b - 12*I*b^3)*c^2*d*f^2 + (72*
I*a*b^2*c*d^2 + (-72*I*a^2*b + 24*I*b^3)*c*d^2*e)*f)*(f*x + e))*sin(4*f*x + 4*e) + ((-24*I*a^2*b + 8*I*b^3)*(f
*x + e)^3*d^3 + (-72*I*a*b^2*d^3 + (72*I*a^2*b - 24*I*b^3)*d^3*e + (-72*I*a^2*b + 24*I*b^3)*c*d^2*f)*(f*x + e)
^2 + (144*I*a*b^2*d^3*e - 24*I*b^3*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e^2 + (-72*I*a^2*b + 24*I*b^3)*c^2*d*f^2
+ (-144*I*a*b^2*c*d^2 + (144*I*a^2*b - 48*I*b^3)*c*d^2*e)*f)*(f*x + e))*sin(2*f*x + 2*e))*arctan2(sin(f*x + e
), -cos(f*x + e) + 1) + ((3*a^2*b - 3*I*a*b^2 - b^3)*(f*x + e)^4*d^3 + (24*a*b^2*d^3 - (12*a^2*b - 12*I*a*b^2
- 4*b^3)*d^3*e + (12*a^2*b - 12*I*a*b^2 - 4*b^3)*c*d^2*f)*(f*x + e)^3 - (72*a*b^2*d^3*e - 12*b^3*d^3 - (18*a^2
*b - 18*I*a*b^2 - 6*b^3)*d^3*e^2 - (18*a^2*b - 18*I*a*b^2 - 6*b^3)*c^2*d*f^2 - (72*a*b^2*c*d^2 - (36*a^2*b - 3
6*I*a*b^2 - 12*b^3)*c*d^2*e)*f)*(f*x + e)^2 + (72*a*b^2*d^3*e^2 - 24*b^3*d^3*e + (12*I*a*b^2 + 4*b^3)*d^3*e^3
+ (-12*I*a*b^2 - 4*b^3)*c^3*f^3 + (72*a*b^2*c^2*d + (36*I*a*b^2 + 12*b^3)*c^2*d*e)*f^2 - (144*a*b^2*c*d^2*e -
24*b^3*c*d^2 - (-36*I*a*b^2 - 12*b^3)*c*d^2*e^2)*f)*(f*x + e))*cos(4*f*x + 4*e) - ((6*a^2*b - 6*I*a*b^2 - 2*b^
3)*(f*x + e)^4*d^3 - 12*b^3*d^3*e^2 + 8*(3*a*b^2 + I*b^3)*d^3*e^3 - 8*(3*a*b^2 + I*b^3)*c^3*f^3 - ((24*a^2*b -
24*I*a*b^2 - 8*b^3)*d^3*e - (24*a^2*b - 24*I*a*b^2 - 8*b^3)*c*d^2*f - 8*(3*a*b^2 - I*b^3)*d^3)*(f*x + e)^3 +
(12*b^3*d^3 + (36*a^2*b - 36*I*a*b^2 - 12*b^3)*d^3*e^2 + (36*a^2*b - 36*I*a*b^2 - 12*b^3)*c^2*d*f^2 - 24*(3*a*
b^2 - I*b^3)*d^3*e - ((72*a^2*b - 72*I*a*b^2 - 24*b^3)*c*d^2*e - 24*(3*a*b^2 - I*b^3)*c*d^2)*f)*(f*x + e)^2 -
12*(b^3*c^2*d - 2*(3*a*b^2 + I*b^3)*c^2*d*e)*f^2 - (24*b^3*d^3*e + (-24*I*a*b^2 - 8*b^3)*d^3*e^3 + (24*I*a*b^2
+ 8*b^3)*c^3*f^3 - 24*(3*a*b^2 - I*b^3)*d^3*e^2 + ((-72*I*a*b^2 - 24*b^3)*c^2*d*e - 24*(3*a*b^2 - I*b^3)*c^2*
d)*f^2 - (24*b^3*c*d^2 - (72*I*a*b^2 + 24*b^3)*c*d^2*e^2 - 48*(3*a*b^2 - I*b^3)*c*d^2*e)*f)*(f*x + e) + 24*(b^
3*c*d^2*e - (3*a*b^2 + I*b^3)*c*d^2*e^2)*f)*cos(2*f*x + 2*e) - (72*a*b^2*d^3*e - 12*(3*a^2*b - b^3)*(f*x + e)^
2*d^3 - 12*b^3*d^3 - 12*(3*a^2*b - b^3)*d^3*e^2 - 12*(3*a^2*b - b^3)*c^2*d*f^2 - 24*(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) - 24*(3*a*b^2*c*d^2 - (3*a^2*b - b^3)*c*d^2*e)*f + 12*(6*a*b^2
*d^3*e - (3*a^2*b - b^3)*(f*x + e)^2*d^3 - 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*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*a*b^2*c*d^2 - (3*a^2*b - b^3)*
c*d^2*e)*f)*cos(4*f*x + 4*e) - 24*(6*a*b^2*d^3*e - (3*a^2*b - b^3)*(f*x + e)^2*d^3 - 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*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*a*b^2*c*d^2 - (3*a^2*b - b^3)*c*d^2*e)*f)*cos(2*f*x + 2*e) - (-72*I*a*b^2*d^3*e + (36*I*a^2*b - 12
*I*b^3)*(f*x + e)^2*d^3 + 12*I*b^3*d^3 + (36*I*a^2*b - 12*I*b^3)*d^3*e^2 + (36*I*a^2*b - 12*I*b^3)*c^2*d*f^2 +
(72*I*a*b^2*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f)*(f*x + e) + (72*I*a*b^2*c
*d^2 + (-72*I*a^2*b + 24*I*b^3)*c*d^2*e)*f)*sin(4*f*x + 4*e) - (144*I*a*b^2*d^3*e + (-72*I*a^2*b + 24*I*b^3)*(
f*x + e)^2*d^3 - 24*I*b^3*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e^2 + (-72*I*a^2*b + 24*I*b^3)*c^2*d*f^2 + (-144*
I*a*b^2*d^3 + (144*I*a^2*b - 48*I*b^3)*d^3*e + (-144*I*a^2*b + 48*I*b^3)*c*d^2*f)*(f*x + e) + (-144*I*a*b^2*c*
d^2 + (144*I*a^2*b - 48*I*b^3)*c*d^2*e)*f)*sin(2*f*x + 2*e))*dilog(-e^(I*f*x + I*e)) - (72*a*b^2*d^3*e - 12*(3
*a^2*b - b^3)*(f*x + e)^2*d^3 - 12*b^3*d^3 - 12*(3*a^2*b - b^3)*d^3*e^2 - 12*(3*a^2*b - b^3)*c^2*d*f^2 - 24*(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) - 24*(3*a*b^2*c*d^2 - (3*a^2*b - b^3)*
c*d^2*e)*f + 12*(6*a*b^2*d^3*e - (3*a^2*b - b^3)*(f*x + e)^2*d^3 - 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*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*a*b^2*
c*d^2 - (3*a^2*b - b^3)*c*d^2*e)*f)*cos(4*f*x + 4*e) - 24*(6*a*b^2*d^3*e - (3*a^2*b - b^3)*(f*x + e)^2*d^3 - 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*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*a*b^2*c*d^2 - (3*a^2*b - b^3)*c*d^2*e)*f)*cos(2*f*x + 2*e) - (-72*I*a*b^2*
d^3*e + (36*I*a^2*b - 12*I*b^3)*(f*x + e)^2*d^3 + 12*I*b^3*d^3 + (36*I*a^2*b - 12*I*b^3)*d^3*e^2 + (36*I*a^2*b
- 12*I*b^3)*c^2*d*f^2 + (72*I*a*b^2*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f)*(
f*x + e) + (72*I*a*b^2*c*d^2 + (-72*I*a^2*b + 24*I*b^3)*c*d^2*e)*f)*sin(4*f*x + 4*e) - (144*I*a*b^2*d^3*e + (-
72*I*a^2*b + 24*I*b^3)*(f*x + e)^2*d^3 - 24*I*b^3*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e^2 + (-72*I*a^2*b + 24*I
*b^3)*c^2*d*f^2 + (-144*I*a*b^2*d^3 + (144*I*a^2*b - 48*I*b^3)*d^3*e + (-144*I*a^2*b + 48*I*b^3)*c*d^2*f)*(f*x
+ e) + (-144*I*a*b^2*c*d^2 + (144*I*a^2*b - 48*I*b^3)*c*d^2*e)*f)*sin(2*f*x + 2*e))*dilog(e^(I*f*x + I*e)) +
(2*I*b^3*d^3*e^3 - 2*I*b^3*c^3*f^3 + 18*I*a*b^2*d^3*e^2 + (6*I*a^2*b - 2*I*b^3)*(f*x + e)^3*d^3 - 6*I*b^3*d^3*
e + (18*I*a*b^2*d^3 + (-18*I*a^2*b + 6*I*b^3)*d^3*e + (18*I*a^2*b - 6*I*b^3)*c*d^2*f)*(f*x + e)^2 + (6*I*b^3*c
^2*d*e + 18*I*a*b^2*c^2*d)*f^2 + (-36*I*a*b^2*d^3*e + 6*I*b^3*d^3 + (18*I*a^2*b - 6*I*b^3)*d^3*e^2 + (18*I*a^2
*b - 6*I*b^3)*c^2*d*f^2 + (36*I*a*b^2*c*d^2 + (-36*I*a^2*b + 12*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-6*I*b^3*c*d^2
*e^2 - 36*I*a*b^2*c*d^2*e + 6*I*b^3*c*d^2)*f + (2*I*b^3*d^3*e^3 - 2*I*b^3*c^3*f^3 + 18*I*a*b^2*d^3*e^2 + (6*I*
a^2*b - 2*I*b^3)*(f*x + e)^3*d^3 - 6*I*b^3*d^3*e + (18*I*a*b^2*d^3 + (-18*I*a^2*b + 6*I*b^3)*d^3*e + (18*I*a^2
*b - 6*I*b^3)*c*d^2*f)*(f*x + e)^2 + (6*I*b^3*c^2*d*e + 18*I*a*b^2*c^2*d)*f^2 + (-36*I*a*b^2*d^3*e + 6*I*b^3*d
^3 + (18*I*a^2*b - 6*I*b^3)*d^3*e^2 + (18*I*a^2*b - 6*I*b^3)*c^2*d*f^2 + (36*I*a*b^2*c*d^2 + (-36*I*a^2*b + 12
*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-6*I*b^3*c*d^2*e^2 - 36*I*a*b^2*c*d^2*e + 6*I*b^3*c*d^2)*f)*cos(4*f*x + 4*e)
+ (-4*I*b^3*d^3*e^3 + 4*I*b^3*c^3*f^3 - 36*I*a*b^2*d^3*e^2 + (-12*I*a^2*b + 4*I*b^3)*(f*x + e)^3*d^3 + 12*I*b^
3*d^3*e + (-36*I*a*b^2*d^3 + (36*I*a^2*b - 12*I*b^3)*d^3*e + (-36*I*a^2*b + 12*I*b^3)*c*d^2*f)*(f*x + e)^2 + (
-12*I*b^3*c^2*d*e - 36*I*a*b^2*c^2*d)*f^2 + (72*I*a*b^2*d^3*e - 12*I*b^3*d^3 + (-36*I*a^2*b + 12*I*b^3)*d^3*e^
2 + (-36*I*a^2*b + 12*I*b^3)*c^2*d*f^2 + (-72*I*a*b^2*c*d^2 + (72*I*a^2*b - 24*I*b^3)*c*d^2*e)*f)*(f*x + e) +
(12*I*b^3*c*d^2*e^2 + 72*I*a*b^2*c*d^2*e - 12*I*b^3*c*d^2)*f)*cos(2*f*x + 2*e) - 2*(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)*sin(4*f*x + 4*e) + 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)*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 + 2*cos(f*x + e) + 1
) + (2*I*b^3*d^3*e^3 - 2*I*b^3*c^3*f^3 + 18*I*a*b^2*d^3*e^2 + (6*I*a^2*b - 2*I*b^3)*(f*x + e)^3*d^3 - 6*I*b^3*
d^3*e + (18*I*a*b^2*d^3 + (-18*I*a^2*b + 6*I*b^3)*d^3*e + (18*I*a^2*b - 6*I*b^3)*c*d^2*f)*(f*x + e)^2 + (6*I*b
^3*c^2*d*e + 18*I*a*b^2*c^2*d)*f^2 + (-36*I*a*b^2*d^3*e + 6*I*b^3*d^3 + (18*I*a^2*b - 6*I*b^3)*d^3*e^2 + (18*I
*a^2*b - 6*I*b^3)*c^2*d*f^2 + (36*I*a*b^2*c*d^2 + (-36*I*a^2*b + 12*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-6*I*b^3*c
*d^2*e^2 - 36*I*a*b^2*c*d^2*e + 6*I*b^3*c*d^2)*f + (2*I*b^3*d^3*e^3 - 2*I*b^3*c^3*f^3 + 18*I*a*b^2*d^3*e^2 + (
6*I*a^2*b - 2*I*b^3)*(f*x + e)^3*d^3 - 6*I*b^3*d^3*e + (18*I*a*b^2*d^3 + (-18*I*a^2*b + 6*I*b^3)*d^3*e + (18*I
*a^2*b - 6*I*b^3)*c*d^2*f)*(f*x + e)^2 + (6*I*b^3*c^2*d*e + 18*I*a*b^2*c^2*d)*f^2 + (-36*I*a*b^2*d^3*e + 6*I*b
^3*d^3 + (18*I*a^2*b - 6*I*b^3)*d^3*e^2 + (18*I*a^2*b - 6*I*b^3)*c^2*d*f^2 + (36*I*a*b^2*c*d^2 + (-36*I*a^2*b
+ 12*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-6*I*b^3*c*d^2*e^2 - 36*I*a*b^2*c*d^2*e + 6*I*b^3*c*d^2)*f)*cos(4*f*x + 4
*e) + (-4*I*b^3*d^3*e^3 + 4*I*b^3*c^3*f^3 - 36*I*a*b^2*d^3*e^2 + (-12*I*a^2*b + 4*I*b^3)*(f*x + e)^3*d^3 + 12*
I*b^3*d^3*e + (-36*I*a*b^2*d^3 + (36*I*a^2*b - 12*I*b^3)*d^3*e + (-36*I*a^2*b + 12*I*b^3)*c*d^2*f)*(f*x + e)^2
+ (-12*I*b^3*c^2*d*e - 36*I*a*b^2*c^2*d)*f^2 + (72*I*a*b^2*d^3*e - 12*I*b^3*d^3 + (-36*I*a^2*b + 12*I*b^3)*d^
3*e^2 + (-36*I*a^2*b + 12*I*b^3)*c^2*d*f^2 + (-72*I*a*b^2*c*d^2 + (72*I*a^2*b - 24*I*b^3)*c*d^2*e)*f)*(f*x + e
) + (12*I*b^3*c*d^2*e^2 + 72*I*a*b^2*c*d^2*e - 12*I*b^3*c*d^2)*f)*cos(2*f*x + 2*e) - 2*(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)*sin(4*f*x + 4*e) + 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)*sin(2*f*x + 2*e))*log(cos(f*x + e)^2 + sin(f*x + e)^2 - 2*cos(f*x + e)
+ 1) - (24*(3*a^2*b - b^3)*d^3*cos(4*f*x + 4*e) - 48*(3*a^2*b - b^3)*d^3*cos(2*f*x + 2*e) - (-72*I*a^2*b + 24
*I*b^3)*d^3*sin(4*f*x + 4*e) - (144*I*a^2*b - 48*I*b^3)*d^3*sin(2*f*x + 2*e) + 24*(3*a^2*b - b^3)*d^3)*polylog
(4, -e^(I*f*x + I*e)) - (24*(3*a^2*b - b^3)*d^3*cos(4*f*x + 4*e) - 48*(3*a^2*b - b^3)*d^3*cos(2*f*x + 2*e) - (
-72*I*a^2*b + 24*I*b^3)*d^3*sin(4*f*x + 4*e) - (144*I*a^2*b - 48*I*b^3)*d^3*sin(2*f*x + 2*e) + 24*(3*a^2*b - b
^3)*d^3)*polylog(4, e^(I*f*x + I*e)) + (72*I*a*b^2*d^3 + (72*I*a^2*b - 24*I*b^3)*(f*x + e)*d^3 + (-72*I*a^2*b
+ 24*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f + (72*I*a*b^2*d^3 + (72*I*a^2*b - 24*I*b^3)*(f*x + e)*d^3
+ (-72*I*a^2*b + 24*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f)*cos(4*f*x + 4*e) + (-144*I*a*b^2*d^3 + (-1
44*I*a^2*b + 48*I*b^3)*(f*x + e)*d^3 + (144*I*a^2*b - 48*I*b^3)*d^3*e + (-144*I*a^2*b + 48*I*b^3)*c*d^2*f)*cos
(2*f*x + 2*e) - 24*(3*a*b^2*d^3 + (3*a^2*b - b^3)*(f*x + e)*d^3 - (3*a^2*b - b^3)*d^3*e + (3*a^2*b - b^3)*c*d^
2*f)*sin(4*f*x + 4*e) + 48*(3*a*b^2*d^3 + (3*a^2*b - b^3)*(f*x + e)*d^3 - (3*a^2*b - b^3)*d^3*e + (3*a^2*b - b
^3)*c*d^2*f)*sin(2*f*x + 2*e))*polylog(3, -e^(I*f*x + I*e)) + (72*I*a*b^2*d^3 + (72*I*a^2*b - 24*I*b^3)*(f*x +
e)*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f + (72*I*a*b^2*d^3 + (72*I*a^2*b - 2
4*I*b^3)*(f*x + e)*d^3 + (-72*I*a^2*b + 24*I*b^3)*d^3*e + (72*I*a^2*b - 24*I*b^3)*c*d^2*f)*cos(4*f*x + 4*e) +
(-144*I*a*b^2*d^3 + (-144*I*a^2*b + 48*I*b^3)*(f*x + e)*d^3 + (144*I*a^2*b - 48*I*b^3)*d^3*e + (-144*I*a^2*b +
48*I*b^3)*c*d^2*f)*cos(2*f*x + 2*e) - 24*(3*a*b^2*d^3 + (3*a^2*b - b^3)*(f*x + e)*d^3 - (3*a^2*b - b^3)*d^3*e
+ (3*a^2*b - b^3)*c*d^2*f)*sin(4*f*x + 4*e) + 48*(3*a*b^2*d^3 + (3*a^2*b - b^3)*(f*x + e)*d^3 - (3*a^2*b - b^
3)*d^3*e + (3*a^2*b - b^3)*c*d^2*f)*sin(2*f*x + 2*e))*polylog(3, e^(I*f*x + I*e)) + ((3*I*a^2*b + 3*a*b^2 - I*
b^3)*(f*x + e)^4*d^3 + (24*I*a*b^2*d^3 + (-12*I*a^2*b - 12*a*b^2 + 4*I*b^3)*d^3*e + (12*I*a^2*b + 12*a*b^2 - 4
*I*b^3)*c*d^2*f)*(f*x + e)^3 + (-72*I*a*b^2*d^3*e + 12*I*b^3*d^3 + (18*I*a^2*b + 18*a*b^2 - 6*I*b^3)*d^3*e^2 +
(18*I*a^2*b + 18*a*b^2 - 6*I*b^3)*c^2*d*f^2 + (72*I*a*b^2*c*d^2 + (-36*I*a^2*b - 36*a*b^2 + 12*I*b^3)*c*d^2*e
)*f)*(f*x + e)^2 + (72*I*a*b^2*d^3*e^2 - 24*I*b^3*d^3*e - 4*(3*a*b^2 - I*b^3)*d^3*e^3 + 4*(3*a*b^2 - I*b^3)*c^
3*f^3 - 12*(-6*I*a*b^2*c^2*d + (3*a*b^2 - I*b^3)*c^2*d*e)*f^2 + (-144*I*a*b^2*c*d^2*e + 24*I*b^3*c*d^2 + 12*(3
*a*b^2 - I*b^3)*c*d^2*e^2)*f)*(f*x + e))*sin(4*f*x + 4*e) + ((-6*I*a^2*b - 6*a*b^2 + 2*I*b^3)*(f*x + e)^4*d^3
+ 12*I*b^3*d^3*e^2 + (-24*I*a*b^2 + 8*b^3)*d^3*e^3 + (24*I*a*b^2 - 8*b^3)*c^3*f^3 + ((24*I*a^2*b + 24*a*b^2 -
8*I*b^3)*d^3*e + (-24*I*a^2*b - 24*a*b^2 + 8*I*b^3)*c*d^2*f + (-24*I*a*b^2 - 8*b^3)*d^3)*(f*x + e)^3 + (-12*I*
b^3*d^3 + (-36*I*a^2*b - 36*a*b^2 + 12*I*b^3)*d^3*e^2 + (-36*I*a^2*b - 36*a*b^2 + 12*I*b^3)*c^2*d*f^2 + (72*I*
a*b^2 + 24*b^3)*d^3*e + ((72*I*a^2*b + 72*a*b^2 - 24*I*b^3)*c*d^2*e + (-72*I*a*b^2 - 24*b^3)*c*d^2)*f)*(f*x +
e)^2 + (12*I*b^3*c^2*d + (-72*I*a*b^2 + 24*b^3)*c^2*d*e)*f^2 + (24*I*b^3*d^3*e + 8*(3*a*b^2 - I*b^3)*d^3*e^3 -
8*(3*a*b^2 - I*b^3)*c^3*f^3 + (-72*I*a*b^2 - 24*b^3)*d^3*e^2 + (24*(3*a*b^2 - I*b^3)*c^2*d*e + (-72*I*a*b^2 -
24*b^3)*c^2*d)*f^2 + (-24*I*b^3*c*d^2 - 24*(3*a*b^2 - I*b^3)*c*d^2*e^2 + (144*I*a*b^2 + 48*b^3)*c*d^2*e)*f)*(
f*x + e) + (-24*I*b^3*c*d^2*e + (72*I*a*b^2 - 24*b^3)*c*d^2*e^2)*f)*sin(2*f*x + 2*e))/(-4*I*f^3*cos(4*f*x + 4*
e) + 8*I*f^3*cos(2*f*x + 2*e) + 4*f^3*sin(4*f*x + 4*e) - 8*f^3*sin(2*f*x + 2*e) - 4*I*f^3))/f

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Fricas [C]  time = 2.82744, size = 5689, normalized size = 9.43 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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 + 36*I*a*b^2*c*d^2*f + 6*I*b^3*d^3 + 6*I*(3*a^2*b
- b^3)*c^2*d*f^2 + 12*I*(3*a*b^2*d^3*f + (3*a^2*b - b^3)*c*d^2*f^2)*x + (-6*I*(3*a^2*b - b^3)*d^3*f^2*x^2 - 3
6*I*a*b^2*c*d^2*f - 6*I*b^3*d^3 - 6*I*(3*a^2*b - b^3)*c^2*d*f^2 - 12*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 - 3
6*I*a*b^2*c*d^2*f - 6*I*b^3*d^3 - 6*I*(3*a^2*b - b^3)*c^2*d*f^2 - 12*I*(3*a*b^2*d^3*f + (3*a^2*b - b^3)*c*d^2*
f^2)*x + (6*I*(3*a^2*b - b^3)*d^3*f^2*x^2 + 36*I*a*b^2*c*d^2*f + 6*I*b^3*d^3 + 6*I*(3*a^2*b - b^3)*c^2*d*f^2 +
12*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)) + 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)*cos(2*f*x + 2*e))*log(-1/2*cos(2*
f*x + 2*e) + 1/2*I*sin(2*f*x + 2*e) + 1/2) + 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)*cos(2*f*x + 2*e))*log(-1/2*cos(2*f*x + 2*e) - 1/2*I*sin(2*f*x + 2*e) + 1/2) + 4*((3*a^2*b - b^3)*d^3*f^3*x
^3 - 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*(3*a^2*b - b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*
f^2 + (3*a^2*b - b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b - b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^
2 + b^3*d^3*f + (3*a^2*b - b^3)*c^2*d*f^3)*x - ((3*a^2*b - b^3)*d^3*f^3*x^3 - 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*(3*a^2*b - b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b - b^3)*c*d^2*f^3)*x^2
+ 3*(6*a*b^2*c*d^2*e - (3*a^2*b - b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b - b^3)*c^2*d
*f^3)*x)*cos(2*f*x + 2*e))*log(-cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e) + 1) + 4*((3*a^2*b - b^3)*d^3*f^3*x^3 -
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*(3*a^2*b - b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 +
(3*a^2*b - b^3)*c*d^2*f^3)*x^2 + 3*(6*a*b^2*c*d^2*e - (3*a^2*b - b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b
^3*d^3*f + (3*a^2*b - b^3)*c^2*d*f^3)*x - ((3*a^2*b - b^3)*d^3*f^3*x^3 - 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*(3*a^2*b - b^3)*c^2*d*e*f^2 + 3*(3*a*b^2*d^3*f^2 + (3*a^2*b - b^3)*c*d^2*f^3)*x^2 + 3*(
6*a*b^2*c*d^2*e - (3*a^2*b - b^3)*c*d^2*e^2)*f + 3*(6*a*b^2*c*d^2*f^2 + b^3*d^3*f + (3*a^2*b - b^3)*c^2*d*f^3)
*x)*cos(2*f*x + 2*e))*log(-cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e) + 1) - (3*I*(3*a^2*b - b^3)*d^3*cos(2*f*x + 2
*e) - 3*I*(3*a^2*b - b^3)*d^3)*polylog(4, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) - (-3*I*(3*a^2*b - b^3)*d^3*c
os(2*f*x + 2*e) + 3*I*(3*a^2*b - b^3)*d^3)*polylog(4, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) + 6*(3*a*b^2*d^3
+ (3*a^2*b - b^3)*d^3*f*x + (3*a^2*b - b^3)*c*d^2*f - (3*a*b^2*d^3 + (3*a^2*b - b^3)*d^3*f*x + (3*a^2*b - b^3)
*c*d^2*f)*cos(2*f*x + 2*e))*polylog(3, cos(2*f*x + 2*e) + I*sin(2*f*x + 2*e)) + 6*(3*a*b^2*d^3 + (3*a^2*b - b^
3)*d^3*f*x + (3*a^2*b - b^3)*c*d^2*f - (3*a*b^2*d^3 + (3*a^2*b - b^3)*d^3*f*x + (3*a^2*b - b^3)*c*d^2*f)*cos(2
*f*x + 2*e))*polylog(3, cos(2*f*x + 2*e) - I*sin(2*f*x + 2*e)) - 12*(2*a*b^2*d^3*f^3*x^3 + 2*a*b^2*c^3*f^3 + b
^3*c^2*d*f^2 + (6*a*b^2*c*d^2*f^3 + b^3*d^3*f^2)*x^2 + 2*(3*a*b^2*c^2*d*f^3 + b^3*c*d^2*f^2)*x)*sin(2*f*x + 2*
e))/(f^4*cos(2*f*x + 2*e) - f^4)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \cot{\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{3}{\left (b \cot \left (f x + e\right ) + a\right )}^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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