### 3.49 $$\int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx$$

Optimal. Leaf size=612 $-\frac{9 a^2 b d^2 (c+d x) \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{9 i a^2 b d (c+d x)^2 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{9 i a^2 b d^3 \text{PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac{9 i a b^2 d^2 (c+d x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac{9 a b^2 d^3 \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac{3 b^3 d^2 (c+d x) \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 i b^3 d (c+d x)^2 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac{3 i b^3 d^3 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac{3 i b^3 d^3 \text{PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac{3 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i a^2 b (c+d x)^4}{4 d}+\frac{a^3 (c+d x)^4}{4 d}+\frac{9 a b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^3}{f}-\frac{3 a b^2 (c+d x)^4}{4 d}-\frac{3 b^3 d^2 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^3}-\frac{3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{b^3 (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}+\frac{3 i b^3 d (c+d x)^2}{2 f^2}+\frac{b^3 (c+d x)^3}{2 f}-\frac{i b^3 (c+d x)^4}{4 d}$

[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^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*Po
lyLog[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*b^3*d*(c + d*x)^2*Tan[e + f*x])/(2*f^2) + (3*a*b^2*(c + d*x)^3*Tan[e + f*x])/f + (b^3*(c +
d*x)^3*Tan[e + f*x]^2)/(2*f)

________________________________________________________________________________________

Rubi [A]  time = 0.981887, antiderivative size = 612, 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, 3719, 2190, 2531, 6609, 2282, 6589, 3720, 32, 2279, 2391} $-\frac{9 a^2 b d^2 (c+d x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{9 i a^2 b d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{3 i a^2 b (c+d x)^4}{4 d}-\frac{9 i a^2 b d^3 \text{Li}_4\left (-e^{2 i (e+f x)}\right )}{4 f^4}+\frac{a^3 (c+d x)^4}{4 d}-\frac{9 i a b^2 d^2 (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac{9 a b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^3}{f}-\frac{3 a b^2 (c+d x)^4}{4 d}+\frac{9 a b^2 d^3 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4}+\frac{3 b^3 d^2 (c+d x) \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 b^3 d^2 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^3}-\frac{3 i b^3 d (c+d x)^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^2}-\frac{3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{b^3 (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}+\frac{3 i b^3 d (c+d x)^2}{2 f^2}+\frac{b^3 (c+d x)^3}{2 f}-\frac{i b^3 (c+d x)^4}{4 d}+\frac{3 i b^3 d^3 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{2 f^4}+\frac{3 i b^3 d^3 \text{Li}_4\left (-e^{2 i (e+f x)}\right )}{4 f^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(c + d*x)^3*(a + b*Tan[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^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*Po
lyLog[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*b^3*d*(c + d*x)^2*Tan[e + f*x])/(2*f^2) + (3*a*b^2*(c + d*x)^3*Tan[e + f*x])/f + (b^3*(c +
d*x)^3*Tan[e + f*x]^2)/(2*f)

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 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
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 \tan (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^3+3 a^2 b (c+d x)^3 \tan (e+f x)+3 a b^2 (c+d x)^3 \tan ^2(e+f x)+b^3 (c+d x)^3 \tan ^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 \tan (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^3 \tan ^2(e+f x) \, dx+b^3 \int (c+d x)^3 \tan ^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 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^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 \tan (e+f x) \, dx-\frac{\left (9 a b^2 d\right ) \int (c+d x)^2 \tan (e+f x) \, dx}{f}-\frac{\left (3 b^3 d\right ) \int (c+d x)^2 \tan ^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 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 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) \tan (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{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{3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}-\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^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 (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}+\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^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{3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}-\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^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 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}+\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^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}-\frac{3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac{3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}\\ \end{align*}

Mathematica [B]  time = 8.3375, size = 2572, normalized size = 4.2 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(((3*I)/4)*a*b^2*d^3*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2
*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec
[e])/(E^(I*e)*f^4) - (((3*I)/4)*a^2*b*c*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e +
f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^(
(-2*I)*(e + f*x))])*Sec[e])/(E^(I*e)*f^3) + ((I/4)*b^3*c*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1
+ E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e)
)*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e])/(E^(I*e)*f^3) - ((3*I)/8)*a^2*b*d^3*E^(I*e)*((2*x^4)/E^((2*I)*e)
- ((4*I)*(1 + E^((-2*I)*e))*x^3*Log[1 + E^((-2*I)*(e + f*x))])/f + (3*(1 + E^((2*I)*e))*(2*f^2*x^2*PolyLog[2,
-E^((-2*I)*(e + f*x))] - (2*I)*f*x*PolyLog[3, -E^((-2*I)*(e + f*x))] - PolyLog[4, -E^((-2*I)*(e + f*x))]))/(E^
((2*I)*e)*f^4))*Sec[e] + (I/8)*b^3*d^3*E^(I*e)*((2*x^4)/E^((2*I)*e) - ((4*I)*(1 + E^((-2*I)*e))*x^3*Log[1 + E^
((-2*I)*(e + f*x))])/f + (3*(1 + E^((2*I)*e))*(2*f^2*x^2*PolyLog[2, -E^((-2*I)*(e + f*x))] - (2*I)*f*x*PolyLog
[3, -E^((-2*I)*(e + f*x))] - PolyLog[4, -E^((-2*I)*(e + f*x))]))/(E^((2*I)*e)*f^4))*Sec[e] + ((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)*Sec[e + f*x]^2)/(2*f) - (3*b^3*c*d^2*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f
*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^3*(Cos[e]^2 + Sin[e]^2)) + (9*a*b^2*c^2*d*Sec[e]*(Cos[e]*Log[Cos[e]*C
os[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^2*(Cos[e]^2 + Sin[e]^2)) - (3*a^2*b*c^3*Sec[e]*(Cos[e]*Log[Cos[e]
*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) + (b^3*c^3*Sec[e]*(Cos[e]*Log[Cos[e]*Cos
[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f*(Cos[e]^2 + Sin[e]^2)) - (3*b^3*d^3*Csc[e]*((f^2*x^2)/E^(I*ArcTan[C
ot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[
1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I
*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(2*f^4*Sqrt[Csc[e]^2*(Cos[e]^2 + S
in[e]^2)]) + (9*a*b^2*c*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) -
Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[
f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqr
t[1 + Cot[e]^2])*Sec[e])/(f^3*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (9*a^2*b*c^2*d*Csc[e]*((f^2*x^2)/E^(I*Ar
cTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]]
)*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]
]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(2*f^2*Sqrt[Csc[e]^2*(Cos[e]
^2 + Sin[e]^2)]) + (3*b^3*c^2*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]
]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[
Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))
/Sqrt[1 + Cot[e]^2])*Sec[e])/(2*f^2*Sqrt[Csc[e]^2*(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*Si
n[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*Sec[e]*Sec[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)

________________________________________________________________________________________

Maple [B]  time = 0.243, size = 1882, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/2*a*b^2*d^3*polylog(3,-exp(2*I*(f*x+e)))/f^4-6*b^3/f^4*d^3*e*ln(exp(I*(f*x+e)))+2*b^3/f^4*d^3*e^3*ln(exp(I*(
f*x+e)))-3*b/f*a^2*c^3*ln(exp(2*I*(f*x+e))+1)+b^3/f*c^3*ln(exp(2*I*(f*x+e))+1)-2*b^3/f*c^3*ln(exp(I*(f*x+e)))-
1/4*I*b^3*d^3*x^4-3/4*a*b^2*d^3*x^4+3/2*a^3*c^2*d*x^2-3*b^2*a*c^3*x+9*I*b/f^2*polylog(2,-exp(2*I*(f*x+e)))*a^2
*c*d^2*x+18*I*b/f*a^2*c^2*d*e*x-18*I*b/f^2*a^2*c*d^2*e^2*x-36*I*b^2/f^2*a*c*d^2*e*x+a^3*c*d^2*x^3+3/2*I*b^3*d^
3*polylog(2,-exp(2*I*(f*x+e)))/f^4+3/4*I*b^3*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4+b^2*(-6*I*b*c*d^2*x*exp(2*I*
(f*x+e))+6*I*a*d^3*f*x^3*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*exp(2*I*(f*x+e))-3*I*b*d^
3*x^2*exp(2*I*(f*x+e))-3*I*b*c^2*d*exp(2*I*(f*x+e))+6*b*c*d^2*f*x^2*exp(2*I*(f*x+e))+18*I*a*c*d^2*f*x^2*exp(2*
I*(f*x+e))+6*I*a*c^3*f+18*I*a*c^2*d*f*x+6*b*c^2*d*f*x*exp(2*I*(f*x+e))-6*I*b*c*d^2*x+18*I*a*c*d^2*f*x^2-3*I*b*
c^2*d+2*b*c^3*f*exp(2*I*(f*x+e))-3*I*b*d^3*x^2+6*I*a*d^3*f*x^3+18*I*a*c^2*d*f*x*exp(2*I*(f*x+e)))/f^2/(exp(2*I
*(f*x+e))+1)^2+I*b^3*c^3*x-9/2*b/f^3*a^2*d^3*polylog(3,-exp(2*I*(f*x+e)))*x+9*b^2/f^2*a*d^3*ln(exp(2*I*(f*x+e)
)+1)*x^2+3*b^3/f*ln(exp(2*I*(f*x+e))+1)*c*d^2*x^2+3*b^3/f*ln(exp(2*I*(f*x+e))+1)*c^2*d*x-2*I*b^3/f^3*d^3*e^3*x
-3/2*I*b^3/f^2*c^2*d*polylog(2,-exp(2*I*(f*x+e)))+6*I*b^3/f^3*d^3*e*x+4*I*b^3/f^3*c*d^2*e^3+9/2*I*b/f^4*a^2*d^
3*e^4-3*I*b^3/f^2*c^2*d*e^2+12*I*b^2/f^4*a*d^3*e^3-6*I*b^2/f*a*d^3*x^3-3/2*I*b^3/f^2*d^3*polylog(2,-exp(2*I*(f
*x+e)))*x^2-6*b^3/f^3*c*d^2*e^2*ln(exp(I*(f*x+e)))-6*b/f^4*a^2*d^3*e^3*ln(exp(I*(f*x+e)))-9/2*b/f^3*a^2*c*d^2*
polylog(3,-exp(2*I*(f*x+e)))-18*b^2/f^4*a*d^3*e^2*ln(exp(I*(f*x+e)))+6*b^3/f^2*c^2*d*e*ln(exp(I*(f*x+e)))+9*b^
2/f^2*a*c^2*d*ln(exp(2*I*(f*x+e))+1)-18*b^2/f^2*a*c^2*d*ln(exp(I*(f*x+e)))+b^3/f*d^3*ln(exp(2*I*(f*x+e))+1)*x^
3+1/4*a^3*d^3*x^4+a^3*c^3*x-3/2*I*b^3*c^2*d*x^2+3/4*I*a^2*b*d^3*x^4-I*b^3*c*d^2*x^3+6*b/f*a^2*c^3*ln(exp(I*(f*
x+e)))-3*b^3/f^3*c*d^2*ln(exp(2*I*(f*x+e))+1)+6*b^3/f^3*c*d^2*ln(exp(I*(f*x+e)))+3/2*b^3/f^3*c*d^2*polylog(3,-
exp(2*I*(f*x+e)))+3/2*b^3/f^3*d^3*polylog(3,-exp(2*I*(f*x+e)))*x-3*b^3/f^3*d^3*ln(exp(2*I*(f*x+e))+1)*x+3*I*b^
3/f^2*d^3*x^2-3/2*I*b^3/f^4*d^3*e^4+3*I*b^3/f^4*d^3*e^2-3*a*b^2*c*d^2*x^3-9/2*a*b^2*c^2*d*x^2+3*I*a^2*b*c*d^2*
x^3+9/2*I*a^2*b*c^2*d*x^2-3*I*a^2*b*c^3*x+6*I*b/f^3*a^2*d^3*e^3*x-18*I*b^2/f^3*a*c*d^2*e^2+9*I*b/f^2*a^2*c^2*d
*e^2+18*I*b^2/f^3*a*d^3*e^2*x+9/2*I*b/f^2*a^2*d^3*polylog(2,-exp(2*I*(f*x+e)))*x^2-12*I*b/f^3*a^2*c*d^2*e^3-9*
I*b^2/f^3*a*d^3*polylog(2,-exp(2*I*(f*x+e)))*x-3*I*b^3/f^2*polylog(2,-exp(2*I*(f*x+e)))*c*d^2*x+6*I*b^3/f^2*c*
d^2*e^2*x-6*I*b^3/f*c^2*d*e*x-18*I*b^2/f*a*c*d^2*x^2-9*I*b^2/f^3*a*c*d^2*polylog(2,-exp(2*I*(f*x+e)))+9/2*I*b/
f^2*a^2*c^2*d*polylog(2,-exp(2*I*(f*x+e)))-9*b/f*ln(exp(2*I*(f*x+e))+1)*a^2*c*d^2*x^2-9*b/f*ln(exp(2*I*(f*x+e)
)+1)*a^2*c^2*d*x+18*b^2/f^2*ln(exp(2*I*(f*x+e))+1)*a*c*d^2*x-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)))-3*b/f*a^2*d^3*ln(exp(2*I*(f*x+e))+1)*x^3+18*b/f^3*a^2*c*d^2*e^2*ln(exp(I*(f*x+e
)))-9/4*I*a^2*b*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4

________________________________________________________________________________________

Maxima [B]  time = 51.5616, size = 9189, normalized size = 15.01 \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*tan(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(sec(f*x + e)) -
12*a^2*b*d^3*e^3*log(sec(f*x + e))/f^3 + 36*a^2*b*c*d^2*e^2*log(sec(f*x + e))/f^2 - 36*a^2*b*c^2*d*e*log(sec(
f*x + e))/f - 4*(72*a*b^2*d^3*e^3 - 72*a*b^2*c^3*f^3 - 3*(3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^4*d^3 + 36*b^3*
d^3*e^2 + 12*((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 - 18*((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 + 36*(6*a*b^2*c^2*d*e + b^3*c^2*d)*f^2 + 12*((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) - 72*(3*a*b^2*c*d^2*e^2
+ b^3*c*d^2*e)*f + (12*b^3*d^3*e^3 - 12*b^3*c^3*f^3 - 108*a*b^2*d^3*e^2 + 16*(3*a^2*b - b^3)*(f*x + e)^3*d^3 -
36*b^3*d^3*e - 36*(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 + 36*(b^3*c^2*d
*e - 3*a*b^2*c^2*d)*f^2 + 36*(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) - 36*(b^3*c*d^2*e^2 - 6*a*b^2*c*d^2*e - b^3*c*d^2)*f
+ 4*(3*b^3*d^3*e^3 - 3*b^3*c^3*f^3 - 27*a*b^2*d^3*e^2 + 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*b^3*d^3*e - 9*(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 + 9*(b^3*c^2*d*e - 3*a*b^2*c^2*d)*f^
2 + 9*(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) - 9*(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*(3
*b^3*d^3*e^3 - 3*b^3*c^3*f^3 - 27*a*b^2*d^3*e^2 + 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*b^3*d^3*e - 9*(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 + 9*(b^3*c^2*d*e - 3*a*b^2*c^2*d)*f^2 + 9*
(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) - 9*(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) - (-12*I*b^3
*d^3*e^3 + 12*I*b^3*c^3*f^3 + 108*I*a*b^2*d^3*e^2 + (-48*I*a^2*b + 16*I*b^3)*(f*x + e)^3*d^3 + 36*I*b^3*d^3*e
+ (108*I*a*b^2*d^3 + (108*I*a^2*b - 36*I*b^3)*d^3*e + (-108*I*a^2*b + 36*I*b^3)*c*d^2*f)*(f*x + e)^2 + (-36*I*
b^3*c^2*d*e + 108*I*a*b^2*c^2*d)*f^2 + (-216*I*a*b^2*d^3*e - 36*I*b^3*d^3 + (-108*I*a^2*b + 36*I*b^3)*d^3*e^2
+ (-108*I*a^2*b + 36*I*b^3)*c^2*d*f^2 + (216*I*a*b^2*c*d^2 + (216*I*a^2*b - 72*I*b^3)*c*d^2*e)*f)*(f*x + e) +
(36*I*b^3*c*d^2*e^2 - 216*I*a*b^2*c*d^2*e - 36*I*b^3*c*d^2)*f)*sin(4*f*x + 4*e) - (-24*I*b^3*d^3*e^3 + 24*I*b^
3*c^3*f^3 + 216*I*a*b^2*d^3*e^2 + (-96*I*a^2*b + 32*I*b^3)*(f*x + e)^3*d^3 + 72*I*b^3*d^3*e + (216*I*a*b^2*d^3
+ (216*I*a^2*b - 72*I*b^3)*d^3*e + (-216*I*a^2*b + 72*I*b^3)*c*d^2*f)*(f*x + e)^2 + (-72*I*b^3*c^2*d*e + 216*
I*a*b^2*c^2*d)*f^2 + (-432*I*a*b^2*d^3*e - 72*I*b^3*d^3 + (-216*I*a^2*b + 72*I*b^3)*d^3*e^2 + (-216*I*a^2*b +
72*I*b^3)*c^2*d*f^2 + (432*I*a*b^2*c*d^2 + (432*I*a^2*b - 144*I*b^3)*c*d^2*e)*f)*(f*x + e) + (72*I*b^3*c*d^2*e
^2 - 432*I*a*b^2*c*d^2*e - 72*I*b^3*c*d^2)*f)*sin(2*f*x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1
) - 3*((3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^4*d^3 - 4*(6*a*b^2*d^3 + (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*(12*a*b^2*d^3*e + 2*b^3*d^3 + (3*a^2*b + 3*I*a*b^2 - b^3)*d^3
*e^2 + (3*a^2*b + 3*I*a*b^2 - b^3)*c^2*d*f^2 - 2*(6*a*b^2*c*d^2 + (3*a^2*b + 3*I*a*b^2 - b^3)*c*d^2*e)*f)*(f*x
+ e)^2 - 4*(18*a*b^2*d^3*e^2 + 6*b^3*d^3*e + (3*I*a*b^2 - b^3)*d^3*e^3 + (-3*I*a*b^2 + b^3)*c^3*f^3 + 3*(6*a*
b^2*c^2*d + (3*I*a*b^2 - b^3)*c^2*d*e)*f^2 - 3*(12*a*b^2*c*d^2*e + 2*b^3*c*d^2 - (-3*I*a*b^2 + b^3)*c*d^2*e^2)
*f)*(f*x + e))*cos(4*f*x + 4*e) - (6*(3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^4*d^3 - 36*b^3*d^3*e^2 - (72*a*b^2
- 24*I*b^3)*d^3*e^3 + (72*a*b^2 - 24*I*b^3)*c^3*f^3 - (24*(3*a^2*b + 3*I*a*b^2 - b^3)*d^3*e - 24*(3*a^2*b + 3*
I*a*b^2 - b^3)*c*d^2*f + (72*a*b^2 + 24*I*b^3)*d^3)*(f*x + e)^3 + (36*b^3*d^3 + 36*(3*a^2*b + 3*I*a*b^2 - b^3)
*d^3*e^2 + 36*(3*a^2*b + 3*I*a*b^2 - b^3)*c^2*d*f^2 + (216*a*b^2 + 72*I*b^3)*d^3*e - (72*(3*a^2*b + 3*I*a*b^2
- b^3)*c*d^2*e + (216*a*b^2 + 72*I*b^3)*c*d^2)*f)*(f*x + e)^2 - (36*b^3*c^2*d + (216*a*b^2 - 72*I*b^3)*c^2*d*e
)*f^2 - (72*b^3*d^3*e + 24*(3*I*a*b^2 - b^3)*d^3*e^3 + 24*(-3*I*a*b^2 + b^3)*c^3*f^3 + (216*a*b^2 + 72*I*b^3)*
d^3*e^2 + (72*(3*I*a*b^2 - b^3)*c^2*d*e + (216*a*b^2 + 72*I*b^3)*c^2*d)*f^2 - (72*b^3*c*d^2 - 72*(-3*I*a*b^2 +
b^3)*c*d^2*e^2 + (432*a*b^2 + 144*I*b^3)*c*d^2*e)*f)*(f*x + e) + (72*b^3*c*d^2*e + (216*a*b^2 - 72*I*b^3)*c*d
^2*e^2)*f)*cos(2*f*x + 2*e) - (108*a*b^2*d^3*e + 24*(3*a^2*b - b^3)*(f*x + e)^2*d^3 + 18*b^3*d^3 + 18*(3*a^2*b
- b^3)*d^3*e^2 + 18*(3*a^2*b - b^3)*c^2*d*f^2 - 36*(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) - 36*(3*a*b^2*c*d^2 + (3*a^2*b - b^3)*c*d^2*e)*f + 6*(18*a*b^2*d^3*e + 4*(3*a^2*b - b^3)*(f*x
+ e)^2*d^3 + 3*b^3*d^3 + 3*(3*a^2*b - b^3)*d^3*e^2 + 3*(3*a^2*b - b^3)*c^2*d*f^2 - 6*(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) - 6*(3*a*b^2*c*d^2 + (3*a^2*b - b^3)*c*d^2*e)*f)*cos(4*f*x +
4*e) + 12*(18*a*b^2*d^3*e + 4*(3*a^2*b - b^3)*(f*x + e)^2*d^3 + 3*b^3*d^3 + 3*(3*a^2*b - b^3)*d^3*e^2 + 3*(3*a
^2*b - b^3)*c^2*d*f^2 - 6*(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) - 6*(3*a*b
^2*c*d^2 + (3*a^2*b - b^3)*c*d^2*e)*f)*cos(2*f*x + 2*e) + (108*I*a*b^2*d^3*e + (72*I*a^2*b - 24*I*b^3)*(f*x +
e)^2*d^3 + 18*I*b^3*d^3 + (54*I*a^2*b - 18*I*b^3)*d^3*e^2 + (54*I*a^2*b - 18*I*b^3)*c^2*d*f^2 + (-108*I*a*b^2*
d^3 + (-108*I*a^2*b + 36*I*b^3)*d^3*e + (108*I*a^2*b - 36*I*b^3)*c*d^2*f)*(f*x + e) + (-108*I*a*b^2*c*d^2 + (-
108*I*a^2*b + 36*I*b^3)*c*d^2*e)*f)*sin(4*f*x + 4*e) + (216*I*a*b^2*d^3*e + (144*I*a^2*b - 48*I*b^3)*(f*x + e)
^2*d^3 + 36*I*b^3*d^3 + (108*I*a^2*b - 36*I*b^3)*d^3*e^2 + (108*I*a^2*b - 36*I*b^3)*c^2*d*f^2 + (-216*I*a*b^2*
d^3 + (-216*I*a^2*b + 72*I*b^3)*d^3*e + (216*I*a^2*b - 72*I*b^3)*c*d^2*f)*(f*x + e) + (-216*I*a*b^2*c*d^2 + (-
216*I*a^2*b + 72*I*b^3)*c*d^2*e)*f)*sin(2*f*x + 2*e))*dilog(-e^(2*I*f*x + 2*I*e)) - (6*I*b^3*d^3*e^3 - 6*I*b^3
*c^3*f^3 - 54*I*a*b^2*d^3*e^2 + (24*I*a^2*b - 8*I*b^3)*(f*x + e)^3*d^3 - 18*I*b^3*d^3*e + (-54*I*a*b^2*d^3 + (
-54*I*a^2*b + 18*I*b^3)*d^3*e + (54*I*a^2*b - 18*I*b^3)*c*d^2*f)*(f*x + e)^2 + (18*I*b^3*c^2*d*e - 54*I*a*b^2*
c^2*d)*f^2 + (108*I*a*b^2*d^3*e + 18*I*b^3*d^3 + (54*I*a^2*b - 18*I*b^3)*d^3*e^2 + (54*I*a^2*b - 18*I*b^3)*c^2
*d*f^2 + (-108*I*a*b^2*c*d^2 + (-108*I*a^2*b + 36*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-18*I*b^3*c*d^2*e^2 + 108*I*
a*b^2*c*d^2*e + 18*I*b^3*c*d^2)*f + (6*I*b^3*d^3*e^3 - 6*I*b^3*c^3*f^3 - 54*I*a*b^2*d^3*e^2 + (24*I*a^2*b - 8*
I*b^3)*(f*x + e)^3*d^3 - 18*I*b^3*d^3*e + (-54*I*a*b^2*d^3 + (-54*I*a^2*b + 18*I*b^3)*d^3*e + (54*I*a^2*b - 18
*I*b^3)*c*d^2*f)*(f*x + e)^2 + (18*I*b^3*c^2*d*e - 54*I*a*b^2*c^2*d)*f^2 + (108*I*a*b^2*d^3*e + 18*I*b^3*d^3 +
(54*I*a^2*b - 18*I*b^3)*d^3*e^2 + (54*I*a^2*b - 18*I*b^3)*c^2*d*f^2 + (-108*I*a*b^2*c*d^2 + (-108*I*a^2*b + 3
6*I*b^3)*c*d^2*e)*f)*(f*x + e) + (-18*I*b^3*c*d^2*e^2 + 108*I*a*b^2*c*d^2*e + 18*I*b^3*c*d^2)*f)*cos(4*f*x + 4
*e) + (12*I*b^3*d^3*e^3 - 12*I*b^3*c^3*f^3 - 108*I*a*b^2*d^3*e^2 + (48*I*a^2*b - 16*I*b^3)*(f*x + e)^3*d^3 - 3
6*I*b^3*d^3*e + (-108*I*a*b^2*d^3 + (-108*I*a^2*b + 36*I*b^3)*d^3*e + (108*I*a^2*b - 36*I*b^3)*c*d^2*f)*(f*x +
e)^2 + (36*I*b^3*c^2*d*e - 108*I*a*b^2*c^2*d)*f^2 + (216*I*a*b^2*d^3*e + 36*I*b^3*d^3 + (108*I*a^2*b - 36*I*b
^3)*d^3*e^2 + (108*I*a^2*b - 36*I*b^3)*c^2*d*f^2 + (-216*I*a*b^2*c*d^2 + (-216*I*a^2*b + 72*I*b^3)*c*d^2*e)*f)
*(f*x + e) + (-36*I*b^3*c*d^2*e^2 + 216*I*a*b^2*c*d^2*e + 36*I*b^3*c*d^2)*f)*cos(2*f*x + 2*e) - 2*(3*b^3*d^3*e
^3 - 3*b^3*c^3*f^3 - 27*a*b^2*d^3*e^2 + 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*b^3*d^3*e - 9*(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 + 9*(b^3*c^2*d*e - 3*a*b^2*c^2*d)*f^2 + 9*(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) - 9*(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*(3*b^3*d^3*e^3 - 3
*b^3*c^3*f^3 - 27*a*b^2*d^3*e^2 + 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*b^3*d^3*e - 9*(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 + 9*(b^3*c^2*d*e - 3*a*b^2*c^2*d)*f^2 + 9*(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) - 9*(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(2*f*x + 2*e)^2 + si
n(2*f*x + 2*e)^2 + 2*cos(2*f*x + 2*e) + 1) + (12*(3*a^2*b - b^3)*d^3*cos(4*f*x + 4*e) + 24*(3*a^2*b - b^3)*d^3
*cos(2*f*x + 2*e) - (-36*I*a^2*b + 12*I*b^3)*d^3*sin(4*f*x + 4*e) - (-72*I*a^2*b + 24*I*b^3)*d^3*sin(2*f*x + 2
*e) + 12*(3*a^2*b - b^3)*d^3)*polylog(4, -e^(2*I*f*x + 2*I*e)) - (-54*I*a*b^2*d^3 + (72*I*a^2*b - 24*I*b^3)*(f
*x + e)*d^3 + (-54*I*a^2*b + 18*I*b^3)*d^3*e + (54*I*a^2*b - 18*I*b^3)*c*d^2*f + (-54*I*a*b^2*d^3 + (72*I*a^2*
b - 24*I*b^3)*(f*x + e)*d^3 + (-54*I*a^2*b + 18*I*b^3)*d^3*e + (54*I*a^2*b - 18*I*b^3)*c*d^2*f)*cos(4*f*x + 4*
e) + (-108*I*a*b^2*d^3 + (144*I*a^2*b - 48*I*b^3)*(f*x + e)*d^3 + (-108*I*a^2*b + 36*I*b^3)*d^3*e + (108*I*a^2
*b - 36*I*b^3)*c*d^2*f)*cos(2*f*x + 2*e) + 6*(9*a*b^2*d^3 - 4*(3*a^2*b - b^3)*(f*x + e)*d^3 + 3*(3*a^2*b - b^3
)*d^3*e - 3*(3*a^2*b - b^3)*c*d^2*f)*sin(4*f*x + 4*e) + 12*(9*a*b^2*d^3 - 4*(3*a^2*b - b^3)*(f*x + e)*d^3 + 3*
(3*a^2*b - b^3)*d^3*e - 3*(3*a^2*b - b^3)*c*d^2*f)*sin(2*f*x + 2*e))*polylog(3, -e^(2*I*f*x + 2*I*e)) - ((9*I*
a^2*b - 9*a*b^2 - 3*I*b^3)*(f*x + e)^4*d^3 + (-72*I*a*b^2*d^3 + (-36*I*a^2*b + 36*a*b^2 + 12*I*b^3)*d^3*e + (3
6*I*a^2*b - 36*a*b^2 - 12*I*b^3)*c*d^2*f)*(f*x + e)^3 + (216*I*a*b^2*d^3*e + 36*I*b^3*d^3 + (54*I*a^2*b - 54*a
*b^2 - 18*I*b^3)*d^3*e^2 + (54*I*a^2*b - 54*a*b^2 - 18*I*b^3)*c^2*d*f^2 + (-216*I*a*b^2*c*d^2 + (-108*I*a^2*b
+ 108*a*b^2 + 36*I*b^3)*c*d^2*e)*f)*(f*x + e)^2 + (-216*I*a*b^2*d^3*e^2 - 72*I*b^3*d^3*e + (36*a*b^2 + 12*I*b^
3)*d^3*e^3 - (36*a*b^2 + 12*I*b^3)*c^3*f^3 + (-216*I*a*b^2*c^2*d + (108*a*b^2 + 36*I*b^3)*c^2*d*e)*f^2 + (432*
I*a*b^2*c*d^2*e + 72*I*b^3*c*d^2 - (108*a*b^2 + 36*I*b^3)*c*d^2*e^2)*f)*(f*x + e))*sin(4*f*x + 4*e) - ((18*I*a
^2*b - 18*a*b^2 - 6*I*b^3)*(f*x + e)^4*d^3 - 36*I*b^3*d^3*e^2 - 24*(3*I*a*b^2 + b^3)*d^3*e^3 - 24*(-3*I*a*b^2
- b^3)*c^3*f^3 + ((-72*I*a^2*b + 72*a*b^2 + 24*I*b^3)*d^3*e + (72*I*a^2*b - 72*a*b^2 - 24*I*b^3)*c*d^2*f - 24*
(3*I*a*b^2 - b^3)*d^3)*(f*x + e)^3 + (36*I*b^3*d^3 + (108*I*a^2*b - 108*a*b^2 - 36*I*b^3)*d^3*e^2 + (108*I*a^2
*b - 108*a*b^2 - 36*I*b^3)*c^2*d*f^2 - 72*(-3*I*a*b^2 + b^3)*d^3*e + ((-216*I*a^2*b + 216*a*b^2 + 72*I*b^3)*c*
d^2*e - 72*(3*I*a*b^2 - b^3)*c*d^2)*f)*(f*x + e)^2 + (-36*I*b^3*c^2*d - 72*(3*I*a*b^2 + b^3)*c^2*d*e)*f^2 + (-
72*I*b^3*d^3*e + (72*a*b^2 + 24*I*b^3)*d^3*e^3 - (72*a*b^2 + 24*I*b^3)*c^3*f^3 - 72*(3*I*a*b^2 - b^3)*d^3*e^2
+ ((216*a*b^2 + 72*I*b^3)*c^2*d*e - 72*(3*I*a*b^2 - b^3)*c^2*d)*f^2 + (72*I*b^3*c*d^2 - (216*a*b^2 + 72*I*b^3)
*c*d^2*e^2 - 144*(-3*I*a*b^2 + b^3)*c*d^2*e)*f)*(f*x + e) - 72*(-I*b^3*c*d^2*e + (-3*I*a*b^2 - b^3)*c*d^2*e^2)
*f)*sin(2*f*x + 2*e))/(-12*I*f^3*cos(4*f*x + 4*e) - 24*I*f^3*cos(2*f*x + 2*e) + 12*f^3*sin(4*f*x + 4*e) + 24*f
^3*sin(2*f*x + 2*e) - 12*I*f^3))/f

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Fricas [C]  time = 2.06165, size = 2549, normalized size = 4.17 \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*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/8*(2*(a^3 - 3*a*b^2)*d^3*f^4*x^4 + 3*I*(3*a^2*b - b^3)*d^3*polylog(4, (tan(f*x + e)^2 + 2*I*tan(f*x + e) - 1
)/(tan(f*x + e)^2 + 1)) - 3*I*(3*a^2*b - b^3)*d^3*polylog(4, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x
+ e)^2 + 1)) + 4*(b^3*d^3*f^3 + 2*(a^3 - 3*a*b^2)*c*d^2*f^4)*x^3 + 12*(b^3*c*d^2*f^3 + (a^3 - 3*a*b^2)*c^2*d*f
^4)*x^2 + 4*(b^3*d^3*f^3*x^3 + 3*b^3*c*d^2*f^3*x^2 + 3*b^3*c^2*d*f^3*x + b^3*c^3*f^3)*tan(f*x + e)^2 + 4*(3*b^
3*c^2*d*f^3 + 2*(a^3 - 3*a*b^2)*c^3*f^4)*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)*dilog(2*(I*tan(f*x +
e) - 1)/(tan(f*x + e)^2 + 1) + 1) + (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)*dilog(2*(-I*tan(f*x + e) - 1)
/(tan(f*x + e)^2 + 1) + 1) - 4*((3*a^2*b - b^3)*d^3*f^3*x^3 - 9*a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b - b
^3)*c^3*f^3 - 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*f^2 - b^3*d^3*f - (3*a^2*
b - b^3)*c^2*d*f^3)*x)*log(-2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 4*((3*a^2*b - b^3)*d^3*f^3*x^3 - 9*
a*b^2*c^2*d*f^2 + 3*b^3*c*d^2*f + (3*a^2*b - b^3)*c^3*f^3 - 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*f^2 - b^3*d^3*f - (3*a^2*b - b^3)*c^2*d*f^3)*x)*log(-2*(-I*tan(f*x + e) - 1)/(tan(f*x + e
)^2 + 1)) + 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)*polylog(3, (tan(f*x + e)^2 + 2
*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) + 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)*polylog(3, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) + 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)*t
an(f*x + e))/f^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\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*tan(f*x+e))**3,x)

[Out]

Integral((a + b*tan(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 \tan \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*tan(f*x+e))^3,x, algorithm="giac")

[Out]

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