\(\int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx\) [49]

Optimal result

Integrand size = 20, antiderivative size = 612 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\frac {3 i b^3 d (c+d x)^2}{2 f^2}-\frac {3 i a b^2 (c+d x)^3}{f}+\frac {b^3 (c+d x)^3}{2 f}+\frac {a^3 (c+d x)^4}{4 d}+\frac {3 i a^2 b (c+d x)^4}{4 d}-\frac {3 a b^2 (c+d x)^4}{4 d}-\frac {i b^3 (c+d x)^4}{4 d}-\frac {3 b^3 d^2 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac {3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac {3 a b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3800, 2221, 2611, 6744, 2320, 6724, 3801, 32, 2317, 2438} \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\frac {a^3 (c+d x)^4}{4 d}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 a^2 b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i a^2 b (c+d x)^4}{4 d}-\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 a b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^3}{f}-\frac {3 a b^2 (c+d x)^4}{4 d}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d^2 (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 b^3 d (c+d x)^2 \tan (e+f x)}{2 f^2}+\frac {b^3 (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^3 \tan ^2(e+f x)}{2 f}+\frac {3 i b^3 d (c+d x)^2}{2 f^2}+\frac {b^3 (c+d x)^3}{2 f}-\frac {i b^3 (c+d x)^4}{4 d}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4} \]

[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 32

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

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2438

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]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3800

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 3801

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

Rule 3803

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Tan[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^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 \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {3 b^3 d (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 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right ) \, dx}{2 f^3}+\frac {\left (9 i a b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{4 f^4}+\frac {\left (9 a b^2 d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}-\frac {\left (3 i b^3 d^3\right ) \text {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 \operatorname {PolyLog}\left (3,-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 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\frac {3 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 ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(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 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 i a b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {9 i a^2 b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}-\frac {3 i b^3 d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{2 f^2}+\frac {9 a b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {9 a^2 b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {3 b^3 d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {9 i a^2 b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}+\frac {3 i b^3 d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{4 f^4}-\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] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2594\) vs. \(2(612)=1224\).

Time = 7.57 (sec) , antiderivative size = 2594, normalized size of antiderivative = 4.24 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\text {Result too large to show} \]

[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*f^4*x^4)/E^((2*I
)*e) - (4*I)*(1 + E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-2*I)*(e + f*x))] + 6*(1 + E^((-2*I)*e))*f^2*x^2*PolyLog[2
, -E^((-2*I)*(e + f*x))] - (6*I)*(1 + E^((-2*I)*e))*f*x*PolyLog[3, -E^((-2*I)*(e + f*x))] - 3*(1 + E^((-2*I)*e
))*PolyLog[4, -E^((-2*I)*(e + f*x))])*Sec[e])/f^4 + ((I/8)*b^3*d^3*E^(I*e)*((2*f^4*x^4)/E^((2*I)*e) - (4*I)*(1
+ E^((-2*I)*e))*f^3*x^3*Log[1 + E^((-2*I)*(e + f*x))] + 6*(1 + E^((-2*I)*e))*f^2*x^2*PolyLog[2, -E^((-2*I)*(e
+ f*x))] - (6*I)*(1 + E^((-2*I)*e))*f*x*PolyLog[3, -E^((-2*I)*(e + f*x))] - 3*(1 + E^((-2*I)*e))*PolyLog[4, -
E^((-2*I)*(e + f*x))])*Sec[e])/f^4 + ((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 + S
in[e]^2)) + (9*a*b^2*c^2*d*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[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 + S
in[e]^2)) - (3*b^3*d^3*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*L
og[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 + Sin[e]^2)]) + (9*a*b^2*c*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTa
n[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]])*L
og[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])/(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*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]]))]))/Sq
rt[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*A
rcTan[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*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*Sec[e]*Se
c[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] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1929 vs. \(2 (544 ) = 1088\).

Time = 1.21 (sec) , antiderivative size = 1930, normalized size of antiderivative = 3.15

method	result	size
risch	\(\text {Expression too large to display}\)	\(1930\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1177 vs. \(2 (530) = 1060\).

Time = 0.27 (sec) , antiderivative size = 1177, normalized size of antiderivative = 1.92 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\text {Too large to display} \]

[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 - 6*I*a*b^2*c*d^2*f + I*b^3*d^3
+ I*(3*a^2*b - b^3)*c^2*d*f^2 - 2*I*(3*a*b^2*d^3*f - (3*a^2*b - b^3)*c*d^2*f^2)*x)*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 + 6*I*a*b^2*c*d^2*f - I*b^3*d^3 - I*(3*a^2*b -
b^3)*c^2*d*f^2 + 2*I*(3*a*b^2*d^3*f - (3*a^2*b - b^3)*c*d^2*f^2)*x)*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)*tan(f*x + e))
/f^4

Sympy [F]

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

[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)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6861 vs. \(2 (530) = 1060\).

Time = 10.88 (sec) , antiderivative size = 6861, normalized size of antiderivative = 11.21 \[ \int (c+d x)^3 (a+b \tan (e+f x))^3 \, dx=\text {Too large to display} \]

[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 + 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 + (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) + 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)*cos(2*f*x + 2*e) + (3*I*b^3*d^3*e^3 -
3*I*b^3*c^3*f^3 - 27*I*a*b^2*d^3*e^2 + 4*(3*I*a^2*b - I*b^3)*(f*x + e)^3*d^3 - 9*I*b^3*d^3*e + 9*(-3*I*a*b^2*d
^3 + (-3*I*a^2*b + I*b^3)*d^3*e + (3*I*a^2*b - I*b^3)*c*d^2*f)*(f*x + e)^2 + 9*(I*b^3*c^2*d*e - 3*I*a*b^2*c^2*
d)*f^2 + 9*(6*I*a*b^2*d^3*e + I*b^3*d^3 + (3*I*a^2*b - I*b^3)*d^3*e^2 + (3*I*a^2*b - I*b^3)*c^2*d*f^2 + 2*(-3*
I*a*b^2*c*d^2 + (-3*I*a^2*b + I*b^3)*c*d^2*e)*f)*(f*x + e) + 9*(-I*b^3*c*d^2*e^2 + 6*I*a*b^2*c*d^2*e + I*b^3*c
*d^2)*f)*sin(4*f*x + 4*e) + 2*(3*I*b^3*d^3*e^3 - 3*I*b^3*c^3*f^3 - 27*I*a*b^2*d^3*e^2 + 4*(3*I*a^2*b - I*b^3)*
(f*x + e)^3*d^3 - 9*I*b^3*d^3*e + 9*(-3*I*a*b^2*d^3 + (-3*I*a^2*b + I*b^3)*d^3*e + (3*I*a^2*b - I*b^3)*c*d^2*f
)*(f*x + e)^2 + 9*(I*b^3*c^2*d*e - 3*I*a*b^2*c^2*d)*f^2 + 9*(6*I*a*b^2*d^3*e + I*b^3*d^3 + (3*I*a^2*b - I*b^3)
*d^3*e^2 + (3*I*a^2*b - I*b^3)*c^2*d*f^2 + 2*(-3*I*a*b^2*c*d^2 + (-3*I*a^2*b + I*b^3)*c*d^2*e)*f)*(f*x + e) +
9*(-I*b^3*c*d^2*e^2 + 6*I*a*b^2*c*d^2*e + 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 - 6*b^3*d^3*e^2 -
4*(3*a*b^2 - I*b^3)*d^3*e^3 + 4*(3*a*b^2 - I*b^3)*c^3*f^3 - 4*((3*a^2*b + 3*I*a*b^2 - b^3)*d^3*e - (3*a^2*b +
3*I*a*b^2 - b^3)*c*d^2*f + (3*a*b^2 + I*b^3)*d^3)*(f*x + e)^3 + 6*(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*(3*a*b^2 + I*b^3)*d^3*e - 2*((3*a^2*b + 3*I*a*b^2 - b^3)*c*d^2
*e + (3*a*b^2 + I*b^3)*c*d^2)*f)*(f*x + e)^2 - 6*(b^3*c^2*d + 2*(3*a*b^2 - I*b^3)*c^2*d*e)*f^2 - 4*(3*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*(3*a*b^2 + I*b^3)*d^3*e^2 + 3*((3*I*a*b^2 - b^3
)*c^2*d*e + (3*a*b^2 + I*b^3)*c^2*d)*f^2 - 3*(b^3*c*d^2 - (-3*I*a*b^2 + b^3)*c*d^2*e^2 + 2*(3*a*b^2 + I*b^3)*c
*d^2*e)*f)*(f*x + e) + 12*(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) - 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 + (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) + 2*(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)
- (-18*I*a*b^2*d^3*e + 4*(-3*I*a^2*b + I*b^3)*(f*x + e)^2*d^3 - 3*I*b^3*d^3 + 3*(-3*I*a^2*b + I*b^3)*d^3*e^2 +
3*(-3*I*a^2*b + I*b^3)*c^2*d*f^2 + 6*(3*I*a*b^2*d^3 + (3*I*a^2*b - I*b^3)*d^3*e + (-3*I*a^2*b + I*b^3)*c*d^2*
f)*(f*x + e) + 6*(3*I*a*b^2*c*d^2 + (3*I*a^2*b - I*b^3)*c*d^2*e)*f)*sin(4*f*x + 4*e) - 2*(-18*I*a*b^2*d^3*e +
4*(-3*I*a^2*b + I*b^3)*(f*x + e)^2*d^3 - 3*I*b^3*d^3 + 3*(-3*I*a^2*b + I*b^3)*d^3*e^2 + 3*(-3*I*a^2*b + I*b^3)
*c^2*d*f^2 + 6*(3*I*a*b^2*d^3 + (3*I*a^2*b - I*b^3)*d^3*e + (-3*I*a^2*b + I*b^3)*c*d^2*f)*(f*x + e) + 6*(3*I*a
*b^2*c*d^2 + (3*I*a^2*b - I*b^3)*c*d^2*e)*f)*sin(2*f*x + 2*e))*dilog(-e^(2*I*f*x + 2*I*e)) + 2*(-3*I*b^3*d^3*e
^3 + 3*I*b^3*c^3*f^3 + 27*I*a*b^2*d^3*e^2 + 4*(-3*I*a^2*b + I*b^3)*(f*x + e)^3*d^3 + 9*I*b^3*d^3*e + 9*(3*I*a*
b^2*d^3 + (3*I*a^2*b - I*b^3)*d^3*e + (-3*I*a^2*b + I*b^3)*c*d^2*f)*(f*x + e)^2 + 9*(-I*b^3*c^2*d*e + 3*I*a*b^
2*c^2*d)*f^2 + 9*(-6*I*a*b^2*d^3*e - I*b^3*d^3 + (-3*I*a^2*b + I*b^3)*d^3*e^2 + (-3*I*a^2*b + I*b^3)*c^2*d*f^2
+ 2*(3*I*a*b^2*c*d^2 + (3*I*a^2*b - I*b^3)*c*d^2*e)*f)*(f*x + e) + 9*(I*b^3*c*d^2*e^2 - 6*I*a*b^2*c*d^2*e - I
*b^3*c*d^2)*f + (-3*I*b^3*d^3*e^3 + 3*I*b^3*c^3*f^3 + 27*I*a*b^2*d^3*e^2 + 4*(-3*I*a^2*b + I*b^3)*(f*x + e)^3*
d^3 + 9*I*b^3*d^3*e + 9*(3*I*a*b^2*d^3 + (3*I*a^2*b - I*b^3)*d^3*e + (-3*I*a^2*b + I*b^3)*c*d^2*f)*(f*x + e)^2
+ 9*(-I*b^3*c^2*d*e + 3*I*a*b^2*c^2*d)*f^2 + 9*(-6*I*a*b^2*d^3*e - I*b^3*d^3 + (-3*I*a^2*b + I*b^3)*d^3*e^2 +
(-3*I*a^2*b + I*b^3)*c^2*d*f^2 + 2*(3*I*a*b^2*c*d^2 + (3*I*a^2*b - I*b^3)*c*d^2*e)*f)*(f*x + e) + 9*(I*b^3*c*
d^2*e^2 - 6*I*a*b^2*c*d^2*e - I*b^3*c*d^2)*f)*cos(4*f*x + 4*e) + 2*(-3*I*b^3*d^3*e^3 + 3*I*b^3*c^3*f^3 + 27*I*
a*b^2*d^3*e^2 + 4*(-3*I*a^2*b + I*b^3)*(f*x + e)^3*d^3 + 9*I*b^3*d^3*e + 9*(3*I*a*b^2*d^3 + (3*I*a^2*b - I*b^3
)*d^3*e + (-3*I*a^2*b + I*b^3)*c*d^2*f)*(f*x + e)^2 + 9*(-I*b^3*c^2*d*e + 3*I*a*b^2*c^2*d)*f^2 + 9*(-6*I*a*b^2
*d^3*e - I*b^3*d^3 + (-3*I*a^2*b + I*b^3)*d^3*e^2 + (-3*I*a^2*b + I*b^3)*c^2*d*f^2 + 2*(3*I*a*b^2*c*d^2 + (3*I
*a^2*b - I*b^3)*c*d^2*e)*f)*(f*x + e) + 9*(I*b^3*c*d^2*e^2 - 6*I*a*b^2*c*d^2*e - I*b^3*c*d^2)*f)*cos(2*f*x + 2
*e) + (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) + 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(2*f*x + 2*e))*log(cos(2
*f*x + 2*e)^2 + sin(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) + 2*(3
*a^2*b - b^3)*d^3*cos(2*f*x + 2*e) + (3*I*a^2*b - I*b^3)*d^3*sin(4*f*x + 4*e) + 2*(3*I*a^2*b - I*b^3)*d^3*sin(
2*f*x + 2*e) + (3*a^2*b - b^3)*d^3)*polylog(4, -e^(2*I*f*x + 2*I*e)) + 6*(9*I*a*b^2*d^3 + 4*(-3*I*a^2*b + I*b^
3)*(f*x + e)*d^3 + 3*(3*I*a^2*b - I*b^3)*d^3*e + 3*(-3*I*a^2*b + I*b^3)*c*d^2*f + (9*I*a*b^2*d^3 + 4*(-3*I*a^2
*b + I*b^3)*(f*x + e)*d^3 + 3*(3*I*a^2*b - I*b^3)*d^3*e + 3*(-3*I*a^2*b + I*b^3)*c*d^2*f)*cos(4*f*x + 4*e) + 2
*(9*I*a*b^2*d^3 + 4*(-3*I*a^2*b + I*b^3)*(f*x + e)*d^3 + 3*(3*I*a^2*b - I*b^3)*d^3*e + 3*(-3*I*a^2*b + I*b^3)*
c*d^2*f)*cos(2*f*x + 2*e) - (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) - 2*(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)) + 3*((-3*I*a^2*b + 3*a*b^2
+ I*b^3)*(f*x + e)^4*d^3 + 4*(6*I*a*b^2*d^3 + (3*I*a^2*b - 3*a*b^2 - I*b^3)*d^3*e + (-3*I*a^2*b + 3*a*b^2 + I
*b^3)*c*d^2*f)*(f*x + e)^3 + 6*(-12*I*a*b^2*d^3*e - 2*I*b^3*d^3 + (-3*I*a^2*b + 3*a*b^2 + I*b^3)*d^3*e^2 + (-3
*I*a^2*b + 3*a*b^2 + I*b^3)*c^2*d*f^2 + 2*(6*I*a*b^2*c*d^2 + (3*I*a^2*b - 3*a*b^2 - I*b^3)*c*d^2*e)*f)*(f*x +
e)^2 + 4*(18*I*a*b^2*d^3*e^2 + 6*I*b^3*d^3*e - (3*a*b^2 + I*b^3)*d^3*e^3 + (3*a*b^2 + I*b^3)*c^3*f^3 + 3*(6*I*
a*b^2*c^2*d - (3*a*b^2 + I*b^3)*c^2*d*e)*f^2 + 3*(-12*I*a*b^2*c*d^2*e - 2*I*b^3*c*d^2 + (3*a*b^2 + I*b^3)*c*d^
2*e^2)*f)*(f*x + e))*sin(4*f*x + 4*e) + 6*((-3*I*a^2*b + 3*a*b^2 + I*b^3)*(f*x + e)^4*d^3 + 6*I*b^3*d^3*e^2 +
4*(3*I*a*b^2 + b^3)*d^3*e^3 + 4*(-3*I*a*b^2 - b^3)*c^3*f^3 + 4*((3*I*a^2*b - 3*a*b^2 - I*b^3)*d^3*e + (-3*I*a^
2*b + 3*a*b^2 + I*b^3)*c*d^2*f + (3*I*a*b^2 - b^3)*d^3)*(f*x + e)^3 + 6*(-I*b^3*d^3 + (-3*I*a^2*b + 3*a*b^2 +
I*b^3)*d^3*e^2 + (-3*I*a^2*b + 3*a*b^2 + I*b^3)*c^2*d*f^2 + 2*(-3*I*a*b^2 + b^3)*d^3*e + 2*((3*I*a^2*b - 3*a*b
^2 - I*b^3)*c*d^2*e + (3*I*a*b^2 - b^3)*c*d^2)*f)*(f*x + e)^2 + 6*(I*b^3*c^2*d + 2*(3*I*a*b^2 + b^3)*c^2*d*e)*
f^2 + 4*(3*I*b^3*d^3*e - (3*a*b^2 + I*b^3)*d^3*e^3 + (3*a*b^2 + I*b^3)*c^3*f^3 + 3*(3*I*a*b^2 - b^3)*d^3*e^2 -
3*((3*a*b^2 + I*b^3)*c^2*d*e - (3*I*a*b^2 - b^3)*c^2*d)*f^2 + 3*(-I*b^3*c*d^2 + (3*a*b^2 + I*b^3)*c*d^2*e^2 +
2*(-3*I*a*b^2 + b^3)*c*d^2*e)*f)*(f*x + e) + 12*(-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

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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