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

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

[Out]

3/2*I*b^3*d*(d*x+c)^2/f^2-3/2*I*b^3*d*(d*x+c)^2*polylog(2,-exp(2*I*(f*x+e)))/f^2+1/2*b^3*(d*x+c)^3/f+1/4*a^3*(
d*x+c)^4/d-3*I*a*b^2*(d*x+c)^3/f-3/4*a*b^2*(d*x+c)^4/d+3/4*I*a^2*b*(d*x+c)^4/d-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+9/2*I*a^2*b*d*(d*x+c)^2*polylog(2,-exp(2*I*(f*x+e)))/f^2+3/2*I*b^3*d^3*polylo
g(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^4-9/4*I*a^2*b*d^3*polylog(4,-exp(2*I*(
f*x+e)))/f^4+9/2*a*b^2*d^3*polylog(3,-exp(2*I*(f*x+e)))/f^4-9/2*a^2*b*d^2*(d*x+c)*polylog(3,-exp(2*I*(f*x+e)))
/f^3+3/2*b^3*d^2*(d*x+c)*polylog(3,-exp(2*I*(f*x+e)))/f^3-9*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-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*t
an(f*x+e)^2/f

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 612, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {3803, 3800, 2221, 2611, 6744, 2320, 6724, 3801, 32, 2317, 2438} \begin {gather*} \frac {a^3 (c+d x)^4}{4 d}-\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 {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} \end {gather*}

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 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*} \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 ) \text {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 ) \text {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 ) \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 \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 ) \text {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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2607\) vs. \(2(612)=1224\).
time = 7.86, size = 2607, normalized size = 4.26 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*a*b^2*d^3*((2*I)*f^2*x^2*(2*E^((2*I)*e)*f*x + (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((2*I)*(e + f*x))]) + (6*I
)*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((2*I)*(e + f*x))] - 3*(1 + E^((2*I)*e))*PolyLog[3, -E^((2*I)*(e + f*x))
])*Sec[e])/(4*E^(I*e)*f^4) + (3*a^2*b*c*d^2*((2*I)*f^2*x^2*(2*E^((2*I)*e)*f*x + (3*I)*(1 + E^((2*I)*e))*Log[1
+ E^((2*I)*(e + f*x))]) + (6*I)*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((2*I)*(e + f*x))] - 3*(1 + E^((2*I)*e))*P
olyLog[3, -E^((2*I)*(e + f*x))])*Sec[e])/(4*E^(I*e)*f^3) - (b^3*c*d^2*((2*I)*f^2*x^2*(2*E^((2*I)*e)*f*x + (3*I
)*(1 + E^((2*I)*e))*Log[1 + E^((2*I)*(e + f*x))]) + (6*I)*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((2*I)*(e + f*x)
)] - 3*(1 + E^((2*I)*e))*PolyLog[3, -E^((2*I)*(e + f*x))])*Sec[e])/(4*E^(I*e)*f^3) - ((3*I)/4)*a^2*b*d^3*E^(I*
e)*(-x^4 + (1 + E^((-2*I)*e))*x^4 - ((1 + E^((2*I)*e))*(2*f^4*x^4 + (4*I)*f^3*x^3*Log[1 + E^((2*I)*(e + f*x))]
+ 6*f^2*x^2*PolyLog[2, -E^((2*I)*(e + f*x))] + (6*I)*f*x*PolyLog[3, -E^((2*I)*(e + f*x))] - 3*PolyLog[4, -E^(
(2*I)*(e + f*x))]))/(2*E^((2*I)*e)*f^4))*Sec[e] + (I/4)*b^3*d^3*E^(I*e)*(-x^4 + (1 + E^((-2*I)*e))*x^4 - ((1 +
E^((2*I)*e))*(2*f^4*x^4 + (4*I)*f^3*x^3*Log[1 + E^((2*I)*(e + f*x))] + 6*f^2*x^2*PolyLog[2, -E^((2*I)*(e + f*
x))] + (6*I)*f*x*PolyLog[3, -E^((2*I)*(e + f*x))] - 3*PolyLog[4, -E^((2*I)*(e + f*x))]))/(2*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*S
ec[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 + Sin[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*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*S
qrt[Csc[e]^2*(Cos[e]^2 + Sin[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 - ArcTa
n[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]]*Lo
g[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 - A
rcTan[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*S
in[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^...

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Maple [B] 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 = 0.44, size = 1930, normalized size = 3.15

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 6424 vs. \(2 (546) = 1092\).
time = 9.42, size = 6424, normalized size = 10.50 \begin {gather*} \text {Too large to display} \end {gather*}

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*c*d^2/f^2 + 6*(f*x + e)^2*a^3*c^2*d/f -
4*(f*x + e)^3*a^3*d^3*e/f^3 - 12*(f*x + e)^2*a^3*c*d^2*e/f^2 - 12*(f*x + e)*a^3*c^2*d*e/f + 12*a^2*b*c^3*log(
sec(f*x + e)) - 36*a^2*b*c^2*d*e*log(sec(f*x + e))/f + 6*(f*x + e)^2*a^3*d^3*e^2/f^3 + 12*(f*x + e)*a^3*c*d^2*
e^2/f^2 + 36*a^2*b*c*d^2*e^2*log(sec(f*x + e))/f^2 - 4*(f*x + e)*a^3*d^3*e^3/f^3 - 12*a^2*b*d^3*e^3*log(sec(f*
x + e))/f^3 + 4*(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*(6*a*b^2*e + b^3)*c^2*d
*f^2 + 72*(3*a*b^2*e^2 + b^3*e)*c*d^2*f + 12*((3*a^2*b + 3*I*a*b^2 - b^3)*c*d^2*f - (3*a^2*b*e + 3*I*a*b^2*e -
b^3*e)*d^3)*(f*x + e)^3 - 36*(2*a*b^2*e^3 + b^3*e^2)*d^3 + 18*((3*a^2*b + 3*I*a*b^2 - b^3)*c^2*d*f^2 - 2*(3*a
^2*b*e + 3*I*a*b^2*e - b^3*e)*c*d^2*f + (3*a^2*b*e^2 + 3*I*a*b^2*e^2 - b^3*e^2)*d^3)*(f*x + e)^2 - 12*((-3*I*a
*b^2 + b^3)*c^3*f^3 + 3*(3*I*a*b^2*e - b^3*e)*c^2*d*f^2 + 3*(-3*I*a*b^2*e^2 + b^3*e^2)*c*d^2*f + (3*I*a*b^2*e^
3 - b^3*e^3)*d^3)*(f*x + e) + 4*(3*b^3*c^3*f^3 - 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*(b^3*e - 3*a*b^2)*c^2*d
*f^2 + 9*(b^3*(e^2 - 1) - 6*a*b^2*e)*c*d^2*f - 3*(b^3*(e^3 - 3*e) - 9*a*b^2*e^2)*d^3 - 9*((3*a^2*b - b^3)*c*d^
2*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^3)*(f*x + e)^2 - 9*((3*a^2*b - b^3)*c^2*d*f^2 - 2*(3*a^2*b*e - b^3*e + 3
*a*b^2)*c*d^2*f - (b^3*(e^2 - 1) - 3*a^2*b*e^2 - 6*a*b^2*e)*d^3)*(f*x + e) + (3*b^3*c^3*f^3 - 4*(3*a^2*b - b^3
)*(f*x + e)^3*d^3 - 9*(b^3*e - 3*a*b^2)*c^2*d*f^2 + 9*(b^3*(e^2 - 1) - 6*a*b^2*e)*c*d^2*f - 3*(b^3*(e^3 - 3*e)
- 9*a*b^2*e^2)*d^3 - 9*((3*a^2*b - b^3)*c*d^2*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^3)*(f*x + e)^2 - 9*((3*a^2*
b - b^3)*c^2*d*f^2 - 2*(3*a^2*b*e - b^3*e + 3*a*b^2)*c*d^2*f - (b^3*(e^2 - 1) - 3*a^2*b*e^2 - 6*a*b^2*e)*d^3)*
(f*x + e))*cos(4*f*x + 4*e) + 2*(3*b^3*c^3*f^3 - 4*(3*a^2*b - b^3)*(f*x + e)^3*d^3 - 9*(b^3*e - 3*a*b^2)*c^2*d
*f^2 + 9*(b^3*(e^2 - 1) - 6*a*b^2*e)*c*d^2*f - 3*(b^3*(e^3 - 3*e) - 9*a*b^2*e^2)*d^3 - 9*((3*a^2*b - b^3)*c*d^
2*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^3)*(f*x + e)^2 - 9*((3*a^2*b - b^3)*c^2*d*f^2 - 2*(3*a^2*b*e - b^3*e + 3
*a*b^2)*c*d^2*f - (b^3*(e^2 - 1) - 3*a^2*b*e^2 - 6*a*b^2*e)*d^3)*(f*x + e))*cos(2*f*x + 2*e) - (-3*I*b^3*c^3*f
^3 + 4*(3*I*a^2*b - I*b^3)*(f*x + e)^3*d^3 + 9*(I*b^3*e - 3*I*a*b^2)*c^2*d*f^2 + 9*(b^3*(-I*e^2 + I) + 6*I*a*b
^2*e)*c*d^2*f + 3*(b^3*(I*e^3 - 3*I*e) - 9*I*a*b^2*e^2)*d^3 + 9*((3*I*a^2*b - I*b^3)*c*d^2*f + (-3*I*a^2*b*e +
I*b^3*e - 3*I*a*b^2)*d^3)*(f*x + e)^2 + 9*((3*I*a^2*b - I*b^3)*c^2*d*f^2 + 2*(-3*I*a^2*b*e + I*b^3*e - 3*I*a*
b^2)*c*d^2*f + (b^3*(-I*e^2 + I) + 3*I*a^2*b*e^2 + 6*I*a*b^2*e)*d^3)*(f*x + e))*sin(4*f*x + 4*e) - 2*(-3*I*b^3
*c^3*f^3 + 4*(3*I*a^2*b - I*b^3)*(f*x + e)^3*d^3 + 9*(I*b^3*e - 3*I*a*b^2)*c^2*d*f^2 + 9*(b^3*(-I*e^2 + I) + 6
*I*a*b^2*e)*c*d^2*f + 3*(b^3*(I*e^3 - 3*I*e) - 9*I*a*b^2*e^2)*d^3 + 9*((3*I*a^2*b - I*b^3)*c*d^2*f + (-3*I*a^2
*b*e + I*b^3*e - 3*I*a*b^2)*d^3)*(f*x + e)^2 + 9*((3*I*a^2*b - I*b^3)*c^2*d*f^2 + 2*(-3*I*a^2*b*e + I*b^3*e -
3*I*a*b^2)*c*d^2*f + (b^3*(-I*e^2 + I) + 3*I*a^2*b*e^2 + 6*I*a*b^2*e)*d^3)*(f*x + e))*sin(2*f*x + 2*e))*arctan
2(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*((3*a^2*b + 3*I
*a*b^2 - b^3)*c*d^2*f - (3*a*b^2*(I*e + 2) + 3*a^2*b*e - b^3*e)*d^3)*(f*x + e)^3 + 6*((3*a^2*b + 3*I*a*b^2 - b
^3)*c^2*d*f^2 - 2*(3*a*b^2*(I*e + 2) + 3*a^2*b*e - b^3*e)*c*d^2*f - (b^3*(e^2 - 2) + 3*a*b^2*(-I*e^2 - 4*e) -
3*a^2*b*e^2)*d^3)*(f*x + e)^2 - 4*((-3*I*a*b^2 + b^3)*c^3*f^3 + 3*(3*a*b^2*(I*e + 2) - b^3*e)*c^2*d*f^2 + 3*(b
^3*(e^2 - 2) + 3*a*b^2*(-I*e^2 - 4*e))*c*d^2*f - (b^3*(e^3 - 6*e) - 3*a*b^2*(I*e^3 + 6*e^2))*d^3)*(f*x + e))*c
os(4*f*x + 4*e) + 6*((3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^4*d^3 + 4*(3*a*b^2 - I*b^3)*c^3*f^3 - 6*(b^3*(-2*I*
e + 1) + 6*a*b^2*e)*c^2*d*f^2 - 12*(b^3*(I*e^2 - e) - 3*a*b^2*e^2)*c*d^2*f + 4*((3*a^2*b + 3*I*a*b^2 - b^3)*c*
d^2*f + (b^3*(e - I) - 3*a*b^2*(I*e + 1) - 3*a^2*b*e)*d^3)*(f*x + e)^3 - 2*(b^3*(-2*I*e^3 + 3*e^2) + 6*a*b^2*e
^3)*d^3 + 6*((3*a^2*b + 3*I*a*b^2 - b^3)*c^2*d*f^2 + 2*(b^3*(e - I) - 3*a*b^2*(I*e + 1) - 3*a^2*b*e)*c*d^2*f -
(b^3*(e^2 - 2*I*e - 1) + 3*a*b^2*(-I*e^2 - 2*e) - 3*a^2*b*e^2)*d^3)*(f*x + e)^2 - 4*((-3*I*a*b^2 + b^3)*c^3*f
^3 - 3*(b^3*(e - I) - 3*a*b^2*(I*e + 1))*c^2*d*f^2 + 3*(b^3*(e^2 - 2*I*e - 1) + 3*a*b^2*(-I*e^2 - 2*e))*c*d^2*
f - (b^3*(e^3 - 3*I*e^2 - 3*e) - 3*a*b^2*(I*e^3 + 3*e^2))*d^3)*(f*x + e))*cos(2*f*x + 2*e) + 6*(4*(3*a^2*b - b
^3)*(f*x + e)^2*d^3 + 3*(3*a^2*b - b^3)*c^2*d*f^2 - 6*(3*a^2*b*e - b^3*e + 3*a*b^2)*c*d^2*f - 3*(b^3*(e^2 - 1)
- 3*a^2*b*e^2 - 6*a*b^2*e)*d^3 + 6*((3*a^2*b - b^3)*c*d^2*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^3)*(f*x + e) +
(4*(3*a^2*b - b^3)*(f*x + e)^2*d^3 + 3*(3*a^2*b - b^3)*c^2*d*f^2 - 6*(3*a^2*b*e - b^3*e + 3*a*b^2)*c*d^2*f - 3
*(b^3*(e^2 - 1) - 3*a^2*b*e^2 - 6*a*b^2*e)*d^3 + 6*((3*a^2*b - b^3)*c*d^2*f - (3*a^2*b*e - b^3*e + 3*a*b^2)*d^
3)*(f*x + e))*cos(4*f*x + 4*e) + 2*(4*(3*a^2*b - b^3)*(f*x + e)^2*d^3 + 3*(3*a^2*b - b^3)*c^2*d*f^2 - 6*(3*a^2
*b*e - b^3*e + 3*a*b^2)*c*d^2*f - 3*(b^3*(e^2 -...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1199 vs. \(2 (546) = 1092\).
time = 0.40, size = 1199, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (a^{3} - 3 \, a b^{2}\right )} d^{3} f^{4} x^{4} + 3 i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 3 i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{3} {\rm polylog}\left (4, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 4 \, {\left (b^{3} d^{3} f^{3} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d^{2} f^{4}\right )} x^{3} + 12 \, {\left (b^{3} c d^{2} f^{3} + {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d f^{4}\right )} x^{2} + 4 \, {\left (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}\right )} \tan \left (f x + e\right )^{2} + 4 \, {\left (3 \, b^{3} c^{2} d f^{3} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c^{3} f^{4}\right )} x - 6 \, {\left (i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{3} f^{2} x^{2} - 6 i \, a b^{2} c d^{2} f + i \, b^{3} d^{3} + i \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d f^{2} - 2 i \, {\left (3 \, a b^{2} d^{3} f - {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} f^{2}\right )} x\right )} {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 6 \, {\left (-i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{3} f^{2} x^{2} + 6 i \, a b^{2} c d^{2} f - i \, b^{3} d^{3} - i \, {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d f^{2} + 2 i \, {\left (3 \, a b^{2} d^{3} f - {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} f^{2}\right )} x\right )} {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1} + 1\right ) - 4 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} d^{3} f^{3} x^{3} - 9 \, a b^{2} c^{2} d f^{2} + 3 \, b^{3} c d^{2} f + {\left (3 \, a^{2} b - b^{3}\right )} c^{3} f^{3} - 3 \, {\left (3 \, a b^{2} d^{3} f^{2} - {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} f^{3}\right )} x^{2} - 3 \, {\left (6 \, a b^{2} c d^{2} f^{2} - b^{3} d^{3} f - {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d f^{3}\right )} x\right )} \log \left (-\frac {2 \, {\left (i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 4 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} d^{3} f^{3} x^{3} - 9 \, a b^{2} c^{2} d f^{2} + 3 \, b^{3} c d^{2} f + {\left (3 \, a^{2} b - b^{3}\right )} c^{3} f^{3} - 3 \, {\left (3 \, a b^{2} d^{3} f^{2} - {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} f^{3}\right )} x^{2} - 3 \, {\left (6 \, a b^{2} c d^{2} f^{2} - b^{3} d^{3} f - {\left (3 \, a^{2} b - b^{3}\right )} c^{2} d f^{3}\right )} x\right )} \log \left (-\frac {2 \, {\left (-i \, \tan \left (f x + e\right ) - 1\right )}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (3 \, a b^{2} d^{3} - {\left (3 \, a^{2} b - b^{3}\right )} d^{3} f x - {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} f\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} + 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (3 \, a b^{2} d^{3} - {\left (3 \, a^{2} b - b^{3}\right )} d^{3} f x - {\left (3 \, a^{2} b - b^{3}\right )} c d^{2} f\right )} {\rm polylog}\left (3, \frac {\tan \left (f x + e\right )^{2} - 2 i \, \tan \left (f x + e\right ) - 1}{\tan \left (f x + e\right )^{2} + 1}\right ) + 12 \, {\left (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} + {\left (6 \, a b^{2} c d^{2} f^{3} - b^{3} d^{3} f^{2}\right )} x^{2} + 2 \, {\left (3 \, a b^{2} c^{2} d f^{3} - b^{3} c d^{2} f^{2}\right )} x\right )} \tan \left (f x + e\right )}{8 \, f^{4}} \end {gather*}

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 - 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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{3}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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