### 3.50 $$\int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx$$

Optimal. Leaf size=436 $\frac{3 i a^2 b d (c+d x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b d^2 \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 i a b^2 d^2 \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}-\frac{i b^3 d (c+d x) \text{PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac{b^3 d^2 \text{PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i a^2 b (c+d x)^3}{d}+\frac{a^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^2}{f}-\frac{a b^2 (c+d x)^3}{d}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac{b^3 c d x}{f}-\frac{i b^3 (c+d x)^3}{3 d}-\frac{b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac{b^3 d^2 x^2}{2 f}$

[Out]

(b^3*c*d*x)/f + (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) + (I*a^2*b*(c + d*
x)^3)/d - (a*b^2*(c + d*x)^3)/d - ((I/3)*b^3*(c + d*x)^3)/d + (6*a*b^2*d*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))
])/f^2 - (3*a^2*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])
/f - (b^3*d^2*Log[Cos[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a^2*b*d
*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (I*b^3*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 -
(3*a^2*b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) + (b^3*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) -
(b^3*d*(c + d*x)*Tan[e + f*x])/f^2 + (3*a*b^2*(c + d*x)^2*Tan[e + f*x])/f + (b^3*(c + d*x)^2*Tan[e + f*x]^2)/(
2*f)

________________________________________________________________________________________

Rubi [A]  time = 0.719831, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 22, number of rules used = 11, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.55, Rules used = {3722, 3719, 2190, 2531, 2282, 6589, 3720, 2279, 2391, 32, 3475} $\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{i a^2 b (c+d x)^3}{d}-\frac{3 a^2 b d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{a^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac{3 i a b^2 (c+d x)^2}{f}-\frac{a b^2 (c+d x)^3}{d}-\frac{3 i a b^2 d^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac{i b^3 d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac{b^3 c d x}{f}-\frac{i b^3 (c+d x)^3}{3 d}+\frac{b^3 d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac{b^3 d^2 x^2}{2 f}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(b^3*c*d*x)/f + (b^3*d^2*x^2)/(2*f) - ((3*I)*a*b^2*(c + d*x)^2)/f + (a^3*(c + d*x)^3)/(3*d) + (I*a^2*b*(c + d*
x)^3)/d - (a*b^2*(c + d*x)^3)/d - ((I/3)*b^3*(c + d*x)^3)/d + (6*a*b^2*d*(c + d*x)*Log[1 + E^((2*I)*(e + f*x))
])/f^2 - (3*a^2*b*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f + (b^3*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])
/f - (b^3*d^2*Log[Cos[e + f*x]])/f^3 - ((3*I)*a*b^2*d^2*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a^2*b*d
*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 - (I*b^3*d*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^2 -
(3*a^2*b*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) + (b^3*d^2*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^3) -
(b^3*d*(c + d*x)*Tan[e + f*x])/f^2 + (3*a*b^2*(c + d*x)^2*Tan[e + f*x])/f + (b^3*(c + d*x)^2*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 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 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]

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (c+d x)^2 (a+b \tan (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^2+3 a^2 b (c+d x)^2 \tan (e+f x)+3 a b^2 (c+d x)^2 \tan ^2(e+f x)+b^3 (c+d x)^2 \tan ^3(e+f x)\right ) \, dx\\ &=\frac{a^3 (c+d x)^3}{3 d}+\left (3 a^2 b\right ) \int (c+d x)^2 \tan (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^2 \tan ^2(e+f x) \, dx+b^3 \int (c+d x)^2 \tan ^3(e+f x) \, dx\\ &=\frac{a^3 (c+d x)^3}{3 d}+\frac{i a^2 b (c+d x)^3}{d}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^2 \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)^2}{1+e^{2 i (e+f x)}} \, dx-\left (3 a b^2\right ) \int (c+d x)^2 \, dx-b^3 \int (c+d x)^2 \tan (e+f x) \, dx-\frac{\left (6 a b^2 d\right ) \int (c+d x) \tan (e+f x) \, dx}{f}-\frac{\left (b^3 d\right ) \int (c+d x) \tan ^2(e+f x) \, dx}{f}\\ &=-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}+\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}-\frac{i b^3 (c+d x)^3}{3 d}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\left (2 i b^3\right ) \int \frac{e^{2 i (e+f x)} (c+d x)^2}{1+e^{2 i (e+f x)}} \, dx+\frac{\left (b^3 d^2\right ) \int \tan (e+f x) \, dx}{f^2}+\frac{\left (6 a^2 b d\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}+\frac{\left (12 i a b^2 d\right ) \int \frac{e^{2 i (e+f x)} (c+d x)}{1+e^{2 i (e+f x)}} \, dx}{f}+\frac{\left (b^3 d\right ) \int (c+d x) \, dx}{f}\\ &=\frac{b^3 c d x}{f}+\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}+\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}-\frac{i b^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}-\frac{\left (3 i a^2 b d^2\right ) \int \text{Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (6 a b^2 d^2\right ) \int \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac{\left (2 b^3 d\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}\\ &=\frac{b^3 c d x}{f}+\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}+\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}-\frac{i b^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{i b^3 d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}-\frac{\left (3 a^2 b d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}+\frac{\left (3 i a b^2 d^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{f^3}+\frac{\left (i b^3 d^2\right ) \int \text{Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=\frac{b^3 c d x}{f}+\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}+\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}-\frac{i b^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{b^3 d^2 \log (\cos (e+f x))}{f^3}-\frac{3 i a b^2 d^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{i b^3 d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac{\left (b^3 d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^3}\\ &=\frac{b^3 c d x}{f}+\frac{b^3 d^2 x^2}{2 f}-\frac{3 i a b^2 (c+d x)^2}{f}+\frac{a^3 (c+d x)^3}{3 d}+\frac{i a^2 b (c+d x)^3}{d}-\frac{a b^2 (c+d x)^3}{d}-\frac{i b^3 (c+d x)^3}{3 d}+\frac{6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac{b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac{b^3 d^2 \log (\cos (e+f x))}{f^3}-\frac{3 i a b^2 d^2 \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac{3 i a^2 b d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{i b^3 d (c+d x) \text{Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac{3 a^2 b d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac{b^3 d^2 \text{Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac{b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac{3 a b^2 (c+d x)^2 \tan (e+f x)}{f}+\frac{b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}\\ \end{align*}

Mathematica [B]  time = 7.62137, size = 1846, normalized size = 4.23 $\text{result too large to display}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-I/4)*a^2*b*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/12)*b^3*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) - (b^3*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)) + (6*a*b^2*c*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^2*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^2*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*a*b^2*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*Ar
cTan[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[C
ot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^3*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a^2*b*c*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 - A
rcTan[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 - A
rcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^2*Sqrt[Csc[e
]^2*(Cos[e]^2 + Sin[e]^2)]) + (b^3*c*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])/(f^2*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) + (Sec[e]*Sec[e + f*x]^2*(6*b^3
*c^2*f*Cos[e] + 12*b^3*c*d*f*x*Cos[e] + 6*a^3*c^2*f^2*x*Cos[e] - 18*a*b^2*c^2*f^2*x*Cos[e] + 6*b^3*d^2*f*x^2*C
os[e] + 6*a^3*c*d*f^2*x^2*Cos[e] - 18*a*b^2*c*d*f^2*x^2*Cos[e] + 2*a^3*d^2*f^2*x^3*Cos[e] - 6*a*b^2*d^2*f^2*x^
3*Cos[e] + 3*a^3*c^2*f^2*x*Cos[e + 2*f*x] - 9*a*b^2*c^2*f^2*x*Cos[e + 2*f*x] + 3*a^3*c*d*f^2*x^2*Cos[e + 2*f*x
] - 9*a*b^2*c*d*f^2*x^2*Cos[e + 2*f*x] + a^3*d^2*f^2*x^3*Cos[e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3*Cos[e + 2*f*x] +
3*a^3*c^2*f^2*x*Cos[3*e + 2*f*x] - 9*a*b^2*c^2*f^2*x*Cos[3*e + 2*f*x] + 3*a^3*c*d*f^2*x^2*Cos[3*e + 2*f*x] -
9*a*b^2*c*d*f^2*x^2*Cos[3*e + 2*f*x] + a^3*d^2*f^2*x^3*Cos[3*e + 2*f*x] - 3*a*b^2*d^2*f^2*x^3*Cos[3*e + 2*f*x]
+ 6*b^3*c*d*Sin[e] - 18*a*b^2*c^2*f*Sin[e] + 6*b^3*d^2*x*Sin[e] - 36*a*b^2*c*d*f*x*Sin[e] + 18*a^2*b*c^2*f^2*
x*Sin[e] - 6*b^3*c^2*f^2*x*Sin[e] - 18*a*b^2*d^2*f*x^2*Sin[e] + 18*a^2*b*c*d*f^2*x^2*Sin[e] - 6*b^3*c*d*f^2*x^
2*Sin[e] + 6*a^2*b*d^2*f^2*x^3*Sin[e] - 2*b^3*d^2*f^2*x^3*Sin[e] - 6*b^3*c*d*Sin[e + 2*f*x] + 18*a*b^2*c^2*f*S
in[e + 2*f*x] - 6*b^3*d^2*x*Sin[e + 2*f*x] + 36*a*b^2*c*d*f*x*Sin[e + 2*f*x] - 9*a^2*b*c^2*f^2*x*Sin[e + 2*f*x
] + 3*b^3*c^2*f^2*x*Sin[e + 2*f*x] + 18*a*b^2*d^2*f*x^2*Sin[e + 2*f*x] - 9*a^2*b*c*d*f^2*x^2*Sin[e + 2*f*x] +
3*b^3*c*d*f^2*x^2*Sin[e + 2*f*x] - 3*a^2*b*d^2*f^2*x^3*Sin[e + 2*f*x] + b^3*d^2*f^2*x^3*Sin[e + 2*f*x] + 9*a^2
*b*c^2*f^2*x*Sin[3*e + 2*f*x] - 3*b^3*c^2*f^2*x*Sin[3*e + 2*f*x] + 9*a^2*b*c*d*f^2*x^2*Sin[3*e + 2*f*x] - 3*b^
3*c*d*f^2*x^2*Sin[3*e + 2*f*x] + 3*a^2*b*d^2*f^2*x^3*Sin[3*e + 2*f*x] - b^3*d^2*f^2*x^3*Sin[3*e + 2*f*x]))/(12
*f^2)

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Maple [B]  time = 0.181, size = 1090, normalized size = 2.5 \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)^2*(a+b*tan(f*x+e))^3,x)

[Out]

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

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Maxima [B]  time = 10.1779, size = 4578, normalized size = 10.5 \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)^2*(a+b*tan(f*x+e))^3,x, algorithm="maxima")

[Out]

1/3*(3*(f*x + e)*a^3*c^2 + (f*x + e)^3*a^3*d^2/f^2 - 3*(f*x + e)^2*a^3*d^2*e/f^2 + 3*(f*x + e)*a^3*d^2*e^2/f^2
+ 3*(f*x + e)^2*a^3*c*d/f - 6*(f*x + e)*a^3*c*d*e/f + 9*a^2*b*c^2*log(sec(f*x + e)) + 9*a^2*b*d^2*e^2*log(sec
(f*x + e))/f^2 - 18*a^2*b*c*d*e*log(sec(f*x + e))/f + 3*(36*a*b^2*d^2*e^2 + 36*a*b^2*c^2*f^2 + 2*(3*a^2*b + 3*
I*a*b^2 - b^3)*(f*x + e)^3*d^2 + 12*b^3*d^2*e - 6*((3*a^2*b + 3*I*a*b^2 - b^3)*d^2*e - (3*a^2*b + 3*I*a*b^2 -
b^3)*c*d*f)*(f*x + e)^2 - 6*((-3*I*a*b^2 + b^3)*d^2*e^2 + 2*(3*I*a*b^2 - b^3)*c*d*e*f + (-3*I*a*b^2 + b^3)*c^2
*f^2)*(f*x + e) - 12*(6*a*b^2*c*d*e + b^3*c*d)*f + (6*b^3*d^2*e^2 + 6*b^3*c^2*f^2 - 36*a*b^2*d^2*e - 6*(3*a^2*
b - b^3)*(f*x + e)^2*d^2 - 6*b^3*d^2 + 12*(3*a*b^2*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b - b^3)*c*d*f)*(f*x +
e) - 12*(b^3*c*d*e - 3*a*b^2*c*d)*f + 6*(b^3*d^2*e^2 + b^3*c^2*f^2 - 6*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e
)^2*d^2 - b^3*d^2 + 2*(3*a*b^2*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e -
3*a*b^2*c*d)*f)*cos(4*f*x + 4*e) + 12*(b^3*d^2*e^2 + b^3*c^2*f^2 - 6*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)^
2*d^2 - b^3*d^2 + 2*(3*a*b^2*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b - b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e - 3
*a*b^2*c*d)*f)*cos(2*f*x + 2*e) + (6*I*b^3*d^2*e^2 + 6*I*b^3*c^2*f^2 - 36*I*a*b^2*d^2*e + (-18*I*a^2*b + 6*I*b
^3)*(f*x + e)^2*d^2 - 6*I*b^3*d^2 + (36*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*d^2*e + (-36*I*a^2*b + 12*I*b^3)
*c*d*f)*(f*x + e) + (-12*I*b^3*c*d*e + 36*I*a*b^2*c*d)*f)*sin(4*f*x + 4*e) + (12*I*b^3*d^2*e^2 + 12*I*b^3*c^2*
f^2 - 72*I*a*b^2*d^2*e + (-36*I*a^2*b + 12*I*b^3)*(f*x + e)^2*d^2 - 12*I*b^3*d^2 + (72*I*a*b^2*d^2 + (72*I*a^2
*b - 24*I*b^3)*d^2*e + (-72*I*a^2*b + 24*I*b^3)*c*d*f)*(f*x + e) + (-24*I*b^3*c*d*e + 72*I*a*b^2*c*d)*f)*sin(2
*f*x + 2*e))*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e) + 1) + 2*((3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^3*d^2
- 3*(6*a*b^2*d^2 + (3*a^2*b + 3*I*a*b^2 - b^3)*d^2*e - (3*a^2*b + 3*I*a*b^2 - b^3)*c*d*f)*(f*x + e)^2 + 3*(12*
a*b^2*d^2*e + 2*b^3*d^2 - (-3*I*a*b^2 + b^3)*d^2*e^2 - (-3*I*a*b^2 + b^3)*c^2*f^2 - 2*(6*a*b^2*c*d + (3*I*a*b^
2 - b^3)*c*d*e)*f)*(f*x + e))*cos(4*f*x + 4*e) + (4*(3*a^2*b + 3*I*a*b^2 - b^3)*(f*x + e)^3*d^2 + 12*b^3*d^2*e
+ (36*a*b^2 - 12*I*b^3)*d^2*e^2 + (36*a*b^2 - 12*I*b^3)*c^2*f^2 - (12*(3*a^2*b + 3*I*a*b^2 - b^3)*d^2*e - 12*
(3*a^2*b + 3*I*a*b^2 - b^3)*c*d*f + (36*a*b^2 + 12*I*b^3)*d^2)*(f*x + e)^2 + (12*b^3*d^2 - 12*(-3*I*a*b^2 + b^
3)*d^2*e^2 - 12*(-3*I*a*b^2 + b^3)*c^2*f^2 + (72*a*b^2 + 24*I*b^3)*d^2*e - (24*(3*I*a*b^2 - b^3)*c*d*e + (72*a
*b^2 + 24*I*b^3)*c*d)*f)*(f*x + e) - (12*b^3*c*d + (72*a*b^2 - 24*I*b^3)*c*d*e)*f)*cos(2*f*x + 2*e) - (18*a*b^
2*d^2 - 6*(3*a^2*b - b^3)*(f*x + e)*d^2 + 6*(3*a^2*b - b^3)*d^2*e - 6*(3*a^2*b - b^3)*c*d*f + 6*(3*a*b^2*d^2 -
(3*a^2*b - b^3)*(f*x + e)*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b - b^3)*c*d*f)*cos(4*f*x + 4*e) + 12*(3*a*b^2
*d^2 - (3*a^2*b - b^3)*(f*x + e)*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b - b^3)*c*d*f)*cos(2*f*x + 2*e) - (-18*
I*a*b^2*d^2 + (18*I*a^2*b - 6*I*b^3)*(f*x + e)*d^2 + (-18*I*a^2*b + 6*I*b^3)*d^2*e + (18*I*a^2*b - 6*I*b^3)*c*
d*f)*sin(4*f*x + 4*e) - (-36*I*a*b^2*d^2 + (36*I*a^2*b - 12*I*b^3)*(f*x + e)*d^2 + (-36*I*a^2*b + 12*I*b^3)*d^
2*e + (36*I*a^2*b - 12*I*b^3)*c*d*f)*sin(2*f*x + 2*e))*dilog(-e^(2*I*f*x + 2*I*e)) + (-3*I*b^3*d^2*e^2 - 3*I*b
^3*c^2*f^2 + 18*I*a*b^2*d^2*e + (9*I*a^2*b - 3*I*b^3)*(f*x + e)^2*d^2 + 3*I*b^3*d^2 + (-18*I*a*b^2*d^2 + (-18*
I*a^2*b + 6*I*b^3)*d^2*e + (18*I*a^2*b - 6*I*b^3)*c*d*f)*(f*x + e) + (6*I*b^3*c*d*e - 18*I*a*b^2*c*d)*f + (-3*
I*b^3*d^2*e^2 - 3*I*b^3*c^2*f^2 + 18*I*a*b^2*d^2*e + (9*I*a^2*b - 3*I*b^3)*(f*x + e)^2*d^2 + 3*I*b^3*d^2 + (-1
8*I*a*b^2*d^2 + (-18*I*a^2*b + 6*I*b^3)*d^2*e + (18*I*a^2*b - 6*I*b^3)*c*d*f)*(f*x + e) + (6*I*b^3*c*d*e - 18*
I*a*b^2*c*d)*f)*cos(4*f*x + 4*e) + (-6*I*b^3*d^2*e^2 - 6*I*b^3*c^2*f^2 + 36*I*a*b^2*d^2*e + (18*I*a^2*b - 6*I*
b^3)*(f*x + e)^2*d^2 + 6*I*b^3*d^2 + (-36*I*a*b^2*d^2 + (-36*I*a^2*b + 12*I*b^3)*d^2*e + (36*I*a^2*b - 12*I*b^
3)*c*d*f)*(f*x + e) + (12*I*b^3*c*d*e - 36*I*a*b^2*c*d)*f)*cos(2*f*x + 2*e) + 3*(b^3*d^2*e^2 + b^3*c^2*f^2 - 6
*a*b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - b^3*d^2 + 2*(3*a*b^2*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b -
b^3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e - 3*a*b^2*c*d)*f)*sin(4*f*x + 4*e) + 6*(b^3*d^2*e^2 + b^3*c^2*f^2 - 6*a*
b^2*d^2*e - (3*a^2*b - b^3)*(f*x + e)^2*d^2 - b^3*d^2 + 2*(3*a*b^2*d^2 + (3*a^2*b - b^3)*d^2*e - (3*a^2*b - b^
3)*c*d*f)*(f*x + e) - 2*(b^3*c*d*e - 3*a*b^2*c*d)*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) + ((9*I*a^2*b - 3*I*b^3)*d^2*cos(4*f*x + 4*e) + (18*I*a^2*b - 6*I*b^3)*d^2*cos(
2*f*x + 2*e) - 3*(3*a^2*b - b^3)*d^2*sin(4*f*x + 4*e) - 6*(3*a^2*b - b^3)*d^2*sin(2*f*x + 2*e) + (9*I*a^2*b -
3*I*b^3)*d^2)*polylog(3, -e^(2*I*f*x + 2*I*e)) + ((6*I*a^2*b - 6*a*b^2 - 2*I*b^3)*(f*x + e)^3*d^2 + (-36*I*a*b
^2*d^2 + (-18*I*a^2*b + 18*a*b^2 + 6*I*b^3)*d^2*e + (18*I*a^2*b - 18*a*b^2 - 6*I*b^3)*c*d*f)*(f*x + e)^2 + (72
*I*a*b^2*d^2*e + 12*I*b^3*d^2 - (18*a*b^2 + 6*I*b^3)*d^2*e^2 - (18*a*b^2 + 6*I*b^3)*c^2*f^2 + (-72*I*a*b^2*c*d
+ (36*a*b^2 + 12*I*b^3)*c*d*e)*f)*(f*x + e))*sin(4*f*x + 4*e) + ((12*I*a^2*b - 12*a*b^2 - 4*I*b^3)*(f*x + e)^
3*d^2 + 12*I*b^3*d^2*e - 12*(-3*I*a*b^2 - b^3)*d^2*e^2 - 12*(-3*I*a*b^2 - b^3)*c^2*f^2 + ((-36*I*a^2*b + 36*a*
b^2 + 12*I*b^3)*d^2*e + (36*I*a^2*b - 36*a*b^2 - 12*I*b^3)*c*d*f - 12*(3*I*a*b^2 - b^3)*d^2)*(f*x + e)^2 + (12
*I*b^3*d^2 - (36*a*b^2 + 12*I*b^3)*d^2*e^2 - (36*a*b^2 + 12*I*b^3)*c^2*f^2 - 24*(-3*I*a*b^2 + b^3)*d^2*e + ((7
2*a*b^2 + 24*I*b^3)*c*d*e - 24*(3*I*a*b^2 - b^3)*c*d)*f)*(f*x + e) + (-12*I*b^3*c*d - 24*(3*I*a*b^2 + b^3)*c*d
*e)*f)*sin(2*f*x + 2*e))/(-6*I*f^2*cos(4*f*x + 4*e) - 12*I*f^2*cos(2*f*x + 2*e) + 6*f^2*sin(4*f*x + 4*e) + 12*
f^2*sin(2*f*x + 2*e) - 6*I*f^2))/f

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Fricas [C]  time = 1.82858, size = 1540, normalized size = 3.53 \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)^2*(a+b*tan(f*x+e))^3,x, algorithm="fricas")

[Out]

1/12*(4*(a^3 - 3*a*b^2)*d^2*f^3*x^3 - 3*(3*a^2*b - b^3)*d^2*polylog(3, (tan(f*x + e)^2 + 2*I*tan(f*x + e) - 1)
/(tan(f*x + e)^2 + 1)) - 3*(3*a^2*b - b^3)*d^2*polylog(3, (tan(f*x + e)^2 - 2*I*tan(f*x + e) - 1)/(tan(f*x + e
)^2 + 1)) + 6*(b^3*d^2*f^2 + 2*(a^3 - 3*a*b^2)*c*d*f^3)*x^2 + 6*(b^3*d^2*f^2*x^2 + 2*b^3*c*d*f^2*x + b^3*c^2*f
^2)*tan(f*x + e)^2 + 12*(b^3*c*d*f^2 + (a^3 - 3*a*b^2)*c^2*f^3)*x + (18*I*a*b^2*d^2 - 6*I*(3*a^2*b - b^3)*d^2*
f*x - 6*I*(3*a^2*b - b^3)*c*d*f)*dilog(2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1) + 1) + (-18*I*a*b^2*d^2 + 6
*I*(3*a^2*b - b^3)*d^2*f*x + 6*I*(3*a^2*b - b^3)*c*d*f)*dilog(2*(-I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1) + 1
) - 6*((3*a^2*b - b^3)*d^2*f^2*x^2 - 6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b - b^3)*c^2*f^2 - 2*(3*a*b^2*d^2*f - (3
*a^2*b - b^3)*c*d*f^2)*x)*log(-2*(I*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) - 6*((3*a^2*b - b^3)*d^2*f^2*x^2 -
6*a*b^2*c*d*f + b^3*d^2 + (3*a^2*b - b^3)*c^2*f^2 - 2*(3*a*b^2*d^2*f - (3*a^2*b - b^3)*c*d*f^2)*x)*log(-2*(-I
*tan(f*x + e) - 1)/(tan(f*x + e)^2 + 1)) + 12*(3*a*b^2*d^2*f^2*x^2 + 3*a*b^2*c^2*f^2 - b^3*c*d*f + (6*a*b^2*c*
d*f^2 - b^3*d^2*f)*x)*tan(f*x + e))/f^3

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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 )^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*tan(e + f*x))**3*(c + d*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{2}{\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)^2*(a+b*tan(f*x+e))^3,x, algorithm="giac")

[Out]

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