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

Optimal. Leaf size=436 \[ \frac {a^3 (c+d x)^3}{3 d}+\frac {3 i a^2 b d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a^2 b (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a^2 b (c+d x)^3}{d}-\frac {3 a^2 b d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}+\frac {6 a b^2 d (c+d x) \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {3 a b^2 (c+d x)^2 \tan (e+f x)}{f}-\frac {3 i a b^2 (c+d x)^2}{f}-\frac {a b^2 (c+d x)^3}{d}-\frac {3 i a b^2 d^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {i b^3 d (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {b^3 d (c+d x) \tan (e+f x)}{f^2}+\frac {b^3 (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^3 (c+d x)^2 \tan ^2(e+f x)}{2 f}+\frac {b^3 c d x}{f}-\frac {i b^3 (c+d x)^3}{3 d}+\frac {b^3 d^2 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^3}-\frac {b^3 d^2 \log (\cos (e+f x))}{f^3}+\frac {b^3 d^2 x^2}{2 f} \]

[Out]

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

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Rubi [A]  time = 0.72, antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, 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 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 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 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 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 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 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 3475

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

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 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 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 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]

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*}

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Mathematica [B]  time = 7.85, 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]

((-1/4*I)*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*S
in[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*
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^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 -
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)]) + (b^3*c*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcT
an[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
*Cos[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
*Sin[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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fricas [C]  time = 0.54, size = 686, normalized size = 1.57 \[ \frac {4 \, {\left (a^{3} - 3 \, a b^{2}\right )} d^{2} f^{3} x^{3} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} {\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 ) - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} {\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 (b^{3} d^{2} f^{2} + 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d f^{3}\right )} x^{2} + 6 \, {\left (b^{3} d^{2} f^{2} x^{2} + 2 \, b^{3} c d f^{2} x + b^{3} c^{2} f^{2}\right )} \tan \left (f x + e\right )^{2} + 12 \, {\left (b^{3} c d f^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} c^{2} f^{3}\right )} x + {\left (18 i \, a b^{2} d^{2} - 6 i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} f x - 6 i \, {\left (3 \, a^{2} b - b^{3}\right )} c d f\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 ) + {\left (-18 i \, a b^{2} d^{2} + 6 i \, {\left (3 \, a^{2} b - b^{3}\right )} d^{2} f x + 6 i \, {\left (3 \, a^{2} b - b^{3}\right )} c d f\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 ({\left (3 \, a^{2} b - b^{3}\right )} d^{2} f^{2} x^{2} - 6 \, a b^{2} c d f + b^{3} d^{2} + {\left (3 \, a^{2} b - b^{3}\right )} c^{2} f^{2} - 2 \, {\left (3 \, a b^{2} d^{2} f - {\left (3 \, a^{2} b - b^{3}\right )} c d f^{2}\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 ({\left (3 \, a^{2} b - b^{3}\right )} d^{2} f^{2} x^{2} - 6 \, a b^{2} c d f + b^{3} d^{2} + {\left (3 \, a^{2} b - b^{3}\right )} c^{2} f^{2} - 2 \, {\left (3 \, a b^{2} d^{2} f - {\left (3 \, a^{2} b - b^{3}\right )} c d f^{2}\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 ) + 12 \, {\left (3 \, a b^{2} d^{2} f^{2} x^{2} + 3 \, a b^{2} c^{2} f^{2} - b^{3} c d f + {\left (6 \, a b^{2} c d f^{2} - b^{3} d^{2} f\right )} x\right )} \tan \left (f x + e\right )}{12 \, f^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{2} {\left (b \tan \left (f x + e\right ) + a\right )}^{3}\,{d x} \]

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)

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maple [B]  time = 0.84, size = 1090, normalized size = 2.50 \[ \text {result too large to display} \]

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]

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

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maxima [B]  time = 3.98, size = 3396, normalized size = 7.79 \[ \text {result too large to display} \]

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 + (6*a^2*b + 6*I*
a*b^2 - 2*b^3)*(f*x + e)^3*d^2 + 12*b^3*d^2*e - ((18*a^2*b + 18*I*a*b^2 - 6*b^3)*d^2*e - (18*a^2*b + 18*I*a*b^
2 - 6*b^3)*c*d*f)*(f*x + e)^2 + ((18*I*a*b^2 - 6*b^3)*d^2*e^2 + (-36*I*a*b^2 + 12*b^3)*c*d*e*f + (18*I*a*b^2 -
6*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) + ((6*a^2*b + 6*I*a*b^2 - 2*b^3)*(f*x
+ e)^3*d^2 - (36*a*b^2*d^2 + (18*a^2*b + 18*I*a*b^2 - 6*b^3)*d^2*e - (18*a^2*b + 18*I*a*b^2 - 6*b^3)*c*d*f)*(f
*x + e)^2 + (72*a*b^2*d^2*e + 12*b^3*d^2 + (18*I*a*b^2 - 6*b^3)*d^2*e^2 + (18*I*a*b^2 - 6*b^3)*c^2*f^2 - (72*a
*b^2*c*d - (-36*I*a*b^2 + 12*b^3)*c*d*e)*f)*(f*x + e))*cos(4*f*x + 4*e) + ((12*a^2*b + 12*I*a*b^2 - 4*b^3)*(f*
x + e)^3*d^2 + 12*b^3*d^2*e + 12*(3*a*b^2 - I*b^3)*d^2*e^2 + 12*(3*a*b^2 - I*b^3)*c^2*f^2 - ((36*a^2*b + 36*I*
a*b^2 - 12*b^3)*d^2*e - (36*a^2*b + 36*I*a*b^2 - 12*b^3)*c*d*f + 12*(3*a*b^2 + I*b^3)*d^2)*(f*x + e)^2 + (12*b
^3*d^2 + (36*I*a*b^2 - 12*b^3)*d^2*e^2 + (36*I*a*b^2 - 12*b^3)*c^2*f^2 + 24*(3*a*b^2 + I*b^3)*d^2*e + ((-72*I*
a*b^2 + 24*b^3)*c*d*e - 24*(3*a*b^2 + I*b^3)*c*d)*f)*(f*x + e) - 12*(b^3*c*d + 2*(3*a*b^2 - I*b^3)*c*d*e)*f)*c
os(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 + (-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)*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 - 6*(3*a*b^2 + I*b^3)*d^2*e^2 - 6*(3*a*b^2 + I*b^3)*c^
2*f^2 + (-72*I*a*b^2*c*d + 12*(3*a*b^2 + 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 + (36*I*a*b^2 + 12*b^3)*d^2*e^2 + (36*I*a*b^2 + 12*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 + (-36*I*a*b^2 + 12*b^
3)*d^2)*(f*x + e)^2 + (12*I*b^3*d^2 - 12*(3*a*b^2 + I*b^3)*d^2*e^2 - 12*(3*a*b^2 + I*b^3)*c^2*f^2 + (72*I*a*b^
2 - 24*b^3)*d^2*e + (24*(3*a*b^2 + I*b^3)*c*d*e + (-72*I*a*b^2 + 24*b^3)*c*d)*f)*(f*x + e) + (-12*I*b^3*c*d +
(-72*I*a*b^2 - 24*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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (e + f x \right )}\right )^{3} \left (c + d x\right )^{2}\, dx \]

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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