3.1.44 \(\int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx\) [44]

Optimal. Leaf size=300 \[ -\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d^3 \text {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f} \]

[Out]

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

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Rubi [A]
time = 0.37, antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3803, 3800, 2221, 2611, 6744, 2320, 6724, 3801, 32} \begin {gather*} \frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a b (c+d x)^4}{2 d}-\frac {3 i a b d^3 \text {Li}_4\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*b^2*(c + d*x)^3)/f + (a^2*(c + d*x)^4)/(4*d) + ((I/2)*a*b*(c + d*x)^4)/d - (b^2*(c + d*x)^4)/(4*d) + (3*
b^2*d*(c + d*x)^2*Log[1 + E^((2*I)*(e + f*x))])/f^2 - (2*a*b*(c + d*x)^3*Log[1 + E^((2*I)*(e + f*x))])/f - ((3
*I)*b^2*d^2*(c + d*x)*PolyLog[2, -E^((2*I)*(e + f*x))])/f^3 + ((3*I)*a*b*d*(c + d*x)^2*PolyLog[2, -E^((2*I)*(e
+ f*x))])/f^2 + (3*b^2*d^3*PolyLog[3, -E^((2*I)*(e + f*x))])/(2*f^4) - (3*a*b*d^2*(c + d*x)*PolyLog[3, -E^((2
*I)*(e + f*x))])/f^3 - (((3*I)/2)*a*b*d^3*PolyLog[4, -E^((2*I)*(e + f*x))])/f^4 + (b^2*(c + d*x)^3*Tan[e + f*x
])/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 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 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))^2 \, dx &=\int \left (a^2 (c+d x)^3+2 a b (c+d x)^3 \tan (e+f x)+b^2 (c+d x)^3 \tan ^2(e+f x)\right ) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+(2 a b) \int (c+d x)^3 \tan (e+f x) \, dx+b^2 \int (c+d x)^3 \tan ^2(e+f x) \, dx\\ &=\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-(4 i a b) \int \frac {e^{2 i (e+f x)} (c+d x)^3}{1+e^{2 i (e+f x)}} \, dx-b^2 \int (c+d x)^3 \, dx-\frac {\left (3 b^2 d\right ) \int (c+d x)^2 \tan (e+f x) \, dx}{f}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {(6 a b d) \int (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f}+\frac {\left (6 i 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 {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (6 i a b d^2\right ) \int (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+e^{2 i (e+f x)}\right ) \, dx}{f^2}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}+\frac {\left (3 a b d^3\right ) \int \text {Li}_3\left (-e^{2 i (e+f x)}\right ) \, dx}{f^3}+\frac {\left (3 i b^2 d^3\right ) \int \text {Li}_2\left (-e^{2 i (e+f x)}\right ) \, dx}{f^3}\\ &=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {\left (3 i a b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 f^4}+\frac {\left (3 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 {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \text {Li}_2\left (-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \text {Li}_3\left (-e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d^3 \text {Li}_4\left (-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{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. \(1347\) vs. \(2(300)=600\).
time = 6.97, size = 1347, normalized size = 4.49 \begin {gather*} -\frac {b^2 d^3 e^{-i e} \left (2 i f^2 x^2 \left (2 e^{2 i e} f x+3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{2 i (e+f x)}\right )\right )+6 i \left (1+e^{2 i e}\right ) f x \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )-3 \left (1+e^{2 i e}\right ) \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )\right ) \sec (e)}{4 f^4}+\frac {a b c d^2 e^{-i e} \left (2 i f^2 x^2 \left (2 e^{2 i e} f x+3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{2 i (e+f x)}\right )\right )+6 i \left (1+e^{2 i e}\right ) f x \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )-3 \left (1+e^{2 i e}\right ) \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )\right ) \sec (e)}{2 f^3}-\frac {1}{2} i a b d^3 e^{i e} \left (-x^4+\left (1+e^{-2 i e}\right ) x^4-\frac {e^{-2 i e} \left (1+e^{2 i e}\right ) \left (2 f^4 x^4+4 i f^3 x^3 \log \left (1+e^{2 i (e+f x)}\right )+6 f^2 x^2 \text {PolyLog}\left (2,-e^{2 i (e+f x)}\right )+6 i f x \text {PolyLog}\left (3,-e^{2 i (e+f x)}\right )-3 \text {PolyLog}\left (4,-e^{2 i (e+f x)}\right )\right )}{2 f^4}\right ) \sec (e)+\frac {3 b^2 c^2 d \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f^2 \left (\cos ^2(e)+\sin ^2(e)\right )}-\frac {2 a b c^3 \sec (e) (\cos (e) \log (\cos (e) \cos (f x)-\sin (e) \sin (f x))+f x \sin (e))}{f \left (\cos ^2(e)+\sin ^2(e)\right )}+\frac {3 b^2 c d^2 \csc (e) \left (e^{-i \text {ArcTan}(\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \text {ArcTan}(\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\text {ArcTan}(\cot (e))) \log \left (1-e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )+\pi \log (\cos (f x))-2 \text {ArcTan}(\cot (e)) \log (\sin (f x-\text {ArcTan}(\cot (e))))+i \text {PolyLog}\left (2,e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^3 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}-\frac {3 a b c^2 d \csc (e) \left (e^{-i \text {ArcTan}(\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \text {ArcTan}(\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\text {ArcTan}(\cot (e))) \log \left (1-e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )+\pi \log (\cos (f x))-2 \text {ArcTan}(\cot (e)) \log (\sin (f x-\text {ArcTan}(\cot (e))))+i \text {PolyLog}\left (2,e^{2 i (f x-\text {ArcTan}(\cot (e)))}\right )\right )}{\sqrt {1+\cot ^2(e)}}\right ) \sec (e)}{f^2 \sqrt {\csc ^2(e) \left (\cos ^2(e)+\sin ^2(e)\right )}}+\frac {\sec (e) \sec (e+f x) \left (4 a^2 c^3 f x \cos (f x)-4 b^2 c^3 f x \cos (f x)+6 a^2 c^2 d f x^2 \cos (f x)-6 b^2 c^2 d f x^2 \cos (f x)+4 a^2 c d^2 f x^3 \cos (f x)-4 b^2 c d^2 f x^3 \cos (f x)+a^2 d^3 f x^4 \cos (f x)-b^2 d^3 f x^4 \cos (f x)+4 a^2 c^3 f x \cos (2 e+f x)-4 b^2 c^3 f x \cos (2 e+f x)+6 a^2 c^2 d f x^2 \cos (2 e+f x)-6 b^2 c^2 d f x^2 \cos (2 e+f x)+4 a^2 c d^2 f x^3 \cos (2 e+f x)-4 b^2 c d^2 f x^3 \cos (2 e+f x)+a^2 d^3 f x^4 \cos (2 e+f x)-b^2 d^3 f x^4 \cos (2 e+f x)+8 b^2 c^3 \sin (f x)+24 b^2 c^2 d x \sin (f x)-8 a b c^3 f x \sin (f x)+24 b^2 c d^2 x^2 \sin (f x)-12 a b c^2 d f x^2 \sin (f x)+8 b^2 d^3 x^3 \sin (f x)-8 a b c d^2 f x^3 \sin (f x)-2 a b d^3 f x^4 \sin (f x)+8 a b c^3 f x \sin (2 e+f x)+12 a b c^2 d f x^2 \sin (2 e+f x)+8 a b c d^2 f x^3 \sin (2 e+f x)+2 a b d^3 f x^4 \sin (2 e+f x)\right )}{8 f} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/4*(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])/(E^(I*e)*f^4) + (a*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))*PolyLog
[3, -E^((2*I)*(e + f*x))])*Sec[e])/(2*E^(I*e)*f^3) - (I/2)*a*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] + (3*b^2*c^2*d*Sec[e]*(Cos[e]*Log[Cos[e]*Cos[f*x] - Sin[e]*Sin[f*x]] + f*x*Sin[e]))/(f^2*(Cos[e]^2 +
Sin[e]^2)) - (2*a*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 + S
in[e]^2)) + (3*b^2*c*d^2*Csc[e]*((f^2*x^2)/E^(I*ArcTan[Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi
*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x
]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1
+ Cot[e]^2])*Sec[e])/(f^3*Sqrt[Csc[e]^2*(Cos[e]^2 + Sin[e]^2)]) - (3*a*b*c^2*d*Csc[e]*((f^2*x^2)/E^(I*ArcTan[
Cot[e]]) - (Cot[e]*(I*f*x*(-Pi - 2*ArcTan[Cot[e]]) - Pi*Log[1 + E^((-2*I)*f*x)] - 2*(f*x - ArcTan[Cot[e]])*Log
[1 - E^((2*I)*(f*x - ArcTan[Cot[e]]))] + Pi*Log[Cos[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] +
I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/Sqrt[1 + Cot[e]^2])*Sec[e])/(f^2*Sqrt[Csc[e]^2*(Cos[e]^2 + Si
n[e]^2)]) + (Sec[e]*Sec[e + f*x]*(4*a^2*c^3*f*x*Cos[f*x] - 4*b^2*c^3*f*x*Cos[f*x] + 6*a^2*c^2*d*f*x^2*Cos[f*x]
- 6*b^2*c^2*d*f*x^2*Cos[f*x] + 4*a^2*c*d^2*f*x^3*Cos[f*x] - 4*b^2*c*d^2*f*x^3*Cos[f*x] + a^2*d^3*f*x^4*Cos[f*
x] - b^2*d^3*f*x^4*Cos[f*x] + 4*a^2*c^3*f*x*Cos[2*e + f*x] - 4*b^2*c^3*f*x*Cos[2*e + f*x] + 6*a^2*c^2*d*f*x^2*
Cos[2*e + f*x] - 6*b^2*c^2*d*f*x^2*Cos[2*e + f*x] + 4*a^2*c*d^2*f*x^3*Cos[2*e + f*x] - 4*b^2*c*d^2*f*x^3*Cos[2
*e + f*x] + a^2*d^3*f*x^4*Cos[2*e + f*x] - b^2*d^3*f*x^4*Cos[2*e + f*x] + 8*b^2*c^3*Sin[f*x] + 24*b^2*c^2*d*x*
Sin[f*x] - 8*a*b*c^3*f*x*Sin[f*x] + 24*b^2*c*d^2*x^2*Sin[f*x] - 12*a*b*c^2*d*f*x^2*Sin[f*x] + 8*b^2*d^3*x^3*Si
n[f*x] - 8*a*b*c*d^2*f*x^3*Sin[f*x] - 2*a*b*d^3*f*x^4*Sin[f*x] + 8*a*b*c^3*f*x*Sin[2*e + f*x] + 12*a*b*c^2*d*f
*x^2*Sin[2*e + f*x] + 8*a*b*c*d^2*f*x^3*Sin[2*e + f*x] + 2*a*b*d^3*f*x^4*Sin[2*e + f*x]))/(8*f)

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (271 ) = 542\).
time = 0.37, size = 952, normalized size = 3.17

method	result	size
risch	\(\frac {12 b a c \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {6 b a c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{2}}{f}-\frac {6 b \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) a \,c^{2} d x}{f}+\frac {d^{3} a^{2} x^{4}}{4}+\frac {a^{2} c^{4}}{4 d}-\frac {d^{3} b^{2} x^{4}}{4}-b^{2} c^{3} x -\frac {b^{2} c^{4}}{4 d}-\frac {12 i b^{2} c \,d^{2} e x}{f^{2}}-\frac {8 i b a c \,d^{2} e^{3}}{f^{3}}+\frac {6 i b a \,c^{2} d \,e^{2}}{f^{2}}+\frac {4 i b a \,d^{3} e^{3} x}{f^{3}}+\frac {3 i b a \,c^{2} d \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{2}}+\frac {3 i b a \,d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x^{2}}{f^{2}}-\frac {12 b a \,c^{2} d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-d^{2} b^{2} c \,x^{3}-\frac {3 d \,b^{2} c^{2} x^{2}}{2}+\frac {2 i b^{2} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )}{f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}+\frac {i d^{3} a b \,x^{4}}{2}-2 i a b \,c^{3} x -\frac {i a b \,c^{4}}{2 d}+\frac {3 b^{2} d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{2}}{f^{2}}-\frac {2 b a \,c^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f}+\frac {4 b a \,c^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}+\frac {3 b^{2} c^{2} d \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{f^{2}}-\frac {6 b^{2} c^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {6 b^{2} d^{3} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{4}}-\frac {2 i b^{2} d^{3} x^{3}}{f}+d^{2} a^{2} c \,x^{3}+\frac {3 d \,a^{2} c^{2} x^{2}}{2}+a^{2} c^{3} x +\frac {3 b^{2} d^{3} \polylog \left (3, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{4}}+\frac {4 i b^{2} d^{3} e^{3}}{f^{4}}-\frac {2 b a \,d^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x^{3}}{f}+2 i d^{2} a b c \,x^{3}+3 i d a b \,c^{2} x^{2}-\frac {3 b a \,d^{3} \polylog \left (3, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{3}}-\frac {3 b a c \,d^{2} \polylog \left (3, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}-\frac {3 i a b \,d^{3} \polylog \left (4, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{2 f^{4}}-\frac {3 i b^{2} d^{3} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{3}}-\frac {3 i b^{2} c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right )}{f^{3}}-\frac {4 b a \,d^{3} e^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{4}}+\frac {12 b^{2} c \,d^{2} e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {6 b^{2} c \,d^{2} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) x}{f^{2}}-\frac {12 i b a c \,d^{2} e^{2} x}{f^{2}}+\frac {12 i b a \,c^{2} d e x}{f}+\frac {6 i b a c \,d^{2} \polylog \left (2, -{\mathrm e}^{2 i \left (f x +e \right )}\right ) x}{f^{2}}-\frac {6 i b^{2} c \,d^{2} x^{2}}{f}-\frac {6 i b^{2} c \,d^{2} e^{2}}{f^{3}}+\frac {6 i b^{2} d^{3} e^{2} x}{f^{3}}+\frac {3 i b a \,d^{3} e^{4}}{f^{4}}\)	\(952\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

12/f^3*b*a*c*d^2*e^2*ln(exp(I*(f*x+e)))-2/f*b*a*d^3*ln(exp(2*I*(f*x+e))+1)*x^3-6/f*b*a*c*d^2*ln(exp(2*I*(f*x+e
))+1)*x^2-6/f*b*ln(exp(2*I*(f*x+e))+1)*a*c^2*d*x-12*I/f^2*b^2*c*d^2*e*x-8*I/f^3*b*a*c*d^2*e^3+6*I/f^2*b*a*c^2*
d*e^2+3*I/f^2*b*a*c^2*d*polylog(2,-exp(2*I*(f*x+e)))+3*I/f^2*b*a*d^3*polylog(2,-exp(2*I*(f*x+e)))*x^2+4*I/f^3*
b*a*d^3*e^3*x+2*I*b^2*(d^3*x^3+3*c*d^2*x^2+3*c^2*d*x+c^3)/f/(exp(2*I*(f*x+e))+1)+1/4*d^3*a^2*x^4+1/4/d*a^2*c^4
-1/4*d^3*b^2*x^4-b^2*c^3*x-1/4/d*b^2*c^4+2*I*d^2*a*b*c*x^3+3*I*d*a*b*c^2*x^2-d^2*b^2*c*x^3-3/2*d*b^2*c^2*x^2-3
/f^3*b*a*d^3*polylog(3,-exp(2*I*(f*x+e)))*x-4/f^4*b*a*d^3*e^3*ln(exp(I*(f*x+e)))-3/f^3*b*a*c*d^2*polylog(3,-ex
p(2*I*(f*x+e)))+12/f^3*b^2*c*d^2*e*ln(exp(I*(f*x+e)))+6/f^2*b^2*c*d^2*ln(exp(2*I*(f*x+e))+1)*x-3/2*I*a*b*d^3*p
olylog(4,-exp(2*I*(f*x+e)))/f^4+1/2*I*d^3*a*b*x^4-2*I*a*b*c^3*x-1/2*I/d*a*b*c^4+d^2*a^2*c*x^3+3/2*d*a^2*c^2*x^
2+a^2*c^3*x+3/f^2*b^2*d^3*ln(exp(2*I*(f*x+e))+1)*x^2-2/f*b*a*c^3*ln(exp(2*I*(f*x+e))+1)+4/f*b*a*c^3*ln(exp(I*(
f*x+e)))+3/f^2*b^2*c^2*d*ln(exp(2*I*(f*x+e))+1)-6/f^2*b^2*c^2*d*ln(exp(I*(f*x+e)))-6/f^4*b^2*d^3*e^2*ln(exp(I*
(f*x+e)))-2*I/f*b^2*d^3*x^3+4*I/f^4*b^2*d^3*e^3-12/f^2*b*a*c^2*d*e*ln(exp(I*(f*x+e)))+6*I/f^2*b*a*c*d^2*polylo
g(2,-exp(2*I*(f*x+e)))*x-12*I/f^2*b*a*c*d^2*e^2*x+12*I/f*b*a*c^2*d*e*x+3*I/f^4*b*a*d^3*e^4-3*I/f^3*b^2*d^3*pol
ylog(2,-exp(2*I*(f*x+e)))*x-6*I/f*b^2*c*d^2*x^2-6*I/f^3*b^2*c*d^2*e^2+6*I/f^3*b^2*d^3*e^2*x-3*I/f^3*b^2*c*d^2*
polylog(2,-exp(2*I*(f*x+e)))+3/2*b^2*d^3*polylog(3,-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. 2500 vs. \(2 (272) = 544\).
time = 1.26, size = 2500, normalized size = 8.33 \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))^2,x, algorithm="maxima")

[Out]

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

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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. 780 vs. \(2 (272) = 544\).
time = 0.39, size = 780, normalized size = 2.60 \begin {gather*} \frac {{\left (a^{2} - b^{2}\right )} d^{3} f^{4} x^{4} + 4 \, {\left (a^{2} - b^{2}\right )} c d^{2} f^{4} x^{3} + 6 \, {\left (a^{2} - b^{2}\right )} c^{2} d f^{4} x^{2} + 4 \, {\left (a^{2} - b^{2}\right )} c^{3} f^{4} x + 3 i \, a b 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 \, a b 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 ) - 6 \, {\left (i \, a b d^{3} f^{2} x^{2} + i \, a b c^{2} d f^{2} - i \, b^{2} c d^{2} f + i \, {\left (2 \, a b c d^{2} f^{2} - b^{2} d^{3} f\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 \, a b d^{3} f^{2} x^{2} - i \, a b c^{2} d f^{2} + i \, b^{2} c d^{2} f - i \, {\left (2 \, a b c d^{2} f^{2} - b^{2} d^{3} f\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 ) - 2 \, {\left (2 \, a b d^{3} f^{3} x^{3} + 2 \, a b c^{3} f^{3} - 3 \, b^{2} c^{2} d f^{2} + 3 \, {\left (2 \, a b c d^{2} f^{3} - b^{2} d^{3} f^{2}\right )} x^{2} + 6 \, {\left (a b c^{2} d f^{3} - b^{2} c d^{2} 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 ) - 2 \, {\left (2 \, a b d^{3} f^{3} x^{3} + 2 \, a b c^{3} f^{3} - 3 \, b^{2} c^{2} d f^{2} + 3 \, {\left (2 \, a b c d^{2} f^{3} - b^{2} d^{3} f^{2}\right )} x^{2} + 6 \, {\left (a b c^{2} d f^{3} - b^{2} c d^{2} 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 ) - 3 \, {\left (2 \, a b d^{3} f x + 2 \, a b c d^{2} f - b^{2} d^{3}\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 ) - 3 \, {\left (2 \, a b d^{3} f x + 2 \, a b c d^{2} f - b^{2} d^{3}\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 ) + 4 \, {\left (b^{2} d^{3} f^{3} x^{3} + 3 \, b^{2} c d^{2} f^{3} x^{2} + 3 \, b^{2} c^{2} d f^{3} x + b^{2} c^{3} f^{3}\right )} \tan \left (f x + e\right )}{4 \, 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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*tan(e + f*x))**2*(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))^2,x, algorithm="giac")

[Out]

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

[Out]

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

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