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

Optimal result

Integrand size = 20, antiderivative size = 300 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=-\frac {i b^2 (c+d x)^3}{f}+\frac {a^2 (c+d x)^4}{4 d}+\frac {i a b (c+d x)^4}{2 d}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{2 f^4}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f} \]

[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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3803, 3800, 2221, 2611, 6744, 2320, 6724, 3801, 32} \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\frac {a^2 (c+d x)^4}{4 d}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {2 a b (c+d x)^3 \log \left (1+e^{2 i (e+f x)}\right )}{f}+\frac {i a b (c+d x)^4}{2 d}-\frac {3 i a b d^3 \operatorname {PolyLog}\left (4,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 i b^2 d^2 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 b^2 d (c+d x)^2 \log \left (1+e^{2 i (e+f x)}\right )}{f^2}+\frac {b^2 (c+d x)^3 \tan (e+f x)}{f}-\frac {i b^2 (c+d x)^3}{f}-\frac {b^2 (c+d x)^4}{4 d}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4} \]

[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*} \text {integral}& = \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 \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-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 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right ) \, dx}{f^3}+\frac {\left (3 i b^2 d^3\right ) \int \operatorname {PolyLog}\left (2,-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) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-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 {\operatorname {PolyLog}(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 {\operatorname {PolyLog}(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) \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^3}+\frac {3 i a b d (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{2 i (e+f x)}\right )}{f^2}+\frac {3 b^2 d^3 \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{2 f^4}-\frac {3 a b d^2 (c+d x) \operatorname {PolyLog}\left (3,-e^{2 i (e+f x)}\right )}{f^3}-\frac {3 i a b d^3 \operatorname {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} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1337\) vs. \(2(300)=600\).

Time = 7.33 (sec) , antiderivative size = 1337, normalized size of antiderivative = 4.46 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\frac {i b^2 d^3 e^{-i e} \left (2 f^2 x^2 \left (2 f x-3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{-2 i (e+f x)}\right )\right )+6 \left (1+e^{2 i e}\right ) f x \operatorname {PolyLog}\left (2,-e^{-2 i (e+f x)}\right )-3 i \left (1+e^{2 i e}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (e+f x)}\right )\right ) \sec (e)}{4 f^4}-\frac {i a b c d^2 e^{-i e} \left (2 f^2 x^2 \left (2 f x-3 i \left (1+e^{2 i e}\right ) \log \left (1+e^{-2 i (e+f x)}\right )\right )+6 \left (1+e^{2 i e}\right ) f x \operatorname {PolyLog}\left (2,-e^{-2 i (e+f x)}\right )-3 i \left (1+e^{2 i e}\right ) \operatorname {PolyLog}\left (3,-e^{-2 i (e+f x)}\right )\right ) \sec (e)}{2 f^3}-\frac {i a b d^3 e^{i e} \left (2 e^{-2 i e} f^4 x^4-4 i \left (1+e^{-2 i e}\right ) f^3 x^3 \log \left (1+e^{-2 i (e+f x)}\right )+6 \left (1+e^{-2 i e}\right ) f^2 x^2 \operatorname {PolyLog}\left (2,-e^{-2 i (e+f x)}\right )-6 i \left (1+e^{-2 i e}\right ) f x \operatorname {PolyLog}\left (3,-e^{-2 i (e+f x)}\right )-3 \left (1+e^{-2 i e}\right ) \operatorname {PolyLog}\left (4,-e^{-2 i (e+f x)}\right )\right ) \sec (e)}{4 f^4}+\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 \arctan (\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \arctan (\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\arctan (\cot (e))) \log \left (1-e^{2 i (f x-\arctan (\cot (e)))}\right )+\pi \log (\cos (f x))-2 \arctan (\cot (e)) \log (\sin (f x-\arctan (\cot (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x-\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 \arctan (\cot (e))} f^2 x^2-\frac {\cot (e) \left (i f x (-\pi -2 \arctan (\cot (e)))-\pi \log \left (1+e^{-2 i f x}\right )-2 (f x-\arctan (\cot (e))) \log \left (1-e^{2 i (f x-\arctan (\cot (e)))}\right )+\pi \log (\cos (f x))-2 \arctan (\cot (e)) \log (\sin (f x-\arctan (\cot (e))))+i \operatorname {PolyLog}\left (2,e^{2 i (f x-\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} \]

[In]

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

[Out]

((I/4)*b^2*d^3*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6*(1 + E^((2*I)*e)
)*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e + f*x))])*Sec[e])/(
E^(I*e)*f^4) - ((I/2)*a*b*c*d^2*(2*f^2*x^2*(2*f*x - (3*I)*(1 + E^((2*I)*e))*Log[1 + E^((-2*I)*(e + f*x))]) + 6
*(1 + E^((2*I)*e))*f*x*PolyLog[2, -E^((-2*I)*(e + f*x))] - (3*I)*(1 + E^((2*I)*e))*PolyLog[3, -E^((-2*I)*(e +
f*x))])*Sec[e])/(E^(I*e)*f^3) - ((I/4)*a*b*d^3*E^(I*e)*((2*f^4*x^4)/E^((2*I)*e) - (4*I)*(1 + E^((-2*I)*e))*f^3
*x^3*Log[1 + E^((-2*I)*(e + f*x))] + 6*(1 + E^((-2*I)*e))*f^2*x^2*PolyLog[2, -E^((-2*I)*(e + f*x))] - (6*I)*(1
+ E^((-2*I)*e))*f*x*PolyLog[3, -E^((-2*I)*(e + f*x))] - 3*(1 + E^((-2*I)*e))*PolyLog[4, -E^((-2*I)*(e + f*x))
])*Sec[e])/f^4 + (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 + Sin[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[Co
s[f*x]] - 2*ArcTan[Cot[e]]*Log[Sin[f*x - ArcTan[Cot[e]]]] + I*PolyLog[2, E^((2*I)*(f*x - ArcTan[Cot[e]]))]))/S
qrt[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*Ar
cTan[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]*(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*C
os[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]))/(8*f)

Maple [B] (verified)

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 = 1.60 (sec) , antiderivative size = 952, normalized size of antiderivative = 3.17

method	result	size
risch	\(\text {Expression too large to display}\)	\(952\)

[In]

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

[Out]

-12*I/f^2*b*e^2*d^2*c*a*x+6*I/f^2*b*a*c*d^2*polylog(2,-exp(2*I*(f*x+e)))*x+12*I/f*b*d*c^2*a*e*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)+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)))+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(ex
p(I*(f*x+e)))-6/f^4*b^2*e^2*d^3*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+1/2*I*d^3*a*b*x^4-3/2
*d*b^2*c^2*x^2+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*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-d^2*b^2*c*x^3-3/2*I*a*b*d^3*polylog(4,-exp(2*I*(f*x+e)))/f^4-4/
f^4*b*e^3*a*d^3*ln(exp(I*(f*x+e)))-3/f^3*b*d^3*a*polylog(3,-exp(2*I*(f*x+e)))*x+6/f^2*b^2*d^2*c*ln(exp(2*I*(f*
x+e))+1)*x-2/f*b*d^3*a*ln(exp(2*I*(f*x+e))+1)*x^3-3/f^3*b*a*c*d^2*polylog(3,-exp(2*I*(f*x+e)))+12/f^3*b^2*e*c*
d^2*ln(exp(I*(f*x+e)))-3*I/f^3*b^2*d^3*polylog(2,-exp(2*I*(f*x+e)))*x+3*I/f^4*b*e^4*a*d^3-6*I/f*b^2*d^2*c*x^2-
6*I/f^3*b^2*d^2*c*e^2-3*I/f^3*b^2*d^2*c*polylog(2,-exp(2*I*(f*x+e)))+6*I/f^3*b^2*d^3*e^2*x+2*I*d^2*a*b*c*x^3+3
*I*d*a*b*c^2*x^2+3/2*b^2*d^3*polylog(3,-exp(2*I*(f*x+e)))/f^4-12/f^2*b*e*a*c^2*d*ln(exp(I*(f*x+e)))-6/f*b*d*c^
2*a*ln(exp(2*I*(f*x+e))+1)*x-6/f*b*a*c*d^2*ln(exp(2*I*(f*x+e))+1)*x^2+12/f^3*b*e^2*a*c*d^2*ln(exp(I*(f*x+e)))+
6*I/f^2*b*d*c^2*a*e^2+3*I/f^2*b*d*c^2*a*polylog(2,-exp(2*I*(f*x+e)))+3*I/f^2*b*d^3*a*polylog(2,-exp(2*I*(f*x+e
)))*x^2+4*I/f^3*b*e^3*a*d^3*x-8*I/f^3*b*e^3*d^2*c*a-12*I/f^2*b^2*d^2*c*e*x

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (264) = 528\).

Time = 0.28 (sec) , antiderivative size = 759, normalized size of antiderivative = 2.53 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\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}} \]

[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

Sympy [F]

\[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\int \left (a + b \tan {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{3}\, dx \]

[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)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2525 vs. \(2 (264) = 528\).

Time = 1.73 (sec) , antiderivative size = 2525, normalized size of antiderivative = 8.42 \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\text {Too large to display} \]

[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*d^3*e/f^3 + 6*(f*x + e)^2*a^2*d^3*e^2/f
^3 - 4*(f*x + e)*a^2*d^3*e^3/f^3 + 4*(f*x + e)^3*a^2*c*d^2/f^2 - 12*(f*x + e)^2*a^2*c*d^2*e/f^2 + 12*(f*x + e)
*a^2*c*d^2*e^2/f^2 + 6*(f*x + e)^2*a^2*c^2*d/f - 12*(f*x + e)*a^2*c^2*d*e/f + 8*a*b*c^3*log(sec(f*x + e)) - 8*
a*b*d^3*e^3*log(sec(f*x + e))/f^3 + 24*a*b*c*d^2*e^2*log(sec(f*x + e))/f^2 - 24*a*b*c^2*d*e*log(sec(f*x + e))/
f + 4*(3*(2*a*b + I*b^2)*(f*x + e)^4*d^3 - 24*b^2*d^3*e^3 + 72*b^2*c*d^2*e^2*f - 72*b^2*c^2*d*e*f^2 + 24*b^2*c
^3*f^3 - 12*((2*a*b + I*b^2)*d^3*e - (2*a*b + I*b^2)*c*d^2*f)*(f*x + e)^3 + 18*((2*a*b + I*b^2)*d^3*e^2 - 2*(2
*a*b + I*b^2)*c*d^2*e*f + (2*a*b + I*b^2)*c^2*d*f^2)*(f*x + e)^2 + 12*(-I*b^2*d^3*e^3 + 3*I*b^2*c*d^2*e^2*f -
3*I*b^2*c^2*d*e*f^2 + I*b^2*c^3*f^3)*(f*x + e) - 4*(8*(f*x + e)^3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*c*d^2*e*f -
9*b^2*c^2*d*f^2 - 9*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 18*(a*b*d^3*e^2 + a*b*c^2*d*f^2 + b
^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f*x + e) + (8*(f*x + e)^3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*c*d^2*e*
f - 9*b^2*c^2*d*f^2 - 9*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 18*(a*b*d^3*e^2 + a*b*c^2*d*f^2
+ b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^2)*f)*(f*x + e))*cos(2*f*x + 2*e) - (-8*I*(f*x + e)^3*a*b*d^3 + 9*I*b^2
*d^3*e^2 - 18*I*b^2*c*d^2*e*f + 9*I*b^2*c^2*d*f^2 + 9*(2*I*a*b*d^3*e - 2*I*a*b*c*d^2*f + I*b^2*d^3)*(f*x + e)^
2 + 18*(-I*a*b*d^3*e^2 - I*a*b*c^2*d*f^2 - I*b^2*d^3*e + (2*I*a*b*c*d^2*e + I*b^2*c*d^2)*f)*(f*x + e))*sin(2*f
*x + 2*e))*arctan2(sin(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*b^2*d^3
+ (2*a*b + I*b^2)*d^3*e - (2*a*b + I*b^2)*c*d^2*f)*(f*x + e)^3 + 6*(4*b^2*d^3*e + (2*a*b + I*b^2)*d^3*e^2 + (
2*a*b + I*b^2)*c^2*d*f^2 - 2*(2*b^2*c*d^2 + (2*a*b + I*b^2)*c*d^2*e)*f)*(f*x + e)^2 + 4*(-I*b^2*d^3*e^3 + I*b^
2*c^3*f^3 - 6*b^2*d^3*e^2 + 3*(-I*b^2*c^2*d*e - 2*b^2*c^2*d)*f^2 + 3*(I*b^2*c*d^2*e^2 + 4*b^2*c*d^2*e)*f)*(f*x
+ e))*cos(2*f*x + 2*e) + 12*(4*(f*x + e)^2*a*b*d^3 + 3*a*b*d^3*e^2 + 3*a*b*c^2*d*f^2 + 3*b^2*d^3*e - 3*(2*a*b
*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e) - 3*(2*a*b*c*d^2*e + b^2*c*d^2)*f + (4*(f*x + e)^2*a*b*d^3 + 3*a*b
*d^3*e^2 + 3*a*b*c^2*d*f^2 + 3*b^2*d^3*e - 3*(2*a*b*d^3*e - 2*a*b*c*d^2*f + b^2*d^3)*(f*x + e) - 3*(2*a*b*c*d^
2*e + b^2*c*d^2)*f)*cos(2*f*x + 2*e) + (4*I*(f*x + e)^2*a*b*d^3 + 3*I*a*b*d^3*e^2 + 3*I*a*b*c^2*d*f^2 + 3*I*b^
2*d^3*e + 3*(-2*I*a*b*d^3*e + 2*I*a*b*c*d^2*f - I*b^2*d^3)*(f*x + e) + 3*(-2*I*a*b*c*d^2*e - I*b^2*c*d^2)*f)*s
in(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*d^3*e^2 + 18*I*b^2*c*d^2*e
*f - 9*I*b^2*c^2*d*f^2 + 9*(-2*I*a*b*d^3*e + 2*I*a*b*c*d^2*f - I*b^2*d^3)*(f*x + e)^2 + 18*(I*a*b*d^3*e^2 + I*
a*b*c^2*d*f^2 + I*b^2*d^3*e + (-2*I*a*b*c*d^2*e - I*b^2*c*d^2)*f)*(f*x + e) + (8*I*(f*x + e)^3*a*b*d^3 - 9*I*b
^2*d^3*e^2 + 18*I*b^2*c*d^2*e*f - 9*I*b^2*c^2*d*f^2 + 9*(-2*I*a*b*d^3*e + 2*I*a*b*c*d^2*f - I*b^2*d^3)*(f*x +
e)^2 + 18*(I*a*b*d^3*e^2 + I*a*b*c^2*d*f^2 + I*b^2*d^3*e + (-2*I*a*b*c*d^2*e - I*b^2*c*d^2)*f)*(f*x + e))*cos(
2*f*x + 2*e) - (8*(f*x + e)^3*a*b*d^3 - 9*b^2*d^3*e^2 + 18*b^2*c*d^2*e*f - 9*b^2*c^2*d*f^2 - 9*(2*a*b*d^3*e -
2*a*b*c*d^2*f + b^2*d^3)*(f*x + e)^2 + 18*(a*b*d^3*e^2 + a*b*c^2*d*f^2 + b^2*d^3*e - (2*a*b*c*d^2*e + b^2*c*d^
2)*f)*(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*d^3*e + 6*I*a*b*c*d^2*f - 3*I*b^2*d^3 + (8*I*(f*x + e)*a*b*d^3 - 6*I*a*b*d^3*e + 6*I
*a*b*c*d^2*f - 3*I*b^2*d^3)*cos(2*f*x + 2*e) - (8*(f*x + e)*a*b*d^3 - 6*a*b*d^3*e + 6*a*b*c*d^2*f - 3*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*b^2*d^3 + (
-2*I*a*b + b^2)*d^3*e + (2*I*a*b - b^2)*c*d^2*f)*(f*x + e)^3 + 6*(4*I*b^2*d^3*e + (2*I*a*b - b^2)*d^3*e^2 + (2
*I*a*b - b^2)*c^2*d*f^2 + 2*(-2*I*b^2*c*d^2 + (-2*I*a*b + b^2)*c*d^2*e)*f)*(f*x + e)^2 + 4*(b^2*d^3*e^3 - b^2*
c^3*f^3 - 6*I*b^2*d^3*e^2 + 3*(b^2*c^2*d*e - 2*I*b^2*c^2*d)*f^2 - 3*(b^2*c*d^2*e^2 - 4*I*b^2*c*d^2*e)*f)*(f*x
+ e))*sin(2*f*x + 2*e))/(-12*I*f^3*cos(2*f*x + 2*e) + 12*f^3*sin(2*f*x + 2*e) - 12*I*f^3))/f

Giac [F]

\[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\int { {\left (d x + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^3 (a+b \tan (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^3 \,d x \]

[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)