3.24 \(\int \frac{(c+d x)^3}{(a+i a \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=270 \[ -\frac{3 i d^2 (c+d x) e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac{3 i d^2 (c+d x) e^{-4 i e-4 i f x}}{128 a^2 f^3}+\frac{3 d (c+d x)^2 e^{-2 i e-2 i f x}}{8 a^2 f^2}+\frac{3 d (c+d x)^2 e^{-4 i e-4 i f x}}{64 a^2 f^2}+\frac{i (c+d x)^3 e^{-2 i e-2 i f x}}{4 a^2 f}+\frac{i (c+d x)^3 e^{-4 i e-4 i f x}}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{3 d^3 e^{-2 i e-2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{-4 i e-4 i f x}}{512 a^2 f^4} \]

[Out]

(-3*d^3*E^((-2*I)*e - (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((-4*I)*e - (4*I)*f*x))/(512*a^2*f^4) - (((3*I)/8)*d
^2*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(a^2*f^3) - (((3*I)/128)*d^2*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(a^2*f
^3) + (3*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(64*
a^2*f^2) + ((I/4)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^3)/(a^2*f) + ((I/16)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^3
)/(a^2*f) + (c + d*x)^4/(16*a^2*d)

________________________________________________________________________________________

Rubi [A]  time = 0.292698, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3729, 2176, 2194} \[ -\frac{3 i d^2 (c+d x) e^{-2 i e-2 i f x}}{8 a^2 f^3}-\frac{3 i d^2 (c+d x) e^{-4 i e-4 i f x}}{128 a^2 f^3}+\frac{3 d (c+d x)^2 e^{-2 i e-2 i f x}}{8 a^2 f^2}+\frac{3 d (c+d x)^2 e^{-4 i e-4 i f x}}{64 a^2 f^2}+\frac{i (c+d x)^3 e^{-2 i e-2 i f x}}{4 a^2 f}+\frac{i (c+d x)^3 e^{-4 i e-4 i f x}}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{3 d^3 e^{-2 i e-2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{-4 i e-4 i f x}}{512 a^2 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*d^3*E^((-2*I)*e - (2*I)*f*x))/(16*a^2*f^4) - (3*d^3*E^((-4*I)*e - (4*I)*f*x))/(512*a^2*f^4) - (((3*I)/8)*d
^2*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(a^2*f^3) - (((3*I)/128)*d^2*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(a^2*f
^3) + (3*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(8*a^2*f^2) + (3*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(64*
a^2*f^2) + ((I/4)*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^3)/(a^2*f) + ((I/16)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^3
)/(a^2*f) + (c + d*x)^4/(16*a^2*d)

Rule 3729

Int[((c_.) + (d_.)*(x_))^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandIntegrand[(c
+ d*x)^m, (1/(2*a) + E^((2*a*(e + f*x))/b)/(2*a))^(-n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
+ b^2, 0] && ILtQ[n, 0]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \frac{(c+d x)^3}{(a+i a \tan (e+f x))^2} \, dx &=\int \left (\frac{(c+d x)^3}{4 a^2}+\frac{e^{-2 i e-2 i f x} (c+d x)^3}{2 a^2}+\frac{e^{-4 i e-4 i f x} (c+d x)^3}{4 a^2}\right ) \, dx\\ &=\frac{(c+d x)^4}{16 a^2 d}+\frac{\int e^{-4 i e-4 i f x} (c+d x)^3 \, dx}{4 a^2}+\frac{\int e^{-2 i e-2 i f x} (c+d x)^3 \, dx}{2 a^2}\\ &=\frac{i e^{-2 i e-2 i f x} (c+d x)^3}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{(3 i d) \int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{16 a^2 f}-\frac{(3 i d) \int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{4 a^2 f}\\ &=\frac{3 d e^{-2 i e-2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac{3 d e^{-4 i e-4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac{i e^{-2 i e-2 i f x} (c+d x)^3}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}-\frac{\left (3 d^2\right ) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{32 a^2 f^2}-\frac{\left (3 d^2\right ) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{4 a^2 f^2}\\ &=-\frac{3 i d^2 e^{-2 i e-2 i f x} (c+d x)}{8 a^2 f^3}-\frac{3 i d^2 e^{-4 i e-4 i f x} (c+d x)}{128 a^2 f^3}+\frac{3 d e^{-2 i e-2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac{3 d e^{-4 i e-4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac{i e^{-2 i e-2 i f x} (c+d x)^3}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}+\frac{\left (3 i d^3\right ) \int e^{-4 i e-4 i f x} \, dx}{128 a^2 f^3}+\frac{\left (3 i d^3\right ) \int e^{-2 i e-2 i f x} \, dx}{8 a^2 f^3}\\ &=-\frac{3 d^3 e^{-2 i e-2 i f x}}{16 a^2 f^4}-\frac{3 d^3 e^{-4 i e-4 i f x}}{512 a^2 f^4}-\frac{3 i d^2 e^{-2 i e-2 i f x} (c+d x)}{8 a^2 f^3}-\frac{3 i d^2 e^{-4 i e-4 i f x} (c+d x)}{128 a^2 f^3}+\frac{3 d e^{-2 i e-2 i f x} (c+d x)^2}{8 a^2 f^2}+\frac{3 d e^{-4 i e-4 i f x} (c+d x)^2}{64 a^2 f^2}+\frac{i e^{-2 i e-2 i f x} (c+d x)^3}{4 a^2 f}+\frac{i e^{-4 i e-4 i f x} (c+d x)^3}{16 a^2 f}+\frac{(c+d x)^4}{16 a^2 d}\\ \end{align*}

Mathematica [A]  time = 1.24597, size = 473, normalized size = 1.75 \[ \frac{\sec ^2(e+f x) (\cos (f x)+i \sin (f x))^2 \left (f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right ) (\cos (2 e)+i \sin (2 e))+\frac{1}{32} (\cos (2 e)-i \sin (2 e)) \cos (4 f x) \left (24 c^2 d f^2 (1+4 i f x)+32 i c^3 f^3+12 c d^2 f \left (8 i f^2 x^2+4 f x-i\right )+d^3 \left (32 i f^3 x^3+24 f^2 x^2-12 i f x-3\right )\right )+\frac{1}{32} (\cos (2 e)-i \sin (2 e)) \sin (4 f x) \left (24 c^2 d f^2 (4 f x-i)+32 c^3 f^3+12 c d^2 f \left (8 f^2 x^2-4 i f x-1\right )+d^3 \left (32 f^3 x^3-24 i f^2 x^2-12 f x+3 i\right )\right )+\sin (2 f x) \left (6 c^2 d f^2 (2 f x-i)+4 c^3 f^3+6 c d^2 f \left (2 f^2 x^2-2 i f x-1\right )+d^3 \left (4 f^3 x^3-6 i f^2 x^2-6 f x+3 i\right )\right )+\cos (2 f x) \left (6 c^2 d f^2 (1+2 i f x)+4 i c^3 f^3+6 c d^2 f \left (2 i f^2 x^2+2 f x-i\right )+d^3 \left (4 i f^3 x^3+6 f^2 x^2-6 i f x-3\right )\right )\right )}{16 f^4 (a+i a \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]^2*(Cos[f*x] + I*Sin[f*x])^2*(((4*I)*c^3*f^3 + 6*c^2*d*f^2*(1 + (2*I)*f*x) + 6*c*d^2*f*(-I + 2*f*
x + (2*I)*f^2*x^2) + d^3*(-3 - (6*I)*f*x + 6*f^2*x^2 + (4*I)*f^3*x^3))*Cos[2*f*x] + (((32*I)*c^3*f^3 + 24*c^2*
d*f^2*(1 + (4*I)*f*x) + 12*c*d^2*f*(-I + 4*f*x + (8*I)*f^2*x^2) + d^3*(-3 - (12*I)*f*x + 24*f^2*x^2 + (32*I)*f
^3*x^3))*Cos[4*f*x]*(Cos[2*e] - I*Sin[2*e]))/32 + f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3)*(Cos[2*e]
+ I*Sin[2*e]) + (4*c^3*f^3 + 6*c^2*d*f^2*(-I + 2*f*x) + 6*c*d^2*f*(-1 - (2*I)*f*x + 2*f^2*x^2) + d^3*(3*I - 6*
f*x - (6*I)*f^2*x^2 + 4*f^3*x^3))*Sin[2*f*x] + ((32*c^3*f^3 + 24*c^2*d*f^2*(-I + 4*f*x) + 12*c*d^2*f*(-1 - (4*
I)*f*x + 8*f^2*x^2) + d^3*(3*I - 12*f*x - (24*I)*f^2*x^2 + 32*f^3*x^3))*(Cos[2*e] - I*Sin[2*e])*Sin[4*f*x])/32
))/(16*f^4*(a + I*a*Tan[e + f*x])^2)

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Maple [A]  time = 0.298, size = 272, normalized size = 1. \begin{align*}{\frac{{d}^{3}{x}^{4}}{16\,{a}^{2}}}+{\frac{c{d}^{2}{x}^{3}}{4\,{a}^{2}}}+{\frac{3\,{c}^{2}d{x}^{2}}{8\,{a}^{2}}}+{\frac{{c}^{3}x}{4\,{a}^{2}}}+{\frac{{\frac{i}{16}} \left ( 4\,{d}^{3}{x}^{3}{f}^{3}-6\,i{d}^{3}{f}^{2}{x}^{2}+12\,c{d}^{2}{f}^{3}{x}^{2}-12\,ic{d}^{2}{f}^{2}x+12\,{c}^{2}d{f}^{3}x-6\,i{c}^{2}d{f}^{2}+4\,{c}^{3}{f}^{3}-6\,{d}^{3}fx+3\,i{d}^{3}-6\,c{d}^{2}f \right ){{\rm e}^{-2\,i \left ( fx+e \right ) }}}{{a}^{2}{f}^{4}}}+{\frac{{\frac{i}{512}} \left ( 32\,{d}^{3}{x}^{3}{f}^{3}-24\,i{d}^{3}{f}^{2}{x}^{2}+96\,c{d}^{2}{f}^{3}{x}^{2}-48\,ic{d}^{2}{f}^{2}x+96\,{c}^{2}d{f}^{3}x-24\,i{c}^{2}d{f}^{2}+32\,{c}^{3}{f}^{3}-12\,{d}^{3}fx+3\,i{d}^{3}-12\,c{d}^{2}f \right ){{\rm e}^{-4\,i \left ( fx+e \right ) }}}{{a}^{2}{f}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/a^2*d^3*x^4+1/4/a^2*c*d^2*x^3+3/8/a^2*c^2*d*x^2+1/4/a^2*c^3*x+1/16*I*(4*d^3*x^3*f^3-6*I*d^3*f^2*x^2+12*c*
d^2*f^3*x^2-12*I*c*d^2*f^2*x+12*c^2*d*f^3*x-6*I*c^2*d*f^2+4*c^3*f^3-6*d^3*f*x+3*I*d^3-6*c*d^2*f)/a^2/f^4*exp(-
2*I*(f*x+e))+1/512*I*(32*d^3*x^3*f^3-24*I*d^3*f^2*x^2+96*c*d^2*f^3*x^2-48*I*c*d^2*f^2*x+96*c^2*d*f^3*x-24*I*c^
2*d*f^2+32*c^3*f^3-12*d^3*f*x+3*I*d^3-12*c*d^2*f)/a^2/f^4*exp(-4*I*(f*x+e))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+I*a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [A]  time = 1.62439, size = 639, normalized size = 2.37 \begin{align*} \frac{{\left (32 i \, d^{3} f^{3} x^{3} + 32 i \, c^{3} f^{3} + 24 \, c^{2} d f^{2} - 12 i \, c d^{2} f - 3 \, d^{3} +{\left (96 i \, c d^{2} f^{3} + 24 \, d^{3} f^{2}\right )} x^{2} +{\left (96 i \, c^{2} d f^{3} + 48 \, c d^{2} f^{2} - 12 i \, d^{3} f\right )} x + 32 \,{\left (d^{3} f^{4} x^{4} + 4 \, c d^{2} f^{4} x^{3} + 6 \, c^{2} d f^{4} x^{2} + 4 \, c^{3} f^{4} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (128 i \, d^{3} f^{3} x^{3} + 128 i \, c^{3} f^{3} + 192 \, c^{2} d f^{2} - 192 i \, c d^{2} f - 96 \, d^{3} +{\left (384 i \, c d^{2} f^{3} + 192 \, d^{3} f^{2}\right )} x^{2} +{\left (384 i \, c^{2} d f^{3} + 384 \, c d^{2} f^{2} - 192 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{512 \, a^{2} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+I*a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

1/512*(32*I*d^3*f^3*x^3 + 32*I*c^3*f^3 + 24*c^2*d*f^2 - 12*I*c*d^2*f - 3*d^3 + (96*I*c*d^2*f^3 + 24*d^3*f^2)*x
^2 + (96*I*c^2*d*f^3 + 48*c*d^2*f^2 - 12*I*d^3*f)*x + 32*(d^3*f^4*x^4 + 4*c*d^2*f^4*x^3 + 6*c^2*d*f^4*x^2 + 4*
c^3*f^4*x)*e^(4*I*f*x + 4*I*e) + (128*I*d^3*f^3*x^3 + 128*I*c^3*f^3 + 192*c^2*d*f^2 - 192*I*c*d^2*f - 96*d^3 +
(384*I*c*d^2*f^3 + 192*d^3*f^2)*x^2 + (384*I*c^2*d*f^3 + 384*c*d^2*f^2 - 192*I*d^3*f)*x)*e^(2*I*f*x + 2*I*e))
*e^(-4*I*f*x - 4*I*e)/(a^2*f^4)

________________________________________________________________________________________

Sympy [A]  time = 1.86125, size = 666, normalized size = 2.47 \begin{align*} \begin{cases} \frac{\left (\left (512 i a^{14} c^{3} f^{19} e^{20 i e} + 1536 i a^{14} c^{2} d f^{19} x e^{20 i e} + 384 a^{14} c^{2} d f^{18} e^{20 i e} + 1536 i a^{14} c d^{2} f^{19} x^{2} e^{20 i e} + 768 a^{14} c d^{2} f^{18} x e^{20 i e} - 192 i a^{14} c d^{2} f^{17} e^{20 i e} + 512 i a^{14} d^{3} f^{19} x^{3} e^{20 i e} + 384 a^{14} d^{3} f^{18} x^{2} e^{20 i e} - 192 i a^{14} d^{3} f^{17} x e^{20 i e} - 48 a^{14} d^{3} f^{16} e^{20 i e}\right ) e^{- 4 i f x} + \left (2048 i a^{14} c^{3} f^{19} e^{22 i e} + 6144 i a^{14} c^{2} d f^{19} x e^{22 i e} + 3072 a^{14} c^{2} d f^{18} e^{22 i e} + 6144 i a^{14} c d^{2} f^{19} x^{2} e^{22 i e} + 6144 a^{14} c d^{2} f^{18} x e^{22 i e} - 3072 i a^{14} c d^{2} f^{17} e^{22 i e} + 2048 i a^{14} d^{3} f^{19} x^{3} e^{22 i e} + 3072 a^{14} d^{3} f^{18} x^{2} e^{22 i e} - 3072 i a^{14} d^{3} f^{17} x e^{22 i e} - 1536 a^{14} d^{3} f^{16} e^{22 i e}\right ) e^{- 2 i f x}\right ) e^{- 24 i e}}{8192 a^{16} f^{20}} & \text{for}\: 8192 a^{16} f^{20} e^{24 i e} \neq 0 \\\frac{x^{4} \left (2 d^{3} e^{2 i e} + d^{3}\right ) e^{- 4 i e}}{16 a^{2}} + \frac{x^{3} \left (2 c d^{2} e^{2 i e} + c d^{2}\right ) e^{- 4 i e}}{4 a^{2}} + \frac{x^{2} \left (6 c^{2} d e^{2 i e} + 3 c^{2} d\right ) e^{- 4 i e}}{8 a^{2}} + \frac{x \left (2 c^{3} e^{2 i e} + c^{3}\right ) e^{- 4 i e}}{4 a^{2}} & \text{otherwise} \end{cases} + \frac{c^{3} x}{4 a^{2}} + \frac{3 c^{2} d x^{2}}{8 a^{2}} + \frac{c d^{2} x^{3}}{4 a^{2}} + \frac{d^{3} x^{4}}{16 a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((((512*I*a**14*c**3*f**19*exp(20*I*e) + 1536*I*a**14*c**2*d*f**19*x*exp(20*I*e) + 384*a**14*c**2*d*f
**18*exp(20*I*e) + 1536*I*a**14*c*d**2*f**19*x**2*exp(20*I*e) + 768*a**14*c*d**2*f**18*x*exp(20*I*e) - 192*I*a
**14*c*d**2*f**17*exp(20*I*e) + 512*I*a**14*d**3*f**19*x**3*exp(20*I*e) + 384*a**14*d**3*f**18*x**2*exp(20*I*e
) - 192*I*a**14*d**3*f**17*x*exp(20*I*e) - 48*a**14*d**3*f**16*exp(20*I*e))*exp(-4*I*f*x) + (2048*I*a**14*c**3
*f**19*exp(22*I*e) + 6144*I*a**14*c**2*d*f**19*x*exp(22*I*e) + 3072*a**14*c**2*d*f**18*exp(22*I*e) + 6144*I*a*
*14*c*d**2*f**19*x**2*exp(22*I*e) + 6144*a**14*c*d**2*f**18*x*exp(22*I*e) - 3072*I*a**14*c*d**2*f**17*exp(22*I
*e) + 2048*I*a**14*d**3*f**19*x**3*exp(22*I*e) + 3072*a**14*d**3*f**18*x**2*exp(22*I*e) - 3072*I*a**14*d**3*f*
*17*x*exp(22*I*e) - 1536*a**14*d**3*f**16*exp(22*I*e))*exp(-2*I*f*x))*exp(-24*I*e)/(8192*a**16*f**20), Ne(8192
*a**16*f**20*exp(24*I*e), 0)), (x**4*(2*d**3*exp(2*I*e) + d**3)*exp(-4*I*e)/(16*a**2) + x**3*(2*c*d**2*exp(2*I
*e) + c*d**2)*exp(-4*I*e)/(4*a**2) + x**2*(6*c**2*d*exp(2*I*e) + 3*c**2*d)*exp(-4*I*e)/(8*a**2) + x*(2*c**3*ex
p(2*I*e) + c**3)*exp(-4*I*e)/(4*a**2), True)) + c**3*x/(4*a**2) + 3*c**2*d*x**2/(8*a**2) + c*d**2*x**3/(4*a**2
) + d**3*x**4/(16*a**2)

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Giac [A]  time = 1.19268, size = 517, normalized size = 1.91 \begin{align*} \frac{{\left (32 \, d^{3} f^{4} x^{4} e^{\left (4 i \, f x + 4 i \, e\right )} + 128 \, c d^{2} f^{4} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 192 \, c^{2} d f^{4} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 128 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, d^{3} f^{3} x^{3} + 128 \, c^{3} f^{4} x e^{\left (4 i \, f x + 4 i \, e\right )} + 384 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 i \, c d^{2} f^{3} x^{2} + 384 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} + 192 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 96 i \, c^{2} d f^{3} x + 24 \, d^{3} f^{2} x^{2} + 128 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 384 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 32 i \, c^{3} f^{3} + 48 \, c d^{2} f^{2} x + 192 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 192 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 24 \, c^{2} d f^{2} - 12 i \, d^{3} f x - 192 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 12 i \, c d^{2} f - 96 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 \, d^{3}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{512 \, a^{2} f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^3/(a+I*a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

1/512*(32*d^3*f^4*x^4*e^(4*I*f*x + 4*I*e) + 128*c*d^2*f^4*x^3*e^(4*I*f*x + 4*I*e) + 192*c^2*d*f^4*x^2*e^(4*I*f
*x + 4*I*e) + 128*I*d^3*f^3*x^3*e^(2*I*f*x + 2*I*e) + 32*I*d^3*f^3*x^3 + 128*c^3*f^4*x*e^(4*I*f*x + 4*I*e) + 3
84*I*c*d^2*f^3*x^2*e^(2*I*f*x + 2*I*e) + 96*I*c*d^2*f^3*x^2 + 384*I*c^2*d*f^3*x*e^(2*I*f*x + 2*I*e) + 192*d^3*
f^2*x^2*e^(2*I*f*x + 2*I*e) + 96*I*c^2*d*f^3*x + 24*d^3*f^2*x^2 + 128*I*c^3*f^3*e^(2*I*f*x + 2*I*e) + 384*c*d^
2*f^2*x*e^(2*I*f*x + 2*I*e) + 32*I*c^3*f^3 + 48*c*d^2*f^2*x + 192*c^2*d*f^2*e^(2*I*f*x + 2*I*e) - 192*I*d^3*f*
x*e^(2*I*f*x + 2*I*e) + 24*c^2*d*f^2 - 12*I*d^3*f*x - 192*I*c*d^2*f*e^(2*I*f*x + 2*I*e) - 12*I*c*d^2*f - 96*d^
3*e^(2*I*f*x + 2*I*e) - 3*d^3)*e^(-4*I*f*x - 4*I*e)/(a^2*f^4)