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

Optimal. Leaf size=396 \[ -\frac{9 i d^2 (c+d x) e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac{9 i d^2 (c+d x) e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac{i d^2 (c+d x) e^{-6 i e-6 i f x}}{288 a^3 f^3}+\frac{9 d (c+d x)^2 e^{-2 i e-2 i f x}}{32 a^3 f^2}+\frac{9 d (c+d x)^2 e^{-4 i e-4 i f x}}{128 a^3 f^2}+\frac{d (c+d x)^2 e^{-6 i e-6 i f x}}{96 a^3 f^2}+\frac{3 i (c+d x)^3 e^{-2 i e-2 i f x}}{16 a^3 f}+\frac{3 i (c+d x)^3 e^{-4 i e-4 i f x}}{32 a^3 f}+\frac{i (c+d x)^3 e^{-6 i e-6 i f x}}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{9 d^3 e^{-2 i e-2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{-4 i e-4 i f x}}{1024 a^3 f^4}-\frac{d^3 e^{-6 i e-6 i f x}}{1728 a^3 f^4} \]

[Out]

(-9*d^3*E^((-2*I)*e - (2*I)*f*x))/(64*a^3*f^4) - (9*d^3*E^((-4*I)*e - (4*I)*f*x))/(1024*a^3*f^4) - (d^3*E^((-6
*I)*e - (6*I)*f*x))/(1728*a^3*f^4) - (((9*I)/32)*d^2*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(a^3*f^3) - (((9*I)/2
56)*d^2*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(a^3*f^3) - ((I/288)*d^2*E^((-6*I)*e - (6*I)*f*x)*(c + d*x))/(a^3*
f^3) + (9*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (9*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(1
28*a^3*f^2) + (d*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^2)/(96*a^3*f^2) + (((3*I)/16)*E^((-2*I)*e - (2*I)*f*x)*(c
+ d*x)^3)/(a^3*f) + (((3*I)/32)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^3)/(a^3*f) + ((I/48)*E^((-6*I)*e - (6*I)*f*
x)*(c + d*x)^3)/(a^3*f) + (c + d*x)^4/(32*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.404201, antiderivative size = 396, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3729, 2176, 2194} \[ -\frac{9 i d^2 (c+d x) e^{-2 i e-2 i f x}}{32 a^3 f^3}-\frac{9 i d^2 (c+d x) e^{-4 i e-4 i f x}}{256 a^3 f^3}-\frac{i d^2 (c+d x) e^{-6 i e-6 i f x}}{288 a^3 f^3}+\frac{9 d (c+d x)^2 e^{-2 i e-2 i f x}}{32 a^3 f^2}+\frac{9 d (c+d x)^2 e^{-4 i e-4 i f x}}{128 a^3 f^2}+\frac{d (c+d x)^2 e^{-6 i e-6 i f x}}{96 a^3 f^2}+\frac{3 i (c+d x)^3 e^{-2 i e-2 i f x}}{16 a^3 f}+\frac{3 i (c+d x)^3 e^{-4 i e-4 i f x}}{32 a^3 f}+\frac{i (c+d x)^3 e^{-6 i e-6 i f x}}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{9 d^3 e^{-2 i e-2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{-4 i e-4 i f x}}{1024 a^3 f^4}-\frac{d^3 e^{-6 i e-6 i f x}}{1728 a^3 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*d^3*E^((-2*I)*e - (2*I)*f*x))/(64*a^3*f^4) - (9*d^3*E^((-4*I)*e - (4*I)*f*x))/(1024*a^3*f^4) - (d^3*E^((-6
*I)*e - (6*I)*f*x))/(1728*a^3*f^4) - (((9*I)/32)*d^2*E^((-2*I)*e - (2*I)*f*x)*(c + d*x))/(a^3*f^3) - (((9*I)/2
56)*d^2*E^((-4*I)*e - (4*I)*f*x)*(c + d*x))/(a^3*f^3) - ((I/288)*d^2*E^((-6*I)*e - (6*I)*f*x)*(c + d*x))/(a^3*
f^3) + (9*d*E^((-2*I)*e - (2*I)*f*x)*(c + d*x)^2)/(32*a^3*f^2) + (9*d*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^2)/(1
28*a^3*f^2) + (d*E^((-6*I)*e - (6*I)*f*x)*(c + d*x)^2)/(96*a^3*f^2) + (((3*I)/16)*E^((-2*I)*e - (2*I)*f*x)*(c
+ d*x)^3)/(a^3*f) + (((3*I)/32)*E^((-4*I)*e - (4*I)*f*x)*(c + d*x)^3)/(a^3*f) + ((I/48)*E^((-6*I)*e - (6*I)*f*
x)*(c + d*x)^3)/(a^3*f) + (c + d*x)^4/(32*a^3*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))^3} \, dx &=\int \left (\frac{(c+d x)^3}{8 a^3}+\frac{3 e^{-2 i e-2 i f x} (c+d x)^3}{8 a^3}+\frac{3 e^{-4 i e-4 i f x} (c+d x)^3}{8 a^3}+\frac{e^{-6 i e-6 i f x} (c+d x)^3}{8 a^3}\right ) \, dx\\ &=\frac{(c+d x)^4}{32 a^3 d}+\frac{\int e^{-6 i e-6 i f x} (c+d x)^3 \, dx}{8 a^3}+\frac{3 \int e^{-2 i e-2 i f x} (c+d x)^3 \, dx}{8 a^3}+\frac{3 \int e^{-4 i e-4 i f x} (c+d x)^3 \, dx}{8 a^3}\\ &=\frac{3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{(i d) \int e^{-6 i e-6 i f x} (c+d x)^2 \, dx}{16 a^3 f}-\frac{(9 i d) \int e^{-4 i e-4 i f x} (c+d x)^2 \, dx}{32 a^3 f}-\frac{(9 i d) \int e^{-2 i e-2 i f x} (c+d x)^2 \, dx}{16 a^3 f}\\ &=\frac{9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac{9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac{d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac{3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}-\frac{d^2 \int e^{-6 i e-6 i f x} (c+d x) \, dx}{48 a^3 f^2}-\frac{\left (9 d^2\right ) \int e^{-4 i e-4 i f x} (c+d x) \, dx}{64 a^3 f^2}-\frac{\left (9 d^2\right ) \int e^{-2 i e-2 i f x} (c+d x) \, dx}{16 a^3 f^2}\\ &=-\frac{9 i d^2 e^{-2 i e-2 i f x} (c+d x)}{32 a^3 f^3}-\frac{9 i d^2 e^{-4 i e-4 i f x} (c+d x)}{256 a^3 f^3}-\frac{i d^2 e^{-6 i e-6 i f x} (c+d x)}{288 a^3 f^3}+\frac{9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac{9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac{d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac{3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}+\frac{\left (i d^3\right ) \int e^{-6 i e-6 i f x} \, dx}{288 a^3 f^3}+\frac{\left (9 i d^3\right ) \int e^{-4 i e-4 i f x} \, dx}{256 a^3 f^3}+\frac{\left (9 i d^3\right ) \int e^{-2 i e-2 i f x} \, dx}{32 a^3 f^3}\\ &=-\frac{9 d^3 e^{-2 i e-2 i f x}}{64 a^3 f^4}-\frac{9 d^3 e^{-4 i e-4 i f x}}{1024 a^3 f^4}-\frac{d^3 e^{-6 i e-6 i f x}}{1728 a^3 f^4}-\frac{9 i d^2 e^{-2 i e-2 i f x} (c+d x)}{32 a^3 f^3}-\frac{9 i d^2 e^{-4 i e-4 i f x} (c+d x)}{256 a^3 f^3}-\frac{i d^2 e^{-6 i e-6 i f x} (c+d x)}{288 a^3 f^3}+\frac{9 d e^{-2 i e-2 i f x} (c+d x)^2}{32 a^3 f^2}+\frac{9 d e^{-4 i e-4 i f x} (c+d x)^2}{128 a^3 f^2}+\frac{d e^{-6 i e-6 i f x} (c+d x)^2}{96 a^3 f^2}+\frac{3 i e^{-2 i e-2 i f x} (c+d x)^3}{16 a^3 f}+\frac{3 i e^{-4 i e-4 i f x} (c+d x)^3}{32 a^3 f}+\frac{i e^{-6 i e-6 i f x} (c+d x)^3}{48 a^3 f}+\frac{(c+d x)^4}{32 a^3 d}\\ \end{align*}

Mathematica [A]  time = 2.38415, size = 667, normalized size = 1.68 \[ \frac{i \sec ^3(e+f x) \left (243 \left (8 c^2 d f^2 (5+12 i f x)+32 i c^3 f^3+4 c d^2 f \left (24 i f^2 x^2+20 f x-9 i\right )+d^3 \left (32 i f^3 x^3+40 f^2 x^2-36 i f x-17\right )\right ) \cos (e+f x)+16 \left (18 c^2 d f^2 \left (18 f^2 x^2+6 i f x+1\right )+36 c^3 f^3 (6 f x+i)+6 c d^2 f \left (36 f^3 x^3+18 i f^2 x^2+6 f x-i\right )+d^3 \left (54 f^4 x^4+36 i f^3 x^3+18 f^2 x^2-6 i f x-1\right )\right ) \cos (3 (e+f x))+5184 i c^2 d f^4 x^2 \sin (3 (e+f x))-7776 c^2 d f^3 x \sin (e+f x)+1728 c^2 d f^3 x \sin (3 (e+f x))+5832 i c^2 d f^2 \sin (e+f x)-288 i c^2 d f^2 \sin (3 (e+f x))+3456 i c^3 f^4 x \sin (3 (e+f x))-2592 c^3 f^3 \sin (e+f x)+576 c^3 f^3 \sin (3 (e+f x))+3456 i c d^2 f^4 x^3 \sin (3 (e+f x))-7776 c d^2 f^3 x^2 \sin (e+f x)+1728 c d^2 f^3 x^2 \sin (3 (e+f x))+11664 i c d^2 f^2 x \sin (e+f x)-576 i c d^2 f^2 x \sin (3 (e+f x))+6804 c d^2 f \sin (e+f x)-96 c d^2 f \sin (3 (e+f x))+864 i d^3 f^4 x^4 \sin (3 (e+f x))-2592 d^3 f^3 x^3 \sin (e+f x)+576 d^3 f^3 x^3 \sin (3 (e+f x))+5832 i d^3 f^2 x^2 \sin (e+f x)-288 i d^3 f^2 x^2 \sin (3 (e+f x))+6804 d^3 f x \sin (e+f x)-96 d^3 f x \sin (3 (e+f x))-3645 i d^3 \sin (e+f x)+16 i d^3 \sin (3 (e+f x))\right )}{27648 a^3 f^4 (\tan (e+f x)-i)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/27648)*Sec[e + f*x]^3*(243*((32*I)*c^3*f^3 + 8*c^2*d*f^2*(5 + (12*I)*f*x) + 4*c*d^2*f*(-9*I + 20*f*x + (24
*I)*f^2*x^2) + d^3*(-17 - (36*I)*f*x + 40*f^2*x^2 + (32*I)*f^3*x^3))*Cos[e + f*x] + 16*(36*c^3*f^3*(I + 6*f*x)
+ 18*c^2*d*f^2*(1 + (6*I)*f*x + 18*f^2*x^2) + 6*c*d^2*f*(-I + 6*f*x + (18*I)*f^2*x^2 + 36*f^3*x^3) + d^3*(-1
- (6*I)*f*x + 18*f^2*x^2 + (36*I)*f^3*x^3 + 54*f^4*x^4))*Cos[3*(e + f*x)] - (3645*I)*d^3*Sin[e + f*x] + 6804*c
*d^2*f*Sin[e + f*x] + (5832*I)*c^2*d*f^2*Sin[e + f*x] - 2592*c^3*f^3*Sin[e + f*x] + 6804*d^3*f*x*Sin[e + f*x]
+ (11664*I)*c*d^2*f^2*x*Sin[e + f*x] - 7776*c^2*d*f^3*x*Sin[e + f*x] + (5832*I)*d^3*f^2*x^2*Sin[e + f*x] - 777
6*c*d^2*f^3*x^2*Sin[e + f*x] - 2592*d^3*f^3*x^3*Sin[e + f*x] + (16*I)*d^3*Sin[3*(e + f*x)] - 96*c*d^2*f*Sin[3*
(e + f*x)] - (288*I)*c^2*d*f^2*Sin[3*(e + f*x)] + 576*c^3*f^3*Sin[3*(e + f*x)] - 96*d^3*f*x*Sin[3*(e + f*x)] -
(576*I)*c*d^2*f^2*x*Sin[3*(e + f*x)] + 1728*c^2*d*f^3*x*Sin[3*(e + f*x)] + (3456*I)*c^3*f^4*x*Sin[3*(e + f*x)
] - (288*I)*d^3*f^2*x^2*Sin[3*(e + f*x)] + 1728*c*d^2*f^3*x^2*Sin[3*(e + f*x)] + (5184*I)*c^2*d*f^4*x^2*Sin[3*
(e + f*x)] + 576*d^3*f^3*x^3*Sin[3*(e + f*x)] + (3456*I)*c*d^2*f^4*x^3*Sin[3*(e + f*x)] + (864*I)*d^3*f^4*x^4*
Sin[3*(e + f*x)]))/(a^3*f^4*(-I + Tan[e + f*x])^3)

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Maple [A]  time = 0.386, size = 385, normalized size = 1. \begin{align*}{\frac{{d}^{3}{x}^{4}}{32\,{a}^{3}}}+{\frac{c{d}^{2}{x}^{3}}{8\,{a}^{3}}}+{\frac{3\,{c}^{2}d{x}^{2}}{16\,{a}^{3}}}+{\frac{{c}^{3}x}{8\,{a}^{3}}}+{\frac{{\frac{3\,i}{64}} \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}^{3}{f}^{4}}}+{\frac{{\frac{3\,i}{1024}} \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}^{3}{f}^{4}}}+{\frac{{\frac{i}{1728}} \left ( 36\,{d}^{3}{x}^{3}{f}^{3}-18\,i{d}^{3}{f}^{2}{x}^{2}+108\,c{d}^{2}{f}^{3}{x}^{2}-36\,ic{d}^{2}{f}^{2}x+108\,{c}^{2}d{f}^{3}x-18\,i{c}^{2}d{f}^{2}+36\,{c}^{3}{f}^{3}-6\,{d}^{3}fx+i{d}^{3}-6\,c{d}^{2}f \right ){{\rm e}^{-6\,i \left ( fx+e \right ) }}}{{a}^{3}{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))^3,x)

[Out]

1/32/a^3*d^3*x^4+1/8/a^3*c*d^2*x^3+3/16/a^3*c^2*d*x^2+1/8/a^3*c^3*x+3/64*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^3/f^4*exp(
-2*I*(f*x+e))+3/1024*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^3/f^4*exp(-4*I*(f*x+e))+1/1728*I*(36*d^3*x^3*f^3-18*I*d^
3*f^2*x^2+108*c*d^2*f^3*x^2-36*I*c*d^2*f^2*x+108*c^2*d*f^3*x-18*I*c^2*d*f^2+36*c^3*f^3-6*d^3*f*x+I*d^3-6*c*d^2
*f)/a^3/f^4*exp(-6*I*(f*x+e))

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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))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.63448, size = 941, normalized size = 2.38 \begin{align*} \frac{{\left (576 i \, d^{3} f^{3} x^{3} + 576 i \, c^{3} f^{3} + 288 \, c^{2} d f^{2} - 96 i \, c d^{2} f - 16 \, d^{3} +{\left (1728 i \, c d^{2} f^{3} + 288 \, d^{3} f^{2}\right )} x^{2} +{\left (1728 i \, c^{2} d f^{3} + 576 \, c d^{2} f^{2} - 96 i \, d^{3} f\right )} x + 864 \,{\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 (6 i \, f x + 6 i \, e\right )} +{\left (5184 i \, d^{3} f^{3} x^{3} + 5184 i \, c^{3} f^{3} + 7776 \, c^{2} d f^{2} - 7776 i \, c d^{2} f - 3888 \, d^{3} +{\left (15552 i \, c d^{2} f^{3} + 7776 \, d^{3} f^{2}\right )} x^{2} +{\left (15552 i \, c^{2} d f^{3} + 15552 \, c d^{2} f^{2} - 7776 i \, d^{3} f\right )} x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (2592 i \, d^{3} f^{3} x^{3} + 2592 i \, c^{3} f^{3} + 1944 \, c^{2} d f^{2} - 972 i \, c d^{2} f - 243 \, d^{3} +{\left (7776 i \, c d^{2} f^{3} + 1944 \, d^{3} f^{2}\right )} x^{2} +{\left (7776 i \, c^{2} d f^{3} + 3888 \, c d^{2} f^{2} - 972 i \, d^{3} f\right )} x\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{27648 \, a^{3} 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))^3,x, algorithm="fricas")

[Out]

1/27648*(576*I*d^3*f^3*x^3 + 576*I*c^3*f^3 + 288*c^2*d*f^2 - 96*I*c*d^2*f - 16*d^3 + (1728*I*c*d^2*f^3 + 288*d
^3*f^2)*x^2 + (1728*I*c^2*d*f^3 + 576*c*d^2*f^2 - 96*I*d^3*f)*x + 864*(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^(6*I*f*x + 6*I*e) + (5184*I*d^3*f^3*x^3 + 5184*I*c^3*f^3 + 7776*c^2*d*f^2 - 7776*I*c
*d^2*f - 3888*d^3 + (15552*I*c*d^2*f^3 + 7776*d^3*f^2)*x^2 + (15552*I*c^2*d*f^3 + 15552*c*d^2*f^2 - 7776*I*d^3
*f)*x)*e^(4*I*f*x + 4*I*e) + (2592*I*d^3*f^3*x^3 + 2592*I*c^3*f^3 + 1944*c^2*d*f^2 - 972*I*c*d^2*f - 243*d^3 +
(7776*I*c*d^2*f^3 + 1944*d^3*f^2)*x^2 + (7776*I*c^2*d*f^3 + 3888*c*d^2*f^2 - 972*I*d^3*f)*x)*e^(2*I*f*x + 2*I
*e))*e^(-6*I*f*x - 6*I*e)/(a^3*f^4)

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Sympy [A]  time = 3.43186, size = 947, normalized size = 2.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((((2359296*I*a**33*c**3*f**29*exp(42*I*e) + 7077888*I*a**33*c**2*d*f**29*x*exp(42*I*e) + 1179648*a**
33*c**2*d*f**28*exp(42*I*e) + 7077888*I*a**33*c*d**2*f**29*x**2*exp(42*I*e) + 2359296*a**33*c*d**2*f**28*x*exp
(42*I*e) - 393216*I*a**33*c*d**2*f**27*exp(42*I*e) + 2359296*I*a**33*d**3*f**29*x**3*exp(42*I*e) + 1179648*a**
33*d**3*f**28*x**2*exp(42*I*e) - 393216*I*a**33*d**3*f**27*x*exp(42*I*e) - 65536*a**33*d**3*f**26*exp(42*I*e))
*exp(-6*I*f*x) + (10616832*I*a**33*c**3*f**29*exp(44*I*e) + 31850496*I*a**33*c**2*d*f**29*x*exp(44*I*e) + 7962
624*a**33*c**2*d*f**28*exp(44*I*e) + 31850496*I*a**33*c*d**2*f**29*x**2*exp(44*I*e) + 15925248*a**33*c*d**2*f*
*28*x*exp(44*I*e) - 3981312*I*a**33*c*d**2*f**27*exp(44*I*e) + 10616832*I*a**33*d**3*f**29*x**3*exp(44*I*e) +
7962624*a**33*d**3*f**28*x**2*exp(44*I*e) - 3981312*I*a**33*d**3*f**27*x*exp(44*I*e) - 995328*a**33*d**3*f**26
*exp(44*I*e))*exp(-4*I*f*x) + (21233664*I*a**33*c**3*f**29*exp(46*I*e) + 63700992*I*a**33*c**2*d*f**29*x*exp(4
6*I*e) + 31850496*a**33*c**2*d*f**28*exp(46*I*e) + 63700992*I*a**33*c*d**2*f**29*x**2*exp(46*I*e) + 63700992*a
**33*c*d**2*f**28*x*exp(46*I*e) - 31850496*I*a**33*c*d**2*f**27*exp(46*I*e) + 21233664*I*a**33*d**3*f**29*x**3
*exp(46*I*e) + 31850496*a**33*d**3*f**28*x**2*exp(46*I*e) - 31850496*I*a**33*d**3*f**27*x*exp(46*I*e) - 159252
48*a**33*d**3*f**26*exp(46*I*e))*exp(-2*I*f*x))*exp(-48*I*e)/(113246208*a**36*f**30), Ne(113246208*a**36*f**30
*exp(48*I*e), 0)), (x**4*(3*d**3*exp(4*I*e) + 3*d**3*exp(2*I*e) + d**3)*exp(-6*I*e)/(32*a**3) + x**3*(3*c*d**2
*exp(4*I*e) + 3*c*d**2*exp(2*I*e) + c*d**2)*exp(-6*I*e)/(8*a**3) + x**2*(9*c**2*d*exp(4*I*e) + 9*c**2*d*exp(2*
I*e) + 3*c**2*d)*exp(-6*I*e)/(16*a**3) + x*(3*c**3*exp(4*I*e) + 3*c**3*exp(2*I*e) + c**3)*exp(-6*I*e)/(8*a**3)
, True)) + c**3*x/(8*a**3) + 3*c**2*d*x**2/(16*a**3) + c*d**2*x**3/(8*a**3) + d**3*x**4/(32*a**3)

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Giac [A]  time = 1.23974, size = 774, normalized size = 1.95 \begin{align*} \frac{{\left (864 \, d^{3} f^{4} x^{4} e^{\left (6 i \, f x + 6 i \, e\right )} + 3456 \, c d^{2} f^{4} x^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + 5184 \, c^{2} d f^{4} x^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 5184 i \, d^{3} f^{3} x^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, d^{3} f^{3} x^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, d^{3} f^{3} x^{3} + 3456 \, c^{3} f^{4} x e^{\left (6 i \, f x + 6 i \, e\right )} + 15552 i \, c d^{2} f^{3} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 i \, c d^{2} f^{3} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 1728 i \, c d^{2} f^{3} x^{2} + 15552 i \, c^{2} d f^{3} x e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 \, d^{3} f^{2} x^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 7776 i \, c^{2} d f^{3} x e^{\left (2 i \, f x + 2 i \, e\right )} + 1944 \, d^{3} f^{2} x^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 1728 i \, c^{2} d f^{3} x + 288 \, d^{3} f^{2} x^{2} + 5184 i \, c^{3} f^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 15552 \, c d^{2} f^{2} x e^{\left (4 i \, f x + 4 i \, e\right )} + 2592 i \, c^{3} f^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 3888 \, c d^{2} f^{2} x e^{\left (2 i \, f x + 2 i \, e\right )} + 576 i \, c^{3} f^{3} + 576 \, c d^{2} f^{2} x + 7776 \, c^{2} d f^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 7776 i \, d^{3} f x e^{\left (4 i \, f x + 4 i \, e\right )} + 1944 \, c^{2} d f^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 972 i \, d^{3} f x e^{\left (2 i \, f x + 2 i \, e\right )} + 288 \, c^{2} d f^{2} - 96 i \, d^{3} f x - 7776 i \, c d^{2} f e^{\left (4 i \, f x + 4 i \, e\right )} - 972 i \, c d^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - 96 i \, c d^{2} f - 3888 \, d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 243 \, d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 16 \, d^{3}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{27648 \, a^{3} 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))^3,x, algorithm="giac")

[Out]

1/27648*(864*d^3*f^4*x^4*e^(6*I*f*x + 6*I*e) + 3456*c*d^2*f^4*x^3*e^(6*I*f*x + 6*I*e) + 5184*c^2*d*f^4*x^2*e^(
6*I*f*x + 6*I*e) + 5184*I*d^3*f^3*x^3*e^(4*I*f*x + 4*I*e) + 2592*I*d^3*f^3*x^3*e^(2*I*f*x + 2*I*e) + 576*I*d^3
*f^3*x^3 + 3456*c^3*f^4*x*e^(6*I*f*x + 6*I*e) + 15552*I*c*d^2*f^3*x^2*e^(4*I*f*x + 4*I*e) + 7776*I*c*d^2*f^3*x
^2*e^(2*I*f*x + 2*I*e) + 1728*I*c*d^2*f^3*x^2 + 15552*I*c^2*d*f^3*x*e^(4*I*f*x + 4*I*e) + 7776*d^3*f^2*x^2*e^(
4*I*f*x + 4*I*e) + 7776*I*c^2*d*f^3*x*e^(2*I*f*x + 2*I*e) + 1944*d^3*f^2*x^2*e^(2*I*f*x + 2*I*e) + 1728*I*c^2*
d*f^3*x + 288*d^3*f^2*x^2 + 5184*I*c^3*f^3*e^(4*I*f*x + 4*I*e) + 15552*c*d^2*f^2*x*e^(4*I*f*x + 4*I*e) + 2592*
I*c^3*f^3*e^(2*I*f*x + 2*I*e) + 3888*c*d^2*f^2*x*e^(2*I*f*x + 2*I*e) + 576*I*c^3*f^3 + 576*c*d^2*f^2*x + 7776*
c^2*d*f^2*e^(4*I*f*x + 4*I*e) - 7776*I*d^3*f*x*e^(4*I*f*x + 4*I*e) + 1944*c^2*d*f^2*e^(2*I*f*x + 2*I*e) - 972*
I*d^3*f*x*e^(2*I*f*x + 2*I*e) + 288*c^2*d*f^2 - 96*I*d^3*f*x - 7776*I*c*d^2*f*e^(4*I*f*x + 4*I*e) - 972*I*c*d^
2*f*e^(2*I*f*x + 2*I*e) - 96*I*c*d^2*f - 3888*d^3*e^(4*I*f*x + 4*I*e) - 243*d^3*e^(2*I*f*x + 2*I*e) - 16*d^3)*
e^(-6*I*f*x - 6*I*e)/(a^3*f^4)