\(\int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx\) [1082]

Optimal result

Integrand size = 28, antiderivative size = 140 \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\frac {(c-i d)^3 x}{8 a^3}+\frac {(c+i d) (c-3 i d) (i c+d)}{8 a^3 f (1+i \tan (e+f x))}+\frac {(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3} \]

[Out]

1/8*(c-I*d)^3*x/a^3+1/8*(c+I*d)*(c-3*I*d)*(I*c+d)/a^3/f/(1+I*tan(f*x+e))+1/8*(c+I*d)^2*(I*c+d)/a/f/(a+I*a*tan(
f*x+e))^2+1/6*I*(c+d*tan(f*x+e))^3/f/(a+I*a*tan(f*x+e))^3

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3627, 3621, 3607, 8} \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\frac {(c+i d) (c-3 i d) (d+i c)}{8 a^3 f (1+i \tan (e+f x))}+\frac {x (c-i d)^3}{8 a^3}+\frac {i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac {(c+i d)^2 (d+i c)}{8 a f (a+i a \tan (e+f x))^2} \]

[In]

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

[Out]

((c - I*d)^3*x)/(8*a^3) + ((c + I*d)*(c - (3*I)*d)*(I*c + d))/(8*a^3*f*(1 + I*Tan[e + f*x])) + ((c + I*d)^2*(I
*c + d))/(8*a*f*(a + I*a*Tan[e + f*x])^2) + ((I/6)*(c + d*Tan[e + f*x])^3)/(f*(a + I*a*Tan[e + f*x])^3)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3607

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

Rule 3621

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(
-b)*(a*c + b*d)^2*((a + b*Tan[e + f*x])^m/(2*a^3*f*m)), x] + Dist[1/(2*a^2), Int[(a + b*Tan[e + f*x])^(m + 1)*
Simp[a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a
*d, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]

Rule 3627

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

Rubi steps \begin{align*} \text {integral}& = \frac {i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac {(c-i d) \int \frac {(c+d \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx}{2 a} \\ & = \frac {(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac {(c-i d) \int \frac {a \left (c^2-2 i c d+d^2\right )-2 i a d^2 \tan (e+f x)}{a+i a \tan (e+f x)} \, dx}{4 a^3} \\ & = \frac {(c+i d) (c-3 i d) (i c+d)}{8 a^3 f (1+i \tan (e+f x))}+\frac {(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3}+\frac {(c-i d)^3 \int 1 \, dx}{8 a^3} \\ & = \frac {(c-i d)^3 x}{8 a^3}+\frac {(c+i d) (c-3 i d) (i c+d)}{8 a^3 f (1+i \tan (e+f x))}+\frac {(c+i d)^2 (i c+d)}{8 a f (a+i a \tan (e+f x))^2}+\frac {i (c+d \tan (e+f x))^3}{6 f (a+i a \tan (e+f x))^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.01 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.93 \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\frac {\sec ^3(e+f x) \left (-9 \left (3 c^3-i c^2 d+3 c d^2-i d^3\right ) \cos (e+f x)-13 c^3 \cos (3 (e+f x))+15 i c^2 d \cos (3 (e+f x))+15 c d^2 \cos (3 (e+f x))+3 i d^3 \cos (3 (e+f x))-9 i c^3 \sin (e+f x)-27 c^2 d \sin (e+f x)-9 i c d^2 \sin (e+f x)-27 d^3 \sin (e+f x)-12 (i c+d)^3 \arctan (\tan (e+f x)) (\cos (3 (e+f x))+i \sin (3 (e+f x)))-9 i c^3 \sin (3 (e+f x))-27 c^2 d \sin (3 (e+f x))+3 i c d^2 \sin (3 (e+f x))+d^3 \sin (3 (e+f x))\right )}{96 a^3 f (-i+\tan (e+f x))^3} \]

[In]

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

[Out]

(Sec[e + f*x]^3*(-9*(3*c^3 - I*c^2*d + 3*c*d^2 - I*d^3)*Cos[e + f*x] - 13*c^3*Cos[3*(e + f*x)] + (15*I)*c^2*d*
Cos[3*(e + f*x)] + 15*c*d^2*Cos[3*(e + f*x)] + (3*I)*d^3*Cos[3*(e + f*x)] - (9*I)*c^3*Sin[e + f*x] - 27*c^2*d*
Sin[e + f*x] - (9*I)*c*d^2*Sin[e + f*x] - 27*d^3*Sin[e + f*x] - 12*(I*c + d)^3*ArcTan[Tan[e + f*x]]*(Cos[3*(e
+ f*x)] + I*Sin[3*(e + f*x)]) - (9*I)*c^3*Sin[3*(e + f*x)] - 27*c^2*d*Sin[3*(e + f*x)] + (3*I)*c*d^2*Sin[3*(e
+ f*x)] + d^3*Sin[3*(e + f*x)]))/(96*a^3*f*(-I + Tan[e + f*x])^3)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (122 ) = 244\).

Time = 0.38 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.10

method	result	size
risch	\(\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c \,d^{2}}{32 a^{3} f}-\frac {i {\mathrm e}^{-6 i \left (f x +e \right )} c \,d^{2}}{16 a^{3} f}+\frac {x \,c^{3}}{8 a^{3}}-\frac {3 x c \,d^{2}}{8 a^{3}}+\frac {3 \,{\mathrm e}^{-2 i \left (f x +e \right )} c^{2} d}{16 a^{3} f}+\frac {3 \,{\mathrm e}^{-2 i \left (f x +e \right )} d^{3}}{16 a^{3} f}+\frac {i x \,d^{3}}{8 a^{3}}-\frac {3 i x \,c^{2} d}{8 a^{3}}-\frac {3 \,{\mathrm e}^{-4 i \left (f x +e \right )} c^{2} d}{32 a^{3} f}-\frac {3 \,{\mathrm e}^{-4 i \left (f x +e \right )} d^{3}}{32 a^{3} f}+\frac {3 i {\mathrm e}^{-4 i \left (f x +e \right )} c^{3}}{32 a^{3} f}+\frac {i {\mathrm e}^{-6 i \left (f x +e \right )} c^{3}}{48 a^{3} f}-\frac {{\mathrm e}^{-6 i \left (f x +e \right )} c^{2} d}{16 a^{3} f}+\frac {{\mathrm e}^{-6 i \left (f x +e \right )} d^{3}}{48 a^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c^{3}}{16 a^{3} f}+\frac {3 i {\mathrm e}^{-2 i \left (f x +e \right )} c \,d^{2}}{16 a^{3} f}\)	\(294\)
derivativedivides	\(-\frac {3 c \,d^{2} \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}-\frac {3 c^{2} d}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 i c^{2} d}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {c \,d^{2}}{2 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {9 i c \,d^{2}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 c \,d^{2}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {c^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {c^{3} \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}-\frac {3 i c^{2} d \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}+\frac {5 d^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {c^{3}}{6 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i d^{3}}{6 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i d^{3} \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}-\frac {i c^{2} d}{2 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}\)	\(354\)
default	\(-\frac {3 c \,d^{2} \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}-\frac {3 c^{2} d}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 i c^{2} d}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {c \,d^{2}}{2 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {9 i c \,d^{2}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 c \,d^{2}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}-\frac {i c^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {c^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}+\frac {c^{3} \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}-\frac {3 i c^{2} d \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}+\frac {5 d^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {c^{3}}{6 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i d^{3}}{6 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}+\frac {i d^{3} \arctan \left (\tan \left (f x +e \right )\right )}{8 f \,a^{3}}-\frac {i c^{2} d}{2 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )^{3}}-\frac {7 i d^{3}}{8 f \,a^{3} \left (\tan \left (f x +e \right )-i\right )}\)	\(354\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\frac {{\left (12 \, {\left (c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + 2 i \, c^{3} - 6 \, c^{2} d - 6 i \, c d^{2} + 2 \, d^{3} - 18 \, {\left (-i \, c^{3} - c^{2} d - i \, c d^{2} - d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} - 9 \, {\left (-i \, c^{3} + c^{2} d - i \, c d^{2} + d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{96 \, a^{3} f} \]

[In]

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

[Out]

1/96*(12*(c^3 - 3*I*c^2*d - 3*c*d^2 + I*d^3)*f*x*e^(6*I*f*x + 6*I*e) + 2*I*c^3 - 6*c^2*d - 6*I*c*d^2 + 2*d^3 -
18*(-I*c^3 - c^2*d - I*c*d^2 - d^3)*e^(4*I*f*x + 4*I*e) - 9*(-I*c^3 + c^2*d - I*c*d^2 + d^3)*e^(2*I*f*x + 2*I
*e))*e^(-6*I*f*x - 6*I*e)/(a^3*f)

Sympy [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.94 \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\begin {cases} \frac {\left (\left (512 i a^{6} c^{3} f^{2} e^{6 i e} - 1536 a^{6} c^{2} d f^{2} e^{6 i e} - 1536 i a^{6} c d^{2} f^{2} e^{6 i e} + 512 a^{6} d^{3} f^{2} e^{6 i e}\right ) e^{- 6 i f x} + \left (2304 i a^{6} c^{3} f^{2} e^{8 i e} - 2304 a^{6} c^{2} d f^{2} e^{8 i e} + 2304 i a^{6} c d^{2} f^{2} e^{8 i e} - 2304 a^{6} d^{3} f^{2} e^{8 i e}\right ) e^{- 4 i f x} + \left (4608 i a^{6} c^{3} f^{2} e^{10 i e} + 4608 a^{6} c^{2} d f^{2} e^{10 i e} + 4608 i a^{6} c d^{2} f^{2} e^{10 i e} + 4608 a^{6} d^{3} f^{2} e^{10 i e}\right ) e^{- 2 i f x}\right ) e^{- 12 i e}}{24576 a^{9} f^{3}} & \text {for}\: a^{9} f^{3} e^{12 i e} \neq 0 \\x \left (- \frac {c^{3} - 3 i c^{2} d - 3 c d^{2} + i d^{3}}{8 a^{3}} + \frac {\left (c^{3} e^{6 i e} + 3 c^{3} e^{4 i e} + 3 c^{3} e^{2 i e} + c^{3} - 3 i c^{2} d e^{6 i e} - 3 i c^{2} d e^{4 i e} + 3 i c^{2} d e^{2 i e} + 3 i c^{2} d - 3 c d^{2} e^{6 i e} + 3 c d^{2} e^{4 i e} + 3 c d^{2} e^{2 i e} - 3 c d^{2} + i d^{3} e^{6 i e} - 3 i d^{3} e^{4 i e} + 3 i d^{3} e^{2 i e} - i d^{3}\right ) e^{- 6 i e}}{8 a^{3}}\right ) & \text {otherwise} \end {cases} + \frac {x \left (c^{3} - 3 i c^{2} d - 3 c d^{2} + i d^{3}\right )}{8 a^{3}} \]

[In]

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

[Out]

Piecewise((((512*I*a**6*c**3*f**2*exp(6*I*e) - 1536*a**6*c**2*d*f**2*exp(6*I*e) - 1536*I*a**6*c*d**2*f**2*exp(
6*I*e) + 512*a**6*d**3*f**2*exp(6*I*e))*exp(-6*I*f*x) + (2304*I*a**6*c**3*f**2*exp(8*I*e) - 2304*a**6*c**2*d*f
**2*exp(8*I*e) + 2304*I*a**6*c*d**2*f**2*exp(8*I*e) - 2304*a**6*d**3*f**2*exp(8*I*e))*exp(-4*I*f*x) + (4608*I*
a**6*c**3*f**2*exp(10*I*e) + 4608*a**6*c**2*d*f**2*exp(10*I*e) + 4608*I*a**6*c*d**2*f**2*exp(10*I*e) + 4608*a*
*6*d**3*f**2*exp(10*I*e))*exp(-2*I*f*x))*exp(-12*I*e)/(24576*a**9*f**3), Ne(a**9*f**3*exp(12*I*e), 0)), (x*(-(
c**3 - 3*I*c**2*d - 3*c*d**2 + I*d**3)/(8*a**3) + (c**3*exp(6*I*e) + 3*c**3*exp(4*I*e) + 3*c**3*exp(2*I*e) + c
**3 - 3*I*c**2*d*exp(6*I*e) - 3*I*c**2*d*exp(4*I*e) + 3*I*c**2*d*exp(2*I*e) + 3*I*c**2*d - 3*c*d**2*exp(6*I*e)
+ 3*c*d**2*exp(4*I*e) + 3*c*d**2*exp(2*I*e) - 3*c*d**2 + I*d**3*exp(6*I*e) - 3*I*d**3*exp(4*I*e) + 3*I*d**3*e
xp(2*I*e) - I*d**3)*exp(-6*I*e)/(8*a**3)), True)) + x*(c**3 - 3*I*c**2*d - 3*c*d**2 + I*d**3)/(8*a**3)

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (112) = 224\).

Time = 0.91 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.92 \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=-\frac {\frac {6 \, {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} \log \left (\tan \left (f x + e\right ) + i\right )}{a^{3}} + \frac {6 \, {\left (i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} - d^{3}\right )} \log \left (\tan \left (f x + e\right ) - i\right )}{a^{3}} + \frac {-11 i \, c^{3} \tan \left (f x + e\right )^{3} - 33 \, c^{2} d \tan \left (f x + e\right )^{3} + 33 i \, c d^{2} \tan \left (f x + e\right )^{3} + 11 \, d^{3} \tan \left (f x + e\right )^{3} - 45 \, c^{3} \tan \left (f x + e\right )^{2} + 135 i \, c^{2} d \tan \left (f x + e\right )^{2} + 135 \, c d^{2} \tan \left (f x + e\right )^{2} + 51 i \, d^{3} \tan \left (f x + e\right )^{2} + 69 i \, c^{3} \tan \left (f x + e\right ) + 207 \, c^{2} d \tan \left (f x + e\right ) - 63 i \, c d^{2} \tan \left (f x + e\right ) + 75 \, d^{3} \tan \left (f x + e\right ) + 51 \, c^{3} - 57 i \, c^{2} d - 9 \, c d^{2} - 29 i \, d^{3}}{a^{3} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \]

[In]

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

[Out]

-1/96*(6*(-I*c^3 - 3*c^2*d + 3*I*c*d^2 + d^3)*log(tan(f*x + e) + I)/a^3 + 6*(I*c^3 + 3*c^2*d - 3*I*c*d^2 - d^3
)*log(tan(f*x + e) - I)/a^3 + (-11*I*c^3*tan(f*x + e)^3 - 33*c^2*d*tan(f*x + e)^3 + 33*I*c*d^2*tan(f*x + e)^3
+ 11*d^3*tan(f*x + e)^3 - 45*c^3*tan(f*x + e)^2 + 135*I*c^2*d*tan(f*x + e)^2 + 135*c*d^2*tan(f*x + e)^2 + 51*I
*d^3*tan(f*x + e)^2 + 69*I*c^3*tan(f*x + e) + 207*c^2*d*tan(f*x + e) - 63*I*c*d^2*tan(f*x + e) + 75*d^3*tan(f*
x + e) + 51*c^3 - 57*I*c^2*d - 9*c*d^2 - 29*I*d^3)/(a^3*(tan(f*x + e) - I)^3))/f

Mupad [B] (verification not implemented)

Time = 6.09 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.31 \[ \int \frac {(c+d \tan (e+f x))^3}{(a+i a \tan (e+f x))^3} \, dx=\frac {\frac {5\,d^3}{12\,a^3}-\mathrm {tan}\left (e+f\,x\right )\,\left (\frac {3\,c^3}{8\,a^3}-\frac {d^3\,9{}\mathrm {i}}{8\,a^3}+\frac {3\,c\,d^2}{8\,a^3}-\frac {c^2\,d\,9{}\mathrm {i}}{8\,a^3}\right )-{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {7\,d^3}{8\,a^3}+\frac {3\,c^2\,d}{8\,a^3}+\frac {c^3\,1{}\mathrm {i}}{8\,a^3}-\frac {c\,d^2\,3{}\mathrm {i}}{8\,a^3}\right )+\frac {c^2\,d}{4\,a^3}+\frac {c^3\,5{}\mathrm {i}}{12\,a^3}+\frac {c\,d^2\,1{}\mathrm {i}}{4\,a^3}}{f\,\left (-{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (e+f\,x\right )}^2+\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+1\right )}+\frac {x\,{\left (d+c\,1{}\mathrm {i}\right )}^3\,1{}\mathrm {i}}{8\,a^3} \]

[In]

int((c + d*tan(e + f*x))^3/(a + a*tan(e + f*x)*1i)^3,x)

[Out]

((c^3*5i)/(12*a^3) - tan(e + f*x)*((3*c^3)/(8*a^3) - (d^3*9i)/(8*a^3) + (3*c*d^2)/(8*a^3) - (c^2*d*9i)/(8*a^3)
) + (5*d^3)/(12*a^3) - tan(e + f*x)^2*((c^3*1i)/(8*a^3) + (7*d^3)/(8*a^3) - (c*d^2*3i)/(8*a^3) + (3*c^2*d)/(8*
a^3)) + (c*d^2*1i)/(4*a^3) + (c^2*d)/(4*a^3))/(f*(tan(e + f*x)*3i - 3*tan(e + f*x)^2 - tan(e + f*x)^3*1i + 1))
+ (x*(c*1i + d)^3*1i)/(8*a^3)