3.1094 \(\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=357 \[ \frac{d \left (5 i c^2 d+c^3-11 c d^2+25 i d^3\right )}{8 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))}+\frac{c^2+5 i c d-12 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))}+\frac{x \left (-10 c^3 d^2-10 i c^2 d^3+5 i c^4 d+c^5-35 c d^4+25 i d^5\right )}{8 a^3 (c-i d)^2 (c+i d)^5}+\frac{d^4 (5 c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (-d+i c)^5 (c-i d)^2}+\frac{-11 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \]

[Out]

((c^5 + (5*I)*c^4*d - 10*c^3*d^2 - (10*I)*c^2*d^3 - 35*c*d^4 + (25*I)*d^5)*x)/(8*a^3*(c - I*d)^2*(c + I*d)^5)
+ ((5*c - (3*I)*d)*d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(I*c - d)^5*(c - I*d)^2*f) + (d*(c^3 + (5*I)
*c^2*d - 11*c*d^2 + (25*I)*d^3))/(8*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])) - 1/(6*(I*c - d)*f*(a +
I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])) + ((3*I)*c - 11*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2*(c +
d*Tan[e + f*x])) + (c^2 + (5*I)*c*d - 12*d^2)/(8*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x]
))

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Rubi [A]  time = 0.914741, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3559, 3596, 3529, 3531, 3530} \[ \frac{d \left (5 i c^2 d+c^3-11 c d^2+25 i d^3\right )}{8 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))}+\frac{c^2+5 i c d-12 d^2}{8 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))}+\frac{x \left (-10 c^3 d^2-10 i c^2 d^3+5 i c^4 d+c^5-35 c d^4+25 i d^5\right )}{8 a^3 (c-i d)^2 (c+i d)^5}+\frac{d^4 (5 c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (-d+i c)^5 (c-i d)^2}+\frac{-11 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]

[Out]

((c^5 + (5*I)*c^4*d - 10*c^3*d^2 - (10*I)*c^2*d^3 - 35*c*d^4 + (25*I)*d^5)*x)/(8*a^3*(c - I*d)^2*(c + I*d)^5)
+ ((5*c - (3*I)*d)*d^4*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(I*c - d)^5*(c - I*d)^2*f) + (d*(c^3 + (5*I)
*c^2*d - 11*c*d^2 + (25*I)*d^3))/(8*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*x])) - 1/(6*(I*c - d)*f*(a +
I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])) + ((3*I)*c - 11*d)/(24*a*(c + I*d)^2*f*(a + I*a*Tan[e + f*x])^2*(c +
d*Tan[e + f*x])) + (c^2 + (5*I)*c*d - 12*d^2)/(8*(I*c - d)^3*f*(a^3 + I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x]
))

Rule 3559

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 + 1))/(2*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d))
, Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[b*c*m - a*d*(2*m + n + 1) + b*d*(m + n + 1)*Tan
[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2
+ d^2, 0] && LtQ[m, 0] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3596

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

Rule 3529

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 + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3531

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

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \, dx &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}-\frac{\int \frac{-a (3 i c-7 d)-4 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2} \, dx}{6 a^2 (i c-d)}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}+\frac{3 i c-11 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}-\frac{\int \frac{-3 a^2 \left (2 c^2+7 i c d-13 d^2\right )-3 a^2 (3 c+11 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^2} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}+\frac{3 i c-11 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac{c^2+5 i c d-12 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))}-\frac{\int \frac{6 a^3 \left (i c^3-5 c^2 d-13 i c d^2+25 d^3\right )+12 a^3 d \left (i c^2-5 c d-12 i d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac{d \left (c^3+5 i c^2 d-11 c d^2+25 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}+\frac{3 i c-11 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac{c^2+5 i c d-12 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))}-\frac{\int \frac{-6 a^3 \left (5 c^3 d-i \left (c^4-11 c^2 d^2-15 i c d^3-24 d^4\right )\right )-6 a^3 d \left (5 c^2 d-i \left (c^3-11 c d^2+25 i d^3\right )\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )}\\ &=\frac{\left (c^5+5 i c^4 d-10 c^3 d^2-10 i c^2 d^3-35 c d^4+25 i d^5\right ) x}{8 a^3 (c-i d)^2 (c+i d)^5}+\frac{d \left (c^3+5 i c^2 d-11 c d^2+25 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}+\frac{3 i c-11 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac{c^2+5 i c d-12 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))}-\frac{\left (d^4 (5 i c+3 d)\right ) \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^3 (c-i d)^2 (c+i d)^5}\\ &=\frac{\left (c^5+5 i c^4 d-10 c^3 d^2-10 i c^2 d^3-35 c d^4+25 i d^5\right ) x}{8 a^3 (c-i d)^2 (c+i d)^5}-\frac{d^4 (5 i c+3 d) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (c-i d)^2 (c+i d)^5 f}+\frac{d \left (c^3+5 i c^2 d-11 c d^2+25 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))}+\frac{3 i c-11 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))}+\frac{c^2+5 i c d-12 d^2}{8 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))}\\ \end{align*}

Mathematica [A]  time = 4.79258, size = 633, normalized size = 1.77 \[ \frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\frac{6 i (c+i d) \left (3 c^2+14 i c d-23 d^2\right ) (\cos (e)+i \sin (e)) \cos (2 f x)}{f}+\frac{6 (c+i d) \left (3 c^2+14 i c d-23 d^2\right ) (\cos (e)+i \sin (e)) \sin (2 f x)}{f}+\frac{96 d^4 (5 c-3 i d) \left (\cos \left (\frac{3 e}{2}\right )+i \sin \left (\frac{3 e}{2}\right )\right )^2 \tan ^{-1}\left (\frac{\left (d^3-3 c^2 d\right ) \cos (f x)-c \left (c^2-3 d^2\right ) \sin (f x)}{d \left (d^2-3 c^2\right ) \sin (f x)+c \left (c^2-3 d^2\right ) \cos (f x)}\right )}{f (c-i d)^2}+\frac{12 x \left (-10 c^3 d^2-10 i c^2 d^3+5 i c^4 d+c^5-35 c d^4+25 i d^5\right ) (\cos (3 e)+i \sin (3 e))}{(c-i d)^2}+\frac{96 d^5 (d-i c) (\cos (3 e)+i \sin (3 e)) \sin (f x)}{f (c-i d) (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x))}-\frac{48 i d^4 (5 c-3 i d) \left (\cos \left (\frac{3 e}{2}\right )+i \sin \left (\frac{3 e}{2}\right )\right )^2 \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )}{f (c-i d)^2}+\frac{96 d^4 x (5 c-3 i d) (\cos (3 e)+i \sin (3 e))}{(c-i d)^2}+\frac{3 (c+i d)^2 (3 c+7 i d) (\sin (e)+i \cos (e)) \cos (4 f x)}{f}+\frac{2 (c+i d)^3 (\sin (3 e)+i \cos (3 e)) \cos (6 f x)}{f}+\frac{3 (c+i d)^2 (3 c+7 i d) (\cos (e)-i \sin (e)) \sin (4 f x)}{f}+\frac{2 (c+i d)^3 (\cos (3 e)-i \sin (3 e)) \sin (6 f x)}{f}\right )}{96 (c+i d)^5 (a+i a \tan (e+f x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2),x]

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(((6*I)*(c + I*d)*(3*c^2 + (14*I)*c*d - 23*d^2)*Cos[2*f*x]*(Cos[e] +
I*Sin[e]))/f + (3*(c + I*d)^2*(3*c + (7*I)*d)*Cos[4*f*x]*(I*Cos[e] + Sin[e]))/f + (96*(5*c - (3*I)*d)*d^4*Arc
Tan[((-3*c^2*d + d^3)*Cos[f*x] - c*(c^2 - 3*d^2)*Sin[f*x])/(c*(c^2 - 3*d^2)*Cos[f*x] + d*(-3*c^2 + d^2)*Sin[f*
x])]*(Cos[(3*e)/2] + I*Sin[(3*e)/2])^2)/((c - I*d)^2*f) - ((48*I)*(5*c - (3*I)*d)*d^4*Log[(c*Cos[e + f*x] + d*
Sin[e + f*x])^2]*(Cos[(3*e)/2] + I*Sin[(3*e)/2])^2)/((c - I*d)^2*f) + (96*(5*c - (3*I)*d)*d^4*x*(Cos[3*e] + I*
Sin[3*e]))/(c - I*d)^2 + (12*(c^5 + (5*I)*c^4*d - 10*c^3*d^2 - (10*I)*c^2*d^3 - 35*c*d^4 + (25*I)*d^5)*x*(Cos[
3*e] + I*Sin[3*e]))/(c - I*d)^2 + (2*(c + I*d)^3*Cos[6*f*x]*(I*Cos[3*e] + Sin[3*e]))/f + (6*(c + I*d)*(3*c^2 +
(14*I)*c*d - 23*d^2)*(Cos[e] + I*Sin[e])*Sin[2*f*x])/f + (3*(c + I*d)^2*(3*c + (7*I)*d)*(Cos[e] - I*Sin[e])*S
in[4*f*x])/f + (2*(c + I*d)^3*(Cos[3*e] - I*Sin[3*e])*Sin[6*f*x])/f + (96*d^5*((-I)*c + d)*(Cos[3*e] + I*Sin[3
*e])*Sin[f*x])/((c - I*d)*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x]))))/(96*(c + I*d)^5*(a + I*
a*Tan[e + f*x])^3)

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Maple [B]  time = 0.067, size = 692, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x)

[Out]

I/f/a^3*d^6/(I*d-c)^2/(c+I*d)^5/(c+d*tan(f*x+e))-1/8*I/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^2*c^3+7/8/f/a^3/(c+I*d)^
5/(tan(f*x+e)-I)^2*c^2*d-5/8/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^2*d^3+23/16*I/f/a^3/(c+I*d)^5*ln(tan(f*x+e)-I)*c*d
^2-1/2*I/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^3*c^2*d+7/16/f/a^3/(c+I*d)^5*ln(tan(f*x+e)-I)*c^2*d-49/16/f/a^3/(c+I*d
)^5*ln(tan(f*x+e)-I)*d^3+7/8*I/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)*c^2*d+11/8*I/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^2*c*
d^2+1/8/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)*c^3-23/8/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)*c*d^2-5*I/f/a^3*d^4/(I*d-c)^2/(
c+I*d)^5*ln(c+d*tan(f*x+e))*c-17/8*I/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)*d^3-1/6/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^3*c
^3+1/2/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^3*c*d^2+1/16*I/f/a^3/(I*d-c)^2*ln(tan(f*x+e)+I)+I/f/a^3*d^4/(I*d-c)^2/(c
+I*d)^5/(c+d*tan(f*x+e))*c^2+1/6*I/f/a^3/(c+I*d)^5/(tan(f*x+e)-I)^3*d^3-1/16*I/f/a^3/(c+I*d)^5*ln(tan(f*x+e)-I
)*c^3-3/f/a^3*d^5/(I*d-c)^2/(c+I*d)^5*ln(c+d*tan(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(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 2.37532, size = 1524, normalized size = 4.27 \begin{align*} -\frac{-2 i \, c^{6} + 4 \, c^{5} d - 2 i \, c^{4} d^{2} + 8 \, c^{3} d^{3} + 2 i \, c^{2} d^{4} + 4 \, c d^{5} + 2 i \, d^{6} -{\left (12 \, c^{6} + 48 i \, c^{5} d - 60 \, c^{4} d^{2} - 1020 \, c^{2} d^{4} + 1488 i \, c d^{5} + 588 \, d^{6}\right )} f x e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (-18 i \, c^{6} + 48 \, c^{5} d - 30 i \, c^{4} d^{2} + 240 \, c^{3} d^{3} - 150 i \, c^{2} d^{4} - 330 i \, d^{6} -{\left (12 \, c^{6} + 72 i \, c^{5} d - 180 \, c^{4} d^{2} - 240 i \, c^{3} d^{3} - 780 \, c^{2} d^{4} - 312 i \, c d^{5} - 588 \, d^{6}\right )} f x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} +{\left (-27 i \, c^{6} + 96 \, c^{5} d + 63 i \, c^{4} d^{2} + 192 \, c^{3} d^{3} + 207 i \, c^{2} d^{4} + 96 \, c d^{5} + 117 i \, d^{6}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} +{\left (-11 i \, c^{6} + 30 \, c^{5} d - 3 i \, c^{4} d^{2} + 60 \, c^{3} d^{3} + 27 i \, c^{2} d^{4} + 30 \, c d^{5} + 19 i \, d^{6}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} +{\left ({\left (480 i \, c^{2} d^{4} + 768 \, c d^{5} - 288 i \, d^{6}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (480 i \, c^{2} d^{4} - 192 \, c d^{5} + 288 i \, d^{6}\right )} e^{\left (6 i \, f x + 6 i \, e\right )}\right )} \log \left (\frac{{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right )}{{\left (96 \, a^{3} c^{8} + 192 i \, a^{3} c^{7} d + 192 \, a^{3} c^{6} d^{2} + 576 i \, a^{3} c^{5} d^{3} + 576 i \, a^{3} c^{3} d^{5} - 192 \, a^{3} c^{2} d^{6} + 192 i \, a^{3} c d^{7} - 96 \, a^{3} d^{8}\right )} f e^{\left (8 i \, f x + 8 i \, e\right )} +{\left (96 \, a^{3} c^{8} + 384 i \, a^{3} c^{7} d - 384 \, a^{3} c^{6} d^{2} + 384 i \, a^{3} c^{5} d^{3} - 960 \, a^{3} c^{4} d^{4} - 384 i \, a^{3} c^{3} d^{5} - 384 \, a^{3} c^{2} d^{6} - 384 i \, a^{3} c d^{7} + 96 \, a^{3} d^{8}\right )} f e^{\left (6 i \, f x + 6 i \, e\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

-(-2*I*c^6 + 4*c^5*d - 2*I*c^4*d^2 + 8*c^3*d^3 + 2*I*c^2*d^4 + 4*c*d^5 + 2*I*d^6 - (12*c^6 + 48*I*c^5*d - 60*c
^4*d^2 - 1020*c^2*d^4 + 1488*I*c*d^5 + 588*d^6)*f*x*e^(8*I*f*x + 8*I*e) + (-18*I*c^6 + 48*c^5*d - 30*I*c^4*d^2
+ 240*c^3*d^3 - 150*I*c^2*d^4 - 330*I*d^6 - (12*c^6 + 72*I*c^5*d - 180*c^4*d^2 - 240*I*c^3*d^3 - 780*c^2*d^4
- 312*I*c*d^5 - 588*d^6)*f*x)*e^(6*I*f*x + 6*I*e) + (-27*I*c^6 + 96*c^5*d + 63*I*c^4*d^2 + 192*c^3*d^3 + 207*I
*c^2*d^4 + 96*c*d^5 + 117*I*d^6)*e^(4*I*f*x + 4*I*e) + (-11*I*c^6 + 30*c^5*d - 3*I*c^4*d^2 + 60*c^3*d^3 + 27*I
*c^2*d^4 + 30*c*d^5 + 19*I*d^6)*e^(2*I*f*x + 2*I*e) + ((480*I*c^2*d^4 + 768*c*d^5 - 288*I*d^6)*e^(8*I*f*x + 8*
I*e) + (480*I*c^2*d^4 - 192*c*d^5 + 288*I*d^6)*e^(6*I*f*x + 6*I*e))*log(((I*c + d)*e^(2*I*f*x + 2*I*e) + I*c -
d)/(I*c + d)))/((96*a^3*c^8 + 192*I*a^3*c^7*d + 192*a^3*c^6*d^2 + 576*I*a^3*c^5*d^3 + 576*I*a^3*c^3*d^5 - 192
*a^3*c^2*d^6 + 192*I*a^3*c*d^7 - 96*a^3*d^8)*f*e^(8*I*f*x + 8*I*e) + (96*a^3*c^8 + 384*I*a^3*c^7*d - 384*a^3*c
^6*d^2 + 384*I*a^3*c^5*d^3 - 960*a^3*c^4*d^4 - 384*I*a^3*c^3*d^5 - 384*a^3*c^2*d^6 - 384*I*a^3*c*d^7 + 96*a^3*
d^8)*f*e^(6*I*f*x + 6*I*e))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))**3/(c+d*tan(f*x+e))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.36661, size = 875, normalized size = 2.45 \begin{align*} \frac{2 \,{\left (\frac{{\left (c^{3} + 7 i \, c^{2} d - 23 \, c d^{2} - 49 i \, d^{3}\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\right )}{32 i \, a^{3} c^{5} - 160 \, a^{3} c^{4} d - 320 i \, a^{3} c^{3} d^{2} + 320 \, a^{3} c^{2} d^{3} + 160 i \, a^{3} c d^{4} - 32 \, a^{3} d^{5}} - \frac{{\left (5 \, c d^{5} - 3 i \, d^{6}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{-2 i \, a^{3} c^{7} d + 6 \, a^{3} c^{6} d^{2} + 2 i \, a^{3} c^{5} d^{3} + 10 \, a^{3} c^{4} d^{4} + 10 i \, a^{3} c^{3} d^{5} + 2 \, a^{3} c^{2} d^{6} + 6 i \, a^{3} c d^{7} - 2 \, a^{3} d^{8}} + \frac{\log \left (-i \, \tan \left (f x + e\right ) + 1\right )}{-32 i \, a^{3} c^{2} - 64 \, a^{3} c d + 32 i \, a^{3} d^{2}} + \frac{5 \, c d^{5} \tan \left (f x + e\right ) - 3 i \, d^{6} \tan \left (f x + e\right ) + 6 \, c^{2} d^{4} - 3 i \, c d^{5} + d^{6}}{{\left (-2 i \, a^{3} c^{7} + 6 \, a^{3} c^{6} d + 2 i \, a^{3} c^{5} d^{2} + 10 \, a^{3} c^{4} d^{3} + 10 i \, a^{3} c^{3} d^{4} + 2 \, a^{3} c^{2} d^{5} + 6 i \, a^{3} c d^{6} - 2 \, a^{3} d^{7}\right )}{\left (d \tan \left (f x + e\right ) + c\right )}} - \frac{11 \, c^{3} \tan \left (f x + e\right )^{3} + 77 i \, c^{2} d \tan \left (f x + e\right )^{3} - 253 \, c d^{2} \tan \left (f x + e\right )^{3} - 539 i \, d^{3} \tan \left (f x + e\right )^{3} - 45 i \, c^{3} \tan \left (f x + e\right )^{2} + 315 \, c^{2} d \tan \left (f x + e\right )^{2} + 1035 i \, c d^{2} \tan \left (f x + e\right )^{2} - 1821 \, d^{3} \tan \left (f x + e\right )^{2} - 69 \, c^{3} \tan \left (f x + e\right ) - 483 i \, c^{2} d \tan \left (f x + e\right ) + 1443 \, c d^{2} \tan \left (f x + e\right ) + 2085 i \, d^{3} \tan \left (f x + e\right ) + 51 i \, c^{3} - 293 \, c^{2} d - 709 i \, c d^{2} + 819 \, d^{3}}{{\left (192 i \, a^{3} c^{5} - 960 \, a^{3} c^{4} d - 1920 i \, a^{3} c^{3} d^{2} + 1920 \, a^{3} c^{2} d^{3} + 960 i \, a^{3} c d^{4} - 192 \, a^{3} d^{5}\right )}{\left (\tan \left (f x + e\right ) - i\right )}^{3}}\right )}}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^2,x, algorithm="giac")

[Out]

2*((c^3 + 7*I*c^2*d - 23*c*d^2 - 49*I*d^3)*log(I*tan(f*x + e) + 1)/(32*I*a^3*c^5 - 160*a^3*c^4*d - 320*I*a^3*c
^3*d^2 + 320*a^3*c^2*d^3 + 160*I*a^3*c*d^4 - 32*a^3*d^5) - (5*c*d^5 - 3*I*d^6)*log(abs(d*tan(f*x + e) + c))/(-
2*I*a^3*c^7*d + 6*a^3*c^6*d^2 + 2*I*a^3*c^5*d^3 + 10*a^3*c^4*d^4 + 10*I*a^3*c^3*d^5 + 2*a^3*c^2*d^6 + 6*I*a^3*
c*d^7 - 2*a^3*d^8) + log(-I*tan(f*x + e) + 1)/(-32*I*a^3*c^2 - 64*a^3*c*d + 32*I*a^3*d^2) + (5*c*d^5*tan(f*x +
e) - 3*I*d^6*tan(f*x + e) + 6*c^2*d^4 - 3*I*c*d^5 + d^6)/((-2*I*a^3*c^7 + 6*a^3*c^6*d + 2*I*a^3*c^5*d^2 + 10*
a^3*c^4*d^3 + 10*I*a^3*c^3*d^4 + 2*a^3*c^2*d^5 + 6*I*a^3*c*d^6 - 2*a^3*d^7)*(d*tan(f*x + e) + c)) - (11*c^3*ta
n(f*x + e)^3 + 77*I*c^2*d*tan(f*x + e)^3 - 253*c*d^2*tan(f*x + e)^3 - 539*I*d^3*tan(f*x + e)^3 - 45*I*c^3*tan(
f*x + e)^2 + 315*c^2*d*tan(f*x + e)^2 + 1035*I*c*d^2*tan(f*x + e)^2 - 1821*d^3*tan(f*x + e)^2 - 69*c^3*tan(f*x
+ e) - 483*I*c^2*d*tan(f*x + e) + 1443*c*d^2*tan(f*x + e) + 2085*I*d^3*tan(f*x + e) + 51*I*c^3 - 293*c^2*d -
709*I*c*d^2 + 819*d^3)/((192*I*a^3*c^5 - 960*a^3*c^4*d - 1920*I*a^3*c^3*d^2 + 1920*a^3*c^2*d^3 + 960*I*a^3*c*d
^4 - 192*a^3*d^5)*(tan(f*x + e) - I)^3))/f