3.1100 \(\int \frac{1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=448 \[ \frac{d \left (-16 c^2 d^2+6 i c^3 d+c^4+94 i c d^3+55 d^4\right )}{8 a^3 f (c-i d)^2 (c+i d)^5 (c+d \tan (e+f x))}+\frac{d \left (6 i c^2 d+c^3-17 c d^2+28 i d^3\right )}{8 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}-\frac{d^4 \left (15 c^2-18 i c d-7 d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^6 (d+i c)^3}+\frac{x \left (-15 c^4 d^2-20 i c^3 d^3-105 c^2 d^4+6 i c^5 d+c^6+150 i c d^5+55 d^6\right )}{8 a^3 (c-i d)^3 (c+i d)^6}+\frac{-13 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \]

[Out]

((c^6 + (6*I)*c^5*d - 15*c^4*d^2 - (20*I)*c^3*d^3 - 105*c^2*d^4 + (150*I)*c*d^5 + 55*d^6)*x)/(8*a^3*(c - I*d)^
3*(c + I*d)^6) - (d^4*(15*c^2 - (18*I)*c*d - 7*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(c + I*d)^6*(I*
c + d)^3*f) + (d*(c^3 + (6*I)*c^2*d - 17*c*d^2 + (28*I)*d^3))/(8*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*
x])^2) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2) + ((3*I)*c - 13*d)/(24*a*(c + I*d)^
2*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2) + (3*c^2 + (18*I)*c*d - 55*d^2)/(24*(I*c - d)^3*f*(a^3 +
I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) + (d*(c^4 + (6*I)*c^3*d - 16*c^2*d^2 + (94*I)*c*d^3 + 55*d^4))/(8*
a^3*(c - I*d)^2*(c + I*d)^5*f*(c + d*Tan[e + f*x]))

________________________________________________________________________________________

Rubi [A]  time = 1.14568, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{d \left (-16 c^2 d^2+6 i c^3 d+c^4+94 i c d^3+55 d^4\right )}{8 a^3 f (c-i d)^2 (c+i d)^5 (c+d \tan (e+f x))}+\frac{d \left (6 i c^2 d+c^3-17 c d^2+28 i d^3\right )}{8 a^3 f (c-i d) (c+i d)^4 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 f (-d+i c)^3 \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}-\frac{d^4 \left (15 c^2-18 i c d-7 d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 f (c+i d)^6 (d+i c)^3}+\frac{x \left (-15 c^4 d^2-20 i c^3 d^3-105 c^2 d^4+6 i c^5 d+c^6+150 i c d^5+55 d^6\right )}{8 a^3 (c-i d)^3 (c+i d)^6}+\frac{-13 d+3 i c}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}-\frac{1}{6 f (-d+i c) (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c^6 + (6*I)*c^5*d - 15*c^4*d^2 - (20*I)*c^3*d^3 - 105*c^2*d^4 + (150*I)*c*d^5 + 55*d^6)*x)/(8*a^3*(c - I*d)^
3*(c + I*d)^6) - (d^4*(15*c^2 - (18*I)*c*d - 7*d^2)*Log[c*Cos[e + f*x] + d*Sin[e + f*x]])/(a^3*(c + I*d)^6*(I*
c + d)^3*f) + (d*(c^3 + (6*I)*c^2*d - 17*c*d^2 + (28*I)*d^3))/(8*a^3*(c - I*d)*(c + I*d)^4*f*(c + d*Tan[e + f*
x])^2) - 1/(6*(I*c - d)*f*(a + I*a*Tan[e + f*x])^3*(c + d*Tan[e + f*x])^2) + ((3*I)*c - 13*d)/(24*a*(c + I*d)^
2*f*(a + I*a*Tan[e + f*x])^2*(c + d*Tan[e + f*x])^2) + (3*c^2 + (18*I)*c*d - 55*d^2)/(24*(I*c - d)^3*f*(a^3 +
I*a^3*Tan[e + f*x])*(c + d*Tan[e + f*x])^2) + (d*(c^4 + (6*I)*c^3*d - 16*c^2*d^2 + (94*I)*c*d^3 + 55*d^4))/(8*
a^3*(c - I*d)^2*(c + I*d)^5*f*(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))^3} \, dx &=-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2}-\frac{\int \frac{-a (3 i c-8 d)-5 i a d \tan (e+f x)}{(a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^3} \, 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))^2}+\frac{3 i c-13 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}-\frac{\int \frac{-2 a^2 \left (3 c^2+12 i c d-29 d^2\right )-4 a^2 (3 c+13 i d) d \tan (e+f x)}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, 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))^2}+\frac{3 i c-13 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}-\frac{\int \frac{6 a^3 \left (i c^3-6 c^2 d-21 i c d^2+56 d^3\right )+6 a^3 d \left (3 i c^2-18 c d-55 i d^2\right ) \tan (e+f x)}{(c+d \tan (e+f x))^3} \, dx}{48 a^6 (i c-d)^3}\\ &=\frac{d \left (c^3+6 i c^2 d-17 c d^2+28 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^2}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2}+\frac{3 i c-13 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}-\frac{\int \frac{-6 a^3 \left (6 c^3 d-i \left (c^4-18 c^2 d^2-38 i c d^3-55 d^4\right )\right )-12 a^3 d \left (6 c^2 d-i \left (c^3-17 c d^2+28 i d^3\right )\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx}{48 a^6 (i c-d)^3 \left (c^2+d^2\right )}\\ &=\frac{d \left (c^3+6 i c^2 d-17 c d^2+28 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^2}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2}+\frac{3 i c-13 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}+\frac{d \left (c^4+6 i c^3 d-16 c^2 d^2+94 i c d^3+55 d^4\right )}{8 a^3 (c-i d)^2 (c+i d)^5 f (c+d \tan (e+f x))}-\frac{\int \frac{6 a^3 \left (i c^5-6 c^4 d-16 i c^3 d^2+26 c^2 d^3-89 i c d^4-56 d^5\right )-6 a^3 d \left (6 c^3 d-i \left (c^4-16 c^2 d^2+94 i c d^3+55 d^4\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 )^2}\\ &=\frac{\left (c^6+6 i c^5 d-15 c^4 d^2-20 i c^3 d^3-105 c^2 d^4+150 i c d^5+55 d^6\right ) x}{8 a^3 (c-i d)^3 (c+i d)^6}+\frac{d \left (c^3+6 i c^2 d-17 c d^2+28 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^2}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2}+\frac{3 i c-13 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}+\frac{d \left (c^4+6 i c^3 d-16 c^2 d^2+94 i c d^3+55 d^4\right )}{8 a^3 (c-i d)^2 (c+i d)^5 f (c+d \tan (e+f x))}-\frac{\left (d^4 \left (15 c^2-18 i c d-7 d^2\right )\right ) \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{a^3 (i c-d)^3 \left (c^2+d^2\right )^3}\\ &=\frac{\left (c^6+6 i c^5 d-15 c^4 d^2-20 i c^3 d^3-105 c^2 d^4+150 i c d^5+55 d^6\right ) x}{8 a^3 (c-i d)^3 (c+i d)^6}+\frac{d^4 \left (15 c^2-18 i c d-7 d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a^3 (i c-d)^6 (i c+d)^3 f}+\frac{d \left (c^3+6 i c^2 d-17 c d^2+28 i d^3\right )}{8 a^3 (c-i d) (c+i d)^4 f (c+d \tan (e+f x))^2}-\frac{1}{6 (i c-d) f (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2}+\frac{3 i c-13 d}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2 (c+d \tan (e+f x))^2}+\frac{3 c^2+18 i c d-55 d^2}{24 (i c-d)^3 f \left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^2}+\frac{d \left (c^4+6 i c^3 d-16 c^2 d^2+94 i c d^3+55 d^4\right )}{8 a^3 (c-i d)^2 (c+i d)^5 f (c+d \tan (e+f x))}\\ \end{align*}

Mathematica [B]  time = 8.10251, size = 5726, normalized size = 12.78 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.077, size = 954, normalized size = 2.1 \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))^3,x)

[Out]

1/8/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)*c^3-7/8/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)^2*d^3-1/6/f/a^3/(c+I*d)^6/(tan(f*x+e
)-I)^3*c^3-1/16*I/f/a^3/(I*d-c)^3*ln(tan(f*x+e)+I)-I/f/a^3*d^6/(I*d-c)^3/(c+I*d)^6/(c+d*tan(f*x+e))^2*c^2+15*I
/f/a^3*d^4/(I*d-c)^3/(c+I*d)^6*ln(c+d*tan(f*x+e))*c^2-1/2*I/f/a^3*d^4/(I*d-c)^3/(c+I*d)^6/(c+d*tan(f*x+e))^2*c
^4-5*I/f/a^3*d^4/(I*d-c)^3/(c+I*d)^6/(c+d*tan(f*x+e))*c^3-5*I/f/a^3*d^6/(I*d-c)^3/(c+I*d)^6/(c+d*tan(f*x+e))*c
-111/16/f/a^3/(c+I*d)^6*ln(tan(f*x+e)-I)*d^3-31/8*I/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)*d^3-1/8*I/f/a^3/(c+I*d)^6/(
tan(f*x+e)-I)^2*c^3-1/16*I/f/a^3/(c+I*d)^6*ln(tan(f*x+e)-I)*c^3+18/f/a^3*d^5/(I*d-c)^3/(c+I*d)^6*ln(c+d*tan(f*
x+e))*c-3/f/a^3*d^5/(I*d-c)^3/(c+I*d)^6/(c+d*tan(f*x+e))*c^2-7*I/f/a^3*d^6/(I*d-c)^3/(c+I*d)^6*ln(c+d*tan(f*x+
e))+15/8*I/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)^2*c*d^2-1/2*I/f/a^3*d^8/(I*d-c)^3/(c+I*d)^6/(c+d*tan(f*x+e))^2-1/2*I
/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)^3*c^2*d+39/16*I/f/a^3/(c+I*d)^6*ln(tan(f*x+e)-I)*c*d^2+9/8*I/f/a^3/(c+I*d)^6/(
tan(f*x+e)-I)*c^2*d+1/2/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)^3*c*d^2+9/16/f/a^3/(c+I*d)^6*ln(tan(f*x+e)-I)*c^2*d-39/
8/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)*c*d^2+9/8/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)^2*c^2*d-3/f/a^3*d^7/(I*d-c)^3/(c+I*d
)^6/(c+d*tan(f*x+e))+1/6*I/f/a^3/(c+I*d)^6/(tan(f*x+e)-I)^3*d^3

________________________________________________________________________________________

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))^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

________________________________________________________________________________________

Fricas [B]  time = 2.60723, size = 2930, normalized size = 6.54 \begin{align*} \text{result too large to display} \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))^3,x, algorithm="fricas")

[Out]

(2*c^8 + 4*I*c^7*d + 4*c^6*d^2 + 12*I*c^5*d^3 + 12*I*c^3*d^5 - 4*c^2*d^6 + 4*I*c*d^7 - 2*d^8 + (-12*I*c^8 + 48
*c^7*d + 48*I*c^6*d^2 + 48*c^5*d^3 + 3000*I*c^4*d^4 + 9168*c^3*d^5 - 11088*I*c^2*d^6 - 6192*c*d^7 + 1332*I*d^8
)*f*x*e^(10*I*f*x + 10*I*e) + (18*c^8 + 36*I*c^7*d + 108*c^6*d^2 + 396*I*c^5*d^3 + 1080*c^4*d^4 - 1620*I*c^3*d
^5 + 372*c^2*d^6 - 1980*I*c*d^7 - 618*d^8 + (-24*I*c^8 + 144*c^7*d + 336*I*c^6*d^2 - 336*c^5*d^3 + 5760*I*c^4*
d^4 + 6576*c^3*d^5 + 2736*I*c^2*d^6 + 7056*c*d^7 - 2664*I*d^8)*f*x)*e^(8*I*f*x + 8*I*e) + (45*c^8 + 144*I*c^7*
d + 72*c^6*d^2 + 936*I*c^5*d^3 + 450*c^4*d^4 + 288*I*c^3*d^5 + 2592*c^2*d^6 + 648*I*c*d^7 + 1017*d^8 + (-12*I*
c^8 + 96*c^7*d + 336*I*c^6*d^2 - 672*c^5*d^3 + 2040*I*c^4*d^4 - 1632*c^3*d^5 + 3024*I*c^2*d^6 - 864*c*d^7 + 13
32*I*d^8)*f*x)*e^(6*I*f*x + 6*I*e) + (38*c^8 + 140*I*c^7*d - 68*c^6*d^2 + 420*I*c^5*d^3 - 432*c^4*d^4 + 420*I*
c^3*d^5 - 508*c^2*d^6 + 140*I*c*d^7 - 182*d^8)*e^(4*I*f*x + 4*I*e) + (13*c^8 + 36*I*c^7*d + 16*c^6*d^2 + 108*I
*c^5*d^3 - 30*c^4*d^4 + 108*I*c^3*d^5 - 56*c^2*d^6 + 36*I*c*d^7 - 23*d^8)*e^(2*I*f*x + 2*I*e) - ((1440*c^4*d^4
- 4608*I*c^3*d^5 - 5568*c^2*d^6 + 3072*I*c*d^7 + 672*d^8)*e^(10*I*f*x + 10*I*e) + (2880*c^4*d^4 - 3456*I*c^3*
d^5 + 1536*c^2*d^6 - 3456*I*c*d^7 - 1344*d^8)*e^(8*I*f*x + 8*I*e) + (1440*c^4*d^4 + 1152*I*c^3*d^5 + 1344*c^2*
d^6 + 384*I*c*d^7 + 672*d^8)*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*I*a^3*c^11 + 96*a^3*c^10*d - 480*I*a^3*c^9*d^2 + 480*a^3*c^8*d^3 - 960*I*a^3*c^7*d^4 + 960*a^3*c^6*d^5 - 9
60*I*a^3*c^5*d^6 + 960*a^3*c^4*d^7 - 480*I*a^3*c^3*d^8 + 480*a^3*c^2*d^9 - 96*I*a^3*c*d^10 + 96*a^3*d^11)*f*e^
(10*I*f*x + 10*I*e) + (-192*I*a^3*c^11 + 576*a^3*c^10*d - 192*I*a^3*c^9*d^2 + 2112*a^3*c^8*d^3 + 1152*I*a^3*c^
7*d^4 + 2688*a^3*c^6*d^5 + 2688*I*a^3*c^5*d^6 + 1152*a^3*c^4*d^7 + 2112*I*a^3*c^3*d^8 - 192*a^3*c^2*d^9 + 576*
I*a^3*c*d^10 - 192*a^3*d^11)*f*e^(8*I*f*x + 8*I*e) + (-96*I*a^3*c^11 + 480*a^3*c^10*d + 672*I*a^3*c^9*d^2 + 48
0*a^3*c^8*d^3 + 2112*I*a^3*c^7*d^4 - 1344*a^3*c^6*d^5 + 1344*I*a^3*c^5*d^6 - 2112*a^3*c^4*d^7 - 480*I*a^3*c^3*
d^8 - 672*a^3*c^2*d^9 - 480*I*a^3*c*d^10 + 96*a^3*d^11)*f*e^(6*I*f*x + 6*I*e))

________________________________________________________________________________________

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))**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.42151, size = 1057, normalized size = 2.36 \begin{align*} \text{result too large to display} \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))^3,x, algorithm="giac")

[Out]

2*((-I*c^3 + 9*c^2*d + 39*I*c*d^2 - 111*d^3)*log(I*tan(f*x + e) + 1)/(32*a^3*c^6 + 192*I*a^3*c^5*d - 480*a^3*c
^4*d^2 - 640*I*a^3*c^3*d^3 + 480*a^3*c^2*d^4 + 192*I*a^3*c*d^5 - 32*a^3*d^6) - (15*c^2*d^5 - 18*I*c*d^6 - 7*d^
7)*log(abs(d*tan(f*x + e) + c))/(-2*I*a^3*c^9*d + 6*a^3*c^8*d^2 + 16*a^3*c^6*d^4 + 12*I*a^3*c^5*d^5 + 12*a^3*c
^4*d^6 + 16*I*a^3*c^3*d^7 + 6*I*a^3*c*d^9 - 2*a^3*d^10) + I*log(-I*tan(f*x + e) + 1)/(32*a^3*c^3 - 96*I*a^3*c^
2*d - 96*a^3*c*d^2 + 32*I*a^3*d^3) + (45*c^2*d^6*tan(f*x + e)^2 - 54*I*c*d^7*tan(f*x + e)^2 - 21*d^8*tan(f*x +
e)^2 + 100*c^3*d^5*tan(f*x + e) - 114*I*c^2*d^6*tan(f*x + e) - 32*c*d^7*tan(f*x + e) - 6*I*d^8*tan(f*x + e) +
56*c^4*d^4 - 60*I*c^3*d^5 - 9*c^2*d^6 - 6*I*c*d^7 + d^8)/((-4*I*a^3*c^9 + 12*a^3*c^8*d + 32*a^3*c^6*d^3 + 24*
I*a^3*c^5*d^4 + 24*a^3*c^4*d^5 + 32*I*a^3*c^3*d^6 + 12*I*a^3*c*d^8 - 4*a^3*d^9)*(d*tan(f*x + e) + c)^2) + (11*
I*c^3*tan(f*x + e)^3 - 99*c^2*d*tan(f*x + e)^3 - 429*I*c*d^2*tan(f*x + e)^3 + 1221*d^3*tan(f*x + e)^3 + 45*c^3
*tan(f*x + e)^2 + 405*I*c^2*d*tan(f*x + e)^2 - 1755*c*d^2*tan(f*x + e)^2 - 4035*I*d^3*tan(f*x + e)^2 - 69*I*c^
3*tan(f*x + e) + 621*c^2*d*tan(f*x + e) + 2403*I*c*d^2*tan(f*x + e) - 4491*d^3*tan(f*x + e) - 51*c^3 - 363*I*c
^2*d + 1125*c*d^2 + 1693*I*d^3)/((192*a^3*c^6 + 1152*I*a^3*c^5*d - 2880*a^3*c^4*d^2 - 3840*I*a^3*c^3*d^3 + 288
0*a^3*c^2*d^4 + 1152*I*a^3*c*d^5 - 192*a^3*d^6)*(tan(f*x + e) - I)^3))/f