3.11.100 \(\int \frac {1}{(a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^3} \, dx\) [1100]

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

[Out]

1/8*(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/a^3/(c-I*d)^3/(c+I*d)^6-d^4*(15*c
^2-18*I*c*d-7*d^2)*ln(c*cos(f*x+e)+d*sin(f*x+e))/a^3/(c+I*d)^6/(I*c+d)^3/f+1/8*d*(c^3+6*I*c^2*d-17*c*d^2+28*I*
d^3)/a^3/(c-I*d)/(c+I*d)^4/f/(c+d*tan(f*x+e))^2-1/6/(I*c-d)/f/(a+I*a*tan(f*x+e))^3/(c+d*tan(f*x+e))^2+1/24*(3*
I*c-13*d)/a/(c+I*d)^2/f/(a+I*a*tan(f*x+e))^2/(c+d*tan(f*x+e))^2+1/24*(3*c^2+18*I*c*d-55*d^2)/(I*c-d)^3/f/(a^3+
I*a^3*tan(f*x+e))/(c+d*tan(f*x+e))^2+1/8*d*(c^4+6*I*c^3*d-16*c^2*d^2+94*I*c*d^3+55*d^4)/a^3/(c-I*d)^2/(c+I*d)^
5/f/(c+d*tan(f*x+e))

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Rubi [A]
time = 0.83, antiderivative size = 448, normalized size of antiderivative = 1.00, 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 = {3640, 3677, 3610, 3612, 3611} \begin {gather*} \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 {d \left (c^3+6 i c^2 d-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 {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 f (c-i d)^2 (c+i d)^5 (c+d \tan (e+f x))}+\frac {x \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 )}{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} \end {gather*}

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 3610

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 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[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] && EqQ[a*c + b*d, 0]

Rule 3612

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 3640

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 3677

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]

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*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5726\) vs. \(2(448)=896\).
time = 8.53, size = 5726, normalized size = 12.78 \begin {gather*} \text {Result too large to show} \end {gather*}

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

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Maple [A]
time = 1.69, size = 374, normalized size = 0.83 Too large to display

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1151 vs. \(2 (398) = 796\).
time = 1.24, size = 1151, normalized size = 2.57 \begin {gather*} -\frac {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 \, {\left (i \, c^{8} - 4 \, c^{7} d - 4 i \, c^{6} d^{2} - 4 \, c^{5} d^{3} - 250 i \, c^{4} d^{4} - 764 \, c^{3} d^{5} + 924 i \, c^{2} d^{6} + 516 \, c d^{7} - 111 i \, d^{8}\right )} f x e^{\left (10 i \, f x + 10 i \, e\right )} + 6 \, {\left (3 \, c^{8} + 6 i \, c^{7} d + 18 \, c^{6} d^{2} + 66 i \, c^{5} d^{3} + 180 \, c^{4} d^{4} - 270 i \, c^{3} d^{5} + 62 \, c^{2} d^{6} - 330 i \, c d^{7} - 103 \, d^{8} - 4 \, {\left (i \, c^{8} - 6 \, c^{7} d - 14 i \, c^{6} d^{2} + 14 \, c^{5} d^{3} - 240 i \, c^{4} d^{4} - 274 \, c^{3} d^{5} - 114 i \, c^{2} d^{6} - 294 \, c d^{7} + 111 i \, d^{8}\right )} f x\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 3 \, {\left (15 \, c^{8} + 48 i \, c^{7} d + 24 \, c^{6} d^{2} + 312 i \, c^{5} d^{3} + 150 \, c^{4} d^{4} + 96 i \, c^{3} d^{5} + 864 \, c^{2} d^{6} + 216 i \, c d^{7} + 339 \, d^{8} - 4 \, {\left (i \, c^{8} - 8 \, c^{7} d - 28 i \, c^{6} d^{2} + 56 \, c^{5} d^{3} - 170 i \, c^{4} d^{4} + 136 \, c^{3} d^{5} - 252 i \, c^{2} d^{6} + 72 \, c d^{7} - 111 i \, d^{8}\right )} f x\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 2 \, {\left (19 \, c^{8} + 70 i \, c^{7} d - 34 \, c^{6} d^{2} + 210 i \, c^{5} d^{3} - 216 \, c^{4} d^{4} + 210 i \, c^{3} d^{5} - 254 \, c^{2} d^{6} + 70 i \, c d^{7} - 91 \, d^{8}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (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}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - 96 \, {\left ({\left (15 \, c^{4} d^{4} - 48 i \, c^{3} d^{5} - 58 \, c^{2} d^{6} + 32 i \, c d^{7} + 7 \, d^{8}\right )} e^{\left (10 i \, f x + 10 i \, e\right )} + 2 \, {\left (15 \, c^{4} d^{4} - 18 i \, c^{3} d^{5} + 8 \, c^{2} d^{6} - 18 i \, c d^{7} - 7 \, d^{8}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (15 \, c^{4} d^{4} + 12 i \, c^{3} d^{5} + 14 \, c^{2} d^{6} + 4 i \, c d^{7} + 7 \, d^{8}\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 )}{96 \, {\left ({\left (i \, a^{3} c^{11} - a^{3} c^{10} d + 5 i \, a^{3} c^{9} d^{2} - 5 \, a^{3} c^{8} d^{3} + 10 i \, a^{3} c^{7} d^{4} - 10 \, a^{3} c^{6} d^{5} + 10 i \, a^{3} c^{5} d^{6} - 10 \, a^{3} c^{4} d^{7} + 5 i \, a^{3} c^{3} d^{8} - 5 \, a^{3} c^{2} d^{9} + i \, a^{3} c d^{10} - a^{3} d^{11}\right )} f e^{\left (10 i \, f x + 10 i \, e\right )} + 2 \, {\left (i \, a^{3} c^{11} - 3 \, a^{3} c^{10} d + i \, a^{3} c^{9} d^{2} - 11 \, a^{3} c^{8} d^{3} - 6 i \, a^{3} c^{7} d^{4} - 14 \, a^{3} c^{6} d^{5} - 14 i \, a^{3} c^{5} d^{6} - 6 \, a^{3} c^{4} d^{7} - 11 i \, a^{3} c^{3} d^{8} + a^{3} c^{2} d^{9} - 3 i \, a^{3} c d^{10} + a^{3} d^{11}\right )} f e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (i \, a^{3} c^{11} - 5 \, a^{3} c^{10} d - 7 i \, a^{3} c^{9} d^{2} - 5 \, a^{3} c^{8} d^{3} - 22 i \, a^{3} c^{7} d^{4} + 14 \, a^{3} c^{6} d^{5} - 14 i \, a^{3} c^{5} d^{6} + 22 \, a^{3} c^{4} d^{7} + 5 i \, a^{3} c^{3} d^{8} + 7 \, a^{3} c^{2} d^{9} + 5 i \, a^{3} c d^{10} - a^{3} d^{11}\right )} f e^{\left (6 i \, f x + 6 i \, e\right )}\right )}} \end {gather*}

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]

-1/96*(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
- 4*c^7*d - 4*I*c^6*d^2 - 4*c^5*d^3 - 250*I*c^4*d^4 - 764*c^3*d^5 + 924*I*c^2*d^6 + 516*c*d^7 - 111*I*d^8)*f*
x*e^(10*I*f*x + 10*I*e) + 6*(3*c^8 + 6*I*c^7*d + 18*c^6*d^2 + 66*I*c^5*d^3 + 180*c^4*d^4 - 270*I*c^3*d^5 + 62*
c^2*d^6 - 330*I*c*d^7 - 103*d^8 - 4*(I*c^8 - 6*c^7*d - 14*I*c^6*d^2 + 14*c^5*d^3 - 240*I*c^4*d^4 - 274*c^3*d^5
- 114*I*c^2*d^6 - 294*c*d^7 + 111*I*d^8)*f*x)*e^(8*I*f*x + 8*I*e) + 3*(15*c^8 + 48*I*c^7*d + 24*c^6*d^2 + 312
*I*c^5*d^3 + 150*c^4*d^4 + 96*I*c^3*d^5 + 864*c^2*d^6 + 216*I*c*d^7 + 339*d^8 - 4*(I*c^8 - 8*c^7*d - 28*I*c^6*
d^2 + 56*c^5*d^3 - 170*I*c^4*d^4 + 136*c^3*d^5 - 252*I*c^2*d^6 + 72*c*d^7 - 111*I*d^8)*f*x)*e^(6*I*f*x + 6*I*e
) + 2*(19*c^8 + 70*I*c^7*d - 34*c^6*d^2 + 210*I*c^5*d^3 - 216*c^4*d^4 + 210*I*c^3*d^5 - 254*c^2*d^6 + 70*I*c*d
^7 - 91*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) - 96*((15*c^4*d^4 - 48*I*c^3*d^5 - 58*c^2*d^6 + 32
*I*c*d^7 + 7*d^8)*e^(10*I*f*x + 10*I*e) + 2*(15*c^4*d^4 - 18*I*c^3*d^5 + 8*c^2*d^6 - 18*I*c*d^7 - 7*d^8)*e^(8*
I*f*x + 8*I*e) + (15*c^4*d^4 + 12*I*c^3*d^5 + 14*c^2*d^6 + 4*I*c*d^7 + 7*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)))/((I*a^3*c^11 - a^3*c^10*d + 5*I*a^3*c^9*d^2 - 5*a^3*c^8*d^3 + 1
0*I*a^3*c^7*d^4 - 10*a^3*c^6*d^5 + 10*I*a^3*c^5*d^6 - 10*a^3*c^4*d^7 + 5*I*a^3*c^3*d^8 - 5*a^3*c^2*d^9 + I*a^3
*c*d^10 - a^3*d^11)*f*e^(10*I*f*x + 10*I*e) + 2*(I*a^3*c^11 - 3*a^3*c^10*d + I*a^3*c^9*d^2 - 11*a^3*c^8*d^3 -
6*I*a^3*c^7*d^4 - 14*a^3*c^6*d^5 - 14*I*a^3*c^5*d^6 - 6*a^3*c^4*d^7 - 11*I*a^3*c^3*d^8 + a^3*c^2*d^9 - 3*I*a^3
*c*d^10 + a^3*d^11)*f*e^(8*I*f*x + 8*I*e) + (I*a^3*c^11 - 5*a^3*c^10*d - 7*I*a^3*c^9*d^2 - 5*a^3*c^8*d^3 - 22*
I*a^3*c^7*d^4 + 14*a^3*c^6*d^5 - 14*I*a^3*c^5*d^6 + 22*a^3*c^4*d^7 + 5*I*a^3*c^3*d^8 + 7*a^3*c^2*d^9 + 5*I*a^3
*c*d^10 - a^3*d^11)*f*e^(6*I*f*x + 6*I*e))

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [A]
time = 1.03, size = 782, normalized size = 1.75 \begin {gather*} -\frac {\frac {192 \, {\left (15 \, c^{2} d^{5} - 18 i \, c d^{6} - 7 \, d^{7}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{-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}} - \frac {6 \, {\left (-i \, c^{3} + 9 \, c^{2} d + 39 i \, c d^{2} - 111 \, d^{3}\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\right )}{a^{3} c^{6} + 6 i \, a^{3} c^{5} d - 15 \, a^{3} c^{4} d^{2} - 20 i \, a^{3} c^{3} d^{3} + 15 \, a^{3} c^{2} d^{4} + 6 i \, a^{3} c d^{5} - a^{3} d^{6}} - \frac {6 i \, \log \left (-i \, \tan \left (f x + e\right ) + 1\right )}{a^{3} c^{3} - 3 i \, a^{3} c^{2} d - 3 \, a^{3} c d^{2} + i \, a^{3} d^{3}} - \frac {192 \, {\left (45 \, c^{2} d^{6} \tan \left (f x + e\right )^{2} - 54 i \, c d^{7} \tan \left (f x + e\right )^{2} - 21 \, d^{8} \tan \left (f x + e\right )^{2} + 100 \, c^{3} d^{5} \tan \left (f x + e\right ) - 114 i \, c^{2} d^{6} \tan \left (f x + e\right ) - 32 \, c d^{7} \tan \left (f x + e\right ) - 6 i \, d^{8} \tan \left (f x + e\right ) + 56 \, c^{4} d^{4} - 60 i \, c^{3} d^{5} - 9 \, c^{2} d^{6} - 6 i \, c d^{7} + d^{8}\right )}}{-4 \, {\left (i \, a^{3} c^{9} - 3 \, a^{3} c^{8} d - 8 \, a^{3} c^{6} d^{3} - 6 i \, a^{3} c^{5} d^{4} - 6 \, a^{3} c^{4} d^{5} - 8 i \, a^{3} c^{3} d^{6} - 3 i \, a^{3} c d^{8} + a^{3} d^{9}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}} - \frac {11 i \, c^{3} \tan \left (f x + e\right )^{3} - 99 \, c^{2} d \tan \left (f x + e\right )^{3} - 429 i \, c d^{2} \tan \left (f x + e\right )^{3} + 1221 \, d^{3} \tan \left (f x + e\right )^{3} + 45 \, c^{3} \tan \left (f x + e\right )^{2} + 405 i \, c^{2} d \tan \left (f x + e\right )^{2} - 1755 \, c d^{2} \tan \left (f x + e\right )^{2} - 4035 i \, d^{3} \tan \left (f x + e\right )^{2} - 69 i \, c^{3} \tan \left (f x + e\right ) + 621 \, c^{2} d \tan \left (f x + e\right ) + 2403 i \, c d^{2} \tan \left (f x + e\right ) - 4491 \, d^{3} \tan \left (f x + e\right ) - 51 \, c^{3} - 363 i \, c^{2} d + 1125 \, c d^{2} + 1693 i \, d^{3}}{{\left (a^{3} c^{6} + 6 i \, a^{3} c^{5} d - 15 \, a^{3} c^{4} d^{2} - 20 i \, a^{3} c^{3} d^{3} + 15 \, a^{3} c^{2} d^{4} + 6 i \, a^{3} c d^{5} - a^{3} d^{6}\right )} {\left (\tan \left (f x + e\right ) - i\right )}^{3}}}{96 \, f} \end {gather*}

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]

-1/96*(192*(15*c^2*d^5 - 18*I*c*d^6 - 7*d^7)*log(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) - 6*(-I*c^3 + 9*c
^2*d + 39*I*c*d^2 - 111*d^3)*log(I*tan(f*x + e) + 1)/(a^3*c^6 + 6*I*a^3*c^5*d - 15*a^3*c^4*d^2 - 20*I*a^3*c^3*
d^3 + 15*a^3*c^2*d^4 + 6*I*a^3*c*d^5 - a^3*d^6) - 6*I*log(-I*tan(f*x + e) + 1)/(a^3*c^3 - 3*I*a^3*c^2*d - 3*a^
3*c*d^2 + I*a^3*d^3) - 192*(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 + 10
0*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 + 112
5*c*d^2 + 1693*I*d^3)/((a^3*c^6 + 6*I*a^3*c^5*d - 15*a^3*c^4*d^2 - 20*I*a^3*c^3*d^3 + 15*a^3*c^2*d^4 + 6*I*a^3
*c*d^5 - a^3*d^6)*(tan(f*x + e) - I)^3))/f

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Mupad [B]
time = 13.52, size = 2500, normalized size = 5.58 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*
c^6*d^2)*(10159*c*d^9 - c^9*d - d^10*3080i + c^2*d^8*10692i - 2652*c^3*d^7 + c^4*d^6*1236i + 186*c^5*d^5 + c^6
*d^4*124i + 68*c^7*d^3 - c^8*d^2*12i) - root(a^9*c^9*d^9*e^3*56320i + a^9*c^11*d^7*e^3*36864i + a^9*c^7*d^11*e
^3*36864i + 29696*a^9*c^12*d^6*e^3 - 29696*a^9*c^6*d^12*e^3 + 16896*a^9*c^10*d^8*e^3 - 16896*a^9*c^8*d^10*e^3
+ 15360*a^9*c^14*d^4*e^3 - 15360*a^9*c^4*d^14*e^3 + a^9*c^13*d^5*e^3*6144i + a^9*c^5*d^13*e^3*6144i - a^9*c^15
*d^3*e^3*4096i - a^9*c^3*d^15*e^3*4096i + 2304*a^9*c^16*d^2*e^3 - 2304*a^9*c^2*d^16*e^3 - a^9*c^17*d*e^3*1536i
- a^9*c*d^17*e^3*1536i + 256*a^9*d^18*e^3 - 256*a^9*c^18*e^3 - a^3*c*d^11*e*64884i - a^3*c^11*d*e*12i + a^3*c
^3*d^9*e*137380i + 136578*a^3*c^2*d^10*e - 58575*a^3*c^4*d^8*e - 1060*a^3*c^6*d^6*e + a^3*c^7*d^5*e*360i - a^3
*c^5*d^7*e*360i - 255*a^3*c^8*d^4*e + a^3*c^9*d^3*e*220i + 66*a^3*c^10*d^2*e - 12433*a^3*d^12*e - a^3*c^12*e -
1026*c^2*d^7 + c^3*d^6*430i + 117*c^4*d^5 - c^5*d^4*15i + c*d^8*1725i + 777*d^9, e, k)*(root(a^9*c^9*d^9*e^3*
56320i + a^9*c^11*d^7*e^3*36864i + a^9*c^7*d^11*e^3*36864i + 29696*a^9*c^12*d^6*e^3 - 29696*a^9*c^6*d^12*e^3 +
16896*a^9*c^10*d^8*e^3 - 16896*a^9*c^8*d^10*e^3 + 15360*a^9*c^14*d^4*e^3 - 15360*a^9*c^4*d^14*e^3 + a^9*c^13*
d^5*e^3*6144i + a^9*c^5*d^13*e^3*6144i - a^9*c^15*d^3*e^3*4096i - a^9*c^3*d^15*e^3*4096i + 2304*a^9*c^16*d^2*e
^3 - 2304*a^9*c^2*d^16*e^3 - a^9*c^17*d*e^3*1536i - a^9*c*d^17*e^3*1536i + 256*a^9*d^18*e^3 - 256*a^9*c^18*e^3
- a^3*c*d^11*e*64884i - a^3*c^11*d*e*12i + a^3*c^3*d^9*e*137380i + 136578*a^3*c^2*d^10*e - 58575*a^3*c^4*d^8*
e - 1060*a^3*c^6*d^6*e + a^3*c^7*d^5*e*360i - a^3*c^5*d^7*e*360i - 255*a^3*c^8*d^4*e + a^3*c^9*d^3*e*220i + 66
*a^3*c^10*d^2*e - 12433*a^3*d^12*e - a^3*c^12*e - 1026*c^2*d^7 + c^3*d^6*430i + 117*c^4*d^5 - c^5*d^4*15i + c*
d^8*1725i + 777*d^9, e, k)*((a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*
c^5*d^3*6i - a^3*c^6*d^2)*(512*a^6*c^15*d - 512*a^6*c*d^15 + a^6*c^2*d^14*3072i + 5632*a^6*c^3*d^13 + a^6*c^4*
d^12*2048i + 19968*a^6*c^5*d^11 - a^6*c^6*d^10*19456i + 13824*a^6*c^7*d^9 - a^6*c^8*d^8*36864i - 13824*a^6*c^9
*d^7 - a^6*c^10*d^6*19456i - 19968*a^6*c^11*d^5 + a^6*c^12*d^4*2048i - 5632*a^6*c^13*d^3 + a^6*c^14*d^2*3072i)
+ tan(e + f*x)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i -
a^3*c^6*d^2)*(a^6*c*d^15*2304i - 384*a^6*d^16 - 128*a^6*c^16 - a^6*c^15*d*768i + 4352*a^6*c^2*d^14 + a^6*c^3*
d^13*768i + 13568*a^6*c^4*d^12 - a^6*c^5*d^11*15104i + 5376*a^6*c^6*d^10 - a^6*c^7*d^9*22784i - 13824*a^6*c^8*
d^8 - a^6*c^9*d^7*5376i - 11520*a^6*c^10*d^6 + a^6*c^11*d^5*6400i + 768*a^6*c^12*d^4 + a^6*c^13*d^3*1792i + 17
92*a^6*c^14*d^2)) + (a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*
6i - a^3*c^6*d^2)*(8*a^3*c^13 + a^3*d^13*440i + 1016*a^3*c*d^12 + a^3*c^12*d*72i + a^3*c^2*d^11*1056i + 4000*a
^3*c^3*d^10 - a^3*c^4*d^9*360i + 5592*a^3*c^5*d^8 - a^3*c^6*d^7*2944i + 2944*a^3*c^7*d^6 - a^3*c^8*d^5*2712i +
40*a^3*c^9*d^4 - a^3*c^10*d^3*672i - 288*a^3*c^11*d^2) + tan(e + f*x)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^
6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)*(1344*a^3*d^13 - a^3*c*d^12*1456i + 16*a^
3*c^12*d + 5008*a^3*c^2*d^11 - a^3*c^3*d^10*4912i + 8400*a^3*c^4*d^9 - a^3*c^5*d^8*7904i + 6560*a^3*c^6*d^7 -
a^3*c^7*d^6*6752i + 1248*a^3*c^8*d^5 - a^3*c^9*d^4*2160i - 560*a^3*c^10*d^3 + a^3*c^11*d^2*144i)) + tan(e + f*
x)*(a^3*d^8 + a^3*c*d^7*6i - 15*a^3*c^2*d^6 - a^3*c^3*d^5*20i + 15*a^3*c^4*d^4 + a^3*c^5*d^3*6i - a^3*c^6*d^2)
*(10596*c^2*d^8 - 3025*d^10 - c*d^9*10340i + c^3*d^7*2348i + 762*c^4*d^6 + c^5*d^5*4i + 68*c^6*d^4 - c^7*d^3*1
2i - c^8*d^2))*root(a^9*c^9*d^9*e^3*56320i + a^9*c^11*d^7*e^3*36864i + a^9*c^7*d^11*e^3*36864i + 29696*a^9*c^1
2*d^6*e^3 - 29696*a^9*c^6*d^12*e^3 + 16896*a^9*c^10*d^8*e^3 - 16896*a^9*c^8*d^10*e^3 + 15360*a^9*c^14*d^4*e^3
- 15360*a^9*c^4*d^14*e^3 + a^9*c^13*d^5*e^3*6144i + a^9*c^5*d^13*e^3*6144i - a^9*c^15*d^3*e^3*4096i - a^9*c^3*
d^15*e^3*4096i + 2304*a^9*c^16*d^2*e^3 - 2304*a^9*c^2*d^16*e^3 - a^9*c^17*d*e^3*1536i - a^9*c*d^17*e^3*1536i +
256*a^9*d^18*e^3 - 256*a^9*c^18*e^3 - a^3*c*d^11*e*64884i - a^3*c^11*d*e*12i + a^3*c^3*d^9*e*137380i + 136578
*a^3*c^2*d^10*e - 58575*a^3*c^4*d^8*e - 1060*a^3*c^6*d^6*e + a^3*c^7*d^5*e*360i - a^3*c^5*d^7*e*360i - 255*a^3
*c^8*d^4*e + a^3*c^9*d^3*e*220i + 66*a^3*c^10*d^2*e - 12433*a^3*d^12*e - a^3*c^12*e - 1026*c^2*d^7 + c^3*d^6*4
30i + 117*c^4*d^5 - c^5*d^4*15i + c*d^8*1725i + 777*d^9, e, k), k, 1, 3)/f + ((tan(e + f*x)^2*(c*d^4*872i - 3*
c^4*d + c^5*3i + 298*d^5 - 317*c^2*d^3 + c^3*d^2*47i))/(24*a^3*d^2*(4*c*d^5 - 4*c^5*d + c^6*1i + d^6*1i - c^2*
d^4*5i - c^4*d^2*5i)) - (tan(e + f*x)^4*(c*d^3*94i + c^3*d*6i + c^4 + 55*d^4 - 16*c^2*d^2))/(8*a^3*(3*c*d^6 -
c^6*d*3i - c^7 + d^7*1i - c^2*d^5*1i + 5*c^3*d^4 - c^4*d^3*5i + c^5*d^2)) + (c^5*d*180i - c*d^5*360i + 50*c^6
+ 60*d^6 + 1250*c^2*d^4 + c^3*d^3*900i - 80*c^4...

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