3.1098 \(\int \frac {1}{(a+i a \tan (e+f x)) (c+d \tan (e+f x))^3} \, dx\)

Optimal. Leaf size=273 \[ \frac {d \left (c^2-8 i c d-3 d^2\right )}{2 a f (c-i d)^2 (c+i d)^3 (c+d \tan (e+f x))}+\frac {2 d^2 \left (3 c^2-2 i c d-d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a f (c+i d)^4 (d+i c)^3}+\frac {x \left (c^4+4 i c^3 d+6 c^2 d^2-12 i c d^3-3 d^4\right )}{2 a (c-i d)^3 (c+i d)^4}+\frac {d (c-2 i d)}{2 a f (c-i d) (c+i d)^2 (c+d \tan (e+f x))^2}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^2} \]

[Out]

1/2*(c^4+4*I*c^3*d+6*c^2*d^2-12*I*c*d^3-3*d^4)*x/a/(c-I*d)^3/(c+I*d)^4+2*d^2*(3*c^2-2*I*c*d-d^2)*ln(c*cos(f*x+
e)+d*sin(f*x+e))/a/(c+I*d)^4/(I*c+d)^3/f+1/2*(c-2*I*d)*d/a/(c-I*d)/(c+I*d)^2/f/(c+d*tan(f*x+e))^2-1/2/(I*c-d)/
f/(a+I*a*tan(f*x+e))/(c+d*tan(f*x+e))^2+1/2*d*(c^2-8*I*c*d-3*d^2)/a/(c-I*d)^2/(c+I*d)^3/f/(c+d*tan(f*x+e))

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Rubi [A]  time = 0.49, antiderivative size = 273, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3552, 3529, 3531, 3530} \[ \frac {d \left (c^2-8 i c d-3 d^2\right )}{2 a f (c-i d)^2 (c+i d)^3 (c+d \tan (e+f x))}+\frac {2 d^2 \left (3 c^2-2 i c d-d^2\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{a f (c+i d)^4 (d+i c)^3}+\frac {x \left (6 c^2 d^2+4 i c^3 d+c^4-12 i c d^3-3 d^4\right )}{2 a (c-i d)^3 (c+i d)^4}+\frac {d (c-2 i d)}{2 a f (c-i d) (c+i d)^2 (c+d \tan (e+f x))^2}-\frac {1}{2 f (-d+i c) (a+i a \tan (e+f x)) (c+d \tan (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

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 3552

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(a
*(c + d*Tan[e + f*x])^(n + 1))/(2*f*(b*c - a*d)*(a + b*Tan[e + f*x])), x] + Dist[1/(2*a*(b*c - a*d)), Int[(c +
d*Tan[e + f*x])^n*Simp[b*c + a*d*(n - 1) - b*d*n*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] &&  !GtQ[n, 0]

Rubi steps

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

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Mathematica [B]  time = 7.82, size = 1231, normalized size = 4.51 \[ \frac {\sec (e+f x) \left (-\cos \left (\frac {e}{2}\right ) d^4-i \sin \left (\frac {e}{2}\right ) d^4-2 i c \cos \left (\frac {e}{2}\right ) d^3+2 c \sin \left (\frac {e}{2}\right ) d^3+3 c^2 \cos \left (\frac {e}{2}\right ) d^2+3 i c^2 \sin \left (\frac {e}{2}\right ) d^2\right ) \left (-2 \tan ^{-1}\left (\frac {-d \cos (f x)-c \sin (f x)}{c \cos (f x)-d \sin (f x)}\right ) \cos \left (\frac {e}{2}\right )-2 i \tan ^{-1}\left (\frac {-d \cos (f x)-c \sin (f x)}{c \cos (f x)-d \sin (f x)}\right ) \sin \left (\frac {e}{2}\right )\right ) (\cos (f x)+i \sin (f x))}{(c-i d)^3 (c+i d)^4 f (i \tan (e+f x) a+a)}+\frac {\sec (e+f x) \left (-\cos \left (\frac {e}{2}\right ) d^4-i \sin \left (\frac {e}{2}\right ) d^4-2 i c \cos \left (\frac {e}{2}\right ) d^3+2 c \sin \left (\frac {e}{2}\right ) d^3+3 c^2 \cos \left (\frac {e}{2}\right ) d^2+3 i c^2 \sin \left (\frac {e}{2}\right ) d^2\right ) \left (i \cos \left (\frac {e}{2}\right ) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )-\log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right ) \sin \left (\frac {e}{2}\right )\right ) (\cos (f x)+i \sin (f x))}{(c-i d)^3 (c+i d)^4 f (i \tan (e+f x) a+a)}+\frac {\cos (2 f x) \sec (e+f x) \left (\frac {1}{4} i \cos (e)+\frac {\sin (e)}{4}\right ) (\cos (f x)+i \sin (f x))}{(c+i d)^3 f (i \tan (e+f x) a+a)}+\frac {\left (c^4+4 i d c^3+6 d^2 c^2-12 i d^3 c-3 d^4\right ) \sec (e+f x) \left (\frac {1}{2} f x \cos (e)+\frac {1}{2} i f x \sin (e)\right ) (\cos (f x)+i \sin (f x))}{(c-i d)^3 (c+i d)^4 f (i \tan (e+f x) a+a)}+\frac {x \sec (e+f x) \left (\frac {2 d^4}{(c-i d)^3 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac {4 i c d^3}{(c-i d)^3 (c+i d)^3 (c \cos (e)+d \sin (e))}-\frac {6 c^2 d^2}{(c-i d)^3 (c+i d)^3 (c \cos (e)+d \sin (e))}+\frac {\left (-d^4-2 i c d^3+3 c^2 d^2\right ) (2 \cos (e)+2 i \sin (e)) (-\cos (2 e) c-i \sin (2 e) c+c+i d+i d \cos (2 e)-d \sin (2 e))}{(c-i d)^3 (c+i d)^4 (\cos (2 e) c+i \sin (2 e) c+c+i d-i d \cos (2 e)+d \sin (2 e))}\right ) (\cos (f x)+i \sin (f x))}{i \tan (e+f x) a+a}+\frac {\sec (e+f x) \left (\frac {\cos (e)}{4}-\frac {1}{4} i \sin (e)\right ) \sin (2 f x) (\cos (f x)+i \sin (f x))}{(c+i d)^3 f (i \tan (e+f x) a+a)}+\frac {\sec (e+f x) \left (\frac {1}{2} i \cos (e-f x) d^4-\frac {1}{2} i \cos (e+f x) d^4-\frac {1}{2} \sin (e-f x) d^4+\frac {1}{2} \sin (e+f x) d^4-2 c \cos (e-f x) d^3+2 c \cos (e+f x) d^3-2 i c \sin (e-f x) d^3+2 i c \sin (e+f x) d^3\right ) (\cos (f x)+i \sin (f x))}{(c-i d)^2 (c+i d)^3 f (c \cos (e)+d \sin (e)) (c \cos (e+f x)+d \sin (e+f x)) (i \tan (e+f x) a+a)}+\frac {\sec (e+f x) \left (\frac {1}{2} d^4 \sin (e)-\frac {1}{2} i d^4 \cos (e)\right ) (\cos (f x)+i \sin (f x))}{(c-i d)^2 (c+i d)^3 f (c \cos (e+f x)+d \sin (e+f x))^2 (i \tan (e+f x) a+a)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sec[e + f*x]*(3*c^2*d^2*Cos[e/2] - (2*I)*c*d^3*Cos[e/2] - d^4*Cos[e/2] + (3*I)*c^2*d^2*Sin[e/2] + 2*c*d^3*Sin
[e/2] - I*d^4*Sin[e/2])*(-2*ArcTan[(-(d*Cos[f*x]) - c*Sin[f*x])/(c*Cos[f*x] - d*Sin[f*x])]*Cos[e/2] - (2*I)*Ar
cTan[(-(d*Cos[f*x]) - c*Sin[f*x])/(c*Cos[f*x] - d*Sin[f*x])]*Sin[e/2])*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^3*(
c + I*d)^4*f*(a + I*a*Tan[e + f*x])) + (Sec[e + f*x]*(3*c^2*d^2*Cos[e/2] - (2*I)*c*d^3*Cos[e/2] - d^4*Cos[e/2]
+ (3*I)*c^2*d^2*Sin[e/2] + 2*c*d^3*Sin[e/2] - I*d^4*Sin[e/2])*(I*Cos[e/2]*Log[(c*Cos[e + f*x] + d*Sin[e + f*x
])^2] - Log[(c*Cos[e + f*x] + d*Sin[e + f*x])^2]*Sin[e/2])*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^3*(c + I*d)^4*f
*(a + I*a*Tan[e + f*x])) + (Cos[2*f*x]*Sec[e + f*x]*((I/4)*Cos[e] + Sin[e]/4)*(Cos[f*x] + I*Sin[f*x]))/((c + I
*d)^3*f*(a + I*a*Tan[e + f*x])) + ((c^4 + (4*I)*c^3*d + 6*c^2*d^2 - (12*I)*c*d^3 - 3*d^4)*Sec[e + f*x]*((f*x*C
os[e])/2 + (I/2)*f*x*Sin[e])*(Cos[f*x] + I*Sin[f*x]))/((c - I*d)^3*(c + I*d)^4*f*(a + I*a*Tan[e + f*x])) + (x*
Sec[e + f*x]*((-6*c^2*d^2)/((c - I*d)^3*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + ((4*I)*c*d^3)/((c - I*d)^3*(c + I
*d)^3*(c*Cos[e] + d*Sin[e])) + (2*d^4)/((c - I*d)^3*(c + I*d)^3*(c*Cos[e] + d*Sin[e])) + ((3*c^2*d^2 - (2*I)*c
*d^3 - d^4)*(2*Cos[e] + (2*I)*Sin[e])*(c + I*d - c*Cos[2*e] + I*d*Cos[2*e] - I*c*Sin[2*e] - d*Sin[2*e]))/((c -
I*d)^3*(c + I*d)^4*(c + I*d + c*Cos[2*e] - I*d*Cos[2*e] + I*c*Sin[2*e] + d*Sin[2*e])))*(Cos[f*x] + I*Sin[f*x]
))/(a + I*a*Tan[e + f*x]) + (Sec[e + f*x]*(Cos[e]/4 - (I/4)*Sin[e])*(Cos[f*x] + I*Sin[f*x])*Sin[2*f*x])/((c +
I*d)^3*f*(a + I*a*Tan[e + f*x])) + (Sec[e + f*x]*((-1/2*I)*d^4*Cos[e] + (d^4*Sin[e])/2)*(Cos[f*x] + I*Sin[f*x]
))/((c - I*d)^2*(c + I*d)^3*f*(c*Cos[e + f*x] + d*Sin[e + f*x])^2*(a + I*a*Tan[e + f*x])) + (Sec[e + f*x]*(Cos
[f*x] + I*Sin[f*x])*(-2*c*d^3*Cos[e - f*x] + (I/2)*d^4*Cos[e - f*x] + 2*c*d^3*Cos[e + f*x] - (I/2)*d^4*Cos[e +
f*x] - (2*I)*c*d^3*Sin[e - f*x] - (d^4*Sin[e - f*x])/2 + (2*I)*c*d^3*Sin[e + f*x] + (d^4*Sin[e + f*x])/2))/((
c - I*d)^2*(c + I*d)^3*f*(c*Cos[e] + d*Sin[e])*(c*Cos[e + f*x] + d*Sin[e + f*x])*(a + I*a*Tan[e + f*x]))

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fricas [B]  time = 0.52, size = 715, normalized size = 2.62 \[ -\frac {c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6} - {\left (2 i \, c^{6} - 4 \, c^{5} d + 50 i \, c^{4} d^{2} + 120 \, c^{3} d^{3} - 130 i \, c^{2} d^{4} - 68 \, c d^{5} + 14 i \, d^{6}\right )} f x e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (c^{6} - 4 i \, c^{5} d - 5 \, c^{4} d^{2} + 32 i \, c^{3} d^{3} - 5 \, c^{2} d^{4} + 36 i \, c d^{5} + d^{6} - {\left (4 i \, c^{6} - 16 \, c^{5} d + 76 i \, c^{4} d^{2} + 64 \, c^{3} d^{3} + 44 i \, c^{2} d^{4} + 80 \, c d^{5} - 28 i \, d^{6}\right )} f x\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (2 \, c^{6} - 4 i \, c^{5} d + 2 \, c^{4} d^{2} + 24 i \, c^{3} d^{3} - 58 \, c^{2} d^{4} - 20 i \, c d^{5} - 10 \, d^{6} - {\left (2 i \, c^{6} - 12 \, c^{5} d + 18 i \, c^{4} d^{2} - 24 \, c^{3} d^{3} + 30 i \, c^{2} d^{4} - 12 \, c d^{5} + 14 i \, d^{6}\right )} f x\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left ({\left (24 \, c^{4} d^{2} - 64 i \, c^{3} d^{3} - 64 \, c^{2} d^{4} + 32 i \, c d^{5} + 8 \, d^{6}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (48 \, c^{4} d^{2} - 32 i \, c^{3} d^{3} + 32 \, c^{2} d^{4} - 32 i \, c d^{5} - 16 \, d^{6}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (24 \, c^{4} d^{2} + 32 i \, c^{3} d^{3} + 8 \, d^{6}\right )} e^{\left (2 i \, f x + 2 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 (4 i \, a c^{9} + 4 \, a c^{8} d + 16 i \, a c^{7} d^{2} + 16 \, a c^{6} d^{3} + 24 i \, a c^{5} d^{4} + 24 \, a c^{4} d^{5} + 16 i \, a c^{3} d^{6} + 16 \, a c^{2} d^{7} + 4 i \, a c d^{8} + 4 \, a d^{9}\right )} f e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (8 i \, a c^{9} - 8 \, a c^{8} d + 32 i \, a c^{7} d^{2} - 32 \, a c^{6} d^{3} + 48 i \, a c^{5} d^{4} - 48 \, a c^{4} d^{5} + 32 i \, a c^{3} d^{6} - 32 \, a c^{2} d^{7} + 8 i \, a c d^{8} - 8 \, a d^{9}\right )} f e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (4 i \, a c^{9} - 12 \, a c^{8} d - 32 \, a c^{6} d^{3} - 24 i \, a c^{5} d^{4} - 24 \, a c^{4} d^{5} - 32 i \, a c^{3} d^{6} - 12 i \, a c d^{8} + 4 \, a d^{9}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^6 + 3*c^4*d^2 + 3*c^2*d^4 + d^6 - (2*I*c^6 - 4*c^5*d + 50*I*c^4*d^2 + 120*c^3*d^3 - 130*I*c^2*d^4 - 68*c*d
^5 + 14*I*d^6)*f*x*e^(6*I*f*x + 6*I*e) + (c^6 - 4*I*c^5*d - 5*c^4*d^2 + 32*I*c^3*d^3 - 5*c^2*d^4 + 36*I*c*d^5
+ d^6 - (4*I*c^6 - 16*c^5*d + 76*I*c^4*d^2 + 64*c^3*d^3 + 44*I*c^2*d^4 + 80*c*d^5 - 28*I*d^6)*f*x)*e^(4*I*f*x
+ 4*I*e) + (2*c^6 - 4*I*c^5*d + 2*c^4*d^2 + 24*I*c^3*d^3 - 58*c^2*d^4 - 20*I*c*d^5 - 10*d^6 - (2*I*c^6 - 12*c^
5*d + 18*I*c^4*d^2 - 24*c^3*d^3 + 30*I*c^2*d^4 - 12*c*d^5 + 14*I*d^6)*f*x)*e^(2*I*f*x + 2*I*e) + ((24*c^4*d^2
- 64*I*c^3*d^3 - 64*c^2*d^4 + 32*I*c*d^5 + 8*d^6)*e^(6*I*f*x + 6*I*e) + (48*c^4*d^2 - 32*I*c^3*d^3 + 32*c^2*d^
4 - 32*I*c*d^5 - 16*d^6)*e^(4*I*f*x + 4*I*e) + (24*c^4*d^2 + 32*I*c^3*d^3 + 8*d^6)*e^(2*I*f*x + 2*I*e))*log(((
I*c + d)*e^(2*I*f*x + 2*I*e) + I*c - d)/(I*c + d)))/((4*I*a*c^9 + 4*a*c^8*d + 16*I*a*c^7*d^2 + 16*a*c^6*d^3 +
24*I*a*c^5*d^4 + 24*a*c^4*d^5 + 16*I*a*c^3*d^6 + 16*a*c^2*d^7 + 4*I*a*c*d^8 + 4*a*d^9)*f*e^(6*I*f*x + 6*I*e) +
(8*I*a*c^9 - 8*a*c^8*d + 32*I*a*c^7*d^2 - 32*a*c^6*d^3 + 48*I*a*c^5*d^4 - 48*a*c^4*d^5 + 32*I*a*c^3*d^6 - 32*
a*c^2*d^7 + 8*I*a*c*d^8 - 8*a*d^9)*f*e^(4*I*f*x + 4*I*e) + (4*I*a*c^9 - 12*a*c^8*d - 32*a*c^6*d^3 - 24*I*a*c^5
*d^4 - 24*a*c^4*d^5 - 32*I*a*c^3*d^6 - 12*I*a*c*d^8 + 4*a*d^9)*f*e^(2*I*f*x + 2*I*e))

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giac [B]  time = 0.98, size = 500, normalized size = 1.83 \[ -\frac {2 \, {\left (\frac {{\left (3 \, c^{2} d^{3} - 2 i \, c d^{4} - d^{5}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{i \, a c^{7} d - a c^{6} d^{2} + 3 i \, a c^{5} d^{3} - 3 \, a c^{4} d^{4} + 3 i \, a c^{3} d^{5} - 3 \, a c^{2} d^{6} + i \, a c d^{7} - a d^{8}} + \frac {{\left (i \, c - 7 \, d\right )} \log \left (i \, \tan \left (f x + e\right ) + 1\right )}{8 \, a c^{4} + 32 i \, a c^{3} d - 48 \, a c^{2} d^{2} - 32 i \, a c d^{3} + 8 \, a d^{4}} - \frac {i \, \log \left (-i \, \tan \left (f x + e\right ) + 1\right )}{8 \, a c^{3} - 24 i \, a c^{2} d - 24 \, a c d^{2} + 8 i \, a d^{3}} + \frac {-i \, c \tan \left (f x + e\right ) + 7 \, d \tan \left (f x + e\right ) - 3 \, c - 9 i \, d}{{\left (8 \, a c^{4} + 32 i \, a c^{3} d - 48 \, a c^{2} d^{2} - 32 i \, a c d^{3} + 8 \, a d^{4}\right )} {\left (\tan \left (f x + e\right ) - i\right )}} - \frac {18 \, c^{2} d^{4} \tan \left (f x + e\right )^{2} - 12 i \, c d^{5} \tan \left (f x + e\right )^{2} - 6 \, d^{6} \tan \left (f x + e\right )^{2} + 42 \, c^{3} d^{3} \tan \left (f x + e\right ) - 26 i \, c^{2} d^{4} \tan \left (f x + e\right ) - 6 \, c d^{5} \tan \left (f x + e\right ) - 2 i \, d^{6} \tan \left (f x + e\right ) + 25 \, c^{4} d^{2} - 14 i \, c^{3} d^{3} + 2 \, c^{2} d^{4} - 2 i \, c d^{5} + d^{6}}{{\left (4 i \, a c^{7} - 4 \, a c^{6} d + 12 i \, a c^{5} d^{2} - 12 \, a c^{4} d^{3} + 12 i \, a c^{3} d^{4} - 12 \, a c^{2} d^{5} + 4 i \, a c d^{6} - 4 \, a d^{7}\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{2}}\right )}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*((3*c^2*d^3 - 2*I*c*d^4 - d^5)*log(d*tan(f*x + e) + c)/(I*a*c^7*d - a*c^6*d^2 + 3*I*a*c^5*d^3 - 3*a*c^4*d^4
+ 3*I*a*c^3*d^5 - 3*a*c^2*d^6 + I*a*c*d^7 - a*d^8) + (I*c - 7*d)*log(I*tan(f*x + e) + 1)/(8*a*c^4 + 32*I*a*c^
3*d - 48*a*c^2*d^2 - 32*I*a*c*d^3 + 8*a*d^4) - I*log(-I*tan(f*x + e) + 1)/(8*a*c^3 - 24*I*a*c^2*d - 24*a*c*d^2
+ 8*I*a*d^3) + (-I*c*tan(f*x + e) + 7*d*tan(f*x + e) - 3*c - 9*I*d)/((8*a*c^4 + 32*I*a*c^3*d - 48*a*c^2*d^2 -
32*I*a*c*d^3 + 8*a*d^4)*(tan(f*x + e) - I)) - (18*c^2*d^4*tan(f*x + e)^2 - 12*I*c*d^5*tan(f*x + e)^2 - 6*d^6*
tan(f*x + e)^2 + 42*c^3*d^3*tan(f*x + e) - 26*I*c^2*d^4*tan(f*x + e) - 6*c*d^5*tan(f*x + e) - 2*I*d^6*tan(f*x
+ e) + 25*c^4*d^2 - 14*I*c^3*d^3 + 2*c^2*d^4 - 2*I*c*d^5 + d^6)/((4*I*a*c^7 - 4*a*c^6*d + 12*I*a*c^5*d^2 - 12*
a*c^4*d^3 + 12*I*a*c^3*d^4 - 12*a*c^2*d^5 + 4*I*a*c*d^6 - 4*a*d^7)*(d*tan(f*x + e) + c)^2))/f

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maple [B]  time = 0.34, size = 542, normalized size = 1.99 \[ -\frac {i \ln \left (\tan \left (f x +e \right )+i\right )}{4 f a \left (i d -c \right )^{3}}-\frac {6 i d^{2} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}+\frac {2 i d^{4} \ln \left (c +d \tan \left (f x +e \right )\right )}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}-\frac {4 d^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4}}+\frac {3 i d^{2} c^{3}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {3 i d^{4} c}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{3} c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {d^{5}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )}+\frac {i d^{2} c^{4}}{2 f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {i d^{4} c^{2}}{f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {i d^{6}}{2 f a \left (i d -c \right )^{3} \left (i d +c \right )^{4} \left (c +d \tan \left (f x +e \right )\right )^{2}}+\frac {1}{2 f a \left (i d +c \right )^{3} \left (\tan \left (f x +e \right )-i\right )}-\frac {i \ln \left (\tan \left (f x +e \right )-i\right ) c}{4 f a \left (i d +c \right )^{4}}+\frac {7 \ln \left (\tan \left (f x +e \right )-i\right ) d}{4 f a \left (i d +c \right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*I/f/a/(I*d-c)^3*ln(tan(f*x+e)+I)-6*I/f/a*d^2/(I*d-c)^3/(c+I*d)^4*ln(c+d*tan(f*x+e))*c^2+2*I/f/a*d^4/(I*d-
c)^3/(c+I*d)^4*ln(c+d*tan(f*x+e))-4/f/a*d^3/(I*d-c)^3/(c+I*d)^4*ln(c+d*tan(f*x+e))*c+3*I/f/a*d^2/(I*d-c)^3/(c+
I*d)^4/(c+d*tan(f*x+e))*c^3+3*I/f/a*d^4/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))*c+1/f/a*d^3/(I*d-c)^3/(c+I*d)^4/(
c+d*tan(f*x+e))*c^2+1/f/a*d^5/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))+1/2*I/f/a*d^2/(I*d-c)^3/(c+I*d)^4/(c+d*tan(
f*x+e))^2*c^4+I/f/a*d^4/(I*d-c)^3/(c+I*d)^4/(c+d*tan(f*x+e))^2*c^2+1/2*I/f/a*d^6/(I*d-c)^3/(c+I*d)^4/(c+d*tan(
f*x+e))^2+1/2/f/a/(c+I*d)^3/(tan(f*x+e)-I)-1/4*I/f/a/(c+I*d)^4*ln(tan(f*x+e)-I)*c+7/4/f/a/(c+I*d)^4*ln(tan(f*x
+e)-I)*d

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*tan(f*x+e))/(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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mupad [B]  time = 10.10, size = 1910, normalized size = 7.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log((a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(47*c*d^5 - c^5*d - d^6*12i + c^2*d^4*56i - 34*c^3*d^3 + c^4*d^2*4
i) - root(a^3*c^7*d^7*e^3*640i + a^3*c^9*d^5*e^3*480i + a^3*c^5*d^9*e^3*480i + a^3*c^11*d^3*e^3*192i + a^3*c^3
*d^11*e^3*192i + 144*a^3*c^10*d^4*e^3 - 144*a^3*c^4*d^10*e^3 + 80*a^3*c^12*d^2*e^3 + 80*a^3*c^8*d^6*e^3 - 80*a
^3*c^6*d^8*e^3 - 80*a^3*c^2*d^12*e^3 + a^3*c^13*d*e^3*32i + a^3*c*d^13*e^3*32i - 16*a^3*d^14*e^3 + 16*a^3*c^14
*e^3 - a*c^3*d^5*e*744i - 660*a*c^2*d^6*e + 558*a*c^4*d^4*e + a*c^5*d^3*e*24i - 4*a*c^6*d^2*e + a*c*d^7*e*264i
+ a*c^7*d*e*8i + 57*a*d^8*e + a*c^8*e + 38*c^2*d^3 - c^3*d^2*6i - c*d^4*26i - 14*d^5, e, k)*((a*d^4 - a*c^2*d
^2 + a*c*d^3*2i)*(2*a*c^9 + a*d^9*6i + a*c^2*d^7*28i - 4*a*c^3*d^6 + a*c^4*d^5*48i + a*c^6*d^3*36i + 4*a*c^7*d
^2 - 2*a*c*d^8 + a*c^8*d*10i) + root(a^3*c^7*d^7*e^3*640i + a^3*c^9*d^5*e^3*480i + a^3*c^5*d^9*e^3*480i + a^3*
c^11*d^3*e^3*192i + a^3*c^3*d^11*e^3*192i + 144*a^3*c^10*d^4*e^3 - 144*a^3*c^4*d^10*e^3 + 80*a^3*c^12*d^2*e^3
+ 80*a^3*c^8*d^6*e^3 - 80*a^3*c^6*d^8*e^3 - 80*a^3*c^2*d^12*e^3 + a^3*c^13*d*e^3*32i + a^3*c*d^13*e^3*32i - 16
*a^3*d^14*e^3 + 16*a^3*c^14*e^3 - a*c^3*d^5*e*744i - 660*a*c^2*d^6*e + 558*a*c^4*d^4*e + a*c^5*d^3*e*24i - 4*a
*c^6*d^2*e + a*c*d^7*e*264i + a*c^7*d*e*8i + 57*a*d^8*e + a*c^8*e + 38*c^2*d^3 - c^3*d^2*6i - c*d^4*26i - 14*d
^5, e, k)*((a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(32*a^2*c^11*d - 32*a^2*c*d^11 + a^2*c^2*d^10*64i - 96*a^2*c^3*d^9
+ a^2*c^4*d^8*256i - 64*a^2*c^5*d^7 + a^2*c^6*d^6*384i + 64*a^2*c^7*d^5 + a^2*c^8*d^4*256i + 96*a^2*c^9*d^3 +
a^2*c^10*d^2*64i) - tan(e + f*x)*(a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(8*a^2*c^12 + 24*a^2*d^12 - a^2*c*d^11*48i
+ a^2*c^11*d*16i + 64*a^2*c^2*d^10 - a^2*c^3*d^9*176i + 24*a^2*c^4*d^8 - a^2*c^5*d^7*224i - 64*a^2*c^6*d^6 - a
^2*c^7*d^5*96i - 56*a^2*c^8*d^4 + a^2*c^9*d^3*16i)) + tan(e + f*x)*(a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(24*a*d^9
+ 44*a*c^2*d^7 + a*c^3*d^6*68i + 20*a*c^4*d^5 + a*c^5*d^4*100i + 4*a*c^6*d^3 + a*c^7*d^2*44i + a*c*d^8*12i + 4
*a*c^8*d)) - tan(e + f*x)*(a*d^4 - a*c^2*d^2 + a*c*d^3*2i)*(c*d^5*48i + 9*d^6 - 70*c^2*d^4 - c^3*d^3*16i + c^4
*d^2))*root(a^3*c^7*d^7*e^3*640i + a^3*c^9*d^5*e^3*480i + a^3*c^5*d^9*e^3*480i + a^3*c^11*d^3*e^3*192i + a^3*c
^3*d^11*e^3*192i + 144*a^3*c^10*d^4*e^3 - 144*a^3*c^4*d^10*e^3 + 80*a^3*c^12*d^2*e^3 + 80*a^3*c^8*d^6*e^3 - 80
*a^3*c^6*d^8*e^3 - 80*a^3*c^2*d^12*e^3 + a^3*c^13*d*e^3*32i + a^3*c*d^13*e^3*32i - 16*a^3*d^14*e^3 + 16*a^3*c^
14*e^3 - a*c^3*d^5*e*744i - 660*a*c^2*d^6*e + 558*a*c^4*d^4*e + a*c^5*d^3*e*24i - 4*a*c^6*d^2*e + a*c*d^7*e*26
4i + a*c^7*d*e*8i + 57*a*d^8*e + a*c^8*e + 38*c^2*d^3 - c^3*d^2*6i - c*d^4*26i - 14*d^5, e, k), k, 1, 3)/f - (
(tan(e + f*x)^2*(c*d*8i - c^2 + 3*d^2))/(2*a*(c*d^4 + c^4*d*1i + c^5 + d^5*1i + c^2*d^3*2i + 2*c^3*d^2)) + (ta
n(e + f*x)*(9*c*d + c^2*2i - d^2*1i))/(2*a*d*(2*c*d^3 + 2*c^3*d - c^4*1i + d^4*1i)) + (c^3*d*6i - c*d^3*6i - 3
*c^4 + 3*d^4 + 24*c^2*d^2)/(6*a*d^2*(c^2 + d^2)*(c*d^2 + c^2*d*1i + c^3 + d^3*1i)))/(f*(tan(e + f*x)^2*((2*c)/
d - 1i) - tan(e + f*x)*((c*2i)/d - c^2/d^2) + tan(e + f*x)^3 - (c^2*1i)/d^2))

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sympy [A]  time = 95.61, size = 824, normalized size = 3.02 \[ \frac {x \left (- c - 7 i d\right )}{- 2 a c^{4} - 8 i a c^{3} d + 12 a c^{2} d^{2} + 8 i a c d^{3} - 2 a d^{4}} + \frac {- 8 c^{2} d^{3} - 6 i c d^{4} - 2 d^{5} + \left (- 8 c^{2} d^{3} e^{2 i e} + 8 i c d^{4} e^{2 i e}\right ) e^{2 i f x}}{a c^{8} f + 2 i a c^{7} d f + 2 a c^{6} d^{2} f + 6 i a c^{5} d^{3} f + 6 i a c^{3} d^{5} f - 2 a c^{2} d^{6} f + 2 i a c d^{7} f - a d^{8} f + \left (2 a c^{8} f e^{2 i e} + 8 a c^{6} d^{2} f e^{2 i e} + 12 a c^{4} d^{4} f e^{2 i e} + 8 a c^{2} d^{6} f e^{2 i e} + 2 a d^{8} f e^{2 i e}\right ) e^{2 i f x} + \left (a c^{8} f e^{4 i e} - 2 i a c^{7} d f e^{4 i e} + 2 a c^{6} d^{2} f e^{4 i e} - 6 i a c^{5} d^{3} f e^{4 i e} - 6 i a c^{3} d^{5} f e^{4 i e} - 2 a c^{2} d^{6} f e^{4 i e} - 2 i a c d^{7} f e^{4 i e} - a d^{8} f e^{4 i e}\right ) e^{4 i f x}} + \begin {cases} \frac {i e^{- 2 i f x}}{4 a c^{3} f e^{2 i e} + 12 i a c^{2} d f e^{2 i e} - 12 a c d^{2} f e^{2 i e} - 4 i a d^{3} f e^{2 i e}} & \text {for}\: 4 a c^{3} f e^{2 i e} + 12 i a c^{2} d f e^{2 i e} - 12 a c d^{2} f e^{2 i e} - 4 i a d^{3} f e^{2 i e} \neq 0 \\x \left (- \frac {c + 7 i d}{2 a c^{4} + 8 i a c^{3} d - 12 a c^{2} d^{2} - 8 i a c d^{3} + 2 a d^{4}} + \frac {- i c e^{2 i e} - i c + 7 d e^{2 i e} + d}{- 2 i a c^{4} e^{2 i e} + 8 a c^{3} d e^{2 i e} + 12 i a c^{2} d^{2} e^{2 i e} - 8 a c d^{3} e^{2 i e} - 2 i a d^{4} e^{2 i e}}\right ) & \text {otherwise} \end {cases} + \frac {2 i d^{2} \left (3 c^{2} - 2 i c d - d^{2}\right ) \log {\left (\frac {i c - d}{i c e^{2 i e} + d e^{2 i e}} + e^{2 i f x} \right )}}{a f \left (c - i d\right )^{3} \left (c + i d\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(-c - 7*I*d)/(-2*a*c**4 - 8*I*a*c**3*d + 12*a*c**2*d**2 + 8*I*a*c*d**3 - 2*a*d**4) + (-8*c**2*d**3 - 6*I*c*d
**4 - 2*d**5 + (-8*c**2*d**3*exp(2*I*e) + 8*I*c*d**4*exp(2*I*e))*exp(2*I*f*x))/(a*c**8*f + 2*I*a*c**7*d*f + 2*
a*c**6*d**2*f + 6*I*a*c**5*d**3*f + 6*I*a*c**3*d**5*f - 2*a*c**2*d**6*f + 2*I*a*c*d**7*f - a*d**8*f + (2*a*c**
8*f*exp(2*I*e) + 8*a*c**6*d**2*f*exp(2*I*e) + 12*a*c**4*d**4*f*exp(2*I*e) + 8*a*c**2*d**6*f*exp(2*I*e) + 2*a*d
**8*f*exp(2*I*e))*exp(2*I*f*x) + (a*c**8*f*exp(4*I*e) - 2*I*a*c**7*d*f*exp(4*I*e) + 2*a*c**6*d**2*f*exp(4*I*e)
- 6*I*a*c**5*d**3*f*exp(4*I*e) - 6*I*a*c**3*d**5*f*exp(4*I*e) - 2*a*c**2*d**6*f*exp(4*I*e) - 2*I*a*c*d**7*f*e
xp(4*I*e) - a*d**8*f*exp(4*I*e))*exp(4*I*f*x)) + Piecewise((I*exp(-2*I*f*x)/(4*a*c**3*f*exp(2*I*e) + 12*I*a*c*
*2*d*f*exp(2*I*e) - 12*a*c*d**2*f*exp(2*I*e) - 4*I*a*d**3*f*exp(2*I*e)), Ne(4*a*c**3*f*exp(2*I*e) + 12*I*a*c**
2*d*f*exp(2*I*e) - 12*a*c*d**2*f*exp(2*I*e) - 4*I*a*d**3*f*exp(2*I*e), 0)), (x*(-(c + 7*I*d)/(2*a*c**4 + 8*I*a
*c**3*d - 12*a*c**2*d**2 - 8*I*a*c*d**3 + 2*a*d**4) + (-I*c*exp(2*I*e) - I*c + 7*d*exp(2*I*e) + d)/(-2*I*a*c**
4*exp(2*I*e) + 8*a*c**3*d*exp(2*I*e) + 12*I*a*c**2*d**2*exp(2*I*e) - 8*a*c*d**3*exp(2*I*e) - 2*I*a*d**4*exp(2*
I*e))), True)) + 2*I*d**2*(3*c**2 - 2*I*c*d - d**2)*log((I*c - d)/(I*c*exp(2*I*e) + d*exp(2*I*e)) + exp(2*I*f*
x))/(a*f*(c - I*d)**3*(c + I*d)**4)

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