### 3.32 $$\int \frac{1}{(c+d x) (a+i a \tan (e+f x))^3} \, dx$$

Optimal. Leaf size=449 $-\frac{3 i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{i \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}$

[Out]

(3*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f
)/d + 4*f*x])/(8*a^3*d) + (Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x]/(8*a^
3*d) - ((I/8)*CosIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(4*c*f)/d
+ 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^3*
d) - (((3*I)/8)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^3*d) - (3*Sin[2*e - (2*c*f)/d]*SinInte
gral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (((3*I)/8)*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^3*d) -
(3*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - ((I/8)*Cos[6*e - (6*c*f)/d]*SinIntegral[(
6*c*f)/d + 6*f*x])/(a^3*d) - (Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 1.78286, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 53, number of rules used = 7, integrand size = 23, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.304, Rules used = {3728, 3303, 3299, 3302, 3312, 4406, 4428} $-\frac{3 i \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{i \text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}+\frac{3 \text{CosIntegral}\left (\frac{4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}+\frac{\text{CosIntegral}\left (\frac{6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (2 x f+\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (4 x f+\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (6 x f+\frac{6 c f}{d}\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(8*a^3*d) + (3*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f
)/d + 4*f*x])/(8*a^3*d) + (Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d) + Log[c + d*x]/(8*a^
3*d) - ((I/8)*CosIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(4*c*f)/d
+ 4*f*x]*Sin[4*e - (4*c*f)/d])/(a^3*d) - (((3*I)/8)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^3*
d) - (((3*I)/8)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^3*d) - (3*Sin[2*e - (2*c*f)/d]*SinInte
gral[(2*c*f)/d + 2*f*x])/(8*a^3*d) - (((3*I)/8)*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^3*d) -
(3*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(8*a^3*d) - ((I/8)*Cos[6*e - (6*c*f)/d]*SinIntegral[(
6*c*f)/d + 6*f*x])/(a^3*d) - (Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(8*a^3*d)

Rule 3728

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

Rule 3303

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4428

Int[((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(p_.)*Sin[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int[E
xpandTrigReduce[(e + f*x)^m, Sin[a + b*x]^p*Sin[c + d*x]^q, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[p,
0] && IGtQ[q, 0] && IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{1}{(c+d x) (a+i a \tan (e+f x))^3} \, dx &=\int \left (\frac{1}{8 a^3 (c+d x)}+\frac{3 \cos (2 e+2 f x)}{8 a^3 (c+d x)}+\frac{3 \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)}+\frac{\cos ^3(2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \sin (2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 \sin ^2(2 e+2 f x)}{8 a^3 (c+d x)}+\frac{i \sin ^3(2 e+2 f x)}{8 a^3 (c+d x)}-\frac{3 i \sin (4 e+4 f x)}{8 a^3 (c+d x)}-\frac{3 \sin (2 e+2 f x) \sin (4 e+4 f x)}{16 a^3 (c+d x)}\right ) \, dx\\ &=\frac{\log (c+d x)}{8 a^3 d}+\frac{i \int \frac{\sin ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{(3 i) \int \frac{\sin (4 e+4 f x)}{c+d x} \, dx}{8 a^3}+\frac{\int \frac{\cos ^3(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{3 \int \frac{\sin (2 e+2 f x) \sin (4 e+4 f x)}{c+d x} \, dx}{16 a^3}+\frac{3 \int \frac{\cos (2 e+2 f x)}{c+d x} \, dx}{8 a^3}+\frac{3 \int \frac{\cos ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}-\frac{3 \int \frac{\sin ^2(2 e+2 f x)}{c+d x} \, dx}{8 a^3}\\ &=\frac{\log (c+d x)}{8 a^3 d}+\frac{i \int \left (\frac{3 \sin (2 e+2 f x)}{4 (c+d x)}-\frac{\sin (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac{(3 i) \int \left (\frac{\sin (2 e+2 f x)}{4 (c+d x)}+\frac{\sin (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}+\frac{\int \left (\frac{3 \cos (2 e+2 f x)}{4 (c+d x)}+\frac{\cos (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{8 a^3}-\frac{3 \int \left (\frac{\cos (2 e+2 f x)}{2 (c+d x)}-\frac{\cos (6 e+6 f x)}{2 (c+d x)}\right ) \, dx}{16 a^3}-\frac{3 \int \left (\frac{1}{2 (c+d x)}-\frac{\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}+\frac{3 \int \left (\frac{1}{2 (c+d x)}+\frac{\cos (4 e+4 f x)}{2 (c+d x)}\right ) \, dx}{8 a^3}-\frac{\left (3 i \cos \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 i \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}+\frac{\left (3 \cos \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 i \sin \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 i \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}-\frac{\left (3 \sin \left (2 e-\frac{2 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{2 c f}{d}+2 f x\right )}{c+d x} \, dx}{8 a^3}\\ &=\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac{i \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{32 a^3}-\frac{(3 i) \int \frac{\sin (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+\frac{\int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+\frac{3 \int \frac{\cos (6 e+6 f x)}{c+d x} \, dx}{32 a^3}+2 \frac{3 \int \frac{\cos (4 e+4 f x)}{c+d x} \, dx}{16 a^3}\\ &=\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}-\frac{\left (i \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 i \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac{\cos \left (6 e-\frac{6 c f}{d}\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+\frac{\left (3 \cos \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (i \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 i \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}-\frac{\left (3 \sin \left (6 e-\frac{6 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{6 c f}{d}+6 f x\right )}{c+d x} \, dx}{32 a^3}+2 \left (\frac{\left (3 \cos \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\cos \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{16 a^3}-\frac{\left (3 \sin \left (4 e-\frac{4 c f}{d}\right )\right ) \int \frac{\sin \left (\frac{4 c f}{d}+4 f x\right )}{c+d x} \, dx}{16 a^3}\right )\\ &=\frac{3 \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Ci}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}+\frac{\cos \left (6 e-\frac{6 c f}{d}\right ) \text{Ci}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}+\frac{\log (c+d x)}{8 a^3 d}-\frac{i \text{Ci}\left (\frac{6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac{6 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac{4 c f}{d}\right )}{8 a^3 d}-\frac{3 i \text{Ci}\left (\frac{2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac{2 c f}{d}\right )}{8 a^3 d}-\frac{3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 c f}{d}+2 f x\right )}{8 a^3 d}-\frac{3 i \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{8 a^3 d}+2 \left (\frac{3 \cos \left (4 e-\frac{4 c f}{d}\right ) \text{Ci}\left (\frac{4 c f}{d}+4 f x\right )}{16 a^3 d}-\frac{3 \sin \left (4 e-\frac{4 c f}{d}\right ) \text{Si}\left (\frac{4 c f}{d}+4 f x\right )}{16 a^3 d}\right )-\frac{i \cos \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}-\frac{\sin \left (6 e-\frac{6 c f}{d}\right ) \text{Si}\left (\frac{6 c f}{d}+6 f x\right )}{8 a^3 d}\\ \end{align*}

Mathematica [A]  time = 0.653526, size = 336, normalized size = 0.75 $\frac{\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\left (\cos \left (e-\frac{4 c f}{d}\right )-i \sin \left (e-\frac{4 c f}{d}\right )\right ) \left (-i \text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right ) \sin \left (2 e-\frac{2 c f}{d}\right )+\text{CosIntegral}\left (\frac{6 f (c+d x)}{d}\right ) \cos \left (2 e-\frac{2 c f}{d}\right )+3 \text{CosIntegral}\left (\frac{2 f (c+d x)}{d}\right ) \left (\cos \left (2 e-\frac{2 c f}{d}\right )+i \sin \left (2 e-\frac{2 c f}{d}\right )\right )+3 \text{CosIntegral}\left (\frac{4 f (c+d x)}{d}\right )+3 \sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )-\sin \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )-3 i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{2 f (c+d x)}{d}\right )-i \cos \left (2 e-\frac{2 c f}{d}\right ) \text{Si}\left (\frac{6 f (c+d x)}{d}\right )-3 i \text{Si}\left (\frac{4 f (c+d x)}{d}\right )\right )+i \sin (3 e) \log (f (c+d x))+\cos (3 e) \log (f (c+d x))\right )}{8 d (a+i a \tan (e+f x))^3}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sec[e + f*x]^3*(Cos[f*x] + I*Sin[f*x])^3*(Cos[3*e]*Log[f*(c + d*x)] + I*Log[f*(c + d*x)]*Sin[3*e] + (Cos[e -
(4*c*f)/d] - I*Sin[e - (4*c*f)/d])*(3*CosIntegral[(4*f*(c + d*x))/d] + Cos[2*e - (2*c*f)/d]*CosIntegral[(6*f*(
c + d*x))/d] + 3*CosIntegral[(2*f*(c + d*x))/d]*(Cos[2*e - (2*c*f)/d] + I*Sin[2*e - (2*c*f)/d]) - I*CosIntegra
l[(6*f*(c + d*x))/d]*Sin[2*e - (2*c*f)/d] - (3*I)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] + 3*Sin[
2*e - (2*c*f)/d]*SinIntegral[(2*f*(c + d*x))/d] - (3*I)*SinIntegral[(4*f*(c + d*x))/d] - I*Cos[2*e - (2*c*f)/d
]*SinIntegral[(6*f*(c + d*x))/d] - Sin[2*e - (2*c*f)/d]*SinIntegral[(6*f*(c + d*x))/d])))/(8*d*(a + I*a*Tan[e
+ f*x])^3)

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Maple [A]  time = 0.305, size = 163, normalized size = 0.4 \begin{align*}{\frac{\ln \left ( dx+c \right ) }{8\,{a}^{3}d}}-{\frac{1}{8\,{a}^{3}d}{{\rm e}^{{\frac{6\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,6\,ifx+6\,ie+{\frac{6\,i \left ( cf-de \right ) }{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{{\rm e}^{{\frac{4\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,4\,ifx+4\,ie+{\frac{4\,i \left ( cf-de \right ) }{d}} \right ) }-{\frac{3}{8\,{a}^{3}d}{{\rm e}^{{\frac{2\,i \left ( cf-de \right ) }{d}}}}{\it Ei} \left ( 1,2\,ifx+2\,ie+{\frac{2\,i \left ( cf-de \right ) }{d}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*ln(d*x+c)/a^3/d-1/8/a^3/d*exp(6*I*(c*f-d*e)/d)*Ei(1,6*I*f*x+6*I*e+6*I*(c*f-d*e)/d)-3/8/a^3/d*exp(4*I*(c*f-
d*e)/d)*Ei(1,4*I*f*x+4*I*e+4*I*(c*f-d*e)/d)-3/8/a^3/d*exp(2*I*(c*f-d*e)/d)*Ei(1,2*I*f*x+2*I*e+2*I*(c*f-d*e)/d)

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Maxima [A]  time = 1.37683, size = 365, normalized size = 0.81 \begin{align*} -\frac{f \cos \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) + 3 \, f \cos \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) + 3 \, f \cos \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) + 3 i \, f E_{1}\left (\frac{2 i \,{\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac{2 \,{\left (d e - c f\right )}}{d}\right ) + 3 i \, f E_{1}\left (\frac{4 i \,{\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) \sin \left (-\frac{4 \,{\left (d e - c f\right )}}{d}\right ) + i \, f E_{1}\left (\frac{6 i \,{\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) \sin \left (-\frac{6 \,{\left (d e - c f\right )}}{d}\right ) - f \log \left ({\left (f x + e\right )} d - d e + c f\right )}{8 \, a^{3} d f} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(f*cos(-6*(d*e - c*f)/d)*exp_integral_e(1, (6*I*(f*x + e)*d - 6*I*d*e + 6*I*c*f)/d) + 3*f*cos(-4*(d*e - c
*f)/d)*exp_integral_e(1, (4*I*(f*x + e)*d - 4*I*d*e + 4*I*c*f)/d) + 3*f*cos(-2*(d*e - c*f)/d)*exp_integral_e(1
, (2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d) + 3*I*f*exp_integral_e(1, (2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d)*
sin(-2*(d*e - c*f)/d) + 3*I*f*exp_integral_e(1, (4*I*(f*x + e)*d - 4*I*d*e + 4*I*c*f)/d)*sin(-4*(d*e - c*f)/d)
+ I*f*exp_integral_e(1, (6*I*(f*x + e)*d - 6*I*d*e + 6*I*c*f)/d)*sin(-6*(d*e - c*f)/d) - f*log((f*x + e)*d -
d*e + c*f))/(a^3*d*f)

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Fricas [A]  time = 1.66196, size = 284, normalized size = 0.63 \begin{align*} \frac{3 \,{\rm Ei}\left (\frac{-2 i \, d f x - 2 i \, c f}{d}\right ) e^{\left (\frac{-2 i \, d e + 2 i \, c f}{d}\right )} + 3 \,{\rm Ei}\left (\frac{-4 i \, d f x - 4 i \, c f}{d}\right ) e^{\left (\frac{-4 i \, d e + 4 i \, c f}{d}\right )} +{\rm Ei}\left (\frac{-6 i \, d f x - 6 i \, c f}{d}\right ) e^{\left (\frac{-6 i \, d e + 6 i \, c f}{d}\right )} + \log \left (\frac{d x + c}{d}\right )}{8 \, a^{3} d} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(3*Ei((-2*I*d*f*x - 2*I*c*f)/d)*e^((-2*I*d*e + 2*I*c*f)/d) + 3*Ei((-4*I*d*f*x - 4*I*c*f)/d)*e^((-4*I*d*e +
4*I*c*f)/d) + Ei((-6*I*d*f*x - 6*I*c*f)/d)*e^((-6*I*d*e + 6*I*c*f)/d) + log((d*x + c)/d))/(a^3*d)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.25765, size = 1142, normalized size = 2.54 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*cos(2*c*f/d)*cos(2*e)^2*cos_integral(-2*(d*f*x + c*f)/d) + cos(2*e)^3*log(d*x + c) + 3*I*cos(2*e)^2*cos_int
egral(-2*(d*f*x + c*f)/d)*sin(2*c*f/d) + 6*I*cos(2*c*f/d)*cos(2*e)*cos_integral(-2*(d*f*x + c*f)/d)*sin(2*e) +
3*I*cos(2*e)^2*log(d*x + c)*sin(2*e) - 6*cos(2*e)*cos_integral(-2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(2*e) - 3*
cos(2*c*f/d)*cos_integral(-2*(d*f*x + c*f)/d)*sin(2*e)^2 - 3*cos(2*e)*log(d*x + c)*sin(2*e)^2 - 3*I*cos_integr
al(-2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(2*e)^2 - I*log(d*x + c)*sin(2*e)^3 - 3*I*cos(2*c*f/d)*cos(2*e)^2*sin_i
ntegral(2*(d*f*x + c*f)/d) + 3*cos(2*e)^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 6*cos(2*c*f/d)*cos(2*
e)*sin(2*e)*sin_integral(2*(d*f*x + c*f)/d) + 6*I*cos(2*e)*sin(2*c*f/d)*sin(2*e)*sin_integral(2*(d*f*x + c*f)/
d) + 3*I*cos(2*c*f/d)*sin(2*e)^2*sin_integral(2*(d*f*x + c*f)/d) - 3*sin(2*c*f/d)*sin(2*e)^2*sin_integral(2*(d
*f*x + c*f)/d) + 3*cos(4*c*f/d)*cos(2*e)*cos_integral(-4*(d*f*x + c*f)/d) + 3*I*cos(2*e)*cos_integral(-4*(d*f*
x + c*f)/d)*sin(4*c*f/d) + 3*I*cos(4*c*f/d)*cos_integral(-4*(d*f*x + c*f)/d)*sin(2*e) - 3*cos_integral(-4*(d*f
*x + c*f)/d)*sin(4*c*f/d)*sin(2*e) - 3*I*cos(4*c*f/d)*cos(2*e)*sin_integral(4*(d*f*x + c*f)/d) + 3*cos(2*e)*si
n(4*c*f/d)*sin_integral(4*(d*f*x + c*f)/d) + 3*cos(4*c*f/d)*sin(2*e)*sin_integral(4*(d*f*x + c*f)/d) + 3*I*sin
(4*c*f/d)*sin(2*e)*sin_integral(4*(d*f*x + c*f)/d) + cos(6*c*f/d)*cos_integral(-6*(d*f*x + c*f)/d) + I*cos_int
egral(-6*(d*f*x + c*f)/d)*sin(6*c*f/d) - I*cos(6*c*f/d)*sin_integral(6*(d*f*x + c*f)/d) + sin(6*c*f/d)*sin_int
egral(6*(d*f*x + c*f)/d))/(8*a^3*d*cos(2*e)^3 + 24*I*a^3*d*cos(2*e)^2*sin(2*e) - 24*a^3*d*cos(2*e)*sin(2*e)^2
- 8*I*a^3*d*sin(2*e)^3)