3.1.32 \(\int \frac {1}{(c+d x) (a+i a \tan (e+f x))^3} \, dx\) [32]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.28, antiderivative size = 449, normalized size of antiderivative = 1.00, 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 = {3809, 3384, 3380, 3383, 3393, 4491, 4513} \begin {gather*} -\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} \end {gather*}

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 3380

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 3383

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 3384

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 3393

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 3809

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 4491

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 4513

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 = 0.85, size = 336, normalized size = 0.75 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (f x)+i \sin (f x))^3 \left (\cos (3 e) \log (f (c+d x))+i \log (f (c+d x)) \sin (3 e)+\left (\cos \left (e-\frac {4 c f}{d}\right )-i \sin \left (e-\frac {4 c f}{d}\right )\right ) \left (3 \text {CosIntegral}\left (\frac {4 f (c+d x)}{d}\right )+\cos \left (2 e-\frac {2 c f}{d}\right ) \text {CosIntegral}\left (\frac {6 f (c+d x)}{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 )-i \text {CosIntegral}\left (\frac {6 f (c+d x)}{d}\right ) \sin \left (2 e-\frac {2 c f}{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 )+3 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )-3 i \text {Si}\left (\frac {4 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 )-\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {6 f (c+d x)}{d}\right )\right )\right )}{8 d (a+i a \tan (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

Maple [A]
time = 0.52, size = 550, normalized size = 1.22

method	result	size
risch	\(\frac {\ln \left (d x +c \right )}{8 a^{3} d}-\frac {{\mathrm e}^{\frac {6 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 6 i f x +6 i e +\frac {6 i \left (c f -d e \right )}{d}\right )}{8 a^{3} d}-\frac {3 \,{\mathrm e}^{\frac {4 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 4 i f x +4 i e +\frac {4 i \left (c f -d e \right )}{d}\right )}{8 a^{3} d}-\frac {3 \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{8 a^{3} d}\)	\(163\)
default	\(\frac {-\frac {3 i f \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{16}-\frac {3 i f \left (\frac {4 \sinIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \cos \left (\frac {4 c f -4 d e}{d}\right )}{d}-\frac {4 \cosineIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \sin \left (\frac {4 c f -4 d e}{d}\right )}{d}\right )}{32}-\frac {i f \left (\frac {6 \sinIntegral \left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right ) \cos \left (\frac {6 c f -6 d e}{d}\right )}{d}-\frac {6 \cosineIntegral \left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right ) \sin \left (\frac {6 c f -6 d e}{d}\right )}{d}\right )}{48}+\frac {f \left (\frac {6 \sinIntegral \left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right ) \sin \left (\frac {6 c f -6 d e}{d}\right )}{d}+\frac {6 \cosineIntegral \left (6 f x +6 e +\frac {6 c f -6 d e}{d}\right ) \cos \left (\frac {6 c f -6 d e}{d}\right )}{d}\right )}{48}+\frac {3 f \left (\frac {4 \sinIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \sin \left (\frac {4 c f -4 d e}{d}\right )}{d}+\frac {4 \cosineIntegral \left (4 f x +4 e +\frac {4 c f -4 d e}{d}\right ) \cos \left (\frac {4 c f -4 d e}{d}\right )}{d}\right )}{32}+\frac {3 f \left (\frac {2 \sinIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {2 \cosineIntegral \left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{16}+\frac {f \ln \left (c f -d e +d \left (f x +e \right )\right )}{8 d}}{a^{3} f}\)	\(550\)

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,method=_RETURNVERBOSE)

[Out]

1/a^3/f*(-3/16*I*f*(2*Si(2*f*x+2*e+2*(c*f-d*e)/d)*cos(2*(c*f-d*e)/d)/d-2*Ci(2*f*x+2*e+2*(c*f-d*e)/d)*sin(2*(c*
f-d*e)/d)/d)-3/32*I*f*(4*Si(4*f*x+4*e+4*(c*f-d*e)/d)*cos(4*(c*f-d*e)/d)/d-4*Ci(4*f*x+4*e+4*(c*f-d*e)/d)*sin(4*
(c*f-d*e)/d)/d)-1/48*I*f*(6*Si(6*f*x+6*e+6*(c*f-d*e)/d)*cos(6*(c*f-d*e)/d)/d-6*Ci(6*f*x+6*e+6*(c*f-d*e)/d)*sin
(6*(c*f-d*e)/d)/d)+1/48*f*(6*Si(6*f*x+6*e+6*(c*f-d*e)/d)*sin(6*(c*f-d*e)/d)/d+6*Ci(6*f*x+6*e+6*(c*f-d*e)/d)*co
s(6*(c*f-d*e)/d)/d)+3/32*f*(4*Si(4*f*x+4*e+4*(c*f-d*e)/d)*sin(4*(c*f-d*e)/d)/d+4*Ci(4*f*x+4*e+4*(c*f-d*e)/d)*c
os(4*(c*f-d*e)/d)/d)+3/16*f*(2*Si(2*f*x+2*e+2*(c*f-d*e)/d)*sin(2*(c*f-d*e)/d)/d+2*Ci(2*f*x+2*e+2*(c*f-d*e)/d)*
cos(2*(c*f-d*e)/d)/d)+1/8*f*ln(c*f-d*e+d*(f*x+e))/d)

________________________________________________________________________________________

Maxima [A]
time = 0.39, size = 296, normalized size = 0.66 \begin {gather*} -\frac {3 \, f \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) E_{1}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + 3 \, f \cos \left (\frac {4 \, {\left (c f - d e\right )}}{d}\right ) E_{1}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + f \cos \left (\frac {6 \, {\left (c f - d e\right )}}{d}\right ) E_{1}\left (-\frac {6 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + i \, f E_{1}\left (-\frac {6 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) \sin \left (\frac {6 \, {\left (c f - d e\right )}}{d}\right ) + 3 i \, f E_{1}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) \sin \left (\frac {4 \, {\left (c f - d e\right )}}{d}\right ) + 3 i \, f E_{1}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) \sin \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) - f \log \left ({\left (f x + e\right )} d + c f - d e\right )}{8 \, a^{3} d f} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 120, normalized size = 0.27 \begin {gather*} \frac {3 \, {\rm Ei}\left (-\frac {2 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, c f + i \, d e\right )}}{d}\right )} + 3 \, {\rm Ei}\left (-\frac {4 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (-i \, c f + i \, d e\right )}}{d}\right )} + {\rm Ei}\left (-\frac {6 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {6 \, {\left (-i \, c f + i \, d e\right )}}{d}\right )} + \log \left (\frac {d x + c}{d}\right )}{8 \, a^{3} d} \end {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {i \int \frac {1}{c \tan ^{3}{\left (e + f x \right )} - 3 i c \tan ^{2}{\left (e + f x \right )} - 3 c \tan {\left (e + f x \right )} + i c + d x \tan ^{3}{\left (e + f x \right )} - 3 i d x \tan ^{2}{\left (e + f x \right )} - 3 d x \tan {\left (e + f x \right )} + i d x}\, dx}{a^{3}} \end {gather*}

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]

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

________________________________________________________________________________________

Giac [A]
time = 0.68, size = 846, normalized size = 1.88 \begin {gather*} \frac {3 \, \cos \left (\frac {2 \, c f}{d}\right ) \cos \left (2 \, e\right )^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + \cos \left (2 \, e\right )^{3} \log \left (d x + c\right ) + 3 i \, \cos \left (2 \, e\right )^{2} \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {2 \, c f}{d}\right ) + 6 i \, \cos \left (\frac {2 \, c f}{d}\right ) \cos \left (2 \, e\right ) \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (2 \, e\right ) + 3 i \, \cos \left (2 \, e\right )^{2} \log \left (d x + c\right ) \sin \left (2 \, e\right ) - 6 \, \cos \left (2 \, e\right ) \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right ) - 3 \, \cos \left (\frac {2 \, c f}{d}\right ) \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (2 \, e\right )^{2} - 3 \, \cos \left (2 \, e\right ) \log \left (d x + c\right ) \sin \left (2 \, e\right )^{2} - 3 i \, \operatorname {Ci}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right )^{2} - i \, \log \left (d x + c\right ) \sin \left (2 \, e\right )^{3} - 3 i \, \cos \left (\frac {2 \, c f}{d}\right ) \cos \left (2 \, e\right )^{2} \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 3 \, \cos \left (2 \, e\right )^{2} \sin \left (\frac {2 \, c f}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 6 \, \cos \left (\frac {2 \, c f}{d}\right ) \cos \left (2 \, e\right ) \sin \left (2 \, e\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 6 i \, \cos \left (2 \, e\right ) \sin \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right ) \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 3 i \, \cos \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right )^{2} \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - 3 \, \sin \left (\frac {2 \, c f}{d}\right ) \sin \left (2 \, e\right )^{2} \operatorname {Si}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 3 \, \cos \left (\frac {4 \, c f}{d}\right ) \cos \left (2 \, e\right ) \operatorname {Ci}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 3 i \, \cos \left (2 \, e\right ) \operatorname {Ci}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {4 \, c f}{d}\right ) + 3 i \, \cos \left (\frac {4 \, c f}{d}\right ) \operatorname {Ci}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (2 \, e\right ) - 3 \, \operatorname {Ci}\left (-\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {4 \, c f}{d}\right ) \sin \left (2 \, e\right ) - 3 i \, \cos \left (\frac {4 \, c f}{d}\right ) \cos \left (2 \, e\right ) \operatorname {Si}\left (\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 3 \, \cos \left (2 \, e\right ) \sin \left (\frac {4 \, c f}{d}\right ) \operatorname {Si}\left (\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 3 \, \cos \left (\frac {4 \, c f}{d}\right ) \sin \left (2 \, e\right ) \operatorname {Si}\left (\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + 3 i \, \sin \left (\frac {4 \, c f}{d}\right ) \sin \left (2 \, e\right ) \operatorname {Si}\left (\frac {4 \, {\left (d f x + c f\right )}}{d}\right ) + \cos \left (\frac {6 \, c f}{d}\right ) \operatorname {Ci}\left (-\frac {6 \, {\left (d f x + c f\right )}}{d}\right ) + i \, \operatorname {Ci}\left (-\frac {6 \, {\left (d f x + c f\right )}}{d}\right ) \sin \left (\frac {6 \, c f}{d}\right ) - i \, \cos \left (\frac {6 \, c f}{d}\right ) \operatorname {Si}\left (\frac {6 \, {\left (d f x + c f\right )}}{d}\right ) + \sin \left (\frac {6 \, c f}{d}\right ) \operatorname {Si}\left (\frac {6 \, {\left (d f x + c f\right )}}{d}\right )}{8 \, {\left (a^{3} d \cos \left (2 \, e\right )^{3} + 3 i \, a^{3} d \cos \left (2 \, e\right )^{2} \sin \left (2 \, e\right ) - 3 \, a^{3} d \cos \left (2 \, e\right ) \sin \left (2 \, e\right )^{2} - i \, a^{3} d \sin \left (2 \, e\right )^{3}\right )}} \end {gather*}

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]

1/8*(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
_integral(-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_in
tegral(-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*s
in_integral(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)*co
s(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
)*sin(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
_integral(-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
_integral(6*(d*f*x + c*f)/d))/(a^3*d*cos(2*e)^3 + 3*I*a^3*d*cos(2*e)^2*sin(2*e) - 3*a^3*d*cos(2*e)*sin(2*e)^2
- I*a^3*d*sin(2*e)^3)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,\left (c+d\,x\right )} \,d x \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*x)),x)

[Out]

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

________________________________________________________________________________________