3.31 \(\int \frac {1}{(c+d x)^2 (a+i a \cot (e+f x))^3} \, dx\)

Optimal. Leaf size=712 \[ \frac {3 f \text {Ci}\left (6 x f+\frac {6 c f}{d}\right ) \sin \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {Ci}\left (4 x f+\frac {4 c f}{d}\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 i f \text {Ci}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 i f \text {Ci}\left (4 x f+\frac {4 c f}{d}\right ) \cos \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 i f \text {Ci}\left (6 x f+\frac {6 c f}{d}\right ) \cos \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 i f \sin \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)} \]

[Out]

-1/8/a^3/d/(d*x+c)-1/8*I*sin(2*f*x+2*e)^3/a^3/d/(d*x+c)+3/2*I*f*Ci(4*c*f/d+4*f*x)*cos(-4*e+4*c*f/d)/a^3/d^2+3/
32*I*sin(6*f*x+6*e)/a^3/d/(d*x+c)+9/32*cos(2*f*x+2*e)/a^3/d/(d*x+c)-3/8*cos(2*f*x+2*e)^2/a^3/d/(d*x+c)+1/8*cos
(2*f*x+2*e)^3/a^3/d/(d*x+c)+3/32*cos(6*f*x+6*e)/a^3/d/(d*x+c)+3/4*f*cos(-2*e+2*c*f/d)*Si(2*c*f/d+2*f*x)/a^3/d^
2-3/2*f*cos(-4*e+4*c*f/d)*Si(4*c*f/d+4*f*x)/a^3/d^2+3/4*f*cos(-6*e+6*c*f/d)*Si(6*c*f/d+6*f*x)/a^3/d^2-3/4*f*Ci
(6*c*f/d+6*f*x)*sin(-6*e+6*c*f/d)/a^3/d^2-3/4*I*f*Si(6*c*f/d+6*f*x)*sin(-6*e+6*c*f/d)/a^3/d^2+3/2*f*Ci(4*c*f/d
+4*f*x)*sin(-4*e+4*c*f/d)/a^3/d^2-3/4*I*f*Si(2*c*f/d+2*f*x)*sin(-2*e+2*c*f/d)/a^3/d^2-3/4*f*Ci(2*c*f/d+2*f*x)*
sin(-2*e+2*c*f/d)/a^3/d^2-3/4*I*f*Ci(6*c*f/d+6*f*x)*cos(-6*e+6*c*f/d)/a^3/d^2+15/32*I*sin(2*f*x+2*e)/a^3/d/(d*
x+c)+3/8*sin(2*f*x+2*e)^2/a^3/d/(d*x+c)-3/8*I*sin(4*f*x+4*e)/a^3/d/(d*x+c)-3/4*I*f*Ci(2*c*f/d+2*f*x)*cos(-2*e+
2*c*f/d)/a^3/d^2+3/2*I*f*Si(4*c*f/d+4*f*x)*sin(-4*e+4*c*f/d)/a^3/d^2

________________________________________________________________________________________

Rubi [A]  time = 1.64, antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 60, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3728, 3297, 3303, 3299, 3302, 3313, 12, 4406, 4428} \[ \frac {3 f \text {CosIntegral}\left (\frac {6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \text {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 i f \text {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 i f \text {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \cos \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 i f \text {CosIntegral}\left (\frac {6 c f}{d}+6 f x\right ) \cos \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 i f \sin \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)} \]

Antiderivative was successfully verified.

[In]

Int[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])^3),x]

[Out]

-1/(8*a^3*d*(c + d*x)) + (9*Cos[2*e + 2*f*x])/(32*a^3*d*(c + d*x)) - (3*Cos[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)
) + Cos[2*e + 2*f*x]^3/(8*a^3*d*(c + d*x)) + (3*Cos[6*e + 6*f*x])/(32*a^3*d*(c + d*x)) - (((3*I)/4)*f*Cos[2*e
- (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a^3*d^2) + (((3*I)/2)*f*Cos[4*e - (4*c*f)/d]*CosIntegral[(4*c*f)
/d + 4*f*x])/(a^3*d^2) - (((3*I)/4)*f*Cos[6*e - (6*c*f)/d]*CosIntegral[(6*c*f)/d + 6*f*x])/(a^3*d^2) + (3*f*Co
sIntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(4*a^3*d^2) - (3*f*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e -
(4*c*f)/d])/(2*a^3*d^2) + (3*f*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(4*a^3*d^2) + (((15*I)/32
)*Sin[2*e + 2*f*x])/(a^3*d*(c + d*x)) + (3*Sin[2*e + 2*f*x]^2)/(8*a^3*d*(c + d*x)) - ((I/8)*Sin[2*e + 2*f*x]^3
)/(a^3*d*(c + d*x)) - (((3*I)/8)*Sin[4*e + 4*f*x])/(a^3*d*(c + d*x)) + (((3*I)/32)*Sin[6*e + 6*f*x])/(a^3*d*(c
+ d*x)) + (3*f*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(4*a^3*d^2) + (((3*I)/4)*f*Sin[2*e - (2*c
*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^3*d^2) - (3*f*Cos[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(
2*a^3*d^2) - (((3*I)/2)*f*Sin[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^3*d^2) + (3*f*Cos[6*e - (6*c
*f)/d]*SinIntegral[(6*c*f)/d + 6*f*x])/(4*a^3*d^2) + (((3*I)/4)*f*Sin[6*e - (6*c*f)/d]*SinIntegral[(6*c*f)/d +
6*f*x])/(a^3*d^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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 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 3313

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x]^
n)/(d*(m + 1)), x] - Dist[(f*n)/(d*(m + 1)), Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]
^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] && LtQ[m, -1]

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 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)^2 (a+i a \cot (e+f x))^3} \, dx &=\int \left (\frac {1}{8 a^3 (c+d x)^2}-\frac {3 \cos (2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {\cos ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 i \sin (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 i \cos ^2(2 e+2 f x) \sin (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {i \sin ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 i \sin (4 e+4 f x)}{8 a^3 (c+d x)^2}+\frac {3 \sin (2 e+2 f x) \sin (4 e+4 f x)}{16 a^3 (c+d x)^2}\right ) \, dx\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {i \int \frac {\sin ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac {(3 i) \int \frac {\sin (2 e+2 f x)}{(c+d x)^2} \, 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)^2} \, dx}{8 a^3}+\frac {(3 i) \int \frac {\sin (4 e+4 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac {\int \frac {\cos ^3(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {3 \int \frac {\sin (2 e+2 f x) \sin (4 e+4 f x)}{(c+d x)^2} \, dx}{16 a^3}-\frac {3 \int \frac {\cos (2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}+\frac {3 \int \frac {\cos ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}-\frac {3 \int \frac {\sin ^2(2 e+2 f x)}{(c+d x)^2} \, dx}{8 a^3}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {3 \cos (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {(3 i) \int \left (\frac {\sin (2 e+2 f x)}{4 (c+d x)^2}+\frac {\sin (6 e+6 f x)}{4 (c+d x)^2}\right ) \, dx}{8 a^3}+\frac {3 \int \left (\frac {\cos (2 e+2 f x)}{2 (c+d x)^2}-\frac {\cos (6 e+6 f x)}{2 (c+d x)^2}\right ) \, dx}{16 a^3}-\frac {(3 i f) \int \frac {\cos (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}+\frac {(3 i f) \int \left (\frac {\cos (2 e+2 f x)}{4 (c+d x)}-\frac {\cos (6 e+6 f x)}{4 (c+d x)}\right ) \, dx}{4 a^3 d}+\frac {(3 i f) \int \frac {\cos (4 e+4 f x)}{c+d x} \, dx}{2 a^3 d}+\frac {(3 f) \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{4 a^3 d}-\frac {(3 f) \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}{4 a^3 d}+\frac {(3 f) \int -\frac {\sin (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}-\frac {(3 f) \int \frac {\sin (4 e+4 f x)}{2 (c+d x)} \, dx}{2 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {3 \cos (2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}-\frac {(3 i) \int \frac {\sin (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac {(3 i) \int \frac {\sin (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac {3 \int \frac {\cos (2 e+2 f x)}{(c+d x)^2} \, dx}{32 a^3}-\frac {3 \int \frac {\cos (6 e+6 f x)}{(c+d x)^2} \, dx}{32 a^3}+\frac {(3 i f) \int \frac {\cos (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {(3 i f) \int \frac {\cos (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {(3 f) \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {(3 f) \int \frac {\sin (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-2 \frac {(3 f) \int \frac {\sin (4 e+4 f x)}{c+d x} \, dx}{4 a^3 d}+\frac {\left (3 i f \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}{2 a^3 d}-\frac {\left (3 i f \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}{4 a^3 d}+\frac {\left (3 f \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}{4 a^3 d}-\frac {\left (3 i f \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}{2 a^3 d}+\frac {\left (3 i f \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}{4 a^3 d}+\frac {\left (3 f \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}{4 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 i f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 i f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Ci}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {3 f \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {3 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {(3 i f) \int \frac {\cos (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {(9 i f) \int \frac {\cos (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {(3 f) \int \frac {\sin (2 e+2 f x)}{c+d x} \, dx}{16 a^3 d}+\frac {(9 f) \int \frac {\sin (6 e+6 f x)}{c+d x} \, dx}{16 a^3 d}-\frac {\left (3 i f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}+\frac {\left (3 i f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}+\frac {\left (3 i f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}-2 \left (\frac {\left (3 f \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}{4 a^3 d}+\frac {\left (3 f \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}{4 a^3 d}\right )-\frac {\left (3 i f \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}{16 a^3 d}+\frac {\left (3 f \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}{16 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {9 i f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{16 a^3 d^2}+\frac {3 i f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Ci}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 i f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Ci}\left (\frac {6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac {3 f \text {Ci}\left (\frac {6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac {6 c f}{d}\right )}{16 a^3 d^2}+\frac {15 f \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{16 a^3 d^2}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {15 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{16 a^3 d^2}+\frac {9 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{16 a^3 d^2}-\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-2 \left (\frac {3 f \text {Ci}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )+\frac {3 f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (\frac {6 c f}{d}+6 f x\right )}{16 a^3 d^2}+\frac {3 i f \sin \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (\frac {6 c f}{d}+6 f x\right )}{16 a^3 d^2}-\frac {\left (9 i f \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}{16 a^3 d}+\frac {\left (9 f \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}{16 a^3 d}-\frac {\left (3 i f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}+\frac {\left (9 i f \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}{16 a^3 d}+\frac {\left (9 f \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}{16 a^3 d}+\frac {\left (3 i f \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}{16 a^3 d}-\frac {\left (3 f \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}{16 a^3 d}\\ &=-\frac {1}{8 a^3 d (c+d x)}+\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 i f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Ci}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 i f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Ci}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 i f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Ci}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac {3 f \text {Ci}\left (\frac {6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \text {Ci}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}+\frac {3 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-2 \left (\frac {3 f \text {Ci}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^3 d^2}+\frac {3 f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^3 d^2}\right )+\frac {3 f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.64, size = 292, normalized size = 0.41 \[ \frac {6 f (c+d x) \left (\text {Ci}\left (\frac {2 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right ) \left (\sin \left (2 e-\frac {2 c f}{d}\right )-i \cos \left (2 e-\frac {2 c f}{d}\right )\right )+12 i f (c+d x) \left (\text {Ci}\left (\frac {4 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {4 f (c+d x)}{d}\right )\right ) \left (\cos \left (4 e-\frac {4 c f}{d}\right )+i \sin \left (4 e-\frac {4 c f}{d}\right )\right )+6 f (c+d x) \left (\text {Ci}\left (\frac {6 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {6 f (c+d x)}{d}\right )\right ) \left (\sin \left (6 e-\frac {6 c f}{d}\right )-i \cos \left (6 e-\frac {6 c f}{d}\right )\right )+3 d (\cos (2 (e+f x))+i \sin (2 (e+f x)))-3 d (\cos (4 (e+f x))+i \sin (4 (e+f x)))+d (\cos (6 (e+f x))+i \sin (6 (e+f x)))-d}{8 a^3 d^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c + d*x)^2*(a + I*a*Cot[e + f*x])^3),x]

[Out]

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

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fricas [A]  time = 0.76, size = 180, normalized size = 0.25 \[ \frac {{\left (-6 i \, d f x - 6 i \, c f\right )} {\rm Ei}\left (\frac {6 i \, d f x + 6 i \, c f}{d}\right ) e^{\left (\frac {6 i \, d e - 6 i \, c f}{d}\right )} + {\left (12 i \, d f x + 12 i \, c f\right )} {\rm Ei}\left (\frac {4 i \, d f x + 4 i \, c f}{d}\right ) e^{\left (\frac {4 i \, d e - 4 i \, c f}{d}\right )} + {\left (-6 i \, d f x - 6 i \, c f\right )} {\rm Ei}\left (\frac {2 i \, d f x + 2 i \, c f}{d}\right ) e^{\left (\frac {2 i \, d e - 2 i \, c f}{d}\right )} + d e^{\left (6 i \, f x + 6 i \, e\right )} - 3 \, d e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, d e^{\left (2 i \, f x + 2 i \, e\right )} - d}{8 \, {\left (a^{3} d^{3} x + a^{3} c d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*cot(f*x+e))^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 87.91, size = 4524, normalized size = 6.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*cot(f*x+e))^3,x, algorithm="giac")

[Out]

1/8*(-6*I*d*f*x*cos(6*c*f/d)*cos(e)^6*cos_integral(6*(d*f*x + c*f)/d) - 6*d*f*x*cos(e)^6*cos_integral(6*(d*f*x
+ c*f)/d)*sin(6*c*f/d) + 36*d*f*x*cos(6*c*f/d)*cos(e)^5*cos_integral(6*(d*f*x + c*f)/d)*sin(e) - 36*I*d*f*x*c
os(e)^5*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e) + 90*I*d*f*x*cos(6*c*f/d)*cos(e)^4*cos_integral(6*
(d*f*x + c*f)/d)*sin(e)^2 + 90*d*f*x*cos(e)^4*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e)^2 - 120*d*f*
x*cos(6*c*f/d)*cos(e)^3*cos_integral(6*(d*f*x + c*f)/d)*sin(e)^3 + 120*I*d*f*x*cos(e)^3*cos_integral(6*(d*f*x
+ c*f)/d)*sin(6*c*f/d)*sin(e)^3 - 90*I*d*f*x*cos(6*c*f/d)*cos(e)^2*cos_integral(6*(d*f*x + c*f)/d)*sin(e)^4 -
90*d*f*x*cos(e)^2*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e)^4 + 36*d*f*x*cos(6*c*f/d)*cos(e)*cos_int
egral(6*(d*f*x + c*f)/d)*sin(e)^5 - 36*I*d*f*x*cos(e)*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e)^5 +
6*I*d*f*x*cos(6*c*f/d)*cos_integral(6*(d*f*x + c*f)/d)*sin(e)^6 + 6*d*f*x*cos_integral(6*(d*f*x + c*f)/d)*sin(
6*c*f/d)*sin(e)^6 + 6*d*f*x*cos(6*c*f/d)*cos(e)^6*sin_integral(6*(d*f*x + c*f)/d) - 6*I*d*f*x*cos(e)^6*sin(6*c
*f/d)*sin_integral(6*(d*f*x + c*f)/d) + 36*I*d*f*x*cos(6*c*f/d)*cos(e)^5*sin(e)*sin_integral(6*(d*f*x + c*f)/d
) + 36*d*f*x*cos(e)^5*sin(6*c*f/d)*sin(e)*sin_integral(6*(d*f*x + c*f)/d) - 90*d*f*x*cos(6*c*f/d)*cos(e)^4*sin
(e)^2*sin_integral(6*(d*f*x + c*f)/d) + 90*I*d*f*x*cos(e)^4*sin(6*c*f/d)*sin(e)^2*sin_integral(6*(d*f*x + c*f)
/d) - 120*I*d*f*x*cos(6*c*f/d)*cos(e)^3*sin(e)^3*sin_integral(6*(d*f*x + c*f)/d) - 120*d*f*x*cos(e)^3*sin(6*c*
f/d)*sin(e)^3*sin_integral(6*(d*f*x + c*f)/d) + 90*d*f*x*cos(6*c*f/d)*cos(e)^2*sin(e)^4*sin_integral(6*(d*f*x
+ c*f)/d) - 90*I*d*f*x*cos(e)^2*sin(6*c*f/d)*sin(e)^4*sin_integral(6*(d*f*x + c*f)/d) + 36*I*d*f*x*cos(6*c*f/d
)*cos(e)*sin(e)^5*sin_integral(6*(d*f*x + c*f)/d) + 36*d*f*x*cos(e)*sin(6*c*f/d)*sin(e)^5*sin_integral(6*(d*f*
x + c*f)/d) - 6*d*f*x*cos(6*c*f/d)*sin(e)^6*sin_integral(6*(d*f*x + c*f)/d) + 6*I*d*f*x*sin(6*c*f/d)*sin(e)^6*
sin_integral(6*(d*f*x + c*f)/d) - 6*I*c*f*cos(6*c*f/d)*cos(e)^6*cos_integral(6*(d*f*x + c*f)/d) - 6*c*f*cos(e)
^6*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d) + 36*c*f*cos(6*c*f/d)*cos(e)^5*cos_integral(6*(d*f*x + c*f)/d)
*sin(e) - 36*I*c*f*cos(e)^5*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e) + 90*I*c*f*cos(6*c*f/d)*cos(e)
^4*cos_integral(6*(d*f*x + c*f)/d)*sin(e)^2 + 90*c*f*cos(e)^4*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin
(e)^2 - 120*c*f*cos(6*c*f/d)*cos(e)^3*cos_integral(6*(d*f*x + c*f)/d)*sin(e)^3 + 120*I*c*f*cos(e)^3*cos_integr
al(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e)^3 - 90*I*c*f*cos(6*c*f/d)*cos(e)^2*cos_integral(6*(d*f*x + c*f)/d)*s
in(e)^4 - 90*c*f*cos(e)^2*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e)^4 + 36*c*f*cos(6*c*f/d)*cos(e)*c
os_integral(6*(d*f*x + c*f)/d)*sin(e)^5 - 36*I*c*f*cos(e)*cos_integral(6*(d*f*x + c*f)/d)*sin(6*c*f/d)*sin(e)^
5 + 6*I*c*f*cos(6*c*f/d)*cos_integral(6*(d*f*x + c*f)/d)*sin(e)^6 + 6*c*f*cos_integral(6*(d*f*x + c*f)/d)*sin(
6*c*f/d)*sin(e)^6 + 6*c*f*cos(6*c*f/d)*cos(e)^6*sin_integral(6*(d*f*x + c*f)/d) - 6*I*c*f*cos(e)^6*sin(6*c*f/d
)*sin_integral(6*(d*f*x + c*f)/d) + 36*I*c*f*cos(6*c*f/d)*cos(e)^5*sin(e)*sin_integral(6*(d*f*x + c*f)/d) + 36
*c*f*cos(e)^5*sin(6*c*f/d)*sin(e)*sin_integral(6*(d*f*x + c*f)/d) - 90*c*f*cos(6*c*f/d)*cos(e)^4*sin(e)^2*sin_
integral(6*(d*f*x + c*f)/d) + 90*I*c*f*cos(e)^4*sin(6*c*f/d)*sin(e)^2*sin_integral(6*(d*f*x + c*f)/d) - 120*I*
c*f*cos(6*c*f/d)*cos(e)^3*sin(e)^3*sin_integral(6*(d*f*x + c*f)/d) - 120*c*f*cos(e)^3*sin(6*c*f/d)*sin(e)^3*si
n_integral(6*(d*f*x + c*f)/d) + 90*c*f*cos(6*c*f/d)*cos(e)^2*sin(e)^4*sin_integral(6*(d*f*x + c*f)/d) - 90*I*c
*f*cos(e)^2*sin(6*c*f/d)*sin(e)^4*sin_integral(6*(d*f*x + c*f)/d) + 36*I*c*f*cos(6*c*f/d)*cos(e)*sin(e)^5*sin_
integral(6*(d*f*x + c*f)/d) + 36*c*f*cos(e)*sin(6*c*f/d)*sin(e)^5*sin_integral(6*(d*f*x + c*f)/d) - 6*c*f*cos(
6*c*f/d)*sin(e)^6*sin_integral(6*(d*f*x + c*f)/d) + 6*I*c*f*sin(6*c*f/d)*sin(e)^6*sin_integral(6*(d*f*x + c*f)
/d) + 12*I*d*f*x*cos(4*c*f/d)*cos(e)^4*cos_integral(4*(d*f*x + c*f)/d) + 12*d*f*x*cos(e)^4*cos_integral(4*(d*f
*x + c*f)/d)*sin(4*c*f/d) - 48*d*f*x*cos(4*c*f/d)*cos(e)^3*cos_integral(4*(d*f*x + c*f)/d)*sin(e) + 48*I*d*f*x
*cos(e)^3*cos_integral(4*(d*f*x + c*f)/d)*sin(4*c*f/d)*sin(e) - 72*I*d*f*x*cos(4*c*f/d)*cos(e)^2*cos_integral(
4*(d*f*x + c*f)/d)*sin(e)^2 - 72*d*f*x*cos(e)^2*cos_integral(4*(d*f*x + c*f)/d)*sin(4*c*f/d)*sin(e)^2 + 48*d*f
*x*cos(4*c*f/d)*cos(e)*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^3 - 48*I*d*f*x*cos(e)*cos_integral(4*(d*f*x + c*
f)/d)*sin(4*c*f/d)*sin(e)^3 + 12*I*d*f*x*cos(4*c*f/d)*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^4 + 12*d*f*x*cos_
integral(4*(d*f*x + c*f)/d)*sin(4*c*f/d)*sin(e)^4 - 12*d*f*x*cos(4*c*f/d)*cos(e)^4*sin_integral(4*(d*f*x + c*f
)/d) + 12*I*d*f*x*cos(e)^4*sin(4*c*f/d)*sin_integral(4*(d*f*x + c*f)/d) - 48*I*d*f*x*cos(4*c*f/d)*cos(e)^3*sin
(e)*sin_integral(4*(d*f*x + c*f)/d) - 48*d*f*x*cos(e)^3*sin(4*c*f/d)*sin(e)*sin_integral(4*(d*f*x + c*f)/d) +
72*d*f*x*cos(4*c*f/d)*cos(e)^2*sin(e)^2*sin_integral(4*(d*f*x + c*f)/d) - 72*I*d*f*x*cos(e)^2*sin(4*c*f/d)*sin
(e)^2*sin_integral(4*(d*f*x + c*f)/d) + 48*I*d*f*x*cos(4*c*f/d)*cos(e)*sin(e)^3*sin_integral(4*(d*f*x + c*f)/d
) + 48*d*f*x*cos(e)*sin(4*c*f/d)*sin(e)^3*sin_integral(4*(d*f*x + c*f)/d) - 12*d*f*x*cos(4*c*f/d)*sin(e)^4*sin
_integral(4*(d*f*x + c*f)/d) + 12*I*d*f*x*sin(4*c*f/d)*sin(e)^4*sin_integral(4*(d*f*x + c*f)/d) + d*cos(6*f*x)
*cos(e)^6 + 12*I*c*f*cos(4*c*f/d)*cos(e)^4*cos_integral(4*(d*f*x + c*f)/d) + I*d*cos(e)^6*sin(6*f*x) + 12*c*f*
cos(e)^4*cos_integral(4*(d*f*x + c*f)/d)*sin(4*c*f/d) + 6*I*d*cos(6*f*x)*cos(e)^5*sin(e) - 48*c*f*cos(4*c*f/d)
*cos(e)^3*cos_integral(4*(d*f*x + c*f)/d)*sin(e) - 6*d*cos(e)^5*sin(6*f*x)*sin(e) + 48*I*c*f*cos(e)^3*cos_inte
gral(4*(d*f*x + c*f)/d)*sin(4*c*f/d)*sin(e) - 15*d*cos(6*f*x)*cos(e)^4*sin(e)^2 - 72*I*c*f*cos(4*c*f/d)*cos(e)
^2*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^2 - 15*I*d*cos(e)^4*sin(6*f*x)*sin(e)^2 - 72*c*f*cos(e)^2*cos_integr
al(4*(d*f*x + c*f)/d)*sin(4*c*f/d)*sin(e)^2 - 20*I*d*cos(6*f*x)*cos(e)^3*sin(e)^3 + 48*c*f*cos(4*c*f/d)*cos(e)
*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^3 + 20*d*cos(e)^3*sin(6*f*x)*sin(e)^3 - 48*I*c*f*cos(e)*cos_integral(4
*(d*f*x + c*f)/d)*sin(4*c*f/d)*sin(e)^3 + 15*d*cos(6*f*x)*cos(e)^2*sin(e)^4 + 12*I*c*f*cos(4*c*f/d)*cos_integr
al(4*(d*f*x + c*f)/d)*sin(e)^4 + 15*I*d*cos(e)^2*sin(6*f*x)*sin(e)^4 + 12*c*f*cos_integral(4*(d*f*x + c*f)/d)*
sin(4*c*f/d)*sin(e)^4 + 6*I*d*cos(6*f*x)*cos(e)*sin(e)^5 - 6*d*cos(e)*sin(6*f*x)*sin(e)^5 - d*cos(6*f*x)*sin(e
)^6 - I*d*sin(6*f*x)*sin(e)^6 - 12*c*f*cos(4*c*f/d)*cos(e)^4*sin_integral(4*(d*f*x + c*f)/d) + 12*I*c*f*cos(e)
^4*sin(4*c*f/d)*sin_integral(4*(d*f*x + c*f)/d) - 48*I*c*f*cos(4*c*f/d)*cos(e)^3*sin(e)*sin_integral(4*(d*f*x
+ c*f)/d) - 48*c*f*cos(e)^3*sin(4*c*f/d)*sin(e)*sin_integral(4*(d*f*x + c*f)/d) + 72*c*f*cos(4*c*f/d)*cos(e)^2
*sin(e)^2*sin_integral(4*(d*f*x + c*f)/d) - 72*I*c*f*cos(e)^2*sin(4*c*f/d)*sin(e)^2*sin_integral(4*(d*f*x + c*
f)/d) + 48*I*c*f*cos(4*c*f/d)*cos(e)*sin(e)^3*sin_integral(4*(d*f*x + c*f)/d) + 48*c*f*cos(e)*sin(4*c*f/d)*sin
(e)^3*sin_integral(4*(d*f*x + c*f)/d) - 12*c*f*cos(4*c*f/d)*sin(e)^4*sin_integral(4*(d*f*x + c*f)/d) + 12*I*c*
f*sin(4*c*f/d)*sin(e)^4*sin_integral(4*(d*f*x + c*f)/d) - 6*I*d*f*x*cos(2*c*f/d)*cos(e)^2*cos_integral(2*(d*f*
x + c*f)/d) - 6*d*f*x*cos(e)^2*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d) + 12*d*f*x*cos(2*c*f/d)*cos(e)*cos
_integral(2*(d*f*x + c*f)/d)*sin(e) - 12*I*d*f*x*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e) +
6*I*d*f*x*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)^2 + 6*d*f*x*cos_integral(2*(d*f*x + c*f)/d)*sin(
2*c*f/d)*sin(e)^2 + 6*d*f*x*cos(2*c*f/d)*cos(e)^2*sin_integral(2*(d*f*x + c*f)/d) - 6*I*d*f*x*cos(e)^2*sin(2*c
*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 12*I*d*f*x*cos(2*c*f/d)*cos(e)*sin(e)*sin_integral(2*(d*f*x + c*f)/d)
+ 12*d*f*x*cos(e)*sin(2*c*f/d)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 6*d*f*x*cos(2*c*f/d)*sin(e)^2*sin_inte
gral(2*(d*f*x + c*f)/d) + 6*I*d*f*x*sin(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) - 3*d*cos(4*f*x)*cos
(e)^4 - 6*I*c*f*cos(2*c*f/d)*cos(e)^2*cos_integral(2*(d*f*x + c*f)/d) - 3*I*d*cos(e)^4*sin(4*f*x) - 6*c*f*cos(
e)^2*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d) - 12*I*d*cos(4*f*x)*cos(e)^3*sin(e) + 12*c*f*cos(2*c*f/d)*co
s(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(e) + 12*d*cos(e)^3*sin(4*f*x)*sin(e) - 12*I*c*f*cos(e)*cos_integral(2
*(d*f*x + c*f)/d)*sin(2*c*f/d)*sin(e) + 18*d*cos(4*f*x)*cos(e)^2*sin(e)^2 + 6*I*c*f*cos(2*c*f/d)*cos_integral(
2*(d*f*x + c*f)/d)*sin(e)^2 + 18*I*d*cos(e)^2*sin(4*f*x)*sin(e)^2 + 6*c*f*cos_integral(2*(d*f*x + c*f)/d)*sin(
2*c*f/d)*sin(e)^2 + 12*I*d*cos(4*f*x)*cos(e)*sin(e)^3 - 12*d*cos(e)*sin(4*f*x)*sin(e)^3 - 3*d*cos(4*f*x)*sin(e
)^4 - 3*I*d*sin(4*f*x)*sin(e)^4 + 6*c*f*cos(2*c*f/d)*cos(e)^2*sin_integral(2*(d*f*x + c*f)/d) - 6*I*c*f*cos(e)
^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 12*I*c*f*cos(2*c*f/d)*cos(e)*sin(e)*sin_integral(2*(d*f*x +
c*f)/d) + 12*c*f*cos(e)*sin(2*c*f/d)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 6*c*f*cos(2*c*f/d)*sin(e)^2*sin_
integral(2*(d*f*x + c*f)/d) + 6*I*c*f*sin(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) + 3*d*cos(2*f*x)*c
os(e)^2 + 3*I*d*cos(e)^2*sin(2*f*x) + 6*I*d*cos(2*f*x)*cos(e)*sin(e) - 6*d*cos(e)*sin(2*f*x)*sin(e) - 3*d*cos(
2*f*x)*sin(e)^2 - 3*I*d*sin(2*f*x)*sin(e)^2 - d)/(a^3*d^3*x + a^3*c*d^2)

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maple [A]  time = 0.64, size = 756, normalized size = 1.06 \[ -\frac {f \left (\frac {3 i \left (-\frac {2 \sin \left (2 f x +2 e \right )}{\left (\left (f x +e \right ) d +c f -d e \right ) d}+\frac {\frac {4 \Si \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 {4 \Ci \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}}{d}\right )}{16}+\frac {i \left (-\frac {6 \sin \left (6 f x +6 e \right )}{\left (\left (f x +e \right ) d +c f -d e \right ) d}+\frac {\frac {36 \Si \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 {36 \Ci \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}}{d}\right )}{48}+\frac {3 \cos \left (4 f x +4 e \right )}{8 \left (\left (f x +e \right ) d +c f -d e \right ) d}+\frac {\frac {3 \Si \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 )}{2 d}-\frac {3 \Ci \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 )}{2 d}}{d}+\frac {1}{8 \left (\left (f x +e \right ) d +c f -d e \right ) d}-\frac {3 \cos \left (2 f x +2 e \right )}{8 \left (\left (f x +e \right ) d +c f -d e \right ) d}-\frac {3 \left (\frac {2 \Si \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 \Ci \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 )}{8 d}-\frac {\cos \left (6 f x +6 e \right )}{8 \left (\left (f x +e \right ) d +c f -d e \right ) d}-\frac {\frac {6 \Si \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 \Ci \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}}{8 d}-\frac {3 i \left (-\frac {4 \sin \left (4 f x +4 e \right )}{\left (\left (f x +e \right ) d +c f -d e \right ) d}+\frac {\frac {16 \Si \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 {16 \Ci \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}}{d}\right )}{32}\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2/(a+I*a*cot(f*x+e))^3,x)

[Out]

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

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maxima [A]  time = 1.65, size = 300, normalized size = 0.42 \[ \frac {8192 \, f^{2} \cos \left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac {6 i \, {\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) - 24576 \, f^{2} \cos \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac {4 i \, {\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) + 24576 \, f^{2} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) - 24576 i \, f^{2} E_{2}\left (-\frac {2 i \, {\left (f x + e\right )} d - 2 i \, d e + 2 i \, c f}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 24576 i \, f^{2} E_{2}\left (-\frac {4 i \, {\left (f x + e\right )} d - 4 i \, d e + 4 i \, c f}{d}\right ) \sin \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) - 8192 i \, f^{2} E_{2}\left (-\frac {6 i \, {\left (f x + e\right )} d - 6 i \, d e + 6 i \, c f}{d}\right ) \sin \left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right ) - 8192 \, f^{2}}{65536 \, {\left ({\left (f x + e\right )} a^{3} d^{2} - a^{3} d^{2} e + a^{3} c d f\right )} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)^2/(a+I*a*cot(f*x+e))^3,x, algorithm="maxima")

[Out]

1/65536*(8192*f^2*cos(-6*(d*e - c*f)/d)*exp_integral_e(2, -(6*I*(f*x + e)*d - 6*I*d*e + 6*I*c*f)/d) - 24576*f^
2*cos(-4*(d*e - c*f)/d)*exp_integral_e(2, -(4*I*(f*x + e)*d - 4*I*d*e + 4*I*c*f)/d) + 24576*f^2*cos(-2*(d*e -
c*f)/d)*exp_integral_e(2, -(2*I*(f*x + e)*d - 2*I*d*e + 2*I*c*f)/d) - 24576*I*f^2*exp_integral_e(2, -(2*I*(f*x
+ e)*d - 2*I*d*e + 2*I*c*f)/d)*sin(-2*(d*e - c*f)/d) + 24576*I*f^2*exp_integral_e(2, -(4*I*(f*x + e)*d - 4*I*
d*e + 4*I*c*f)/d)*sin(-4*(d*e - c*f)/d) - 8192*I*f^2*exp_integral_e(2, -(6*I*(f*x + e)*d - 6*I*d*e + 6*I*c*f)/
d)*sin(-6*(d*e - c*f)/d) - 8192*f^2)/(((f*x + e)*a^3*d^2 - a^3*d^2*e + a^3*c*d*f)*f)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + a*cot(e + f*x)*1i)^3*(c + d*x)^2),x)

[Out]

int(1/((a + a*cot(e + f*x)*1i)^3*(c + d*x)^2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {i \int \frac {1}{c^{2} \cot ^{3}{\left (e + f x \right )} - 3 i c^{2} \cot ^{2}{\left (e + f x \right )} - 3 c^{2} \cot {\left (e + f x \right )} + i c^{2} + 2 c d x \cot ^{3}{\left (e + f x \right )} - 6 i c d x \cot ^{2}{\left (e + f x \right )} - 6 c d x \cot {\left (e + f x \right )} + 2 i c d x + d^{2} x^{2} \cot ^{3}{\left (e + f x \right )} - 3 i d^{2} x^{2} \cot ^{2}{\left (e + f x \right )} - 3 d^{2} x^{2} \cot {\left (e + f x \right )} + i d^{2} x^{2}}\, dx}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2/(a+I*a*cot(f*x+e))**3,x)

[Out]

I*Integral(1/(c**2*cot(e + f*x)**3 - 3*I*c**2*cot(e + f*x)**2 - 3*c**2*cot(e + f*x) + I*c**2 + 2*c*d*x*cot(e +
f*x)**3 - 6*I*c*d*x*cot(e + f*x)**2 - 6*c*d*x*cot(e + f*x) + 2*I*c*d*x + d**2*x**2*cot(e + f*x)**3 - 3*I*d**2
*x**2*cot(e + f*x)**2 - 3*d**2*x**2*cot(e + f*x) + I*d**2*x**2), x)/a**3

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