3.1.33 \(\int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx\) [33]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 1.25, 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 = {3809, 3378, 3384, 3380, 3383, 3394, 12, 4491, 4513} \begin {gather*} -\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)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*1/(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*
CosIntegral[(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 3378

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 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 3394

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 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)^2 (a+i a \tan (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 = 3.28, size = 833, normalized size = 1.17 \begin {gather*} \frac {\sec ^3(e+f x) \left (-i \cos \left (\frac {3 c f}{d}\right )+\sin \left (\frac {3 c f}{d}\right )\right ) \left (3 d \cos \left (e+f \left (-\frac {3 c}{d}+x\right )\right )+d \cos \left (3 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )+d \cos \left (3 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+3 d \cos \left (e+f \left (\frac {3 c}{d}+x\right )\right )+6 i c f \cos \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {CosIntegral}\left (\frac {6 f (c+d x)}{d}\right )+6 i d f x \cos \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {CosIntegral}\left (\frac {6 f (c+d x)}{d}\right )+6 i f (c+d x) \text {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {c f}{d}+3 f x\right )+i \sin \left (e-\frac {c f}{d}+3 f x\right )\right )+3 i d \sin \left (e+f \left (-\frac {3 c}{d}+x\right )\right )+i d \sin \left (3 \left (e+f \left (-\frac {c}{d}+x\right )\right )\right )-i d \sin \left (3 \left (e+f \left (\frac {c}{d}+x\right )\right )\right )-3 i d \sin \left (e+f \left (\frac {3 c}{d}+x\right )\right )+6 c f \text {CosIntegral}\left (\frac {6 f (c+d x)}{d}\right ) \sin \left (3 e-\frac {3 f (c+d x)}{d}\right )+6 d f x \text {CosIntegral}\left (\frac {6 f (c+d x)}{d}\right ) \sin \left (3 e-\frac {3 f (c+d x)}{d}\right )+12 f (c+d x) \text {CosIntegral}\left (\frac {4 f (c+d x)}{d}\right ) \left (i \cos \left (e-\frac {f (c+3 d x)}{d}\right )+\sin \left (e-\frac {f (c+3 d x)}{d}\right )\right )+6 c f \cos \left (e-\frac {c f}{d}+3 f x\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+6 d f x \cos \left (e-\frac {c f}{d}+3 f x\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+6 i c f \sin \left (e-\frac {c f}{d}+3 f x\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+6 i d f x \sin \left (e-\frac {c f}{d}+3 f x\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )+12 c f \cos \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Si}\left (\frac {4 f (c+d x)}{d}\right )+12 d f x \cos \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Si}\left (\frac {4 f (c+d x)}{d}\right )-12 i c f \sin \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Si}\left (\frac {4 f (c+d x)}{d}\right )-12 i d f x \sin \left (e-\frac {f (c+3 d x)}{d}\right ) \text {Si}\left (\frac {4 f (c+d x)}{d}\right )+6 c f \cos \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Si}\left (\frac {6 f (c+d x)}{d}\right )+6 d f x \cos \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Si}\left (\frac {6 f (c+d x)}{d}\right )-6 i c f \sin \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Si}\left (\frac {6 f (c+d x)}{d}\right )-6 i d f x \sin \left (3 e-\frac {3 f (c+d x)}{d}\right ) \text {Si}\left (\frac {6 f (c+d x)}{d}\right )\right )}{8 a^3 d^2 (c+d x) (-i+\tan (e+f x))^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]
time = 0.57, size = 787, normalized size = 1.11

method	result	size
risch	\(-\frac {1}{8 a^{3} d \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-6 i \left (f x +e \right )}}{8 a^{3} \left (d x f +c f \right ) d}+\frac {3 i f \,{\mathrm e}^{\frac {6 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 6 i f x +6 i e +\frac {6 i \left (c f -d e \right )}{d}\right )}{4 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-4 i \left (f x +e \right )}}{8 a^{3} \left (d x f +c f \right ) d}+\frac {3 i f \,{\mathrm e}^{\frac {4 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 4 i f x +4 i e +\frac {4 i \left (c f -d e \right )}{d}\right )}{2 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} \left (d x f +c f \right ) d}+\frac {3 i f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \expIntegral \left (1, 2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{4 a^{3} d^{2}}\)	\(254\)
default	\(\frac {-\frac {3 i f^{2} \left (-\frac {2 \sin \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {4 \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 {4 \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}}{d}\right )}{16}-\frac {3 i f^{2} \left (-\frac {4 \sin \left (4 f x +4 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {16 \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 {16 \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}}{d}\right )}{32}-\frac {i f^{2} \left (-\frac {6 \sin \left (6 f x +6 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {36 \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 {36 \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}}{d}\right )}{48}+\frac {f^{2} \left (-\frac {6 \cos \left (6 f x +6 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {6 \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 )}{d}\right )}{48}+\frac {3 f^{2} \left (-\frac {4 \cos \left (4 f x +4 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {4 \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 )}{d}\right )}{32}+\frac {3 f^{2} \left (-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \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 )}{d}\right )}{16}-\frac {f^{2}}{8 \left (c f -d e +d \left (f x +e \right )\right ) d}}{a^{3} f}\)	\(787\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2/(a+I*a*tan(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 317, normalized size = 0.45 \begin {gather*} -\frac {3 \, f^{2} \cos \left (\frac {2 \, {\left (c f - d e\right )}}{d}\right ) E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + 3 \, f^{2} \cos \left (\frac {4 \, {\left (c f - d e\right )}}{d}\right ) E_{2}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + f^{2} \cos \left (\frac {6 \, {\left (c f - d e\right )}}{d}\right ) E_{2}\left (-\frac {6 \, {\left (-i \, {\left (f x + e\right )} d - i \, c f + i \, d e\right )}}{d}\right ) + i \, f^{2} E_{2}\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^{2} E_{2}\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^{2} E_{2}\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^{2}}{8 \, {\left ({\left (f x + e\right )} a^{3} d^{2} + a^{3} c d f - a^{3} d^{2} e\right )} f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.37, size = 204, normalized size = 0.29 \begin {gather*} -\frac {{\left ({\left (6 \, {\left (i \, d f x + i \, c f\right )} {\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 )} + 12 \, {\left (i \, d f x + i \, c f\right )} {\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 )} + 6 \, {\left (i \, d f x + i \, c f\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 )} + d\right )} 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\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{8 \, {\left (a^{3} d^{3} x + a^{3} c d^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3165 vs. \(2 (669) = 1338\).
time = 33.04, size = 3165, normalized size = 4.45 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(6*I*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos(2*(c*f - d*e)/d)*cos_integral(-2*((d*x + c)*(c
*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - 6*I*c*f^3*cos(2*(c*f - d*e)/d)*cos_integral(-2*((d*x + c)*
(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 12*I*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*c
os(4*(c*f - d*e)/d)*cos_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - 12*I*c*f^
3*cos(4*(c*f - d*e)/d)*cos_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 6*I*(d
*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos(6*(c*f - d*e)/d)*cos_integral(-6*((d*x + c)*(c*f/(d*x + c)
- f - d*e/(d*x + c)) - c*f + d*e)/d) - 6*I*c*f^3*cos(6*(c*f - d*e)/d)*cos_integral(-6*((d*x + c)*(c*f/(d*x +
c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 6*I*d*f^2*cos(2*(c*f - d*e)/d)*cos_integral(-2*((d*x + c)*(c*f/(d*x
+ c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e + 12*I*d*f^2*cos(4*(c*f - d*e)/d)*cos_integral(-4*((d*x + c)*(c*f/
(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e + 6*I*d*f^2*cos(6*(c*f - d*e)/d)*cos_integral(-6*((d*x + c)*(
c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e - 6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos
_integral(-6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*sin(6*(c*f - d*e)/d) + 6*c*f^3*cos
_integral(-6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*sin(6*(c*f - d*e)/d) - 6*d*f^2*cos
_integral(-6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e*sin(6*(c*f - d*e)/d) - 12*(d*x +
c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c
*f + d*e)/d)*sin(4*(c*f - d*e)/d) + 12*c*f^3*cos_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) -
c*f + d*e)/d)*sin(4*(c*f - d*e)/d) - 12*d*f^2*cos_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) -
c*f + d*e)/d)*e*sin(4*(c*f - d*e)/d) - 6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*cos_integral(-2*((
d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*sin(2*(c*f - d*e)/d) + 6*c*f^3*cos_integral(-2*((
d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*sin(2*(c*f - d*e)/d) - 6*d*f^2*cos_integral(-2*((
d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)*e*sin(2*(c*f - d*e)/d) + 6*(d*x + c)*(c*f/(d*x +
c) - f - d*e/(d*x + c))*f^2*cos(2*(c*f - d*e)/d)*sin_integral(-2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)
) - c*f + d*e)/d) - 6*c*f^3*cos(2*(c*f - d*e)/d)*sin_integral(-2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)
) - c*f + d*e)/d) + 6*d*f^2*cos(2*(c*f - d*e)/d)*e*sin_integral(-2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x +
c)) - c*f + d*e)/d) + 6*I*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*sin(2*(c*f - d*e)/d)*sin_integral(
-2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - 6*I*c*f^3*sin(2*(c*f - d*e)/d)*sin_integra
l(-2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 6*I*d*f^2*e*sin(2*(c*f - d*e)/d)*sin_int
egral(-2*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 12*(d*x + c)*(c*f/(d*x + c) - f - d*
e/(d*x + c))*f^2*cos(4*(c*f - d*e)/d)*sin_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d
*e)/d) - 12*c*f^3*cos(4*(c*f - d*e)/d)*sin_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f +
d*e)/d) + 12*d*f^2*cos(4*(c*f - d*e)/d)*e*sin_integral(-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f
+ d*e)/d) + 12*I*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*sin(4*(c*f - d*e)/d)*sin_integral(-4*((d*x
+ c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - 12*I*c*f^3*sin(4*(c*f - d*e)/d)*sin_integral(-4*((
d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 12*I*d*f^2*e*sin(4*(c*f - d*e)/d)*sin_integral(
-4*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x
+ c))*f^2*cos(6*(c*f - d*e)/d)*sin_integral(-6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)
- 6*c*f^3*cos(6*(c*f - d*e)/d)*sin_integral(-6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d)
+ 6*d*f^2*cos(6*(c*f - d*e)/d)*e*sin_integral(-6*((d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d
) + 6*I*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))*f^2*sin(6*(c*f - d*e)/d)*sin_integral(-6*((d*x + c)*(c*f
/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - 6*I*c*f^3*sin(6*(c*f - d*e)/d)*sin_integral(-6*((d*x + c)*(c
*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) + 6*I*d*f^2*e*sin(6*(c*f - d*e)/d)*sin_integral(-6*((d*x + c
)*(c*f/(d*x + c) - f - d*e/(d*x + c)) - c*f + d*e)/d) - d*f^2*cos(6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x +
c))/d) - 3*d*f^2*cos(4*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))/d) - 3*d*f^2*cos(2*(d*x + c)*(c*f/(d*x +
c) - f - d*e/(d*x + c))/d) - I*d*f^2*sin(6*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))/d) - 3*I*d*f^2*sin(4*
(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c))/d) - 3*I*d*f^2*sin(2*(d*x + c)*(c*f/(d*x + c) - f - d*e/(d*x + c
))/d) - d*f^2)*d^2/(((d*x + c)*a^3*d^4*(c*f/(d*...

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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 )}^2} \,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)^2),x)

[Out]

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

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