Integrand size = 23, antiderivative size = 712 \[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=-\frac {1}{8 a^3 d (c+d x)}-\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 i f \cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 i f \cos \left (4 e-\frac {4 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 i f \cos \left (6 e-\frac {6 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}-\frac {3 f \operatorname {CosIntegral}\left (\frac {6 c f}{d}+6 f x\right ) \sin \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {3 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{4 a^3 d^2}-\frac {3 f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}+\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{2 a^3 d^2}-\frac {3 f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (\frac {6 c f}{d}+6 f x\right )}{4 a^3 d^2} \]
-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+15/ 32*I*sin(2*f*x+2*e)/a^3/d/(d*x+c)-1/8*I*sin(2*f*x+2*e)^3/a^3/d/(d*x+c)-9/3 2*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^2-3/2*I*f*Si(4*c*f/d+4*f*x)*sin(-4*e+ 4*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+3/32*I* sin(6*f*x+6*e)/a^3/d/(d*x+c)+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*Ci(4*c*f/d+4*f*x)*cos(-4*e+4*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+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^2-3/4*I*f*Si(2*c*f/d+2*f*x)*sin(-2*e+2*c*f/d)/a^3/d^2
Time = 2.85 (sec) , antiderivative size = 833, normalized size of antiderivative = 1.17 \[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=\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 ) \operatorname {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 ) \operatorname {CosIntegral}\left (\frac {6 f (c+d x)}{d}\right )+6 i f (c+d x) \operatorname {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 \operatorname {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 \operatorname {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) \operatorname {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} \]
(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]*Cos Integral[(6*f*(c + d*x))/d] + (6*I)*d*f*x*Cos[3*e - (3*f*(c + d*x))/d]*Cos Integral[(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*C os[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]*SinIntegral[(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]*S inIntegral[(4*f*(c + d*x))/d] - (12*I)*d*f*x*Sin[e - (f*(c + 3*d*x))/d]*Si nIntegral[(4*f*(c + d*x))/d] + 6*c*f*Cos[3*e - (3*f*(c + d*x))/d]*SinInteg ral[(6*f*(c + d*x))/d] + 6*d*f*x*Cos[3*e - (3*f*(c + d*x))/d]*SinIntegr...
Time = 2.06 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3042, 4211, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3}dx\) |
| \(\Big \downarrow \) 4211 |
| \(\displaystyle \int \left (\frac {i \sin ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 i \sin (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 \sin (2 e+2 f x) \sin (4 e+4 f x)}{16 a^3 (c+d x)^2}-\frac {3 i \sin (4 e+4 f x)}{8 a^3 (c+d x)^2}+\frac {\cos ^3(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {3 \cos (2 e+2 f x)}{8 a^3 (c+d x)^2}-\frac {3 i \sin (2 e+2 f x) \cos ^2(2 e+2 f x)}{8 a^3 (c+d x)^2}+\frac {1}{8 a^3 (c+d x)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 f \operatorname {CosIntegral}\left (6 x f+\frac {6 c f}{d}\right ) \sin \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \operatorname {CosIntegral}\left (4 x f+\frac {4 c f}{d}\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 i f \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 i f \operatorname {CosIntegral}\left (4 x f+\frac {4 c f}{d}\right ) \cos \left (4 e-\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 i f \operatorname {CosIntegral}\left (6 x f+\frac {6 c f}{d}\right ) \cos \left (6 e-\frac {6 c f}{d}\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}+\frac {3 i f \sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}+\frac {3 i f \sin \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{4 a^3 d^2}-\frac {3 f \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{2 a^3 d^2}-\frac {3 f \cos \left (6 e-\frac {6 c f}{d}\right ) \text {Si}\left (6 x f+\frac {6 c f}{d}\right )}{4 a^3 d^2}-\frac {i \sin ^3(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {3 \sin ^2(2 e+2 f x)}{8 a^3 d (c+d x)}+\frac {15 i \sin (2 e+2 f x)}{32 a^3 d (c+d x)}+\frac {3 i \sin (4 e+4 f x)}{8 a^3 d (c+d x)}+\frac {3 i \sin (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {\cos ^3(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {3 \cos ^2(2 e+2 f x)}{8 a^3 d (c+d x)}-\frac {9 \cos (2 e+2 f x)}{32 a^3 d (c+d x)}-\frac {3 \cos (6 e+6 f x)}{32 a^3 d (c+d x)}-\frac {1}{8 a^3 d (c+d x)}\) |
-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*CosI ntegral[(6*c*f)/d + 6*f*x]*Sin[6*e - (6*c*f)/d])/(4*a^3*d^2) - (3*f*CosInt egral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(2*a^3*d^2) - (3*f*CosInteg ral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(4*a^3*d^2) + (((15*I)/32)*Si n[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)
3.1.33.3.1 Defintions of rubi rules used
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]
Time = 1.06 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.36
| method | result | size |
| risch | \(-\frac {1}{8 a^{3} d \left (d x +c \right )}-\frac {f \,{\mathrm e}^{-6 i \left (f x +e \right )}}{8 a^{3} \left (d f x +c f \right ) d}+\frac {3 i f \,{\mathrm e}^{\frac {6 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (6 i f x +6 i e +\frac {6 i \left (c f -d e \right )}{d}\right )}{4 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-4 i \left (f x +e \right )}}{8 a^{3} \left (d f x +c f \right ) d}+\frac {3 i f \,{\mathrm e}^{\frac {4 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (4 i f x +4 i e +\frac {4 i \left (c f -d e \right )}{d}\right )}{2 a^{3} d^{2}}-\frac {3 f \,{\mathrm e}^{-2 i \left (f x +e \right )}}{8 a^{3} \left (d f x +c f \right ) d}+\frac {3 i f \,{\mathrm e}^{\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (2 i f x +2 i e +\frac {2 i \left (c f -d e \right )}{d}\right )}{4 a^{3} d^{2}}\) | \(254\) |
-1/8/a^3/d/(d*x+c)-1/8/a^3*f*exp(-6*I*(f*x+e))/(d*f*x+c*f)/d+3/4*I/a^3*f/d ^2*exp(6*I*(c*f-d*e)/d)*Ei(1,6*I*f*x+6*I*e+6*I*(c*f-d*e)/d)-3/8/a^3*f*exp( -4*I*(f*x+e))/(d*f*x+c*f)/d+3/2*I/a^3*f/d^2*exp(4*I*(c*f-d*e)/d)*Ei(1,4*I* f*x+4*I*e+4*I*(c*f-d*e)/d)-3/8/a^3*f*exp(-2*I*(f*x+e))/(d*f*x+c*f)/d+3/4*I /a^3*f/d^2*exp(2*I*(c*f-d*e)/d)*Ei(1,2*I*f*x+2*I*e+2*I*(c*f-d*e)/d)
Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.28 \[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=-\frac {{\left ({\left (6 \, {\left (i \, d f x + i \, c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (i \, d e - i \, c f\right )}}{d}\right )} + 12 \, {\left (i \, d f x + i \, c f\right )} {\rm Ei}\left (-\frac {4 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (i \, d e - i \, c f\right )}}{d}\right )} + 6 \, {\left (i \, d f x + i \, c f\right )} {\rm Ei}\left (-\frac {6 \, {\left (i \, d f x + i \, c f\right )}}{d}\right ) e^{\left (-\frac {6 \, {\left (i \, d e - i \, c f\right )}}{d}\right )} + d\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, d e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, d e^{\left (2 i \, f x + 2 i \, e\right )} + d\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{8 \, {\left (a^{3} d^{3} x + a^{3} c d^{2}\right )}} \]
-1/8*((6*(I*d*f*x + I*c*f)*Ei(-2*(I*d*f*x + I*c*f)/d)*e^(-2*(I*d*e - I*c*f )/d) + 12*(I*d*f*x + I*c*f)*Ei(-4*(I*d*f*x + I*c*f)/d)*e^(-4*(I*d*e - I*c* f)/d) + 6*(I*d*f*x + I*c*f)*Ei(-6*(I*d*f*x + I*c*f)/d)*e^(-6*(I*d*e - 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)*e^(-6*I*f*x - 6*I*e)/(a^3*d^3*x + a^3*c*d^2)
\[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=\frac {i \int \frac {1}{c^{2} \tan ^{3}{\left (e + f x \right )} - 3 i c^{2} \tan ^{2}{\left (e + f x \right )} - 3 c^{2} \tan {\left (e + f x \right )} + i c^{2} + 2 c d x \tan ^{3}{\left (e + f x \right )} - 6 i c d x \tan ^{2}{\left (e + f x \right )} - 6 c d x \tan {\left (e + f x \right )} + 2 i c d x + d^{2} x^{2} \tan ^{3}{\left (e + f x \right )} - 3 i d^{2} x^{2} \tan ^{2}{\left (e + f x \right )} - 3 d^{2} x^{2} \tan {\left (e + f x \right )} + i d^{2} x^{2}}\, dx}{a^{3}} \]
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
Time = 0.43 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.42 \[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=-\frac {3 \, f^{2} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + 3 \, f^{2} \cos \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + f^{2} \cos \left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right ) E_{2}\left (-\frac {6 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + 3 i \, f^{2} E_{2}\left (-\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 3 i \, f^{2} E_{2}\left (-\frac {4 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) + i \, f^{2} E_{2}\left (-\frac {6 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {6 \, {\left (d e - c f\right )}}{d}\right ) + f^{2}}{8 \, {\left ({\left (f x + e\right )} a^{3} d^{2} - a^{3} d^{2} e + a^{3} c d f\right )} f} \]
-1/8*(3*f^2*cos(-2*(d*e - c*f)/d)*exp_integral_e(2, -2*(-I*(f*x + e)*d + I *d*e - I*c*f)/d) + 3*f^2*cos(-4*(d*e - c*f)/d)*exp_integral_e(2, -4*(-I*(f *x + e)*d + I*d*e - I*c*f)/d) + f^2*cos(-6*(d*e - c*f)/d)*exp_integral_e(2 , -6*(-I*(f*x + e)*d + I*d*e - I*c*f)/d) + 3*I*f^2*exp_integral_e(2, -2*(- I*(f*x + e)*d + I*d*e - I*c*f)/d)*sin(-2*(d*e - c*f)/d) + 3*I*f^2*exp_inte gral_e(2, -4*(-I*(f*x + e)*d + I*d*e - I*c*f)/d)*sin(-4*(d*e - c*f)/d) + I *f^2*exp_integral_e(2, -6*(-I*(f*x + e)*d + I*d*e - I*c*f)/d)*sin(-6*(d*e - c*f)/d) + f^2)/(((f*x + e)*a^3*d^2 - a^3*d^2*e + a^3*c*d*f)*f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2915 vs. \(2 (648) = 1296\).
Time = 30.48 (sec) , antiderivative size = 2915, normalized size of antiderivative = 4.09 \[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=\text {Too large to display} \]
1/8*(-6*I*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos(-6*(d*e - c*f)/d)*cos_integral(6*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d* e + c*f)/d) + 6*I*d*e*f^2*cos(-6*(d*e - c*f)/d)*cos_integral(6*((d*x + c)* (d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 6*I*c*f^3*cos(-6*(d* e - c*f)/d)*cos_integral(6*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 12*I*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*c os(-4*(d*e - c*f)/d)*cos_integral(4*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) + 12*I*d*e*f^2*cos(-4*(d*e - c*f)/d)*cos_integral (4*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) - 12*I*c *f^3*cos(-4*(d*e - c*f)/d)*cos_integral(4*((d*x + c)*(d*e/(d*x + c) - c*f/ (d*x + c) + f) - d*e + c*f)/d) - 6*I*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos(-2*(d*e - c*f)/d)*cos_integral(2*((d*x + c)*(d*e/(d*x + c ) - c*f/(d*x + c) + f) - d*e + c*f)/d) + 6*I*d*e*f^2*cos(-2*(d*e - c*f)/d) *cos_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f )/d) - 6*I*c*f^3*cos(-2*(d*e - c*f)/d)*cos_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d) + 6*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*cos_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin(-2*(d*e - c*f)/d) - 6*d*e*f^2*cos_integral( 2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*sin(-2*(d *e - c*f)/d) + 6*c*f^3*cos_integral(2*((d*x + c)*(d*e/(d*x + c) - c*f/(...
Timed out. \[ \int \frac {1}{(c+d x)^2 (a+i a \tan (e+f x))^3} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (c+d\,x\right )}^2} \,d x \]