Integrand size = 23, antiderivative size = 227 \[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=\frac {i f}{2 a d^2 (c+d x)}+\frac {f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}-\frac {i f}{d^2 (c+d x) (a+i a \cot (e+f x))}+\frac {i f^2 \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{a d^3}+\frac {i f^2 \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3}-\frac {f^2 \sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{a d^3} \]
1/2*I*f/a/d^2/(d*x+c)+f^2*Ci(2*c*f/d+2*f*x)*cos(-2*e+2*c*f/d)/a/d^3-1/2/d/ (d*x+c)^2/(a+I*a*cot(f*x+e))-I*f/d^2/(d*x+c)/(a+I*a*cot(f*x+e))+I*f^2*cos( -2*e+2*c*f/d)*Si(2*c*f/d+2*f*x)/a/d^3-I*f^2*Ci(2*c*f/d+2*f*x)*sin(-2*e+2*c *f/d)/a/d^3+f^2*Si(2*c*f/d+2*f*x)*sin(-2*e+2*c*f/d)/a/d^3
Time = 1.66 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=\frac {\left (\cos \left (e+f \left (-\frac {c}{d}+x\right )\right )+i \sin \left (e+f \left (-\frac {c}{d}+x\right )\right )\right ) \left (4 f^2 (c+d x)^2 \operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right ) \left (\cos \left (e-\frac {f (c+d x)}{d}\right )+i \sin \left (e-\frac {f (c+d x)}{d}\right )\right )+i \left (d \left (i d \cos \left (e+f \left (-\frac {c}{d}+x\right )\right )+(-i d+2 c f+2 d f x) \cos \left (e+f \left (\frac {c}{d}+x\right )\right )+d \sin \left (e+f \left (-\frac {c}{d}+x\right )\right )+d \sin \left (e+f \left (\frac {c}{d}+x\right )\right )+2 i c f \sin \left (e+f \left (\frac {c}{d}+x\right )\right )+2 i d f x \sin \left (e+f \left (\frac {c}{d}+x\right )\right )\right )+4 f^2 (c+d x)^2 \left (\cos \left (e-\frac {f (c+d x)}{d}\right )+i \sin \left (e-\frac {f (c+d x)}{d}\right )\right ) \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )\right )}{4 a d^3 (c+d x)^2} \]
((Cos[e + f*(-(c/d) + x)] + I*Sin[e + f*(-(c/d) + x)])*(4*f^2*(c + d*x)^2* CosIntegral[(2*f*(c + d*x))/d]*(Cos[e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))/d]) + I*(d*(I*d*Cos[e + f*(-(c/d) + x)] + ((-I)*d + 2*c*f + 2*d*f *x)*Cos[e + f*(c/d + x)] + d*Sin[e + f*(-(c/d) + x)] + d*Sin[e + f*(c/d + x)] + (2*I)*c*f*Sin[e + f*(c/d + x)] + (2*I)*d*f*x*Sin[e + f*(c/d + x)]) + 4*f^2*(c + d*x)^2*(Cos[e - (f*(c + d*x))/d] + I*Sin[e - (f*(c + d*x))/d]) *SinIntegral[(2*f*(c + d*x))/d])))/(4*a*d^3*(c + d*x)^2)
Time = 0.91 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 4208, 3042, 4207, 25, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c+d x)^3 \left (a-i a \tan \left (e+f x+\frac {\pi }{2}\right )\right )}dx\) |
| \(\Big \downarrow \) 4208 |
| \(\displaystyle \frac {i f \int \frac {1}{(c+d x)^2 (i \cot (e+f x) a+a)}dx}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i f \int \frac {1}{(c+d x)^2 \left (a-i a \tan \left (e+f x+\frac {\pi }{2}\right )\right )}dx}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 4207 |
| \(\displaystyle \frac {i f \left (-\frac {f \int -\frac {\sin (2 e+2 f x)}{c+d x}dx}{a d}+\frac {i f \int -\frac {\cos (2 e+2 f x)}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {i f \left (\frac {f \int \frac {\sin (2 e+2 f x)}{c+d x}dx}{a d}-\frac {i f \int \frac {\cos (2 e+2 f x)}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i f \left (\frac {f \int \frac {\sin (2 e+2 f x)}{c+d x}dx}{a d}-\frac {i f \int \frac {\sin \left (2 e+2 f x+\frac {\pi }{2}\right )}{c+d x}dx}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {i f \left (\frac {f \left (\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cos \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx+\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {i f \left (\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\cos \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx-\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i f \left (-\frac {i f \left (\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx-\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}+\frac {f \left (\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}\right )}{c+d x}dx\right )}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {i f \left (-\frac {i f \left (\cos \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}+\frac {f \left (\sin \left (2 e-\frac {2 c f}{d}\right ) \int \frac {\sin \left (2 x f+\frac {2 c f}{d}+\frac {\pi }{2}\right )}{c+d x}dx+\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {i f \left (\frac {f \left (\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{d}+\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {i f \left (\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{d}-\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{d}\right )}{a d}-\frac {1}{d (c+d x) (a+i a \cot (e+f x))}\right )}{d}+\frac {i f}{2 a d^2 (c+d x)}-\frac {1}{2 d (c+d x)^2 (a+i a \cot (e+f x))}\) |
((I/2)*f)/(a*d^2*(c + d*x)) - 1/(2*d*(c + d*x)^2*(a + I*a*Cot[e + f*x])) + (I*f*(-(1/(d*(c + d*x)*(a + I*a*Cot[e + f*x]))) + (f*((CosIntegral[(2*c*f )/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/d + (Cos[2*e - (2*c*f)/d]*SinIntegral[( 2*c*f)/d + 2*f*x])/d))/(a*d) - (I*f*((Cos[2*e - (2*c*f)/d]*CosIntegral[(2* c*f)/d + 2*f*x])/d - (Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x]) /d))/(a*d)))/d
3.1.21.3.1 Defintions of rubi rules used
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[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]
Int[1/(((c_.) + (d_.)*(x_))^2*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])), x_Sy mbol] :> -Simp[(d*(c + d*x)*(a + b*Tan[e + f*x]))^(-1), x] + (-Simp[f/(a*d) Int[Sin[2*e + 2*f*x]/(c + d*x), x], x] + Simp[f/(b*d) Int[Cos[2*e + 2* f*x]/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0 ]
Int[((c_.) + (d_.)*(x_))^(m_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Sym bol] :> Simp[f*((c + d*x)^(m + 2)/(b*d^2*(m + 1)*(m + 2))), x] + (Simp[2*b* (f/(a*d*(m + 1))) Int[(c + d*x)^(m + 1)/(a + b*Tan[e + f*x]), x], x] + Si mp[(c + d*x)^(m + 1)/(d*(m + 1)*(a + b*Tan[e + f*x])), x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[m, -1] && NeQ[m, -2]
Time = 0.36 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.63
| method | result | size |
| risch | \(-\frac {1}{4 d \left (d x +c \right )^{2} a}-\frac {f^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{4 a \,d^{3} \left (i f x +\frac {i c f}{d}\right )^{2}}-\frac {f^{2} {\mathrm e}^{2 i \left (f x +e \right )}}{2 a \,d^{3} \left (i f x +\frac {i c f}{d}\right )}-\frac {f^{2} {\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{a \,d^{3}}\) | \(143\) |
| derivativedivides | \(\frac {f^{2} \left (-\frac {i \left (-\frac {\sin \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}}{d}\right )}{4}-\frac {1}{4 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {\cos \left (2 f x +2 e \right )}{4 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {2 \sin \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {4 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}}{d}}{4 d}\right )}{a}\) | \(346\) |
| default | \(\frac {f^{2} \left (-\frac {i \left (-\frac {\sin \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {2 \cos \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}-\frac {2 \left (\frac {2 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}-\frac {2 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}\right )}{d}}{d}\right )}{4}-\frac {1}{4 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {\cos \left (2 f x +2 e \right )}{4 \left (c f -d e +d \left (f x +e \right )\right )^{2} d}+\frac {-\frac {2 \sin \left (2 f x +2 e \right )}{\left (c f -d e +d \left (f x +e \right )\right ) d}+\frac {\frac {4 \,\operatorname {Si}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \sin \left (\frac {2 c f -2 d e}{d}\right )}{d}+\frac {4 \,\operatorname {Ci}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right ) \cos \left (\frac {2 c f -2 d e}{d}\right )}{d}}{d}}{4 d}\right )}{a}\) | \(346\) |
-1/4/d/(d*x+c)^2/a-1/4*f^2/a/d^3*exp(2*I*(f*x+e))/(I*f*x+I/d*c*f)^2-1/2*f^ 2/a/d^3*exp(2*I*(f*x+e))/(I*f*x+I/d*c*f)-f^2/a/d^3*exp(-2*I*(c*f-d*e)/d)*E i(1,-2*I*f*x-2*I*e-2*(I*c*f-I*d*e)/d)
Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.53 \[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=\frac {4 \, {\left (d^{2} f^{2} x^{2} + 2 \, c d f^{2} x + c^{2} f^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, d e + i \, c f\right )}}{d}\right )} - d^{2} + {\left (2 i \, d^{2} f x + 2 i \, c d f + d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{4 \, {\left (a d^{5} x^{2} + 2 \, a c d^{4} x + a c^{2} d^{3}\right )}} \]
1/4*(4*(d^2*f^2*x^2 + 2*c*d*f^2*x + c^2*f^2)*Ei(-2*(-I*d*f*x - I*c*f)/d)*e ^(-2*(-I*d*e + I*c*f)/d) - d^2 + (2*I*d^2*f*x + 2*I*c*d*f + d^2)*e^(2*I*f* x + 2*I*e))/(a*d^5*x^2 + 2*a*c*d^4*x + a*c^2*d^3)
\[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=- \frac {i \int \frac {1}{c^{3} \cot {\left (e + f x \right )} - i c^{3} + 3 c^{2} d x \cot {\left (e + f x \right )} - 3 i c^{2} d x + 3 c d^{2} x^{2} \cot {\left (e + f x \right )} - 3 i c d^{2} x^{2} + d^{3} x^{3} \cot {\left (e + f x \right )} - i d^{3} x^{3}}\, dx}{a} \]
-I*Integral(1/(c**3*cot(e + f*x) - I*c**3 + 3*c**2*d*x*cot(e + f*x) - 3*I* c**2*d*x + 3*c*d**2*x**2*cot(e + f*x) - 3*I*c*d**2*x**2 + d**3*x**3*cot(e + f*x) - I*d**3*x**3), x)/a
Time = 0.41 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=\frac {2 \, f^{3} \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{3}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - 2 i \, f^{3} E_{3}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - f^{3}}{4 \, {\left ({\left (f x + e\right )}^{2} a d^{3} + a d^{3} e^{2} - 2 \, a c d^{2} e f + a c^{2} d f^{2} - 2 \, {\left (a d^{3} e - a c d^{2} f\right )} {\left (f x + e\right )}\right )} f} \]
1/4*(2*f^3*cos(-2*(d*e - c*f)/d)*exp_integral_e(3, 2*(-I*(f*x + e)*d + I*d *e - I*c*f)/d) - 2*I*f^3*exp_integral_e(3, 2*(-I*(f*x + e)*d + I*d*e - I*c *f)/d)*sin(-2*(d*e - c*f)/d) - f^3)/(((f*x + e)^2*a*d^3 + a*d^3*e^2 - 2*a* c*d^2*e*f + a*c^2*d*f^2 - 2*(a*d^3*e - a*c*d^2*f)*(f*x + e))*f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1558 vs. \(2 (210) = 420\).
Time = 0.30 (sec) , antiderivative size = 1558, normalized size of antiderivative = 6.86 \[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=\text {Too large to display} \]
1/4*(4*d^2*f^2*x^2*cos(e)^2*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d) + 8*I*d^2*f^2*x^2*cos(e)*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e ) - 4*d^2*f^2*x^2*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)^2 - 4*I*d^2*f^2*x^2*cos(e)^2*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d) + 8* d^2*f^2*x^2*cos(e)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)*sin(2*c*f/d) + 4 *I*d^2*f^2*x^2*cos_integral(2*(d*f*x + c*f)/d)*sin(e)^2*sin(2*c*f/d) + 4*I *d^2*f^2*x^2*cos(e)^2*cos(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 8*d^2 *f^2*x^2*cos(e)*cos(2*c*f/d)*sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 4*I* d^2*f^2*x^2*cos(2*c*f/d)*sin(e)^2*sin_integral(2*(d*f*x + c*f)/d) + 4*d^2* f^2*x^2*cos(e)^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 8*I*d^2*f^ 2*x^2*cos(e)*sin(e)*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 4*d^2*f ^2*x^2*sin(e)^2*sin(2*c*f/d)*sin_integral(2*(d*f*x + c*f)/d) + 8*c*d*f^2*x *cos(e)^2*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d) + 16*I*c*d*f^2*x*co s(e)*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e) - 8*c*d*f^2*x*cos (2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e)^2 - 8*I*c*d*f^2*x*cos(e)^ 2*cos_integral(2*(d*f*x + c*f)/d)*sin(2*c*f/d) + 16*c*d*f^2*x*cos(e)*cos_i ntegral(2*(d*f*x + c*f)/d)*sin(e)*sin(2*c*f/d) + 8*I*c*d*f^2*x*cos_integra l(2*(d*f*x + c*f)/d)*sin(e)^2*sin(2*c*f/d) + 8*I*c*d*f^2*x*cos(e)^2*cos(2* c*f/d)*sin_integral(2*(d*f*x + c*f)/d) - 16*c*d*f^2*x*cos(e)*cos(2*c*f/d)* sin(e)*sin_integral(2*(d*f*x + c*f)/d) - 8*I*c*d*f^2*x*cos(2*c*f/d)*sin...
Timed out. \[ \int \frac {1}{(c+d x)^3 (a+i a \cot (e+f x))} \, dx=\int \frac {1}{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^3} \,d x \]