Integrand size = 23, antiderivative size = 305 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=-\frac {\cos \left (2 e-\frac {2 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\cos \left (4 e-\frac {4 c f}{d}\right ) \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}+\frac {i \operatorname {CosIntegral}\left (\frac {4 c f}{d}+4 f x\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \operatorname {CosIntegral}\left (\frac {2 c f}{d}+2 f x\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (\frac {2 c f}{d}+2 f x\right )}{2 a^2 d}+\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d}-\frac {\sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (\frac {4 c f}{d}+4 f x\right )}{4 a^2 d} \] Output:
-1/2*cos(-2*e+2*c*f/d)*Ci(2*c*f/d+2*f*x)/a^2/d+1/4*cos(-4*e+4*c*f/d)*Ci(4* c*f/d+4*f*x)/a^2/d+1/4*ln(d*x+c)/a^2/d-1/4*I*Ci(4*c*f/d+4*f*x)*sin(-4*e+4* c*f/d)/a^2/d+1/2*I*Ci(2*c*f/d+2*f*x)*sin(-2*e+2*c*f/d)/a^2/d-1/2*I*cos(-2* e+2*c*f/d)*Si(2*c*f/d+2*f*x)/a^2/d-1/2*sin(-2*e+2*c*f/d)*Si(2*c*f/d+2*f*x) /a^2/d+1/4*I*cos(-4*e+4*c*f/d)*Si(4*c*f/d+4*f*x)/a^2/d+1/4*sin(-4*e+4*c*f/ d)*Si(4*c*f/d+4*f*x)/a^2/d
Time = 0.45 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=\frac {\log (c+d x)-2 \left (\cos \left (2 e-\frac {2 c f}{d}\right )+i \sin \left (2 e-\frac {2 c f}{d}\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {2 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {2 f (c+d x)}{d}\right )\right )+\left (\cos \left (4 e-\frac {4 c f}{d}\right )+i \sin \left (4 e-\frac {4 c f}{d}\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {4 f (c+d x)}{d}\right )+i \text {Si}\left (\frac {4 f (c+d x)}{d}\right )\right )}{4 a^2 d} \] Input:
Integrate[1/((c + d*x)*(a + I*a*Cot[e + f*x])^2),x]
Output:
(Log[c + d*x] - 2*(Cos[2*e - (2*c*f)/d] + I*Sin[2*e - (2*c*f)/d])*(CosInte gral[(2*f*(c + d*x))/d] + I*SinIntegral[(2*f*(c + d*x))/d]) + (Cos[4*e - ( 4*c*f)/d] + I*Sin[4*e - (4*c*f)/d])*(CosIntegral[(4*f*(c + d*x))/d] + I*Si nIntegral[(4*f*(c + d*x))/d]))/(4*a^2*d)
Time = 1.08 (sec) , antiderivative size = 305, 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.
| \(\displaystyle \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(c+d x) \left (a-i a \tan \left (e+f x+\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4211 |
| \(\displaystyle \int \left (-\frac {\sin ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {i \sin (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {i \sin (4 e+4 f x)}{4 a^2 (c+d x)}+\frac {\cos ^2(2 e+2 f x)}{4 a^2 (c+d x)}-\frac {\cos (2 e+2 f x)}{2 a^2 (c+d x)}+\frac {1}{4 a^2 (c+d x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {i \operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \sin \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {i \operatorname {CosIntegral}\left (4 x f+\frac {4 c f}{d}\right ) \sin \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {\operatorname {CosIntegral}\left (2 x f+\frac {2 c f}{d}\right ) \cos \left (2 e-\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {\operatorname {CosIntegral}\left (4 x f+\frac {4 c f}{d}\right ) \cos \left (4 e-\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\sin \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}-\frac {\sin \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}-\frac {i \cos \left (2 e-\frac {2 c f}{d}\right ) \text {Si}\left (2 x f+\frac {2 c f}{d}\right )}{2 a^2 d}+\frac {i \cos \left (4 e-\frac {4 c f}{d}\right ) \text {Si}\left (4 x f+\frac {4 c f}{d}\right )}{4 a^2 d}+\frac {\log (c+d x)}{4 a^2 d}\) |
Input:
Int[1/((c + d*x)*(a + I*a*Cot[e + f*x])^2),x]
Output:
-1/2*(Cos[2*e - (2*c*f)/d]*CosIntegral[(2*c*f)/d + 2*f*x])/(a^2*d) + (Cos[ 4*e - (4*c*f)/d]*CosIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d) + Log[c + d*x]/ (4*a^2*d) + ((I/4)*CosIntegral[(4*c*f)/d + 4*f*x]*Sin[4*e - (4*c*f)/d])/(a ^2*d) - ((I/2)*CosIntegral[(2*c*f)/d + 2*f*x]*Sin[2*e - (2*c*f)/d])/(a^2*d ) - ((I/2)*Cos[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(a^2*d) + (Sin[2*e - (2*c*f)/d]*SinIntegral[(2*c*f)/d + 2*f*x])/(2*a^2*d) + ((I/4)*C os[4*e - (4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(a^2*d) - (Sin[4*e - ( 4*c*f)/d]*SinIntegral[(4*c*f)/d + 4*f*x])/(4*a^2*d)
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 = 0.97 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.39
| method | result | size |
| risch | \(\frac {\ln \left (d x +c \right )}{4 a^{2} d}+\frac {{\mathrm e}^{-\frac {2 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 i f x -2 i e -\frac {2 \left (i c f -i d e \right )}{d}\right )}{2 a^{2} d}-\frac {{\mathrm e}^{-\frac {4 i \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-4 i f x -4 i e -\frac {4 \left (i c f -i d e \right )}{d}\right )}{4 a^{2} d}\) | \(118\) |
Input:
int(1/(d*x+c)/(a+I*a*cot(f*x+e))^2,x,method=_RETURNVERBOSE)
Output:
1/4*ln(d*x+c)/a^2/d+1/2/a^2/d*exp(-2*I*(c*f-d*e)/d)*Ei(1,-2*I*f*x-2*I*e-2* (I*c*f-I*d*e)/d)-1/4/a^2/d*exp(-4*I*(c*f-d*e)/d)*Ei(1,-4*I*f*x-4*I*e-4*(I* c*f-I*d*e)/d)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.29 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=-\frac {2 \, {\rm Ei}\left (-\frac {2 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (-i \, d e + i \, c f\right )}}{d}\right )} - {\rm Ei}\left (-\frac {4 \, {\left (-i \, d f x - i \, c f\right )}}{d}\right ) e^{\left (-\frac {4 \, {\left (-i \, d e + i \, c f\right )}}{d}\right )} - \log \left (\frac {d x + c}{d}\right )}{4 \, a^{2} d} \] Input:
integrate(1/(d*x+c)/(a+I*a*cot(f*x+e))^2,x, algorithm="fricas")
Output:
-1/4*(2*Ei(-2*(-I*d*f*x - I*c*f)/d)*e^(-2*(-I*d*e + I*c*f)/d) - Ei(-4*(-I* d*f*x - I*c*f)/d)*e^(-4*(-I*d*e + I*c*f)/d) - log((d*x + c)/d))/(a^2*d)
\[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=- \frac {\int \frac {1}{c \cot ^{2}{\left (e + f x \right )} - 2 i c \cot {\left (e + f x \right )} - c + d x \cot ^{2}{\left (e + f x \right )} - 2 i d x \cot {\left (e + f x \right )} - d x}\, dx}{a^{2}} \] Input:
integrate(1/(d*x+c)/(a+I*a*cot(f*x+e))**2,x)
Output:
-Integral(1/(c*cot(e + f*x)**2 - 2*I*c*cot(e + f*x) - c + d*x*cot(e + f*x) **2 - 2*I*d*x*cot(e + f*x) - d*x), x)/a**2
Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=-\frac {f \cos \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac {4 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) - 2 \, f \cos \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) E_{1}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) + 2 i \, f E_{1}\left (\frac {2 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) - i \, f E_{1}\left (\frac {4 \, {\left (-i \, {\left (f x + e\right )} d + i \, d e - i \, c f\right )}}{d}\right ) \sin \left (-\frac {4 \, {\left (d e - c f\right )}}{d}\right ) - f \log \left ({\left (f x + e\right )} d - d e + c f\right )}{4 \, a^{2} d f} \] Input:
integrate(1/(d*x+c)/(a+I*a*cot(f*x+e))^2,x, algorithm="maxima")
Output:
-1/4*(f*cos(-4*(d*e - c*f)/d)*exp_integral_e(1, 4*(-I*(f*x + e)*d + I*d*e - I*c*f)/d) - 2*f*cos(-2*(d*e - c*f)/d)*exp_integral_e(1, 2*(-I*(f*x + e)* d + I*d*e - I*c*f)/d) + 2*I*f*exp_integral_e(1, 2*(-I*(f*x + e)*d + I*d*e - I*c*f)/d)*sin(-2*(d*e - c*f)/d) - I*f*exp_integral_e(1, 4*(-I*(f*x + e)* d + I*d*e - I*c*f)/d)*sin(-4*(d*e - c*f)/d) - f*log((f*x + e)*d - d*e + c* f))/(a^2*d*f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (279) = 558\).
Time = 0.16 (sec) , antiderivative size = 939, normalized size of antiderivative = 3.08 \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(d*x+c)/(a+I*a*cot(f*x+e))^2,x, algorithm="giac")
Output:
1/4*(cos(e)^4*cos(4*c*f/d)*cos_integral(4*(d*f*x + c*f)/d) + 4*I*cos(e)^3* cos(4*c*f/d)*cos_integral(4*(d*f*x + c*f)/d)*sin(e) - 6*cos(e)^2*cos(4*c*f /d)*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^2 - 4*I*cos(e)*cos(4*c*f/d)*cos _integral(4*(d*f*x + c*f)/d)*sin(e)^3 + cos(4*c*f/d)*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^4 - I*cos(e)^4*cos_integral(4*(d*f*x + c*f)/d)*sin(4*c*f /d) + 4*cos(e)^3*cos_integral(4*(d*f*x + c*f)/d)*sin(e)*sin(4*c*f/d) + 6*I *cos(e)^2*cos_integral(4*(d*f*x + c*f)/d)*sin(e)^2*sin(4*c*f/d) - 4*cos(e) *cos_integral(4*(d*f*x + c*f)/d)*sin(e)^3*sin(4*c*f/d) - I*cos_integral(4* (d*f*x + c*f)/d)*sin(e)^4*sin(4*c*f/d) + I*cos(e)^4*cos(4*c*f/d)*sin_integ ral(4*(d*f*x + c*f)/d) - 4*cos(e)^3*cos(4*c*f/d)*sin(e)*sin_integral(4*(d* f*x + c*f)/d) - 6*I*cos(e)^2*cos(4*c*f/d)*sin(e)^2*sin_integral(4*(d*f*x + c*f)/d) + 4*cos(e)*cos(4*c*f/d)*sin(e)^3*sin_integral(4*(d*f*x + c*f)/d) + I*cos(4*c*f/d)*sin(e)^4*sin_integral(4*(d*f*x + c*f)/d) + cos(e)^4*sin(4 *c*f/d)*sin_integral(4*(d*f*x + c*f)/d) + 4*I*cos(e)^3*sin(e)*sin(4*c*f/d) *sin_integral(4*(d*f*x + c*f)/d) - 6*cos(e)^2*sin(e)^2*sin(4*c*f/d)*sin_in tegral(4*(d*f*x + c*f)/d) - 4*I*cos(e)*sin(e)^3*sin(4*c*f/d)*sin_integral( 4*(d*f*x + c*f)/d) + sin(e)^4*sin(4*c*f/d)*sin_integral(4*(d*f*x + c*f)/d) - 2*cos(e)^2*cos(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d) - 4*I*cos(e)*co s(2*c*f/d)*cos_integral(2*(d*f*x + c*f)/d)*sin(e) + 2*cos(2*c*f/d)*cos_int egral(2*(d*f*x + c*f)/d)*sin(e)^2 + 2*I*cos(e)^2*cos_integral(2*(d*f*x ...
Timed out. \[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=\int \frac {1}{{\left (a+a\,\mathrm {cot}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2\,\left (c+d\,x\right )} \,d x \] Input:
int(1/((a + a*cot(e + f*x)*1i)^2*(c + d*x)),x)
Output:
int(1/((a + a*cot(e + f*x)*1i)^2*(c + d*x)), x)
\[ \int \frac {1}{(c+d x) (a+i a \cot (e+f x))^2} \, dx=-\frac {\int \frac {1}{\cot \left (f x +e \right )^{2} c +\cot \left (f x +e \right )^{2} d x -2 \cot \left (f x +e \right ) c i -2 \cot \left (f x +e \right ) d i x -c -d x}d x}{a^{2}} \] Input:
int(1/(d*x+c)/(a+I*a*cot(f*x+e))^2,x)
Output:
( - int(1/(cot(e + f*x)**2*c + cot(e + f*x)**2*d*x - 2*cot(e + f*x)*c*i - 2*cot(e + f*x)*d*i*x - c - d*x),x))/a**2