3.199 \(\int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx\)

Optimal. Leaf size=64 \[ -\frac {3 x}{-x^2+e^{2 i a}}+\frac {e^{2 i a}}{x \left (-x^2+e^{2 i a}\right )}-2 e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right ) \]

[Out]

exp(2*I*a)/x/(exp(2*I*a)-x^2)-3*x/(exp(2*I*a)-x^2)-2*arctanh(x/exp(I*a))/exp(I*a)

________________________________________________________________________________________

Rubi [F]  time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]^2/x^2,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]^2/x^2, x]

Rubi steps

\begin {align*} \int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx &=\int \frac {\cot ^2(a+i \log (x))}{x^2} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 72, normalized size = 1.12 \[ \frac {2 x (\cos (a)-i \sin (a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-2 \cos (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+2 i \sin (a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+\frac {1}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]^2/x^2,x]

[Out]

x^(-1) - 2*ArcTanh[x*(Cos[a] - I*Sin[a])]*Cos[a] + (2*I)*ArcTanh[x*(Cos[a] - I*Sin[a])]*Sin[a] + (2*x*(Cos[a]
- I*Sin[a]))/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a])

________________________________________________________________________________________

fricas [A]  time = 0.71, size = 74, normalized size = 1.16 \[ -\frac {{\left (x^{3} - x e^{\left (2 i \, a\right )}\right )} e^{\left (-i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - {\left (x^{3} - x e^{\left (2 i \, a\right )}\right )} e^{\left (-i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - 3 \, x^{2} + e^{\left (2 i \, a\right )}}{x^{3} - x e^{\left (2 i \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="fricas")

[Out]

-((x^3 - x*e^(2*I*a))*e^(-I*a)*log(x + e^(I*a)) - (x^3 - x*e^(2*I*a))*e^(-I*a)*log(x - e^(I*a)) - 3*x^2 + e^(2
*I*a))/(x^3 - x*e^(2*I*a))

________________________________________________________________________________________

giac [A]  time = 0.30, size = 87, normalized size = 1.36 \[ 2 \, {\left (\frac {\arctan \left (\frac {x}{\sqrt {-e^{\left (2 i \, a\right )}}}\right ) e^{\left (-2 i \, a\right )}}{\sqrt {-e^{\left (2 i \, a\right )}}} + \frac {x e^{\left (-2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac {5 \, x^{2}}{x^{3} - x e^{\left (2 i \, a\right )}} - \frac {e^{\left (2 i \, a\right )}}{x^{3} - x e^{\left (2 i \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="giac")

[Out]

2*(arctan(x/sqrt(-e^(2*I*a)))*e^(-2*I*a)/sqrt(-e^(2*I*a)) + x*e^(-2*I*a)/(x^2 - e^(2*I*a)))*e^(2*I*a) + 5*x^2/
(x^3 - x*e^(2*I*a)) - e^(2*I*a)/(x^3 - x*e^(2*I*a))

________________________________________________________________________________________

maple [A]  time = 0.06, size = 38, normalized size = 0.59 \[ \frac {1}{x}-\frac {2}{x \left (\frac {{\mathrm e}^{2 i a}}{x^{2}}-1\right )}-2 \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x))^2/x^2,x)

[Out]

1/x-2/x/(exp(2*I*a)/x^2-1)-2*arctanh(x*exp(-I*a))*exp(-I*a)

________________________________________________________________________________________

maxima [B]  time = 0.39, size = 285, normalized size = 4.45 \[ -\frac {2 \, {\left ({\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x^{3} + {\left ({\left (2 \, {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (2 \, {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x - 6 \, x^{2} + {\left (x^{3} {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} - {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \cos \left (2 \, a\right ) + {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - {\left (x^{3} {\left (\cos \relax (a) - i \, \sin \relax (a)\right )} - {\left ({\left (\cos \relax (a) - i \, \sin \relax (a)\right )} \cos \left (2 \, a\right ) - {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \sin \left (2 \, a\right )\right )} x\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + 2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )}{2 \, x^{3} - x {\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="maxima")

[Out]

-(2*((I*cos(a) + sin(a))*arctan2(sin(a), x + cos(a)) + (I*cos(a) + sin(a))*arctan2(sin(a), x - cos(a)))*x^3 +
((2*(-I*cos(a) - sin(a))*cos(2*a) + (2*cos(a) - 2*I*sin(a))*sin(2*a))*arctan2(sin(a), x + cos(a)) + (2*(-I*cos
(a) - sin(a))*cos(2*a) + (2*cos(a) - 2*I*sin(a))*sin(2*a))*arctan2(sin(a), x - cos(a)))*x - 6*x^2 + (x^3*(cos(
a) - I*sin(a)) - ((cos(a) - I*sin(a))*cos(2*a) + (I*cos(a) + sin(a))*sin(2*a))*x)*log(x^2 + 2*x*cos(a) + cos(a
)^2 + sin(a)^2) - (x^3*(cos(a) - I*sin(a)) - ((cos(a) - I*sin(a))*cos(2*a) - (-I*cos(a) - sin(a))*sin(2*a))*x)
*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2) + 2*cos(2*a) + 2*I*sin(2*a))/(2*x^3 - x*(2*cos(2*a) + 2*I*sin(2*a
)))

________________________________________________________________________________________

mupad [B]  time = 2.21, size = 47, normalized size = 0.73 \[ -\frac {2\,\mathrm {atanh}\left (\frac {x}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )}{\sqrt {{\mathrm {e}}^{a\,2{}\mathrm {i}}}}-\frac {{\mathrm {e}}^{a\,2{}\mathrm {i}}-3\,x^2}{x^3-x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + log(x)*1i)^2/x^2,x)

[Out]

- (2*atanh(x/exp(a*2i)^(1/2)))/exp(a*2i)^(1/2) - (exp(a*2i) - 3*x^2)/(x^3 - x*exp(a*2i))

________________________________________________________________________________________

sympy [A]  time = 0.38, size = 46, normalized size = 0.72 \[ - \frac {- 3 x^{2} + e^{2 i a}}{x^{3} - x e^{2 i a}} - \left (- \log {\left (x - e^{i a} \right )} + \log {\left (x + e^{i a} \right )}\right ) e^{- i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*ln(x))**2/x**2,x)

[Out]

-(-3*x**2 + exp(2*I*a))/(x**3 - x*exp(2*I*a)) - (-log(x - exp(I*a)) + log(x + exp(I*a)))*exp(-I*a)

________________________________________________________________________________________