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

Optimal. Leaf size=29 \[ 2 i e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac {i}{x} \]

[Out]

-I/x+2*I*arctanh(x/exp(I*a))/exp(I*a)

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Rubi [F]  time = 0.02, 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 = {} \[ \int \frac {\cot (a+i \log (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]/x^2,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]/x^2, x]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 44, normalized size = 1.52 \[ 2 i \cos (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac {i}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]/x^2,x]

[Out]

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

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fricas [A]  time = 0.91, size = 36, normalized size = 1.24 \[ \frac {i \, x e^{\left (-i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, x e^{\left (-i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - i}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.60, size = 40, normalized size = 1.38 \[ i \, e^{\left (-i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (-i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - \frac {i}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 24, normalized size = 0.83 \[ -\frac {i}{x}+2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{-i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-I/x+2*I*arctanh(x*exp(-I*a))*exp(-I*a)

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maxima [B]  time = 0.38, size = 103, normalized size = 3.55 \[ \frac {x {\left (i \, \cos \relax (a) + \sin \relax (a)\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + x {\left (-i \, \cos \relax (a) - \sin \relax (a)\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - {\left ({\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) + {\left (2 \, \cos \relax (a) - 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right )\right )} x - 2 i}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x*(I*cos(a) + sin(a))*log(x^2 + 2*x*cos(a) + cos(a)^2 + sin(a)^2) + x*(-I*cos(a) - sin(a))*log(x^2 - 2*x*
cos(a) + cos(a)^2 + sin(a)^2) - ((2*cos(a) - 2*I*sin(a))*arctan2(sin(a), x + cos(a)) + (2*cos(a) - 2*I*sin(a))
*arctan2(sin(a), x - cos(a)))*x - 2*I)/x

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mupad [B]  time = 2.21, size = 31, normalized size = 1.07 \[ -\frac {\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,2{}\mathrm {i}}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}-\frac {1{}\mathrm {i}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + log(x)*1i)/x^2,x)

[Out]

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

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sympy [A]  time = 0.22, size = 29, normalized size = 1.00 \[ - \left (i \log {\left (x - e^{i a} \right )} - i \log {\left (x + e^{i a} \right )}\right ) e^{- i a} - \frac {i}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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