3.189 \(\int \cot (a+i \log (x)) \, dx\)

Optimal. Leaf size=27 \[ 2 i e^{i a} \tanh ^{-1}\left (e^{-i a} x\right )-i x \]

[Out]

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

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Rubi [F]  time = 0.01, 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 = {} \[ \int \cot (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]],x]

[Out]

Defer[Int][Cot[a + I*Log[x]], x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 42, normalized size = 1.56 \[ 2 i \cos (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-i x \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]],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 [B]  time = 0.68, size = 49, normalized size = 1.81 \[ -\sqrt {-e^{\left (2 i \, a\right )}} \log \left (x + i \, \sqrt {-e^{\left (2 i \, a\right )}}\right ) + \sqrt {-e^{\left (2 i \, a\right )}} \log \left (x - i \, \sqrt {-e^{\left (2 i \, a\right )}}\right ) - i \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.44, size = 38, normalized size = 1.41 \[ i \, e^{\left (i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x)),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 = 0.05, size = 22, normalized size = 0.81 \[ -i x +2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x)),x)

[Out]

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

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maxima [B]  time = 0.36, size = 98, normalized size = 3.63 \[ -\frac {1}{2} \, {\left (2 \, \cos \relax (a) + 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{2} \, {\left (2 \, \cos \relax (a) + 2 i \, \sin \relax (a)\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) - \frac {1}{2} \, {\left (-i \, \cos \relax (a) + \sin \relax (a)\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - \frac {1}{2} \, {\left (i \, \cos \relax (a) - \sin \relax (a)\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) - i \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.18, size = 29, normalized size = 1.07 \[ -x\,1{}\mathrm {i}+\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a + log(x)*1i),x)

[Out]

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

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sympy [A]  time = 0.18, size = 29, normalized size = 1.07 \[ - i x - \left (i \log {\left (x - e^{i a} \right )} - i \log {\left (x + e^{i a} \right )}\right ) e^{i a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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