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)*E^(I*a)*ArcTanh[x/E^(I*a)]

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Rubi [F]  time = 0.0072593, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.0083278, 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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Maple [A]  time = 0.071, size = 44, normalized size = 1.6 \begin{align*} ix+i \left ( -2\,x+{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}+x \right ) -{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}-x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.0568, size = 132, normalized size = 4.89 \begin{align*} -\frac{1}{2} \,{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - \frac{1}{2} \,{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) - \frac{1}{2} \,{\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) - \frac{1}{2} \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) - i \, x \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((I*e^(2*I*a - 2*log(x)) + I)/(e^(2*I*a - 2*log(x)) - 1), x)

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Sympy [A]  time = 0.494968, size = 29, normalized size = 1.07 \begin{align*} - i x - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + e^{i a} \right )}\right ) e^{i a} \end{align*}

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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Giac [B]  time = 1.34179, size = 51, normalized size = 1.89 \begin{align*} 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 \end{align*}

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