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

Optimal. Leaf size=35 \[ -i e^{2 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{i x^2}{2} \]

[Out]

(-I/2)*x^2 - I*E^((2*I)*a)*Log[E^((2*I)*a) - x^2]

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Rubi [F]  time = 0.014523, 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 = {} \[ \int x \cot (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Cot[a + I*Log[x]],x]

[Out]

Defer[Int][x*Cot[a + I*Log[x]], x]

Rubi steps

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

Mathematica [B]  time = 0.0233421, size = 118, normalized size = 3.37 \[ -\frac{1}{2} i \cos (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\frac{1}{2} \sin (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\cos (2 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-i \sin (2 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-\frac{i x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cot[a + I*Log[x]],x]

[Out]

(-I/2)*x^2 - ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Cos[2*a] - (I/2)*Cos[2*a]*Log[1 + x^4 - 2*x^2*
Cos[2*a]] - I*ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Sin[2*a] + (Log[1 + x^4 - 2*x^2*Cos[2*a]]*Sin
[2*a])/2

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Maple [A]  time = 0.069, size = 53, normalized size = 1.5 \begin{align*}{\frac{i}{2}}{x}^{2}+i \left ( -{x}^{2}- \left ({{\rm e}^{ia}} \right ) ^{2}\ln \left ({{\rm e}^{ia}}-x \right ) - \left ({{\rm e}^{ia}} \right ) ^{2}\ln \left ({{\rm e}^{ia}}+x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.09912, size = 154, normalized size = 4.4 \begin{align*} -\frac{1}{2} i \, x^{2} + \frac{1}{2} \,{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - \frac{1}{2} \,{\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + \frac{1}{2} \,{\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, 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 (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*I*x^2 + 1/2*(2*cos(2*a) + 2*I*sin(2*a))*arctan2(sin(a), x + cos(a)) - 1/2*(2*cos(2*a) + 2*I*sin(2*a))*arc
tan2(sin(a), x - cos(a)) + 1/2*(-I*cos(2*a) + sin(2*a))*log(x^2 + 2*x*cos(a) + cos(a)^2 + sin(a)^2) + 1/2*(-I*
cos(2*a) + sin(2*a))*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2)

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

[Out]

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

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Sympy [A]  time = 0.505025, size = 27, normalized size = 0.77 \begin{align*} - \frac{i x^{2}}{2} - i e^{2 i a} \log{\left (x^{2} - e^{2 i a} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.37946, size = 55, normalized size = 1.57 \begin{align*} -\frac{1}{2} i \, x^{2} + \frac{1}{2} \, \pi e^{\left (2 i \, a\right )} - i \, e^{\left (2 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (2 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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