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]

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

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

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*}

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Mathematica [B]  time = 0.02, size = 118, normalized size = 3.37 \[ -\cos (2 a) \tan ^{-1}\left (\frac {\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-i \sin (2 a) \tan ^{-1}\left (\frac {\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-\frac {1}{2} i \cos (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\frac {1}{2} \sin (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\frac {i x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cot[a + I*Log[x]],x]

[Out]

(-1/2*I)*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]]*S
in[2*a])/2

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fricas [A]  time = 0.59, size = 23, normalized size = 0.66 \[ -\frac {1}{2} i \, x^{2} - i \, e^{\left (2 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 2.50, size = 41, normalized size = 1.17 \[ -\frac {1}{2} i \, x^{2} + \frac {1}{2} \, \pi e^{\left (2 i \, a\right )} - i \, e^{\left (2 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (2 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \]

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))

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maple [A]  time = 0.06, size = 28, normalized size = 0.80 \[ -\frac {i x^{2}}{2}-i {\mathrm e}^{2 i a} \ln \left ({\mathrm e}^{2 i a}-x^{2}\right ) \]

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

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maxima [B]  time = 0.34, size = 114, normalized size = 3.26 \[ -\frac {1}{2} i \, x^{2} + \frac {1}{2} \, {\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{2} \, {\left (2 \, \cos \left (2 \, a\right ) + 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) + \frac {1}{2} \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + \frac {1}{2} \, {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) \]

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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mupad [B]  time = 2.20, size = 27, normalized size = 0.77 \[ -\ln \left (x^2-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,1{}\mathrm {i}-\frac {x^2\,1{}\mathrm {i}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cot(a + log(x)*1i),x)

[Out]

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

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sympy [A]  time = 0.20, size = 27, normalized size = 0.77 \[ - \frac {i x^{2}}{2} - i e^{2 i a} \log {\left (x^{2} - e^{2 i a} \right )} \]

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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