3.192 \(\int \frac{\cot (a+i \log (x))}{x^3} \, dx\)

Optimal. Leaf size=36 \[ -i e^{-2 i a} \log \left (1-\frac{e^{2 i a}}{x^2}\right )-\frac{i}{2 x^2} \]

[Out]

(-I/2)/x^2 - (I*Log[1 - E^((2*I)*a)/x^2])/E^((2*I)*a)

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Rubi [F]  time = 0.0240748, 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 = {} \[ \int \frac{\cot (a+i \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]/x^3,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]/x^3, x]

Rubi steps

\begin{align*} \int \frac{\cot (a+i \log (x))}{x^3} \, dx &=\int \frac{\cot (a+i \log (x))}{x^3} \, dx\\ \end{align*}

Mathematica [B]  time = 0.0297716, size = 136, normalized size = 3.78 \[ -\frac{1}{2} i \cos (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\frac{1}{2} \sin (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\cos (2 a) \left (-\tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )\right )+i \sin (2 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )+2 \sin (2 a) \log (x)+2 i \cos (2 a) \log (x)-\frac{i}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]/x^3,x]

[Out]

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

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Maple [A]  time = 0.066, size = 59, normalized size = 1.6 \begin{align*}{\frac{-{\frac{i}{2}}}{{x}^{2}}}+i \left ( 2\,{\frac{\ln \left ( x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}-{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}-{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x))/x^3,x)

[Out]

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

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Maxima [B]  time = 1.38878, size = 188, normalized size = 5.22 \begin{align*} -\frac{x^{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 ) + x^{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 ) -{\left ({\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 ) -{\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 ) + 4 \,{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x\right )\right )} x^{2} + i}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^3,x, algorithm="maxima")

[Out]

-1/2*(x^2*(I*cos(2*a) + sin(2*a))*log(x^2 + 2*x*cos(a) + cos(a)^2 + sin(a)^2) + x^2*(I*cos(2*a) + sin(2*a))*lo
g(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2) - ((2*cos(2*a) - 2*I*sin(2*a))*arctan2(sin(a), x + cos(a)) - (2*cos(
2*a) - 2*I*sin(2*a))*arctan2(sin(a), x - cos(a)) + 4*(I*cos(2*a) + sin(2*a))*log(x))*x^2 + I)/x^2

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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}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^3,x, algorithm="fricas")

[Out]

integral((I*e^(2*I*a - 2*log(x)) + I)/(x^3*e^(2*I*a - 2*log(x)) - x^3), x)

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Sympy [A]  time = 0.697061, size = 39, normalized size = 1.08 \begin{align*} 2 i e^{- 2 i a} \log{\left (x \right )} - i e^{- 2 i a} \log{\left (x^{2} - e^{2 i a} \right )} - \frac{i}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*ln(x))/x**3,x)

[Out]

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

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Giac [B]  time = 1.3175, size = 66, normalized size = 1.83 \begin{align*} \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 ) + 2 i \, e^{\left (-2 i \, a\right )} \log \left (x\right ) - i \, e^{\left (-2 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) - \frac{i}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^3,x, algorithm="giac")

[Out]

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