3.193 \(\int \frac{\cot (a+i \log (x))}{x^4} \, dx\)

Optimal. Leaf size=45 \[ -\frac{2 i e^{-2 i a}}{x}+2 i e^{-3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i}{3 x^3} \]

[Out]

(-I/3)/x^3 - (2*I)/(E^((2*I)*a)*x) + ((2*I)*ArcTanh[x/E^(I*a)])/E^((3*I)*a)

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Rubi [F]  time = 0.025074, 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 = {} \[ \int \frac{\cot (a+i \log (x))}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]/x^4,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]/x^4, x]

Rubi steps

\begin{align*} \int \frac{\cot (a+i \log (x))}{x^4} \, dx &=\int \frac{\cot (a+i \log (x))}{x^4} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0215919, size = 70, normalized size = 1.56 \[ -\frac{2 \sin (2 a)}{x}-\frac{2 i \cos (2 a)}{x}+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))+2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]/x^4,x]

[Out]

(-I/3)/x^3 - ((2*I)*Cos[2*a])/x + (2*I)*ArcTanh[x*Cos[a] - I*x*Sin[a]]*Cos[3*a] - (2*Sin[2*a])/x + 2*ArcTanh[x
*Cos[a] - I*x*Sin[a]]*Sin[3*a]

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Maple [A]  time = 0.061, size = 59, normalized size = 1.3 \begin{align*}{\frac{-{\frac{i}{3}}}{{x}^{3}}}+i \left ( -2\,{\frac{1}{ \left ({{\rm e}^{ia}} \right ) ^{2}x}}+{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{3}}}-{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{3}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x))/x^4,x)

[Out]

-1/3*I/x^3+I*(-2/exp(I*a)^2/x+1/exp(I*a)^3*ln(exp(I*a)+x)-1/exp(I*a)^3*ln(exp(I*a)-x))

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Maxima [B]  time = 1.14161, size = 192, normalized size = 4.27 \begin{align*} -\frac{3 \, x^{3}{\left (-i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + 3 \, x^{3}{\left (i \, \cos \left (3 \, a\right ) + \sin \left (3 \, 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 (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (6 \, \cos \left (3 \, a\right ) - 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{3} + 12 \, x^{2}{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} + 2 i}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^4,x, algorithm="maxima")

[Out]

-1/6*(3*x^3*(-I*cos(3*a) - sin(3*a))*log(x^2 + 2*x*cos(a) + cos(a)^2 + sin(a)^2) + 3*x^3*(I*cos(3*a) + sin(3*a
))*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2) + ((6*cos(3*a) - 6*I*sin(3*a))*arctan2(sin(a), x + cos(a)) + (6
*cos(3*a) - 6*I*sin(3*a))*arctan2(sin(a), x - cos(a)))*x^3 + 12*x^2*(I*cos(2*a) + sin(2*a)) + 2*I)/x^3

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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^{4} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^4,x, algorithm="fricas")

[Out]

integral((I*e^(2*I*a - 2*log(x)) + I)/(x^4*e^(2*I*a - 2*log(x)) - x^4), x)

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Sympy [A]  time = 0.626059, size = 54, normalized size = 1.2 \begin{align*} - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + e^{i a} \right )}\right ) e^{- 3 i a} - \frac{\left (6 i x^{2} + i e^{2 i a}\right ) e^{- 2 i a}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*ln(x))/x**4,x)

[Out]

-(I*log(x - exp(I*a)) - I*log(x + exp(I*a)))*exp(-3*I*a) - (6*I*x**2 + I*exp(2*I*a))*exp(-2*I*a)/(3*x**3)

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Giac [A]  time = 1.37743, size = 66, normalized size = 1.47 \begin{align*} i \, e^{\left (-3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (-3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) - \frac{2 i \, e^{\left (-2 i \, a\right )}}{x} - \frac{i}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))/x^4,x, algorithm="giac")

[Out]

I*e^(-3*I*a)*log(I*x + I*e^(I*a)) - I*e^(-3*I*a)*log(-I*x + I*e^(I*a)) - 2*I*e^(-2*I*a)/x - 1/3*I/x^3