3.197 \(\int \cot ^2(a+i \log (x)) \, dx\)

Optimal. Leaf size=48 \[ -\frac{2 e^{2 i a} x}{-x^2+e^{2 i a}}+2 e^{i a} \tanh ^{-1}\left (e^{-i a} x\right )-x \]

[Out]

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

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Rubi [F]  time = 0.0105502, 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 = {} \[ \int \cot ^2(a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cot[a + I*Log[x]]^2,x]

[Out]

Defer[Int][Cot[a + I*Log[x]]^2, x]

Rubi steps

\begin{align*} \int \cot ^2(a+i \log (x)) \, dx &=\int \cot ^2(a+i \log (x)) \, dx\\ \end{align*}

Mathematica [A]  time = 0.0845853, size = 70, normalized size = 1.46 \[ \frac{-x \left (x^2-3\right ) \cos (a)+i x \left (x^2+3\right ) \sin (a)}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}+2 (\cos (a)+i \sin (a)) \tanh ^{-1}(x (\cos (a)-i \sin (a))) \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + I*Log[x]]^2,x]

[Out]

2*ArcTanh[x*(Cos[a] - I*Sin[a])]*(Cos[a] + I*Sin[a]) + (-(x*(-3 + x^2)*Cos[a]) + I*x*(3 + x^2)*Sin[a])/((-1 +
x^2)*Cos[a] - I*(1 + x^2)*Sin[a])

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Maple [A]  time = 0.073, size = 56, normalized size = 1.2 \begin{align*} -3\,x-2\,{\frac{x}{ \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1}}-{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}-x \right ) +{{\rm e}^{ia}}\ln \left ({{\rm e}^{ia}}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+I*ln(x))^2,x)

[Out]

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

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Maxima [B]  time = 1.2429, size = 375, normalized size = 7.81 \begin{align*} -\frac{2 \,{\left ({\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{2} + 2 \, x^{3} - x{\left (6 \, \cos \left (2 \, a\right ) + 6 i \, \sin \left (2 \, a\right )\right )} +{\left (2 \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (2 \,{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (2 \, \cos \left (a\right ) + 2 i \, \sin \left (a\right )\right )} \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) -{\left (x^{2}{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} -{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) +{\left (-i \, \cos \left (a\right ) + \sin \left (a\right )\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 (x^{2}{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} -{\left (\cos \left (a\right ) + i \, \sin \left (a\right )\right )} \cos \left (2 \, a\right ) -{\left (i \, \cos \left (a\right ) - \sin \left (a\right )\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 )}{2 \, x^{2} - 2 \, \cos \left (2 \, a\right ) - 2 i \, \sin \left (2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 3}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) - 2 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.480368, size = 42, normalized size = 0.88 \begin{align*} - x + \frac{2 x e^{2 i a}}{x^{2} - e^{2 i a}} - \left (\log{\left (x - e^{i a} \right )} - \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))**2,x)

[Out]

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

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Giac [B]  time = 1.33802, size = 107, normalized size = 2.23 \begin{align*} -\frac{x^{3}}{x^{2} - e^{\left (2 i \, a\right )}} - 2 \,{\left (\frac{\arctan \left (\frac{x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right )}{\sqrt{-e^{\left (2 i \, a\right )}}} - \frac{x}{x^{2} - e^{\left (2 i \, a\right )}}\right )} e^{\left (2 i \, a\right )} + \frac{5 \, x e^{\left (2 i \, a\right )}}{x^{2} - e^{\left (2 i \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+I*log(x))^2,x, algorithm="giac")

[Out]

-x^3/(x^2 - e^(2*I*a)) - 2*(arctan(x/sqrt(-e^(2*I*a)))/sqrt(-e^(2*I*a)) - x/(x^2 - e^(2*I*a)))*e^(2*I*a) + 5*x
*e^(2*I*a)/(x^2 - e^(2*I*a))