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

Optimal. Leaf size=43 \[ -2 i e^{2 i a} x+2 i e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i x^3}{3} \]

[Out]

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

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Rubi [F]  time = 0.0223548, 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 = {} \[ \int x^2 \cot (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Cot[a + I*Log[x]],x]

[Out]

Defer[Int][x^2*Cot[a + I*Log[x]], x]

Rubi steps

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

Mathematica [A]  time = 0.0181679, size = 66, normalized size = 1.53 \[ 2 x \sin (2 a)-2 i x \cos (2 a)+2 i \cos (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-2 \sin (3 a) \tanh ^{-1}(x \cos (a)-i x \sin (a))-\frac{i x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cot[a + I*Log[x]],x]

[Out]

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

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Maple [A]  time = 0.089, size = 62, normalized size = 1.4 \begin{align*}{\frac{i}{3}}{x}^{3}+i \left ( -{\frac{2\,{x}^{3}}{3}}-2\, \left ({{\rm e}^{ia}} \right ) ^{2}x+ \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}+x \right ) - \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}-x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.18713, size = 176, normalized size = 4.09 \begin{align*} -\frac{1}{3} i \, x^{3} + 2 \, x{\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - \frac{1}{6} \,{\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 ) - \frac{1}{6} \,{\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 ) + \frac{1}{2} \,{\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 ) + \frac{1}{2} \,{\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*I*x^3 + 2*x*(-I*cos(2*a) + sin(2*a)) - 1/6*(6*cos(3*a) + 6*I*sin(3*a))*arctan2(sin(a), x + cos(a)) - 1/6*
(6*cos(3*a) + 6*I*sin(3*a))*arctan2(sin(a), x - cos(a)) + 1/2*(I*cos(3*a) - sin(3*a))*log(x^2 + 2*x*cos(a) + c
os(a)^2 + sin(a)^2) + 1/2*(-I*cos(3*a) + sin(3*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^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{2}}{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^2*cot(a+I*log(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.497822, size = 46, normalized size = 1.07 \begin{align*} - \frac{i x^{3}}{3} - 2 i x e^{2 i a} - \left (i \log{\left (x - e^{i a} \right )} - i \log{\left (x + e^{i a} \right )}\right ) e^{3 i a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cot(a+I*ln(x)),x)

[Out]

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

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Giac [A]  time = 1.46495, size = 63, normalized size = 1.47 \begin{align*} -\frac{1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} + 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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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