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

Optimal. Leaf size=64 \[ -\frac{2 e^{2 i a} x^3}{-x^2+e^{2 i a}}-6 e^{2 i a} x+6 e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{x^3}{3} \]

[Out]

-6*E^((2*I)*a)*x - x^3/3 - (2*E^((2*I)*a)*x^3)/(E^((2*I)*a) - x^2) + 6*E^((3*I)*a)*ArcTanh[x/E^(I*a)]

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Rubi [F]  time = 0.0509217, 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 = {} \[ \int x^2 \cot ^2(a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^2*Cot[a + I*Log[x]]^2,x]

[Out]

Defer[Int][x^2*Cot[a + I*Log[x]]^2, x]

Rubi steps

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

Mathematica [A]  time = 0.126752, size = 100, normalized size = 1.56 \[ \frac{2 x (\cos (3 a)+i \sin (3 a))}{\left (x^2-1\right ) \cos (a)-i \left (x^2+1\right ) \sin (a)}-4 i x \sin (2 a)-4 x \cos (2 a)+6 \cos (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))+6 i \sin (3 a) \tanh ^{-1}(x (\cos (a)-i \sin (a)))-\frac{x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cot[a + I*Log[x]]^2,x]

[Out]

-x^3/3 - 4*x*Cos[2*a] + 6*ArcTanh[x*(Cos[a] - I*Sin[a])]*Cos[3*a] - (4*I)*x*Sin[2*a] + (2*x*(Cos[3*a] + I*Sin[
3*a]))/((-1 + x^2)*Cos[a] - I*(1 + x^2)*Sin[a]) + (6*I)*ArcTanh[x*(Cos[a] - I*Sin[a])]*Sin[3*a]

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Maple [A]  time = 0.086, size = 75, normalized size = 1.2 \begin{align*} -{\frac{7\,{x}^{3}}{3}}-2\,{\frac{{x}^{3}}{ \left ({{\rm e}^{i \left ( a+i\ln \left ( x \right ) \right ) }} \right ) ^{2}-1}}-6\, \left ({{\rm e}^{ia}} \right ) ^{2}x-3\, \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}-x \right ) +3\, \left ({{\rm e}^{ia}} \right ) ^{3}\ln \left ({{\rm e}^{ia}}+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7/3*x^3-2*x^3/(exp(I*(a+I*ln(x)))^2-1)-6*exp(I*a)^2*x-3*exp(I*a)^3*ln(exp(I*a)-x)+3*exp(I*a)^3*ln(exp(I*a)+x)

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Maxima [B]  time = 1.23497, size = 475, normalized size = 7.42 \begin{align*} -\frac{2 \, x^{5} + x^{3}{\left (22 \, \cos \left (2 \, a\right ) + 22 i \, \sin \left (2 \, a\right )\right )} + 18 \,{\left ({\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right )\right )} x^{2} - x{\left (36 \, \cos \left (4 \, a\right ) + 36 i \, \sin \left (4 \, a\right )\right )} +{\left (18 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) -{\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) +{\left (18 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) -{\left (18 \, \cos \left (2 \, a\right ) + 18 i \, \sin \left (2 \, a\right )\right )} \sin \left (3 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) -{\left (x^{2}{\left (9 \, \cos \left (3 \, a\right ) + 9 i \, \sin \left (3 \, a\right )\right )} -{\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) - 9 \,{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\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 (x^{2}{\left (9 \, \cos \left (3 \, a\right ) + 9 i \, \sin \left (3 \, a\right )\right )} -{\left (9 \, \cos \left (2 \, a\right ) + 9 i \, \sin \left (2 \, a\right )\right )} \cos \left (3 \, a\right ) + 9 \,{\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\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 )}{6 \, x^{2} - 6 \, \cos \left (2 \, a\right ) - 6 i \, \sin \left (2 \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*x^5 + x^3*(22*cos(2*a) + 22*I*sin(2*a)) + 18*((-I*cos(3*a) + sin(3*a))*arctan2(sin(a), x + cos(a)) + (-I*c
os(3*a) + sin(3*a))*arctan2(sin(a), x - cos(a)))*x^2 - x*(36*cos(4*a) + 36*I*sin(4*a)) + (18*(I*cos(2*a) - sin
(2*a))*cos(3*a) - (18*cos(2*a) + 18*I*sin(2*a))*sin(3*a))*arctan2(sin(a), x + cos(a)) + (18*(I*cos(2*a) - sin(
2*a))*cos(3*a) - (18*cos(2*a) + 18*I*sin(2*a))*sin(3*a))*arctan2(sin(a), x - cos(a)) - (x^2*(9*cos(3*a) + 9*I*
sin(3*a)) - (9*cos(2*a) + 9*I*sin(2*a))*cos(3*a) - 9*(I*cos(2*a) - sin(2*a))*sin(3*a))*log(x^2 + 2*x*cos(a) +
cos(a)^2 + sin(a)^2) + (x^2*(9*cos(3*a) + 9*I*sin(3*a)) - (9*cos(2*a) + 9*I*sin(2*a))*cos(3*a) + 9*(-I*cos(2*a
) + sin(2*a))*sin(3*a))*log(x^2 - 2*x*cos(a) + cos(a)^2 + sin(a)^2))/(6*x^2 - 6*cos(2*a) - 6*I*sin(2*a))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{2 \, x^{3} -{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 7 \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right )}{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(x^2*cot(a+I*log(x))^2,x, algorithm="fricas")

[Out]

-(2*x^3 - (e^(2*I*a - 2*log(x)) - 1)*integral(-(x^2*e^(2*I*a - 2*log(x)) - 7*x^2)/(e^(2*I*a - 2*log(x)) - 1),
x))/(e^(2*I*a - 2*log(x)) - 1)

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Sympy [A]  time = 0.949005, size = 60, normalized size = 0.94 \begin{align*} - \frac{x^{3}}{3} - 4 x e^{2 i a} + \frac{2 x e^{4 i a}}{x^{2} - e^{2 i a}} - 3 \left (\log{\left (x - e^{i a} \right )} - \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))**2,x)

[Out]

-x**3/3 - 4*x*exp(2*I*a) + 2*x*exp(4*I*a)/(x**2 - exp(2*I*a)) - 3*(log(x - exp(I*a)) - log(x + exp(I*a)))*exp(
3*I*a)

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Giac [A]  time = 1.31541, size = 112, normalized size = 1.75 \begin{align*} -\frac{x^{5}}{3 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac{11 \, x^{3} e^{\left (2 i \, a\right )}}{3 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} - \frac{6 \, \arctan \left (\frac{x}{\sqrt{-e^{\left (2 i \, a\right )}}}\right ) e^{\left (4 i \, a\right )}}{\sqrt{-e^{\left (2 i \, a\right )}}} + \frac{10 \, x e^{\left (4 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(x^2*cot(a+I*log(x))^2,x, algorithm="giac")

[Out]

-1/3*x^5/(x^2 - e^(2*I*a)) - 11/3*x^3*e^(2*I*a)/(x^2 - e^(2*I*a)) - 6*arctan(x/sqrt(-e^(2*I*a)))*e^(4*I*a)/sqr
t(-e^(2*I*a)) + 10*x*e^(4*I*a)/(x^2 - e^(2*I*a))