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

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Rubi [F]  time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.02, 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]

(-1/3*I)*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*C
os[a] - I*x*Sin[a]]*Sin[3*a]

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fricas [B]  time = 1.48, size = 82, normalized size = 1.91 \[ -\frac {1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} - \sqrt {-e^{\left (6 i \, a\right )}} \log \left (\frac {1}{2} \, {\left (2 \, x e^{\left (2 i \, a\right )} + 2 i \, \sqrt {-e^{\left (6 i \, a\right )}}\right )} e^{\left (-2 i \, a\right )}\right ) + \sqrt {-e^{\left (6 i \, a\right )}} \log \left (\frac {1}{2} \, {\left (2 \, x e^{\left (2 i \, a\right )} - 2 i \, \sqrt {-e^{\left (6 i \, a\right )}}\right )} e^{\left (-2 i \, a\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*I*x^3 - 2*I*x*e^(2*I*a) - sqrt(-e^(6*I*a))*log(1/2*(2*x*e^(2*I*a) + 2*I*sqrt(-e^(6*I*a)))*e^(-2*I*a)) + s
qrt(-e^(6*I*a))*log(1/2*(2*x*e^(2*I*a) - 2*I*sqrt(-e^(6*I*a)))*e^(-2*I*a))

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giac [A]  time = 1.07, size = 47, normalized size = 1.09 \[ -\frac {1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} + i \, e^{\left (3 i \, a\right )} \log \left (i \, x + i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (3 i \, a\right )} \log \left (-i \, x + i \, e^{\left (i \, a\right )}\right ) \]

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))

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maple [A]  time = 0.06, size = 33, normalized size = 0.77 \[ -\frac {i x^{3}}{3}-2 i {\mathrm e}^{2 i a} x +2 i \arctanh \left (x \,{\mathrm e}^{-i a}\right ) {\mathrm e}^{3 i a} \]

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^3-2*I*exp(2*I*a)*x+2*I*arctanh(x*exp(-I*a))*exp(3*I*a)

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maxima [B]  time = 0.38, size = 130, normalized size = 3.02 \[ -\frac {1}{3} i \, x^{3} + 2 \, x {\left (-i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} - \frac {1}{6} \, {\left (6 \, \cos \left (3 \, a\right ) + 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x + \cos \relax (a)\right ) - \frac {1}{6} \, {\left (6 \, \cos \left (3 \, a\right ) + 6 i \, \sin \left (3 \, a\right )\right )} \arctan \left (\sin \relax (a), x - \cos \relax (a)\right ) + \frac {1}{2} \, {\left (i \, \cos \left (3 \, a\right ) - \sin \left (3 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) + \frac {1}{2} \, {\left (-i \, \cos \left (3 \, a\right ) + \sin \left (3 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \relax (a) + \cos \relax (a)^{2} + \sin \relax (a)^{2}\right ) \]

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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mupad [B]  time = 2.20, size = 40, normalized size = 0.93 \[ -\mathrm {atan}\left (\frac {x}{\sqrt {-{\mathrm {e}}^{a\,2{}\mathrm {i}}}}\right )\,{\left (-{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )}^{3/2}\,2{}\mathrm {i}-\frac {x^3\,1{}\mathrm {i}}{3}-x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cot(a + log(x)*1i),x)

[Out]

- atan(x/(-exp(a*2i))^(1/2))*(-exp(a*2i))^(3/2)*2i - (x^3*1i)/3 - x*exp(a*2i)*2i

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sympy [A]  time = 0.20, size = 63, normalized size = 1.47 \[ - \frac {i x^{3}}{3} - 2 i x e^{2 i a} - \left (i \log {\left (x e^{2 i a} - e^{3 i a} \right )} - i \log {\left (x e^{2 i a} + e^{3 i a} \right )}\right ) e^{3 i a} \]

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

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