3.186 \(\int x^3 \cot (a+i \log (x)) \, dx\)

Optimal. Leaf size=49 \[ -i e^{2 i a} x^2-i e^{4 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{i x^4}{4} \]

[Out]

(-I)*E^((2*I)*a)*x^2 - (I/4)*x^4 - I*E^((4*I)*a)*Log[E^((2*I)*a) - x^2]

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Rubi [F]  time = 0.0260149, 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 = {} \[ \int x^3 \cot (a+i \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^3*Cot[a + I*Log[x]],x]

[Out]

Defer[Int][x^3*Cot[a + I*Log[x]], x]

Rubi steps

\begin{align*} \int x^3 \cot (a+i \log (x)) \, dx &=\int x^3 \cot (a+i \log (x)) \, dx\\ \end{align*}

Mathematica [B]  time = 0.0337094, size = 137, normalized size = 2.8 \[ x^2 \sin (2 a)-i x^2 \cos (2 a)-\frac{1}{2} i \cos (4 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\frac{1}{2} \sin (4 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\cos (4 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-i \sin (4 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )-\frac{i x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Cot[a + I*Log[x]],x]

[Out]

(-I/4)*x^4 - I*x^2*Cos[2*a] - ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Cos[4*a] - (I/2)*Cos[4*a]*Log
[1 + x^4 - 2*x^2*Cos[2*a]] + x^2*Sin[2*a] - I*ArcTan[((-1 + x^2)*Cos[a])/(-Sin[a] - x^2*Sin[a])]*Sin[4*a] + (L
og[1 + x^4 - 2*x^2*Cos[2*a]]*Sin[4*a])/2

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Maple [A]  time = 0.079, size = 65, normalized size = 1.3 \begin{align*}{\frac{i}{4}}{x}^{4}+i \left ( -{\frac{{x}^{4}}{2}}-{x}^{2} \left ({{\rm e}^{ia}} \right ) ^{2}- \left ({{\rm e}^{ia}} \right ) ^{4}\ln \left ({{\rm e}^{ia}}-x \right ) - \left ({{\rm e}^{ia}} \right ) ^{4}\ln \left ({{\rm e}^{ia}}+x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cot(a+I*ln(x)),x)

[Out]

1/4*I*x^4+I*(-1/2*x^4-x^2*exp(I*a)^2-exp(I*a)^4*ln(exp(I*a)-x)-exp(I*a)^4*ln(exp(I*a)+x))

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Maxima [B]  time = 1.08604, size = 184, normalized size = 3.76 \begin{align*} -\frac{1}{4} i \, x^{4} - x^{2}{\left (i \, \cos \left (2 \, a\right ) - \sin \left (2 \, a\right )\right )} + \frac{1}{4} \,{\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) - \frac{1}{4} \,{\left (4 \, \cos \left (4 \, a\right ) + 4 i \, \sin \left (4 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) - \frac{1}{2} \,{\left (i \, \cos \left (4 \, a\right ) - \sin \left (4 \, 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 (4 \, a\right ) - \sin \left (4 \, 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^3*cot(a+I*log(x)),x, algorithm="maxima")

[Out]

-1/4*I*x^4 - x^2*(I*cos(2*a) - sin(2*a)) + 1/4*(4*cos(4*a) + 4*I*sin(4*a))*arctan2(sin(a), x + cos(a)) - 1/4*(
4*cos(4*a) + 4*I*sin(4*a))*arctan2(sin(a), x - cos(a)) - 1/2*(I*cos(4*a) - sin(4*a))*log(x^2 + 2*x*cos(a) + co
s(a)^2 + sin(a)^2) - 1/2*(I*cos(4*a) - sin(4*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^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{3}}{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^3*cot(a+I*log(x)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.606986, size = 39, normalized size = 0.8 \begin{align*} - \frac{i x^{4}}{4} - i x^{2} e^{2 i a} - i e^{4 i a} \log{\left (x^{2} - e^{2 i a} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cot(a+I*ln(x)),x)

[Out]

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

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Giac [A]  time = 1.55213, size = 68, normalized size = 1.39 \begin{align*} -\frac{1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} + \frac{1}{2} \, \pi e^{\left (4 i \, a\right )} - i \, e^{\left (4 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (4 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cot(a+I*log(x)),x, algorithm="giac")

[Out]

-1/4*I*x^4 - I*x^2*e^(2*I*a) + 1/2*pi*e^(4*I*a) - I*e^(4*I*a)*log(x + e^(I*a)) - I*e^(4*I*a)*log(-x + e^(I*a))