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