Optimal. Leaf size=30 \[ \frac{1}{2} e^{-2 a} \log \left (1-e^{2 a} x^4\right )+\frac{x^4}{4} \]
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Rubi [F] time = 0.0285749, 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 = {} \[ \int x^3 \coth (a+2 \log (x)) \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int x^3 \coth (a+2 \log (x)) \, dx &=\int x^3 \coth (a+2 \log (x)) \, dx\\ \end{align*}
Mathematica [B] time = 0.0241018, size = 64, normalized size = 2.13 \[ \frac{1}{2} \cosh (2 a) \log \left (x^4 \sinh (a)+x^4 \cosh (a)+\sinh (a)-\cosh (a)\right )-\frac{1}{2} \sinh (2 a) \log \left (x^4 \sinh (a)+x^4 \cosh (a)+\sinh (a)-\cosh (a)\right )+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 24, normalized size = 0.8 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{{{\rm e}^{-2\,a}}\ln \left ({{\rm e}^{2\,a}}{x}^{4}-1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21452, size = 49, normalized size = 1.63 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.53483, size = 72, normalized size = 2.4 \begin{align*} \frac{1}{4} \,{\left (x^{4} e^{\left (2 \, a\right )} + 2 \, \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right )\right )} e^{\left (-2 \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \coth{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14286, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, x^{4} + \frac{1}{2} \, e^{\left (-2 \, a\right )} \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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