Optimal. Leaf size=47 \[ \frac {x^4}{4}+\frac {e^{-2 a}}{1-e^{2 a} x^4}+e^{-2 a} \log \left (1-e^{2 a} x^4\right ) \]
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Rubi [A]
time = 0.05, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5657, 455, 45}
\begin {gather*} \frac {e^{-2 a}}{1-e^{2 a} x^4}+e^{-2 a} \log \left (1-e^{2 a} x^4\right )+\frac {x^4}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rule 5657
Rubi steps
\begin {align*} \int x^3 \coth ^2(a+2 \log (x)) \, dx &=\int x^3 \coth ^2(a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 86, normalized size = 1.83 \begin {gather*} \frac {x^4}{4}+\cosh (2 a) \log \left (\left (-1+x^4\right ) \cosh (a)+\left (1+x^4\right ) \sinh (a)\right )-\log \left (\left (-1+x^4\right ) \cosh (a)+\left (1+x^4\right ) \sinh (a)\right ) \sinh (2 a)+\frac {-\cosh (3 a)+\sinh (3 a)}{\left (-1+x^4\right ) \cosh (a)+\left (1+x^4\right ) \sinh (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 41, normalized size = 0.87
method | result | size |
risch | \(\frac {x^{4}}{4}-\frac {{\mathrm e}^{-2 a}}{-1+{\mathrm e}^{2 a} x^{4}}+{\mathrm e}^{-2 a} \ln \left (-1+{\mathrm e}^{2 a} x^{4}\right )\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 53, normalized size = 1.13 \begin {gather*} \frac {1}{4} \, x^{4} + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} + 1\right ) + e^{\left (-2 \, a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac {1}{x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 61, normalized size = 1.30 \begin {gather*} \frac {x^{8} e^{\left (4 \, a\right )} - x^{4} e^{\left (2 \, a\right )} + 4 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{4} e^{\left (2 \, a\right )} - 1\right ) - 4}{4 \, {\left (x^{4} e^{\left (4 \, a\right )} - e^{\left (2 \, a\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \coth ^{2}{\left (a + 2 \log {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 40, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, x^{4} - \frac {x^{4}}{x^{4} e^{\left (2 \, a\right )} - 1} + e^{\left (-2 \, a\right )} \log \left ({\left | x^{4} e^{\left (2 \, a\right )} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 40, normalized size = 0.85 \begin {gather*} \ln \left (x^4-{\mathrm {e}}^{-2\,a}\right )\,{\mathrm {e}}^{-2\,a}-\frac {{\mathrm {e}}^{-2\,a}}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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