Optimal. Leaf size=41 \[ \frac {x^2}{2}+\frac {x^2}{1-e^{2 a} x^4}-e^{-a} \tanh ^{-1}\left (e^a x^2\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5657, 474, 470,
281, 212} \begin {gather*} -e^{-a} \tanh ^{-1}\left (e^a x^2\right )+\frac {x^2}{1-e^{2 a} x^4}+\frac {x^2}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rule 470
Rule 474
Rule 5657
Rubi steps
\begin {align*} \int x \coth ^2(a+2 \log (x)) \, dx &=\int x \coth ^2(a+2 \log (x)) \, dx\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 2.14, size = 163, normalized size = 3.98 \begin {gather*} \frac {e^{-4 a} \left (-375-713 e^{2 a} x^4-181 e^{4 a} x^8+61 e^{6 a} x^{12}+\frac {3 \left (125+196 e^{2 a} x^4-14 e^{4 a} x^8-52 e^{6 a} x^{12}+e^{8 a} x^{16}\right ) \tanh ^{-1}\left (\sqrt {e^{2 a} x^4}\right )}{\sqrt {e^{2 a} x^4}}\right )}{96 x^6}+\frac {2}{105} e^{2 a} x^6 \left (1+e^{2 a} x^4\right )^2 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};e^{2 a} x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 54, normalized size = 1.32
method | result | size |
risch | \(\frac {x^{2}}{2}-\frac {x^{2}}{-1+{\mathrm e}^{2 a} x^{4}}-\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}+1\right )}{2}+\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}-1\right )}{2}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 53, normalized size = 1.29 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 74 vs.
\(2 (36) = 72\).
time = 0.35, size = 74, normalized size = 1.80 \begin {gather*} \frac {x^{6} e^{\left (3 \, a\right )} - 3 \, x^{2} e^{a} - {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} + 1\right ) + {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} - 1\right )}{2 \, {\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\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 \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 = 0.42, size = 54, normalized size = 1.32 \begin {gather*} \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.23, size = 42, normalized size = 1.02 \begin {gather*} \frac {x^2}{2}-\frac {x^2}{x^4\,{\mathrm {e}}^{2\,a}-1}-\frac {\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )}{\sqrt {{\mathrm {e}}^{2\,a}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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