Optimal. Leaf size=59 \[ -\frac {1}{2 x^2 \left (1+e^{2 a} x^4\right )}-\frac {3 e^{2 a} x^2}{2 \left (1+e^{2 a} x^4\right )}-e^a \text {ArcTan}\left (e^a x^2\right ) \]
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Rubi [A]
time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5656, 473, 468,
281, 209} \begin {gather*} -e^a \text {ArcTan}\left (e^a x^2\right )-\frac {3 e^{2 a} x^2}{2 \left (e^{2 a} x^4+1\right )}-\frac {1}{2 x^2 \left (e^{2 a} x^4+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 281
Rule 468
Rule 473
Rule 5656
Rubi steps
\begin {align*} \int \frac {\tanh ^2(a+2 \log (x))}{x^3} \, dx &=\int \frac {\tanh ^2(a+2 \log (x))}{x^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 40, normalized size = 0.68 \begin {gather*} \frac {-1-\frac {2}{1+e^{-2 (a+2 \log (x))}}}{2 x^2}+e^a \text {ArcTan}\left (\frac {e^{-a}}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.50, size = 66, normalized size = 1.12
method | result | size |
risch | \(\frac {-\frac {3 \,{\mathrm e}^{2 a} x^{4}}{2}-\frac {1}{2}}{x^{2} \left (1+{\mathrm e}^{2 a} x^{4}\right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{2 a}+\textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \,{\mathrm e}^{2 a}-5 \textit {\_R}^{2}\right ) x^{2}-\textit {\_R} \right )\right )}{2}\) | \(66\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 37, normalized size = 0.63 \begin {gather*} \arctan \left (\frac {e^{\left (-a\right )}}{x^{2}}\right ) e^{a} - \frac {1}{2 \, x^{2}} - \frac {e^{\left (2 \, a\right )}}{x^{2} {\left (\frac {1}{x^{4}} + e^{\left (2 \, a\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 51, normalized size = 0.86 \begin {gather*} -\frac {3 \, x^{4} e^{\left (2 \, a\right )} + 2 \, {\left (x^{6} e^{\left (3 \, a\right )} + x^{2} e^{a}\right )} \arctan \left (x^{2} e^{a}\right ) + 1}{2 \, {\left (x^{6} e^{\left (2 \, a\right )} + x^{2}\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 \frac {\tanh ^{2}{\left (a + 2 \log {\left (x \right )} \right )}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 39, normalized size = 0.66 \begin {gather*} -\arctan \left (x^{2} e^{a}\right ) e^{a} - \frac {3 \, x^{4} e^{\left (2 \, a\right )} + 1}{2 \, {\left (x^{6} e^{\left (2 \, a\right )} + x^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.06, size = 47, normalized size = 0.80 \begin {gather*} -\mathrm {atan}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}}-\frac {\frac {3\,{\mathrm {e}}^{2\,a}\,x^4}{2}+\frac {1}{2}}{{\mathrm {e}}^{2\,a}\,x^6+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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