Optimal. Leaf size=20 \[ \frac {1}{2 x^2}+e^a \text {ArcTan}\left (e^a x^2\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5656, 464, 281,
209} \begin {gather*} e^a \text {ArcTan}\left (e^a x^2\right )+\frac {1}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 281
Rule 464
Rule 5656
Rubi steps
\begin {align*} \int \frac {\tanh (a+2 \log (x))}{x^3} \, dx &=\int \frac {\tanh (a+2 \log (x))}{x^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 40, normalized size = 2.00 \begin {gather*} \frac {1}{2 x^2}-\text {ArcTan}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right ) \cosh (a)-\text {ArcTan}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right ) \sinh (a) \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 = 0.62, size = 44, normalized size = 2.20
method | result | size |
risch | \(\frac {1}{2 x^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{2 a}+\textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (4 \,{\mathrm e}^{2 a}+5 \textit {\_R}^{2}\right ) x^{2}-\textit {\_R} \right )\right )}{2}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 19, normalized size = 0.95 \begin {gather*} -\arctan \left (\frac {e^{\left (-a\right )}}{x^{2}}\right ) e^{a} + \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 21, normalized size = 1.05 \begin {gather*} \frac {2 \, x^{2} \arctan \left (x^{2} e^{a}\right ) e^{a} + 1}{2 \, x^{2}} \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 {\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 = 16, normalized size = 0.80 \begin {gather*} \arctan \left (x^{2} e^{a}\right ) e^{a} + \frac {1}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.05, size = 24, normalized size = 1.20 \begin {gather*} \mathrm {atan}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}}+\frac {1}{2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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