Optimal. Leaf size=15 \[ \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {281, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 281
Rubi steps
\begin {align*} \int \frac {x}{a^4-x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{a^4-x^2} \, dx,x,x^2\right )\\ &=\frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(29\) vs.
\(2(13)=26\).
time = 0.13, size = 30, normalized size = 2.00
method | result | size |
default | \(\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{2}}-\frac {\ln \left (a^{2}-x^{2}\right )}{4 a^{2}}\) | \(30\) |
risch | \(-\frac {\ln \left (-a^{2}+x^{2}\right )}{4 a^{2}}+\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{2}}\) | \(30\) |
norman | \(-\frac {\ln \left (a -x \right )}{4 a^{2}}-\frac {\ln \left (a +x \right )}{4 a^{2}}+\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 29 vs.
\(2 (13) = 26\).
time = 2.98, size = 29, normalized size = 1.93 \begin {gather*} \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac {\log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 26, normalized size = 1.73 \begin {gather*} \frac {\log \left (a^{2} + x^{2}\right ) - \log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 24, normalized size = 1.60 \begin {gather*} - \frac {\frac {\log {\left (- a^{2} + x^{2} \right )}}{4} - \frac {\log {\left (a^{2} + x^{2} \right )}}{4}}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs.
\(2 (13) = 26\).
time = 1.14, size = 30, normalized size = 2.00 \begin {gather*} \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac {\log \left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 13, normalized size = 0.87 \begin {gather*} \frac {\mathrm {atanh}\left (\frac {x^2}{a^2}\right )}{2\,a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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