Optimal. Leaf size=15 \[ \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]
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Rubi [A] time = 0.01, 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 = {275, 206} \[ \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 206
Rule 275
Rubi steps
\begin {align*} \int \frac {x}{a^4-x^4} \, dx &=\frac {1}{2} \operatorname {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 \[ \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {x^2}{a^2}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 26, normalized size = 1.73 \[ \frac {\log \left (a^{2} + x^{2}\right ) - \log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.81, size = 30, normalized size = 2.00 \[ \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac {\log \left ({\left | -a^{2} + x^{2} \right |}\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.34, 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] time = 0.42, size = 29, normalized size = 1.93 \[ \frac {\log \left (a^{2} + x^{2}\right )}{4 \, a^{2}} - \frac {\log \left (-a^{2} + x^{2}\right )}{4 \, a^{2}} \]
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 \[ \frac {\mathrm {atanh}\left (\frac {x^2}{a^2}\right )}{2\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 24, normalized size = 1.60 \[ - \frac {\frac {\log {\left (- a^{2} + x^{2} \right )}}{4} - \frac {\log {\left (a^{2} + x^{2} \right )}}{4}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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