Optimal. Leaf size=24 \[ \frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4} \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 36, 31, 29}
\begin {gather*} \frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x \left (a^4-x^4\right )} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {1}{\left (a^4-x\right ) x} \, dx,x,x^4\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{a^4-x} \, dx,x,x^4\right )}{4 a^4}+\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^4\right )}{4 a^4}\\ &=\frac {\log (x)}{a^4}-\frac {\log \left (a^4-x^4\right )}{4 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{a^4}-\frac {\log \left (-a^4+x^4\right )}{4 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 41, normalized size = 1.71
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a^{4}}-\frac {\ln \left (-a^{4}+x^{4}\right )}{4 a^{4}}\) | \(23\) |
default | \(\frac {\ln \left (x \right )}{a^{4}}-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{4}}-\frac {\ln \left (a +x \right )}{4 a^{4}}-\frac {\ln \left (a -x \right )}{4 a^{4}}\) | \(41\) |
norman | \(\frac {\ln \left (x \right )}{a^{4}}-\frac {\ln \left (a^{2}+x^{2}\right )}{4 a^{4}}-\frac {\ln \left (a +x \right )}{4 a^{4}}-\frac {\ln \left (a -x \right )}{4 a^{4}}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.80, size = 25, normalized size = 1.04 \begin {gather*} -\frac {\log \left (-a^{4} + x^{4}\right )}{4 \, a^{4}} + \frac {\log \left (x^{4}\right )}{4 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 20, normalized size = 0.83 \begin {gather*} -\frac {\log \left (-a^{4} + x^{4}\right ) - 4 \, \log \left (x\right )}{4 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 19, normalized size = 0.79 \begin {gather*} \frac {\log {\left (x \right )}}{a^{4}} - \frac {\log {\left (- a^{4} + x^{4} \right )}}{4 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.94, size = 26, normalized size = 1.08 \begin {gather*} \frac {\log \left (x^{4}\right )}{4 \, a^{4}} - \frac {\log \left ({\left | -a^{4} + x^{4} \right |}\right )}{4 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 20, normalized size = 0.83 \begin {gather*} -\frac {\ln \left (x^4-a^4\right )-4\,\ln \left (x\right )}{4\,a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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