Optimal. Leaf size=22 \[ \frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31}
\begin {gather*} \frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \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^3+x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \left (a^3+x\right )} \, dx,x,x^3\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )}{3 a^3}-\frac {\text {Subst}\left (\int \frac {1}{a^3+x} \, dx,x,x^3\right )}{3 a^3}\\ &=\frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.00 \begin {gather*} \frac {\log (x)}{a^3}-\frac {\log \left (a^3+x^3\right )}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 34, normalized size = 1.55
method | result | size |
risch | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (a^{3}+x^{3}\right )}{3 a^{3}}\) | \(21\) |
default | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{3}}-\frac {\ln \left (a +x \right )}{3 a^{3}}\) | \(34\) |
norman | \(\frac {\ln \left (x \right )}{a^{3}}-\frac {\ln \left (a^{2}-a x +x^{2}\right )}{3 a^{3}}-\frac {\ln \left (a +x \right )}{3 a^{3}}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.02, size = 23, normalized size = 1.05 \begin {gather*} -\frac {\log \left (a^{3} + x^{3}\right )}{3 \, a^{3}} + \frac {\log \left (x^{3}\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\log \left (a^{3} + x^{3}\right ) - 3 \, \log \left (x\right )}{3 \, a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 19, normalized size = 0.86 \begin {gather*} \frac {\log {\left (x \right )}}{a^{3}} - \frac {\log {\left (a^{3} + x^{3} \right )}}{3 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 22, normalized size = 1.00 \begin {gather*} -\frac {\log \left ({\left | a^{3} + x^{3} \right |}\right )}{3 \, a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.25, size = 18, normalized size = 0.82 \begin {gather*} -\frac {\ln \left (a^3+x^3\right )-3\,\ln \left (x\right )}{3\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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