Optimal. Leaf size=17 \[ \frac {\text {Li}_2\left (1-\frac {a}{x^2}\right )}{2 a} \]
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Rubi [A] time = 0.09, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1593, 2343, 2333, 2315} \[ \frac {\text {PolyLog}\left (2,1-\frac {a}{x^2}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 2315
Rule 2333
Rule 2343
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {a}{x^2}\right )}{a x-x^3} \, dx &=\int \frac {\log \left (\frac {a}{x^2}\right )}{x \left (a-x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log (a x)}{\left (a-\frac {1}{x}\right ) x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {\log (a x)}{-1+a x} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {\text {Li}_2\left (1-\frac {a}{x^2}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 21, normalized size = 1.24 \[ \frac {\text {Li}_2\left (-\frac {a-x^2}{x^2}\right )}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 14, normalized size = 0.82 \[ \frac {{\rm Li}_2\left (-\frac {a}{x^{2}} + 1\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\log \left (\frac {a}{x^{2}}\right )}{x^{3} - a x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 12, normalized size = 0.71 \[ \frac {\dilog \left (\frac {a}{x^{2}}\right )}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.61, size = 81, normalized size = 4.76 \[ -\frac {1}{2} \, {\left (\frac {\log \left (x^{2} - a\right )}{a} - \frac {2 \, \log \relax (x)}{a}\right )} \log \left (\frac {a}{x^{2}}\right ) - \frac {\log \left (x^{2} - a\right ) \log \relax (x) - \log \relax (x)^{2}}{a} + \frac {2 \, \log \relax (x) \log \left (-\frac {x^{2}}{a} + 1\right ) + {\rm Li}_2\left (\frac {x^{2}}{a}\right )}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.50, size = 11, normalized size = 0.65 \[ \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {a}{x^2}\right )}{2\,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.31, size = 78, normalized size = 4.59 \[ - \frac {\begin {cases} i \pi \log {\relax (x )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \left |{x}\right | < 1 \\- i \pi \log {\left (\frac {1}{x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- i \pi {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} + i \pi {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} + \frac {\operatorname {Li}_{2}\left (\frac {a}{x^{2}}\right )}{2} & \text {otherwise} \end {cases}}{a} - \frac {\log {\left (\frac {a}{x^{2}} \right )} \log {\left (\frac {a}{x^{2}} - 1 \right )}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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