Optimal. Leaf size=12 \[ \frac {1}{2} \text {Li}_2\left (-\frac {a}{x^2}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2511, 2438}
\begin {gather*} \frac {1}{2} \text {PolyLog}\left (2,-\frac {a}{x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2511
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {a+x^2}{x^2}\right )}{x} \, dx &=\int \frac {\log \left (1+\frac {a}{x^2}\right )}{x} \, dx\\ &=\frac {1}{2} \text {Li}_2\left (-\frac {a}{x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} \frac {1}{2} \text {Li}_2\left (-\frac {a}{x^2}\right ) \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. \(85\) vs.
\(2(10)=20\).
time = 0.09, size = 86, normalized size = 7.17
method | result | size |
risch | \(-\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {a}{x^{2}}\right )+\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {\sqrt {-a}}{x}\right )+\ln \left (\frac {1}{x}\right ) \ln \left (1-\frac {\sqrt {-a}}{x}\right )+\dilog \left (1+\frac {\sqrt {-a}}{x}\right )+\dilog \left (1-\frac {\sqrt {-a}}{x}\right )\) | \(76\) |
derivativedivides | \(-\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {a}{x^{2}}\right )+2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-a}}{x}\right )+\ln \left (1-\frac {\sqrt {-a}}{x}\right )\right )}{2 a}+\frac {\dilog \left (1+\frac {\sqrt {-a}}{x}\right )+\dilog \left (1-\frac {\sqrt {-a}}{x}\right )}{2 a}\right )\) | \(86\) |
default | \(-\ln \left (\frac {1}{x}\right ) \ln \left (1+\frac {a}{x^{2}}\right )+2 a \left (\frac {\ln \left (\frac {1}{x}\right ) \left (\ln \left (1+\frac {\sqrt {-a}}{x}\right )+\ln \left (1-\frac {\sqrt {-a}}{x}\right )\right )}{2 a}+\frac {\dilog \left (1+\frac {\sqrt {-a}}{x}\right )+\dilog \left (1-\frac {\sqrt {-a}}{x}\right )}{2 a}\right )\) | \(86\) |
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. 69 vs.
\(2 (9) = 18\).
time = 0.37, size = 69, normalized size = 5.75 \begin {gather*} -{\left (\log \left (x^{2} + a\right ) - 2 \, \log \left (x\right )\right )} \log \left (x\right ) + \log \left (x^{2} + a\right ) \log \left (x\right ) - \log \left (x\right )^{2} - \log \left (x\right ) \log \left (\frac {x^{2}}{a} + 1\right ) + \log \left (x\right ) \log \left (\frac {x^{2} + a}{x^{2}}\right ) - \frac {1}{2} \, {\rm Li}_2\left (-\frac {x^{2}}{a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 15, normalized size = 1.25 \begin {gather*} \frac {1}{2} \, {\rm Li}_2\left (-\frac {x^{2} + a}{x^{2}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\log {\left (\frac {a}{x^{2}} + 1 \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 10, normalized size = 0.83 \begin {gather*} \frac {\mathrm {polylog}\left (2,-\frac {a}{x^2}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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