Optimal. Leaf size=11 \[ -\log (x)+\log \left (2+x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {457, 78}
\begin {gather*} \log \left (x^2+2\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {-2+x^2}{x \left (2+x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {-2+x}{x (2+x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (-\frac {1}{x}+\frac {2}{2+x}\right ) \, dx,x,x^2\right )\\ &=-\log (x)+\log \left (2+x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 11, normalized size = 1.00 \begin {gather*} -\log (x)+\log \left (2+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 12, normalized size = 1.09
method | result | size |
default | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
norman | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
risch | \(-\ln \left (x \right )+\ln \left (x^{2}+2\right )\) | \(12\) |
meijerg | \(\ln \left (1+\frac {x^{2}}{2}\right )-\ln \left (x \right )+\frac {\ln \left (2\right )}{2}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 13, normalized size = 1.18 \begin {gather*} \log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 11, normalized size = 1.00 \begin {gather*} \log \left (x^{2} + 2\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.73 \begin {gather*} - \log {\left (x \right )} + \log {\left (x^{2} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.85, size = 13, normalized size = 1.18 \begin {gather*} \log \left (x^{2} + 2\right ) - \frac {1}{2} \, \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 11, normalized size = 1.00 \begin {gather*} \ln \left (x^2+2\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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