Optimal. Leaf size=16 \[ \frac {1}{2} \log \left (\frac {1-x^2}{x}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1593, 446, 72} \begin {gather*} \frac {1}{2} \log \left (1-x^2\right )-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 72
Rule 446
Rule 1593
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1+x^2}{x \left (-2+2 x^2\right )} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x}{x (-2+2 x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{-1+x}-\frac {1}{2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 1.19 \begin {gather*} -\frac {\log (x)}{2}+\frac {1}{2} \log \left (1-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 13, normalized size = 0.81 \begin {gather*} \frac {1}{2} \, \log \left (x^{2} - 1\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 16, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \, \log \left (x^{2}\right ) + \frac {1}{2} \, \log \left ({\left | x^{2} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 14, normalized size = 0.88
method | result | size |
risch | \(-\frac {\ln \relax (x )}{2}+\frac {\ln \left (x^{2}-1\right )}{2}\) | \(14\) |
default | \(-\frac {\ln \relax (x )}{2}+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(18\) |
norman | \(-\frac {\ln \relax (x )}{2}+\frac {\ln \left (x -1\right )}{2}+\frac {\ln \left (x +1\right )}{2}\) | \(18\) |
meijerg | \(-\frac {\ln \relax (x )}{2}-\frac {i \pi }{4}+\frac {\ln \left (-x^{2}+1\right )}{2}\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 17, normalized size = 1.06 \begin {gather*} \frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (x - 1\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 13, normalized size = 0.81 \begin {gather*} \frac {\ln \left (x^2-1\right )}{2}-\frac {\ln \relax (x)}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} - \frac {\log {\relax (x )}}{2} + \frac {\log {\left (x^{2} - 1 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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