Optimal. Leaf size=14 \[ \log \left (1-\frac {2+\log ^2(3)}{x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.07, number of steps
used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6, 12, 629}
\begin {gather*} \log \left (-x+2+\log ^2(3)\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 629
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2-\log ^2(3)}{-x^2+x \left (2+\log ^2(3)\right )} \, dx\\ &=\left (-2-\log ^2(3)\right ) \int \frac {1}{-x^2+x \left (2+\log ^2(3)\right )} \, dx\\ &=-\log (x)+\log \left (2-x+\log ^2(3)\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 1.07 \begin {gather*} -\log (x)+\log \left (2-x+\log ^2(3)\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. \(41\) vs.
\(2(14)=28\).
time = 1.78, size = 42, normalized size = 3.00
method | result | size |
norman | \(-\ln \left (x \right )+\ln \left (\ln \left (3\right )^{2}-x +2\right )\) | \(16\) |
default | \(\left (-\ln \left (3\right )^{2}-2\right ) \left (-\frac {\ln \left (-\ln \left (3\right )^{2}+x -2\right )}{2+\ln \left (3\right )^{2}}+\frac {\ln \left (x \right )}{2+\ln \left (3\right )^{2}}\right )\) | \(42\) |
risch | \(\frac {\ln \left (-\ln \left (3\right )^{2}+x -2\right ) \ln \left (3\right )^{2}}{2+\ln \left (3\right )^{2}}+\frac {2 \ln \left (-\ln \left (3\right )^{2}+x -2\right )}{2+\ln \left (3\right )^{2}}-\frac {\ln \left (x \right ) \ln \left (3\right )^{2}}{2+\ln \left (3\right )^{2}}-\frac {2 \ln \left (x \right )}{2+\ln \left (3\right )^{2}}\) | \(73\) |
meijerg | \(\frac {\ln \left (3\right )^{2} \left (-\ln \left (3\right )^{2}-2\right ) \left (-\ln \left (1-\frac {x}{2+\ln \left (3\right )^{2}}\right )+\ln \left (x \right )-\ln \left (2+\ln \left (3\right )^{2}\right )+i \pi \right )}{\left (2+\ln \left (3\right )^{2}\right )^{2}}+\frac {2 \left (-\ln \left (3\right )^{2}-2\right ) \left (-\ln \left (1-\frac {x}{2+\ln \left (3\right )^{2}}\right )+\ln \left (x \right )-\ln \left (2+\ln \left (3\right )^{2}\right )+i \pi \right )}{\left (2+\ln \left (3\right )^{2}\right )^{2}}\) | \(105\) |
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. 39 vs.
\(2 (14) = 28\).
time = 0.27, size = 39, normalized size = 2.79 \begin {gather*} {\left (\log \left (3\right )^{2} + 2\right )} {\left (\frac {\log \left (-\log \left (3\right )^{2} + x - 2\right )}{\log \left (3\right )^{2} + 2} - \frac {\log \left (x\right )}{\log \left (3\right )^{2} + 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 15, normalized size = 1.07 \begin {gather*} \log \left (-\log \left (3\right )^{2} + x - 2\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 114 vs.
\(2 (10) = 20\).
time = 0.14, size = 114, normalized size = 8.14 \begin {gather*} \left (\frac {\log {\left (x - 1 - \frac {2 \log {\left (3 \right )}^{2}}{\log {\left (3 \right )}^{2} + 2} - \frac {2}{\log {\left (3 \right )}^{2} + 2} - \frac {\log {\left (3 \right )}^{2}}{2} - \frac {\log {\left (3 \right )}^{4}}{2 \left (\log {\left (3 \right )}^{2} + 2\right )} \right )}}{\log {\left (3 \right )}^{2} + 2} - \frac {\log {\left (x - 1 - \frac {\log {\left (3 \right )}^{2}}{2} + \frac {\log {\left (3 \right )}^{4}}{2 \left (\log {\left (3 \right )}^{2} + 2\right )} + \frac {2}{\log {\left (3 \right )}^{2} + 2} + \frac {2 \log {\left (3 \right )}^{2}}{\log {\left (3 \right )}^{2} + 2} \right )}}{\log {\left (3 \right )}^{2} + 2}\right ) \left (\log {\left (3 \right )}^{2} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (14) = 28\).
time = 0.42, size = 41, normalized size = 2.93 \begin {gather*} {\left (\log \left (3\right )^{2} + 2\right )} {\left (\frac {\log \left ({\left | -\log \left (3\right )^{2} + x - 2 \right |}\right )}{\log \left (3\right )^{2} + 2} - \frac {\log \left ({\left | x \right |}\right )}{\log \left (3\right )^{2} + 2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.30, size = 18, normalized size = 1.29 \begin {gather*} -2\,\mathrm {atanh}\left (\frac {4\,x}{2\,{\ln \left (3\right )}^2+4}-1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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