Optimal. Leaf size=17 \[ \frac {\log (x)}{2}-\frac {1}{2} \sin (\log (x)) \cos (\log (x)) \]
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Rubi [A] time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2635, 8} \[ \frac {\log (x)}{2}-\frac {1}{2} \sin (\log (x)) \cos (\log (x)) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \frac {\sin ^2(\log (x))}{x} \, dx &=\operatorname {Subst}\left (\int \sin ^2(x) \, dx,x,\log (x)\right )\\ &=-\frac {1}{2} \cos (\log (x)) \sin (\log (x))+\frac {1}{2} \operatorname {Subst}(\int 1 \, dx,x,\log (x))\\ &=\frac {\log (x)}{2}-\frac {1}{2} \cos (\log (x)) \sin (\log (x))\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.94 \[ \frac {\log (x)}{2}-\frac {1}{4} \sin (2 \log (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 13, normalized size = 0.76 \[ -\frac {1}{2} \, \cos \left (\log \relax (x)\right ) \sin \left (\log \relax (x)\right ) + \frac {1}{2} \, \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 12, normalized size = 0.71 \[ \frac {1}{2} \, \log \relax (x) - \frac {1}{4} \, \sin \left (2 \, \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.31, size = 14, normalized size = 0.82 \[ -\frac {\cos \left (\ln \relax (x )\right ) \sin \left (\ln \relax (x )\right )}{2}+\frac {\ln \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 12, normalized size = 0.71 \[ \frac {1}{2} \, \log \relax (x) - \frac {1}{4} \, \sin \left (2 \, \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 12, normalized size = 0.71 \[ \frac {\ln \relax (x)}{2}-\frac {\sin \left (2\,\ln \relax (x)\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 3.37, size = 156, normalized size = 9.18 \[ \frac {\log {\relax (x )} \tan ^{4}{\left (\frac {\log {\relax (x )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\relax (x )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\relax (x )}}{2} \right )} + 2} + \frac {2 \log {\relax (x )} \tan ^{2}{\left (\frac {\log {\relax (x )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\relax (x )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\relax (x )}}{2} \right )} + 2} + \frac {\log {\relax (x )}}{2 \tan ^{4}{\left (\frac {\log {\relax (x )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\relax (x )}}{2} \right )} + 2} + \frac {2 \tan ^{3}{\left (\frac {\log {\relax (x )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\relax (x )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\relax (x )}}{2} \right )} + 2} - \frac {2 \tan {\left (\frac {\log {\relax (x )}}{2} \right )}}{2 \tan ^{4}{\left (\frac {\log {\relax (x )}}{2} \right )} + 4 \tan ^{2}{\left (\frac {\log {\relax (x )}}{2} \right )} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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