Optimal. Leaf size=17 \[ \frac{\log (x)}{2}-\frac{1}{2} \sin (\log (x)) \cos (\log (x)) \]
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Rubi [A] time = 0.0199459, antiderivative size = 17, normalized size of antiderivative = 1., 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 2635
Rule 8
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*}
Mathematica [A] time = 0.013997, 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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Maple [A] time = 0.02, size = 14, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{2}}-{\frac{\cos \left ( \ln \left ( x \right ) \right ) \sin \left ( \ln \left ( x \right ) \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12501, size = 16, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \log \left (x\right ) - \frac{1}{4} \, \sin \left (2 \, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23494, size = 58, normalized size = 3.41 \begin{align*} -\frac{1}{2} \, \cos \left (\log \left (x\right )\right ) \sin \left (\log \left (x\right )\right ) + \frac{1}{2} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.99746, size = 156, normalized size = 9.18 \begin{align*} \frac{\log{\left (x \right )} \tan ^{4}{\left (\frac{\log{\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 2} + \frac{2 \log{\left (x \right )} \tan ^{2}{\left (\frac{\log{\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 2} + \frac{\log{\left (x \right )}}{2 \tan ^{4}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 2} + \frac{2 \tan ^{3}{\left (\frac{\log{\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 2} - \frac{2 \tan{\left (\frac{\log{\left (x \right )}}{2} \right )}}{2 \tan ^{4}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 4 \tan ^{2}{\left (\frac{\log{\left (x \right )}}{2} \right )} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29689, size = 16, normalized size = 0.94 \begin{align*} \frac{1}{2} \, \log \left (x\right ) - \frac{1}{4} \, \sin \left (2 \, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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