Optimal. Leaf size=29 \[ \frac{\log (x)}{8}-\frac{1}{4} \sin (\log (x)) \cos ^3(\log (x))+\frac{1}{8} \sin (\log (x)) \cos (\log (x)) \]
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Rubi [A] time = 0.0563327, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2568, 2635, 8} \[ \frac{\log (x)}{8}-\frac{1}{4} \sin (\log (x)) \cos ^3(\log (x))+\frac{1}{8} \sin (\log (x)) \cos (\log (x)) \]
Antiderivative was successfully verified.
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Rule 2568
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^2(\log (x)) \sin ^2(\log (x))}{x} \, dx &=\operatorname{Subst}\left (\int \cos ^2(x) \sin ^2(x) \, dx,x,\log (x)\right )\\ &=-\frac{1}{4} \cos ^3(\log (x)) \sin (\log (x))+\frac{1}{4} \operatorname{Subst}\left (\int \cos ^2(x) \, dx,x,\log (x)\right )\\ &=\frac{1}{8} \cos (\log (x)) \sin (\log (x))-\frac{1}{4} \cos ^3(\log (x)) \sin (\log (x))+\frac{1}{8} \operatorname{Subst}(\int 1 \, dx,x,\log (x))\\ &=\frac{\log (x)}{8}+\frac{1}{8} \cos (\log (x)) \sin (\log (x))-\frac{1}{4} \cos ^3(\log (x)) \sin (\log (x))\\ \end{align*}
Mathematica [A] time = 0.0164266, size = 16, normalized size = 0.55 \[ \frac{\log (x)}{8}-\frac{1}{32} \sin (4 \log (x)) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 24, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( x \right ) }{8}}+{\frac{\cos \left ( \ln \left ( x \right ) \right ) \sin \left ( \ln \left ( x \right ) \right ) }{8}}-{\frac{ \left ( \cos \left ( \ln \left ( x \right ) \right ) \right ) ^{3}\sin \left ( \ln \left ( x \right ) \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.962125, size = 16, normalized size = 0.55 \begin{align*} \frac{1}{8} \, \log \left (x\right ) - \frac{1}{32} \, \sin \left (4 \, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.1343, size = 85, normalized size = 2.93 \begin{align*} -\frac{1}{8} \,{\left (2 \, \cos \left (\log \left (x\right )\right )^{3} - \cos \left (\log \left (x\right )\right )\right )} \sin \left (\log \left (x\right )\right ) + \frac{1}{8} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 66.3406, size = 476, normalized size = 16.41 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07292, size = 16, normalized size = 0.55 \begin{align*} \frac{1}{8} \, \log \left (x\right ) - \frac{1}{32} \, \sin \left (4 \, \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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