Optimal. Leaf size=12 \[ -\frac{\cos (\log (x))}{\sin (\log (x))+1} \]
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Rubi [A] time = 0.0289448, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {2648} \[ -\frac{\cos (\log (x))}{\sin (\log (x))+1} \]
Antiderivative was successfully verified.
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Rule 2648
Rubi steps
\begin{align*} \int \frac{1}{x (1+\sin (\log (x)))} \, dx &=\operatorname{Subst}\left (\int \frac{1}{1+\sin (x)} \, dx,x,\log (x)\right )\\ &=-\frac{\cos (\log (x))}{1+\sin (\log (x))}\\ \end{align*}
Mathematica [B] time = 0.0196309, size = 26, normalized size = 2.17 \[ \frac{2 \sin \left (\frac{\log (x)}{2}\right )}{\sin \left (\frac{\log (x)}{2}\right )+\cos \left (\frac{\log (x)}{2}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 12, normalized size = 1. \begin{align*} -2\, \left ( 1+\tan \left ( 1/2\,\ln \left ( x \right ) \right ) \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09909, size = 23, normalized size = 1.92 \begin{align*} -\frac{2}{\frac{\sin \left (\log \left (x\right )\right )}{\cos \left (\log \left (x\right )\right ) + 1} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02112, size = 89, normalized size = 7.42 \begin{align*} -\frac{\cos \left (\log \left (x\right )\right ) - \sin \left (\log \left (x\right )\right ) + 1}{\cos \left (\log \left (x\right )\right ) + \sin \left (\log \left (x\right )\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.35437, size = 15, normalized size = 1.25 \begin{align*} \frac{2 \tan{\left (\frac{\log{\left (x \right )}}{2} \right )}}{\tan{\left (\frac{\log{\left (x \right )}}{2} \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11301, size = 15, normalized size = 1.25 \begin{align*} -\frac{2}{\tan \left (\frac{1}{2} \, \log \left (x\right )\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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