Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\cot (x)}{x},x\right )+\text{Ei}(-\log (x))-\frac{\log (\log (x) \sin (x))}{x} \]
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Rubi [A] time = 0.344563, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\log (\log (x) \sin (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\log (\log (x) \sin (x))}{x^2} \, dx &=-\frac{\log (\log (x) \sin (x))}{x}-\int \frac{-1-x \cot (x) \log (x)}{x^2 \log (x)} \, dx\\ &=-\frac{\log (\log (x) \sin (x))}{x}-\int \left (-\frac{\cot (x)}{x}-\frac{1}{x^2 \log (x)}\right ) \, dx\\ &=-\frac{\log (\log (x) \sin (x))}{x}+\int \frac{\cot (x)}{x} \, dx+\int \frac{1}{x^2 \log (x)} \, dx\\ &=-\frac{\log (\log (x) \sin (x))}{x}+\int \frac{\cot (x)}{x} \, dx+\operatorname{Subst}\left (\int \frac{e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\text{Ei}(-\log (x))-\frac{\log (\log (x) \sin (x))}{x}+\int \frac{\cot (x)}{x} \, dx\\ \end{align*}
Mathematica [A] time = 1.93027, size = 0, normalized size = 0. \[ \int \frac{\log (\log (x) \sin (x))}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( \ln \left ( x \right ) \sin \left ( x \right ) \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{x{\left ({\rm Ei}\left (-\log \left (x\right )\right ) + \overline{{\rm Ei}\left (-\log \left (x\right )\right )}\right )} - 2 \, x \int \frac{\sin \left (x\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right )} x}\,{d x} + 2 \, x \int \frac{\sin \left (x\right )}{{\left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right )} x}\,{d x} + 2 \, \log \left (2\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) - \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 \, \log \left (\log \left (x\right )\right )}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (\log \left (x\right ) \sin \left (x\right )\right )}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (\log{\left (x \right )} \sin{\left (x \right )} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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