Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\cot (x)}{x},x\right )+\text{Ei}(-\log (x))+\log (x)-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x} \]
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Rubi [A] time = 0.0688721, 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 = {} \[ \int \frac{\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx &=-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x}+\int \frac{1+\cot (x)+\frac{1}{x \log (x)}}{x} \, dx\\ &=-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x}+\int \left (\frac{1+\cot (x)}{x}+\frac{1}{x^2 \log (x)}\right ) \, dx\\ &=-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x}+\int \frac{1+\cot (x)}{x} \, dx+\int \frac{1}{x^2 \log (x)} \, dx\\ &=-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x}+\int \left (\frac{1}{x}+\frac{\cot (x)}{x}\right ) \, dx+\operatorname{Subst}\left (\int \frac{e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=\text{Ei}(-\log (x))+\log (x)-\frac{\log \left (e^x \log (x) \sin (x)\right )}{x}+\int \frac{\cot (x)}{x} \, dx\\ \end{align*}
Mathematica [A] time = 2.26533, size = 0, normalized size = 0. \[ \int \frac{\log \left (e^x \log (x) \sin (x)\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.622, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ({{\rm e}^{x}}\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 \, x \log \left (x\right ) + 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 (e^{x} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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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