Optimal. Leaf size=13 \[ \text{li}(x)+\log (\log (x))-\log (x+\log (x)) \]
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Rubi [A] time = 0.142422, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6742, 2353, 2298, 2302, 29, 6684} \[ \text{li}(x)+\log (\log (x))-\log (x+\log (x)) \]
Antiderivative was successfully verified.
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Rule 6742
Rule 2353
Rule 2298
Rule 2302
Rule 29
Rule 6684
Rubi steps
\begin{align*} \int \frac{1+x}{\log (x) (x+\log (x))} \, dx &=\int \left (\frac{1+x}{x \log (x)}+\frac{-1-x}{x (x+\log (x))}\right ) \, dx\\ &=\int \frac{1+x}{x \log (x)} \, dx+\int \frac{-1-x}{x (x+\log (x))} \, dx\\ &=-\log (x+\log (x))+\int \left (\frac{1}{\log (x)}+\frac{1}{x \log (x)}\right ) \, dx\\ &=-\log (x+\log (x))+\int \frac{1}{\log (x)} \, dx+\int \frac{1}{x \log (x)} \, dx\\ &=-\log (x+\log (x))+\text{li}(x)+\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log (x)\right )\\ &=\log (\log (x))-\log (x+\log (x))+\text{li}(x)\\ \end{align*}
Mathematica [A] time = 0.0385612, size = 13, normalized size = 1. \[ \text{li}(x)+\log (\log (x))-\log (x+\log (x)) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.013, size = 0, normalized size = 0. \begin{align*} \int{\frac{1+x}{\ln \left ( x \right ) \left ( x+\ln \left ( x \right ) \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{x \log \left (x\right )}\,{d x} - \log \left (x + \log \left (x\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13983, size = 68, normalized size = 5.23 \begin{align*} -\log \left (x + \log \left (x\right )\right ) + \log \left (\log \left (x\right )\right ) + \logintegral \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.14364, size = 15, normalized size = 1.15 \begin{align*} - \log{\left (x + \log{\left (x \right )} \right )} + \log{\left (\log{\left (x \right )} \right )} + \operatorname{Ei}{\left (\log{\left (x \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{{\left (x + \log \left (x\right )\right )} \log \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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