Optimal. Leaf size=13 \[ \text {li}(x)+\log (\log (x))-\log (x+\log (x)) \]
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Rubi [A] time = 0.14, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 29
Rule 2298
Rule 2302
Rule 2353
Rule 6684
Rule 6742
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*}
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Mathematica [A] time = 0.04, size = 13, normalized size = 1.00 \[ \text {li}(x)+\log (\log (x))-\log (x+\log (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 13, normalized size = 1.00 \[ -\log \left (x + \log \relax (x)\right ) + \log \left (\log \relax (x)\right ) + \operatorname {log\_integral}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{{\left (x + \log \relax (x)\right )} \log \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {x +1}{\left (x +\ln \relax (x )\right ) \ln \relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{x \log \relax (x)}\,{d x} - \log \left (x + \log \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 13, normalized size = 1.00 \[ \ln \left (\ln \relax (x)\right )-\ln \left (x+\ln \relax (x)\right )+\mathrm {logint}\relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.30, size = 15, normalized size = 1.15 \[ - \log {\left (x + \log {\relax (x )} \right )} + \log {\left (\log {\relax (x )} \right )} + \operatorname {Ei}{\left (\log {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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