Optimal. Leaf size=122 \[ 2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{x}}{1+\sqrt{5}}\right )-2 \text{PolyLog}\left (2,\frac{2 \sqrt{x}}{1-\sqrt{5}}\right )-2 \log \left (\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (-2 \sqrt{x}+\sqrt{5}+1\right )-2 \log \left (1-\frac{2 \sqrt{x}}{1-\sqrt{5}}\right ) \log \left (\sqrt{x}\right )+2 \log \left (-x+\sqrt{x}+1\right ) \log \left (\sqrt{x}\right ) \]
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Rubi [A] time = 0.14536, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {2530, 2524, 2357, 2317, 2391, 2316, 2315} \[ 2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{x}}{1+\sqrt{5}}\right )-2 \text{PolyLog}\left (2,\frac{2 \sqrt{x}}{1-\sqrt{5}}\right )-2 \log \left (\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (-2 \sqrt{x}+\sqrt{5}+1\right )-2 \log \left (1-\frac{2 \sqrt{x}}{1-\sqrt{5}}\right ) \log \left (\sqrt{x}\right )+2 \log \left (-x+\sqrt{x}+1\right ) \log \left (\sqrt{x}\right ) \]
Antiderivative was successfully verified.
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Rule 2530
Rule 2524
Rule 2357
Rule 2317
Rule 2391
Rule 2316
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log \left (1+\sqrt{x}-x\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{\log \left (1+x-x^2\right )}{x} \, dx,x,\sqrt{x}\right )\\ &=2 \log \left (1+\sqrt{x}-x\right ) \log \left (\sqrt{x}\right )-2 \operatorname{Subst}\left (\int \frac{(1-2 x) \log (x)}{1+x-x^2} \, dx,x,\sqrt{x}\right )\\ &=2 \log \left (1+\sqrt{x}-x\right ) \log \left (\sqrt{x}\right )-2 \operatorname{Subst}\left (\int \left (-\frac{2 \log (x)}{1-\sqrt{5}-2 x}-\frac{2 \log (x)}{1+\sqrt{5}-2 x}\right ) \, dx,x,\sqrt{x}\right )\\ &=2 \log \left (1+\sqrt{x}-x\right ) \log \left (\sqrt{x}\right )+4 \operatorname{Subst}\left (\int \frac{\log (x)}{1-\sqrt{5}-2 x} \, dx,x,\sqrt{x}\right )+4 \operatorname{Subst}\left (\int \frac{\log (x)}{1+\sqrt{5}-2 x} \, dx,x,\sqrt{x}\right )\\ &=-2 \log \left (\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (1+\sqrt{5}-2 \sqrt{x}\right )-2 \log \left (1-\frac{2 \sqrt{x}}{1-\sqrt{5}}\right ) \log \left (\sqrt{x}\right )+2 \log \left (1+\sqrt{x}-x\right ) \log \left (\sqrt{x}\right )+2 \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 x}{1-\sqrt{5}}\right )}{x} \, dx,x,\sqrt{x}\right )+4 \operatorname{Subst}\left (\int \frac{\log \left (\frac{2 x}{1+\sqrt{5}}\right )}{1+\sqrt{5}-2 x} \, dx,x,\sqrt{x}\right )\\ &=-2 \log \left (\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (1+\sqrt{5}-2 \sqrt{x}\right )-2 \log \left (1-\frac{2 \sqrt{x}}{1-\sqrt{5}}\right ) \log \left (\sqrt{x}\right )+2 \log \left (1+\sqrt{x}-x\right ) \log \left (\sqrt{x}\right )+2 \text{Li}_2\left (1-\frac{2 \sqrt{x}}{1+\sqrt{5}}\right )-2 \text{Li}_2\left (\frac{2 \sqrt{x}}{1-\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.075291, size = 121, normalized size = 0.99 \[ 2 \text{PolyLog}\left (2,\frac{-2 \sqrt{x}+\sqrt{5}+1}{1+\sqrt{5}}\right )-2 \text{PolyLog}\left (2,-\frac{2 \sqrt{x}}{\sqrt{5}-1}\right )-2 \log \left (\frac{1}{2} \left (1+\sqrt{5}\right )\right ) \log \left (-2 \sqrt{x}+\sqrt{5}+1\right )+\left (\log \left (\sqrt{5}-1\right )-\log \left (2 \sqrt{x}+\sqrt{5}-1\right )\right ) \log (x)+\log \left (-x+\sqrt{x}+1\right ) \log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 102, normalized size = 0.8 \begin{align*} \ln \left ( x \right ) \ln \left ( 1-x+\sqrt{x} \right ) -\ln \left ( x \right ) \ln \left ({\frac{1}{\sqrt{5}+1} \left ( 1+\sqrt{5}-2\,\sqrt{x} \right ) } \right ) -\ln \left ( x \right ) \ln \left ({\frac{1}{\sqrt{5}-1} \left ( -1+\sqrt{5}+2\,\sqrt{x} \right ) } \right ) -2\,{\it dilog} \left ({\frac{1+\sqrt{5}-2\,\sqrt{x}}{\sqrt{5}+1}} \right ) -2\,{\it dilog} \left ({\frac{-1+\sqrt{5}+2\,\sqrt{x}}{\sqrt{5}-1}} \right ) \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{\log \left (-x + \sqrt{x} + 1\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (-x + \sqrt{x} + 1\right )}{x}, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (-x + \sqrt{x} + 1\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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