Optimal. Leaf size=122 \[ 2 \text {Li}_2\left (1-\frac {2 \sqrt {x}}{1+\sqrt {5}}\right )-2 \text {Li}_2\left (\frac {2 \sqrt {x}}{1-\sqrt {5}}\right )-2 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (-2 \sqrt {x}+\sqrt {5}+1\right )-2 \log \left (1-\frac {2 \sqrt {x}}{1-\sqrt {5}}\right ) \log \left (\sqrt {x}\right )+2 \log \left (-x+\sqrt {x}+1\right ) \log \left (\sqrt {x}\right ) \]
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Rubi [A] time = 0.15, antiderivative size = 122, normalized size of antiderivative = 1.00, 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 2315
Rule 2316
Rule 2317
Rule 2357
Rule 2391
Rule 2524
Rule 2530
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*}
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Mathematica [A] time = 0.08, size = 121, normalized size = 0.99 \[ 2 \text {Li}_2\left (\frac {-2 \sqrt {x}+\sqrt {5}+1}{1+\sqrt {5}}\right )-2 \text {Li}_2\left (-\frac {2 \sqrt {x}}{-1+\sqrt {5}}\right )-2 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (-2 \sqrt {x}+\sqrt {5}+1\right )+\left (\log \left (\sqrt {5}-1\right )-\log \left (2 \sqrt {x}+\sqrt {5}-1\right )\right ) \log (x)+\log \left (-x+\sqrt {x}+1\right ) \log (x) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (-x + \sqrt {x} + 1\right )}{x}, x\right ) \]
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 \[ \int \frac {\log \left (-x + \sqrt {x} + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 102, normalized size = 0.84 \[ -\ln \relax (x ) \ln \left (\frac {2 \sqrt {x}-1+\sqrt {5}}{\sqrt {5}-1}\right )-\ln \relax (x ) \ln \left (\frac {-2 \sqrt {x}+1+\sqrt {5}}{\sqrt {5}+1}\right )+\ln \relax (x ) \ln \left (-x +\sqrt {x}+1\right )-2 \dilog \left (\frac {2 \sqrt {x}-1+\sqrt {5}}{\sqrt {5}-1}\right )-2 \dilog \left (\frac {-2 \sqrt {x}+1+\sqrt {5}}{\sqrt {5}+1}\right ) \]
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 \[ \int \frac {\log \left (-x + \sqrt {x} + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (\sqrt {x}-x+1\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (\sqrt {x} - x + 1 \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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