Optimal. Leaf size=313 \[ -\operatorname {PolyLog}\left (2,\frac {2 \left (1-\sqrt {x+1}\right )}{3-\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (1-\sqrt {x+1}\right )}{3+\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1-\sqrt {5}}\right )-\operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1+\sqrt {5}}\right )+\log \left (\sqrt {x+1}-1\right ) \log \left (x+\sqrt {x+1}\right )+\log \left (\sqrt {x+1}+1\right ) \log \left (x+\sqrt {x+1}\right )-\log \left (\sqrt {x+1}-1\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{3-\sqrt {5}}\right )-\log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+\sqrt {5}}\right )-\log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-\sqrt {5}}\right )-\log \left (\sqrt {x+1}-1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{3+\sqrt {5}}\right ) \]
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Rubi [A] time = 0.38, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2530, 2528, 2524, 2418, 2394, 2393, 2391} \[ -\text {PolyLog}\left (2,\frac {2 \left (1-\sqrt {x+1}\right )}{3-\sqrt {5}}\right )-\text {PolyLog}\left (2,\frac {2 \left (1-\sqrt {x+1}\right )}{3+\sqrt {5}}\right )-\text {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1-\sqrt {5}}\right )-\text {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1+\sqrt {5}}\right )+\log \left (\sqrt {x+1}-1\right ) \log \left (x+\sqrt {x+1}\right )+\log \left (\sqrt {x+1}+1\right ) \log \left (x+\sqrt {x+1}\right )-\log \left (\sqrt {x+1}-1\right ) \log \left (\frac {2 \sqrt {x+1}-\sqrt {5}+1}{3-\sqrt {5}}\right )-\log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+\sqrt {5}}\right )-\log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-\sqrt {5}}\right )-\log \left (\sqrt {x+1}-1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{3+\sqrt {5}}\right ) \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2528
Rule 2530
Rubi steps
\begin {align*} \int \frac {\log \left (x+\sqrt {1+x}\right )}{x} \, dx &=2 \operatorname {Subst}\left (\int \frac {x \log \left (-1+x+x^2\right )}{-1+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {\log \left (-1+x+x^2\right )}{2 (-1+x)}+\frac {\log \left (-1+x+x^2\right )}{2 (1+x)}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=\operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{-1+x} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{1+x} \, dx,x,\sqrt {1+x}\right )\\ &=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\operatorname {Subst}\left (\int \frac {(1+2 x) \log (-1+x)}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )-\operatorname {Subst}\left (\int \frac {(1+2 x) \log (1+x)}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\operatorname {Subst}\left (\int \left (\frac {2 \log (-1+x)}{1-\sqrt {5}+2 x}+\frac {2 \log (-1+x)}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )-\operatorname {Subst}\left (\int \left (\frac {2 \log (1+x)}{1-\sqrt {5}+2 x}+\frac {2 \log (1+x)}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {\log (-1+x)}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {\log (-1+x)}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {\log (1+x)}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {\log (1+x)}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )\\ &=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{3-\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )+\operatorname {Subst}\left (\int \frac {\log \left (\frac {1-\sqrt {5}+2 x}{-1-\sqrt {5}}\right )}{1+x} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (\frac {1-\sqrt {5}+2 x}{3-\sqrt {5}}\right )}{-1+x} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (\frac {1+\sqrt {5}+2 x}{-1+\sqrt {5}}\right )}{1+x} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (\frac {1+\sqrt {5}+2 x}{3+\sqrt {5}}\right )}{-1+x} \, dx,x,\sqrt {1+x}\right )\\ &=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{3-\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )+\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-1-\sqrt {5}}\right )}{x} \, dx,x,1+\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{3-\sqrt {5}}\right )}{x} \, dx,x,-1+\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-1+\sqrt {5}}\right )}{x} \, dx,x,1+\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{3+\sqrt {5}}\right )}{x} \, dx,x,-1+\sqrt {1+x}\right )\\ &=\log \left (-1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )+\log \left (1+\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1-\sqrt {5}+2 \sqrt {1+x}}{3-\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\log \left (1+\sqrt {1+x}\right ) \log \left (-\frac {1+\sqrt {5}+2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\log \left (-1+\sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{3+\sqrt {5}}\right )-\text {Li}_2\left (-\frac {2 \left (-1+\sqrt {1+x}\right )}{3-\sqrt {5}}\right )-\text {Li}_2\left (-\frac {2 \left (-1+\sqrt {1+x}\right )}{3+\sqrt {5}}\right )-\text {Li}_2\left (\frac {2 \left (1+\sqrt {1+x}\right )}{1-\sqrt {5}}\right )-\text {Li}_2\left (\frac {2 \left (1+\sqrt {1+x}\right )}{1+\sqrt {5}}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 303, normalized size = 0.97 \[ -\operatorname {PolyLog}\left (2,\frac {2 \left (\sqrt {x+1}+1\right )}{1-\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}-\sqrt {5}+1}{3-\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{1+\sqrt {5}}\right )+\operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{3+\sqrt {5}}\right )+\log \left (1-\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\log \left (\sqrt {x+1}+1\right ) \log \left (x+\sqrt {x+1}\right )-\log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\log \left (\frac {1}{2} \left (3-\sqrt {5}\right )\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\log \left (\frac {1}{2} \left (3+\sqrt {5}\right )\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\log \left (\sqrt {x+1}+1\right ) \log \left (-\frac {2 \sqrt {x+1}+\sqrt {5}+1}{1-\sqrt {5}}\right ) \]
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 = 0.02, size = 252, normalized size = 0.81 \[ -\ln \left (\frac {-1+\sqrt {5}-2 \sqrt {x +1}}{\sqrt {5}-3}\right ) \ln \left (-1+\sqrt {x +1}\right )-\ln \left (\frac {-1+\sqrt {5}-2 \sqrt {x +1}}{\sqrt {5}+1}\right ) \ln \left (1+\sqrt {x +1}\right )-\ln \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{3+\sqrt {5}}\right ) \ln \left (-1+\sqrt {x +1}\right )-\ln \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{\sqrt {5}-1}\right ) \ln \left (1+\sqrt {x +1}\right )+\ln \left (-1+\sqrt {x +1}\right ) \ln \left (x +\sqrt {x +1}\right )+\ln \left (1+\sqrt {x +1}\right ) \ln \left (x +\sqrt {x +1}\right )-\dilog \left (\frac {-1+\sqrt {5}-2 \sqrt {x +1}}{\sqrt {5}-3}\right )-\dilog \left (\frac {-1+\sqrt {5}-2 \sqrt {x +1}}{\sqrt {5}+1}\right )-\dilog \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{3+\sqrt {5}}\right )-\dilog \left (\frac {1+\sqrt {5}+2 \sqrt {x +1}}{\sqrt {5}-1}\right ) \]
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 = -0.00 \[ \int \frac {\ln \left (x+\sqrt {x+1}\right )}{x} \,d x \]
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 \[ \int \frac {\log {\left (x + \sqrt {x + 1} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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