Optimal. Leaf size=291 \[ \sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1-\sqrt {\sqrt {x+1}+1}\right )}{2-\sqrt {2}}\right )-\sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1-\sqrt {\sqrt {x+1}+1}\right )}{2+\sqrt {2}}\right )-\sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (\sqrt {\sqrt {x+1}+1}+1\right )}{2-\sqrt {2}}\right )+\sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (\sqrt {\sqrt {x+1}+1}+1\right )}{2+\sqrt {2}}\right )-\frac {2 \log (x+1)}{\sqrt {\sqrt {x+1}+1}}-8 \tanh ^{-1}\left (\sqrt {\sqrt {x+1}+1}\right )-\sqrt {2} \log (x+1) \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {2}}\right )+2 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {\sqrt {x+1}+1}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (\sqrt {\sqrt {x+1}+1}+1\right ) \]
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Rubi [F] time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx &=\int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.11, size = 430, normalized size = 1.48 \[ -\sqrt {2} \left (2 \operatorname {PolyLog}\left (2,\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}\right )-\operatorname {PolyLog}\left (2,-\left (\left (\sqrt {2}-2\right ) \left (\frac {1}{\sqrt {\sqrt {x+1}+1}}+1\right )\right )\right )+\operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) \left (\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}-1\right )\right )+\log \left (1+\sqrt {2}\right ) \log \left (\frac {1}{\sqrt {\sqrt {x+1}+1}}+1\right )+\log \left (-\left (\left (2+\sqrt {2}\right ) \left (\frac {1}{\sqrt {\sqrt {x+1}+1}}-1\right )\right )\right ) \log \left (1-\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}\right )\right )+\sqrt {2} \left (2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}\right )-\operatorname {PolyLog}\left (2,\left (\sqrt {2}-2\right ) \left (\frac {1}{\sqrt {\sqrt {x+1}+1}}-1\right )\right )+\operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) \left (\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}+1\right )\right )\right )+\log \left (1+\sqrt {2}\right ) \log \left (1-\frac {1}{\sqrt {\sqrt {x+1}+1}}\right )+\log \left (\left (2+\sqrt {2}\right ) \left (\frac {1}{\sqrt {\sqrt {x+1}+1}}+1\right )\right ) \log \left (\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}+1\right )\right )-\frac {2 \log (x+1)}{\sqrt {\sqrt {x+1}+1}}+\frac {\log (x+1) \log \left (1-\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}\right )}{\sqrt {2}}-\frac {\log (x+1) \log \left (\frac {\sqrt {2}}{\sqrt {\sqrt {x+1}+1}}+1\right )}{\sqrt {2}}-8 \tanh ^{-1}\left (\frac {1}{\sqrt {\sqrt {x+1}+1}}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\log \left (x + 1\right )}{x \sqrt {\sqrt {x + 1} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.17, size = 171, normalized size = 0.59 \[ -8 \arctanh \left (\sqrt {1+\sqrt {x +1}}\right )+4 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (-2 \ln \left (\frac {1+\sqrt {1+\sqrt {x +1}}}{\underline {\hspace {1.25 ex}}\alpha +1}\right ) \ln \left (-\underline {\hspace {1.25 ex}}\alpha +\sqrt {1+\sqrt {x +1}}\right )-2 \ln \left (\frac {\sqrt {1+\sqrt {x +1}}-1}{\underline {\hspace {1.25 ex}}\alpha -1}\right ) \ln \left (-\underline {\hspace {1.25 ex}}\alpha +\sqrt {1+\sqrt {x +1}}\right )+\ln \left (x +1\right ) \ln \left (-\underline {\hspace {1.25 ex}}\alpha +\sqrt {1+\sqrt {x +1}}\right )-2 \dilog \left (\frac {1+\sqrt {1+\sqrt {x +1}}}{\underline {\hspace {1.25 ex}}\alpha +1}\right )-2 \dilog \left (\frac {\sqrt {1+\sqrt {x +1}}-1}{\underline {\hspace {1.25 ex}}\alpha -1}\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )-\frac {2 \ln \left (x +1\right )}{\sqrt {1+\sqrt {x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 366, normalized size = 1.26 \[ \frac {1}{2} \, {\left (\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}\right ) - \frac {4}{\sqrt {\sqrt {x + 1} + 1}}\right )} \log \left (x + 1\right ) + \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} - \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} + \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 4 \, \log \left (\sqrt {\sqrt {x + 1} + 1} + 1\right ) + 4 \, \log \left (\sqrt {\sqrt {x + 1} + 1} - 1\right ) \]
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 \[ \int \frac {\ln \left (x+1\right )}{x\,\sqrt {\sqrt {x+1}+1}} \,d x \]
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 \[ \int \frac {\log {\left (x + 1 \right )}}{x \sqrt {\sqrt {x + 1} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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