Optimal. Leaf size=291 \[ -8 \tanh ^{-1}\left (\sqrt {1+\sqrt {1+x}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}-\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {1+x}}}{\sqrt {2}}\right ) \log (1+x)+2 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+\sqrt {2} \text {Li}_2\left (-\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )-\sqrt {2} \text {Li}_2\left (\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right )-\sqrt {2} \text {Li}_2\left (-\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )+\sqrt {2} \text {Li}_2\left (\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right ) \]
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Rubi [F]
time = 0.03, 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 = {}
\begin {gather*} \int \frac {\log (1+x)}{x \sqrt {1+\sqrt {1+x}}} \, dx \end {gather*}
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 = 0.77, size = 430, normalized size = 1.48 \begin {gather*} -8 \tanh ^{-1}\left (\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )-\frac {2 \log (1+x)}{\sqrt {1+\sqrt {1+x}}}+\frac {\log (1+x) \log \left (1-\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )}{\sqrt {2}}-\frac {\log (1+x) \log \left (1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )}{\sqrt {2}}-\sqrt {2} \left (\log \left (1+\sqrt {2}\right ) \log \left (1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )+\log \left (-\left (\left (2+\sqrt {2}\right ) \left (-1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right ) \log \left (1-\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )+2 \text {Li}_2\left (\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )-\text {Li}_2\left (-\left (\left (-2+\sqrt {2}\right ) \left (1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right )+\text {Li}_2\left (\left (1+\sqrt {2}\right ) \left (-1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right )+\sqrt {2} \left (\log \left (1+\sqrt {2}\right ) \log \left (1-\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )+\log \left (\left (2+\sqrt {2}\right ) \left (1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right ) \log \left (1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )+2 \text {Li}_2\left (-\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )-\text {Li}_2\left (\left (-2+\sqrt {2}\right ) \left (-1+\frac {1}{\sqrt {1+\sqrt {1+x}}}\right )\right )+\text {Li}_2\left (-\left (\left (1+\sqrt {2}\right ) \left (1+\frac {\sqrt {2}}{\sqrt {1+\sqrt {1+x}}}\right )\right )\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.15, size = 172, normalized size = 0.59
method | result | size |
derivativedivides | \(8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\dilog \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )-\frac {2 \ln \left (1+x \right )}{\sqrt {1+\sqrt {1+x}}}-8 \arctanh \left (\sqrt {1+\sqrt {1+x}}\right )\) | \(172\) |
default | \(8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\dilog \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\dilog \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{8}\right )-\frac {2 \ln \left (1+x \right )}{\sqrt {1+\sqrt {1+x}}}-8 \arctanh \left (\sqrt {1+\sqrt {1+x}}\right )\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.92, size = 366, normalized size = 1.26 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\log {\left (x + 1 \right )}}{x \sqrt {\sqrt {x + 1} + 1}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (x+1\right )}{x\,\sqrt {\sqrt {x+1}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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