Integrand size = 21, antiderivative size = 308 \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=-16 \sqrt {1+\sqrt {1+x}}+16 \text {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )+4 \sqrt {1+\sqrt {1+x}} \log (1+x)-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\sqrt {1+x}}}{\sqrt {2}}\right ) \log (1+x)+4 \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-4 \sqrt {2} \text {arctanh}\left (\frac {1}{\sqrt {2}}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+2 \sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )-2 \sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1-\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right )-2 \sqrt {2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2-\sqrt {2}}\right )+2 \sqrt {2} \operatorname {PolyLog}\left (2,\frac {\sqrt {2} \left (1+\sqrt {1+\sqrt {1+x}}\right )}{2+\sqrt {2}}\right ) \]
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\[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=\int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=-16 \sqrt {1+\sqrt {1+x}}+16 \text {arctanh}\left (\sqrt {1+\sqrt {1+x}}\right )+4 \sqrt {1+\sqrt {1+x}} \log (1+x)+\sqrt {2} \log (1+x) \left (\log \left (\sqrt {2}-\sqrt {1+\sqrt {1+x}}\right )-\log \left (\sqrt {2}+\sqrt {1+\sqrt {1+x}}\right )\right )-2 \sqrt {2} \left (\log \left (-1+\sqrt {2}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-\log \left (1+\sqrt {2}\right ) \log \left (1-\sqrt {1+\sqrt {1+x}}\right )-\log \left (-1+\sqrt {2}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+\log \left (1+\sqrt {2}\right ) \log \left (1+\sqrt {1+\sqrt {1+x}}\right )+\operatorname {PolyLog}\left (2,-\left (\left (-1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )-\operatorname {PolyLog}\left (2,\left (1+\sqrt {2}\right ) \left (-1+\sqrt {1+\sqrt {1+x}}\right )\right )-\operatorname {PolyLog}\left (2,\left (-1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )+\operatorname {PolyLog}\left (2,-\left (\left (1+\sqrt {2}\right ) \left (1+\sqrt {1+\sqrt {1+x}}\right )\right )\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.03 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.65
method | result | size |
derivativedivides | \(4 \ln \left (1+x \right ) \sqrt {1+\sqrt {1+x}}-16 \sqrt {1+\sqrt {1+x}}-8 \ln \left (\sqrt {1+\sqrt {1+x}}-1\right )+8 \ln \left (1+\sqrt {1+\sqrt {1+x}}\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\operatorname {dilog}\left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\operatorname {dilog}\left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{4}\right )\) | \(199\) |
default | \(4 \ln \left (1+x \right ) \sqrt {1+\sqrt {1+x}}-16 \sqrt {1+\sqrt {1+x}}-8 \ln \left (\sqrt {1+\sqrt {1+x}}-1\right )+8 \ln \left (1+\sqrt {1+\sqrt {1+x}}\right )+8 \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{\sum }\frac {\left (\frac {\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (1+x \right )}{2}-\operatorname {dilog}\left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {\sqrt {1+\sqrt {1+x}}-1}{-1+\underline {\hspace {1.25 ex}}\alpha }\right )-\operatorname {dilog}\left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )-\ln \left (\sqrt {1+\sqrt {1+x}}-\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (\frac {1+\sqrt {1+\sqrt {1+x}}}{1+\underline {\hspace {1.25 ex}}\alpha }\right )\right ) \underline {\hspace {1.25 ex}}\alpha }{4}\right )\) | \(199\) |
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Exception generated. \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=\text {Timed out} \]
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none
Time = 0.28 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx={\left (\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}\right ) + 4 \, \sqrt {\sqrt {x + 1} + 1}\right )} \log \left (x + 1\right ) + 2 \, \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} - 2 \, \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} + 1}\right )\right )} + 2 \, \sqrt {2} {\left (\log \left (\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 2 \, \sqrt {2} {\left (\log \left (-\sqrt {2} + \sqrt {\sqrt {x + 1} + 1}\right ) \log \left (-\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1} + 1\right ) + {\rm Li}_2\left (\frac {\sqrt {2} - \sqrt {\sqrt {x + 1} + 1}}{\sqrt {2} - 1}\right )\right )} - 16 \, \sqrt {\sqrt {x + 1} + 1} + 8 \, \log \left (\sqrt {\sqrt {x + 1} + 1} + 1\right ) - 8 \, \log \left (\sqrt {\sqrt {x + 1} + 1} - 1\right ) \]
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\[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=\int { \frac {\sqrt {\sqrt {x + 1} + 1} \log \left (x + 1\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {1+\sqrt {1+x}} \log (1+x)}{x} \, dx=\int \frac {\ln \left (x+1\right )\,\sqrt {\sqrt {x+1}+1}}{x} \,d x \]
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