Integrand size = 33, antiderivative size = 130 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\frac {4}{3} \sqrt {1+\sqrt {1+x}}+\frac {4}{3} \sqrt {1+x} \sqrt {1+\sqrt {1+x}}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-20 \text {$\#$1}^7+18 \text {$\#$1}^9-7 \text {$\#$1}^{11}+\text {$\#$1}^{13}}\&\right ] \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+x} \left (-1+x-x^2+x^3\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx \\ & = 2 \text {Subst}\left (\int \frac {x^2 \sqrt {1+x} \left (-4+6 x^2-4 x^4+x^6\right )}{1+\left (-1+x^2\right )^4} \, dx,x,\sqrt {1+x}\right ) \\ & = 4 \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2 \left (-4+6 \left (-1+x^2\right )^2-4 \left (-1+x^2\right )^4+\left (-1+x^2\right )^6\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = 4 \text {Subst}\left (\int \left (x^2-\frac {2 x^2}{1+x^8 \left (-2+x^2\right )^4}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = \frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-8 \text {Subst}\left (\int \frac {x^2}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{8 \text {$\#$1}^5-20 \text {$\#$1}^7+18 \text {$\#$1}^9-7 \text {$\#$1}^{11}+\text {$\#$1}^{13}}\&\right ] \]
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Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(88\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(88\) |
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Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x + 1} + 1}}{{\left (x^{4} + 1\right )} \sqrt {x + 1}} \,d x } \]
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Not integrable
Time = 0.41 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x + 1} + 1}}{{\left (x^{4} + 1\right )} \sqrt {x + 1}} \,d x } \]
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Not integrable
Time = 5.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{\sqrt {1+x} \left (1+x^4\right )} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x+1}+1}}{\left (x^4+1\right )\,\sqrt {x+1}} \,d x \]
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