Integrand size = 26, antiderivative size = 129 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4}{15} \sqrt {1+x} \sqrt {1+\sqrt {1+x}}+\frac {4}{15} (1+3 x) \sqrt {1+\sqrt {1+x}}-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \]
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\[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x \sqrt {1+x} \left (-1+\left (-1+x^2\right )^4\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 ) \left (-1+x^8 \left (-2+x^2\right )^4\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+x^4-\frac {2 x^2 \left (-1+x^2\right )}{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}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \text {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = -\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \text {Subst}\left (\int \left (-\frac {x^2}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}+\frac {x^4}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ & = -\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}+8 \text {Subst}\left (\int \frac {x^2}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-8 \text {Subst}\left (\int \frac {x^4}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right ) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\frac {4}{15} \sqrt {1+\sqrt {1+x}} \left (-2+\sqrt {1+x}+3 (1+x)\right )-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \]
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Time = 0.32 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(105\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{4}-\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(105\) |
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Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\text {Timed out} \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.19 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1} \,d x } \]
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Not integrable
Time = 26.75 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 1} \,d x } \]
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Not integrable
Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.19 \[ \int \frac {\left (-1+x^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx=\int \frac {\left (x^4-1\right )\,\sqrt {\sqrt {x+1}+1}}{x^4+1} \,d x \]
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