Integrand size = 42, antiderivative size = 255 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=-\frac {1}{\left (-1-x+\sqrt {1+x^2}\right ) \left (1+\sqrt {x-\sqrt {1+x^2}}\right )}+\frac {1}{2} \arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \text {RootSum}\left [8-32 \text {$\#$1}+80 \text {$\#$1}^2-128 \text {$\#$1}^3+128 \text {$\#$1}^4-80 \text {$\#$1}^5+32 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^3}{4-16 \text {$\#$1}+32 \text {$\#$1}^2-32 \text {$\#$1}^3+18 \text {$\#$1}^4-6 \text {$\#$1}^5+\text {$\#$1}^6}\&\right ] \]
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\[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )}+\frac {\sqrt {x-\sqrt {1+x^2}}}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )}\right ) \, dx \\ & = -\int \frac {1}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )} \, dx+\int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2 \left (-x^2+2 \sqrt {1+x^2}\right )} \, dx \\ & = -\int \left (\frac {\sqrt {1+x^2}}{2 x^2}+\frac {1}{4+4 x^2-x^4}+\frac {2 \sqrt {1+x^2}}{-4-4 x^2+x^4}-\frac {x^2 \sqrt {1+x^2}}{2 \left (-4-4 x^2+x^4\right )}\right ) \, dx+\int \left (\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 x^2}+\frac {\sqrt {x-\sqrt {1+x^2}}}{4+4 x^2-x^4}+\frac {2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 x^2+x^4}-\frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \left (-4-4 x^2+x^4\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\sqrt {1+x^2}}{x^2} \, dx\right )+\frac {1}{2} \int \frac {x^2 \sqrt {1+x^2}}{-4-4 x^2+x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{x^2} \, dx-\frac {1}{2} \int \frac {x^2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 x^2+x^4} \, dx-2 \int \frac {\sqrt {1+x^2}}{-4-4 x^2+x^4} \, dx+2 \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 x^2+x^4} \, dx-\int \frac {1}{4+4 x^2-x^4} \, dx+\int \frac {\sqrt {x-\sqrt {1+x^2}}}{4+4 x^2-x^4} \, dx \\ & = \frac {\sqrt {1+x^2}}{2 x}-\frac {1}{2} \int \frac {-4-5 x^2}{\sqrt {1+x^2} \left (-4-4 x^2+x^4\right )} \, dx-\frac {1}{2} \int \left (\frac {\left (1+\frac {1}{\sqrt {2}}\right ) \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 \sqrt {2}+2 x^2}+\frac {\left (1-\frac {1}{\sqrt {2}}\right ) \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2}\right ) \, dx-\frac {1}{2} \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{\sqrt {x} \left (-1+x^2\right )^2} \, dx,x,x-\sqrt {1+x^2}\right )+2 \int \left (-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (4+4 \sqrt {2}-2 x^2\right )}-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx+\frac {\int \frac {1}{2-2 \sqrt {2}-x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {1}{2+2 \sqrt {2}-x^2} \, dx}{4 \sqrt {2}}-\frac {\int \frac {\sqrt {1+x^2}}{-4-4 \sqrt {2}+2 x^2} \, dx}{\sqrt {2}}+\frac {\int \frac {\sqrt {1+x^2}}{-4+4 \sqrt {2}+2 x^2} \, dx}{\sqrt {2}}+\int \left (\frac {\sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (4+4 \sqrt {2}-2 x^2\right )}+\frac {\sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {2} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx \\ & = \frac {\sqrt {1+x^2}}{2 x}+\frac {1}{2 x \sqrt {x-\sqrt {1+x^2}}}-\frac {\arctan \left (\frac {x}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{8 \sqrt {-1+\sqrt {2}}}-\frac {\text {arctanh}\left (\frac {x}{\sqrt {2 \left (1+\sqrt {2}\right )}}\right )}{8 \sqrt {1+\sqrt {2}}}-\frac {1}{4} \text {Subst}\left (\int \frac {2 x^{3/2}}{-1+x^2} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \int \left (\frac {-5-\frac {7}{\sqrt {2}}}{\sqrt {1+x^2} \left (-4-4 \sqrt {2}+2 x^2\right )}+\frac {-5+\frac {7}{\sqrt {2}}}{\sqrt {1+x^2} \left (-4+4 \sqrt {2}+2 x^2\right )}\right ) \, dx+\frac {\int \frac {\sqrt {x-\sqrt {1+x^2}}}{4+4 \sqrt {2}-2 x^2} \, dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2} \, dx}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{4+4 \sqrt {2}-2 x^2} \, dx}{\sqrt {2}}-\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2} \, dx}{\sqrt {2}}-\frac {1}{2} \left (4-3 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^2} \left (-4+4 \sqrt {2}+2 x^2\right )} \, dx-\frac {1}{4} \left (2-\sqrt {2}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4+4 \sqrt {2}+2 x^2} \, dx-\frac {1}{4} \left (2+\sqrt {2}\right ) \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-4-4 \sqrt {2}+2 x^2} \, dx-\frac {1}{2} \left (4+3 \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^2} \left (-4-4 \sqrt {2}+2 x^2\right )} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.30 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.13 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\frac {x^2 \left (x^2-2 \sqrt {1+x^2}\right ) \left (-\frac {2}{\left (-1-x+\sqrt {1+x^2}\right ) \left (1+\sqrt {x-\sqrt {1+x^2}}\right )}+\arctan \left (\sqrt {x-\sqrt {1+x^2}}\right )+\text {arctanh}\left (\sqrt {x-\sqrt {1+x^2}}\right )-\text {RootSum}\left [8-32 \text {$\#$1}+80 \text {$\#$1}^2-128 \text {$\#$1}^3+128 \text {$\#$1}^4-80 \text {$\#$1}^5+32 \text {$\#$1}^6-8 \text {$\#$1}^7+\text {$\#$1}^8\&,\frac {\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}-2 \log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^2+\log \left (-1+\sqrt {x-\sqrt {1+x^2}}+\text {$\#$1}\right ) \text {$\#$1}^3}{4-16 \text {$\#$1}+32 \text {$\#$1}^2-32 \text {$\#$1}^3+18 \text {$\#$1}^4-6 \text {$\#$1}^5+\text {$\#$1}^6}\&\right ]\right )}{2 \left (x^4-2 x^2 \sqrt {1+x^2}\right )} \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.14
\[\int \frac {1-\sqrt {x -\sqrt {x^{2}+1}}}{x^{4}-2 x^{2} \sqrt {x^{2}+1}}d x\]
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Timed out. \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\text {Timed out} \]
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Not integrable
Time = 8.06 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.21 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=- \int \frac {\sqrt {x - \sqrt {x^{2} + 1}}}{x^{4} - 2 x^{2} \sqrt {x^{2} + 1}}\, dx - \int \left (- \frac {1}{x^{4} - 2 x^{2} \sqrt {x^{2} + 1}}\right )\, dx \]
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Not integrable
Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.43 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}} - 1}{x^{4} - 2 \, \sqrt {x^{2} + 1} x^{2}} \,d x } \]
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Not integrable
Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.15 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int { -\frac {\sqrt {x - \sqrt {x^{2} + 1}} - 1}{x^{4} - 2 \, \sqrt {x^{2} + 1} x^{2}} \,d x } \]
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Not integrable
Time = 8.67 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.15 \[ \int \frac {1-\sqrt {x-\sqrt {1+x^2}}}{x^4-2 x^2 \sqrt {1+x^2}} \, dx=\int \frac {\sqrt {x-\sqrt {x^2+1}}-1}{2\,x^2\,\sqrt {x^2+1}-x^4} \,d x \]
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