Optimal. Leaf size=105 \[ \text {RootSum}\left [1-2 \text {$\#$1}^2-2 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\& ,\frac {\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-2 \text {$\#$1}^2-3 \text {$\#$1}^4+2 \text {$\#$1}^6}\& \right ] \]
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Rubi [F]
time = 4.87, 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 {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {x-\sqrt {1+x^2}}}{x^2+\sqrt {1+x^2}} \, dx &=\int \left (\frac {x^2 \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4}-\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4}\right ) \, dx\\ &=\int \frac {x^2 \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4} \, dx-\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1-x^2+x^4} \, dx\\ &=\int \left (\frac {\left (1+\frac {1}{\sqrt {5}}\right ) \sqrt {x-\sqrt {1+x^2}}}{-1-\sqrt {5}+2 x^2}+\frac {\left (1-\frac {1}{\sqrt {5}}\right ) \sqrt {x-\sqrt {1+x^2}}}{-1+\sqrt {5}+2 x^2}\right ) \, dx-\int \left (-\frac {2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {5} \left (1+\sqrt {5}-2 x^2\right )}-\frac {2 \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {5} \left (-1+\sqrt {5}+2 x^2\right )}\right ) \, dx\\ &=\frac {2 \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{1+\sqrt {5}-2 x^2} \, dx}{\sqrt {5}}+\frac {2 \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{-1+\sqrt {5}+2 x^2} \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-\sqrt {5}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1+\sqrt {5}+2 x^2} \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{-1-\sqrt {5}+2 x^2} \, dx\\ &=\frac {2 \int \left (\frac {i \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {5}}+\frac {2 \int \left (\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {1+\sqrt {5}} \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {1+\sqrt {5}} \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx}{\sqrt {5}}+\frac {1}{5} \left (5-\sqrt {5}\right ) \int \left (\frac {i \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {i \sqrt {x-\sqrt {1+x^2}}}{2 \sqrt {-1+\sqrt {5}} \left (i \sqrt {-1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx+\frac {1}{5} \left (5+\sqrt {5}\right ) \int \left (\frac {\sqrt {1+\sqrt {5}} \sqrt {x-\sqrt {1+x^2}}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}-\sqrt {2} x\right )}+\frac {\sqrt {1+\sqrt {5}} \sqrt {x-\sqrt {1+x^2}}}{2 \left (-1-\sqrt {5}\right ) \left (\sqrt {1+\sqrt {5}}+\sqrt {2} x\right )}\right ) \, dx\\ &=\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \int \frac {\sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \int \frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \int \frac {\sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (\sqrt {2}+2 \sqrt {1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \left (-\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x-\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x+\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 \sqrt {1+\sqrt {5}} x-\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \left (\frac {1}{\sqrt {2} \sqrt {x}}+\frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x+\sqrt {2} x^2\right )}\right ) \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {1}{2} \left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (\sqrt {2}+2 \sqrt {1+\sqrt {5}} x-\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x}{\sqrt {x} \left (-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x+\sqrt {2} x^2\right )} \, dx,x,x-\sqrt {1+x^2}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2+i \sqrt {2 \left (-1+\sqrt {5}\right )} x^2}{\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x^2-\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\left (i \sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )}\right ) \text {Subst}\left (\int \frac {2-i \sqrt {2 \left (-1+\sqrt {5}\right )} x^2}{-\sqrt {2}+2 i \sqrt {-1+\sqrt {5}} x^2+\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}-\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2+\sqrt {2 \left (1+\sqrt {5}\right )} x^2}{\sqrt {2}+2 \sqrt {1+\sqrt {5}} x^2-\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \text {Subst}\left (\int \frac {2-\sqrt {2 \left (1+\sqrt {5}\right )} x^2}{-\sqrt {2}+2 \sqrt {1+\sqrt {5}} x^2+\sqrt {2} x^4} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ &=\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (1+\sqrt {5}\right )} \left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}-\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}-\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )-\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )+\frac {1}{2} \left (\sqrt {\frac {1}{5} \left (-1+\sqrt {5}\right )} \left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\sqrt {x-\sqrt {1+x^2}}\right )\\ &=\frac {\left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}-\frac {\left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}-\frac {\left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}+\frac {\left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}-\frac {\left (i \left (3-\sqrt {5}\right )+2 \sqrt {-2+\sqrt {5}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}-i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}+\frac {\left (i \left (3-\sqrt {5}\right )-2 \sqrt {-2+\sqrt {5}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}}}\right )}{2 \sqrt [4]{2} \sqrt {\frac {5 \left (\sqrt {3-\sqrt {5}}+i \sqrt {-1+\sqrt {5}}\right )}{1+\sqrt {5}}}}+\frac {\left (3+\sqrt {5}-2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {-\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}-\frac {\left (3+\sqrt {5}+2 \sqrt {2+\sqrt {5}}\right ) \sqrt {\frac {-1+\sqrt {5}}{5 \left (\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}\right )}} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {\sqrt {1+\sqrt {5}}+\sqrt {3+\sqrt {5}}}}\right )}{2 \sqrt [4]{2}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {i \int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{i \sqrt {-1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (-1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}-\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}+\frac {\int \frac {\sqrt {1+x^2} \sqrt {x-\sqrt {1+x^2}}}{\sqrt {1+\sqrt {5}}+\sqrt {2} x} \, dx}{\sqrt {5 \left (1+\sqrt {5}\right )}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 105, normalized size = 1.00 \begin {gather*} \text {RootSum}\left [1-2 \text {$\#$1}^2-2 \text {$\#$1}^4-2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (\sqrt {x-\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^5}{-1-2 \text {$\#$1}^2-3 \text {$\#$1}^4+2 \text {$\#$1}^6}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x -\sqrt {x^{2}+1}}}{x^{2}+\sqrt {x^{2}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.21, size = 2627, normalized size = 25.02 \begin {gather*} \text {Too large to display} \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 {\sqrt {x - \sqrt {x^{2} + 1}}}{x^{2} + \sqrt {x^{2} + 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 = -0.01 \begin {gather*} \int \frac {\sqrt {x-\sqrt {x^2+1}}}{\sqrt {x^2+1}+x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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