Integrand size = 15, antiderivative size = 205 \[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {\log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Time = 0.16 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {714, 1141, 1175, 632, 210, 1178, 642} \[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {x+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \arctan \left (\frac {2 \sqrt {x+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {\log \left (x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Rule 210
Rule 632
Rule 642
Rule 714
Rule 1141
Rule 1175
Rule 1178
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ & = -\text {Subst}\left (\int \frac {\sqrt {2}-x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\text {Subst}\left (\int \frac {\sqrt {2}+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{-\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 x}{-\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x-x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \\ & = \frac {\log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}\right )-\text {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}\right ) \\ & = \frac {\tan ^{-1}\left (\frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{\sqrt {2 \left (-1+\sqrt {2}\right )}}+\frac {\log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.28 \[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=\sqrt {1-i} \arctan \left (\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {1+x}\right )+\sqrt {1+i} \arctan \left (\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {1+x}\right ) \]
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Time = 2.47 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.89
method | result | size |
derivativedivides | \(-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (-\frac {\ln \left (1+x +\sqrt {2}-\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (\frac {\ln \left (1+x +\sqrt {2}+\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\) | \(183\) |
default | \(-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (-\frac {\ln \left (1+x +\sqrt {2}-\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+x}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \left (\sqrt {2}-1\right ) \left (\frac {\ln \left (1+x +\sqrt {2}+\sqrt {1+x}\, \sqrt {2+2 \sqrt {2}}\right )}{2}-\frac {\sqrt {2+2 \sqrt {2}}\, \arctan \left (\frac {2 \sqrt {1+x}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}\right )}{2}\) | \(183\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) \ln \left (-\frac {-24 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{4} x \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) x \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+24 \sqrt {1+x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+30 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right ) x +14 \sqrt {1+x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+2\right )}{4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +x +1}\right )}{2}-\operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) \ln \left (\frac {24 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{5} x +26 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3} x +12 \sqrt {1+x}\, \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2}+30 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{3}+6 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right ) x -\sqrt {1+x}+10 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )}{4 \operatorname {RootOf}\left (8 \textit {\_Z}^{4}+4 \textit {\_Z}^{2}+1\right )^{2} x +x -1}\right )\) | \(409\) |
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Result contains complex when optimal does not.
Time = 0.51 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.34 \[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {i - 1} \log \left (i \, \sqrt {i - 1} + \sqrt {x + 1}\right ) - \frac {1}{2} \, \sqrt {i - 1} \log \left (-i \, \sqrt {i - 1} + \sqrt {x + 1}\right ) - \frac {1}{2} \, \sqrt {-i - 1} \log \left (i \, \sqrt {-i - 1} + \sqrt {x + 1}\right ) + \frac {1}{2} \, \sqrt {-i - 1} \log \left (-i \, \sqrt {-i - 1} + \sqrt {x + 1}\right ) \]
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\[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=\int \frac {\sqrt {x + 1}}{x^{2} + 1}\, dx \]
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\[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=\int { \frac {\sqrt {x + 1}}{x^{2} + 1} \,d x } \]
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Time = 0.66 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=\frac {1}{2} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) + \frac {1}{2} \, \sqrt {2 \, \sqrt {2} + 2} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} - 2} \log \left (-2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) \]
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Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.53 \[ \int \frac {\sqrt {1+x}}{1+x^2} \, dx=\mathrm {atanh}\left (4\,{\left (\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}+\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )}^3\,\sqrt {x+1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}+2\,\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )+\mathrm {atanh}\left (4\,{\left (\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}-\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right )}^3\,\sqrt {x+1}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{8}-\frac {1}{8}}-2\,\sqrt {\frac {\sqrt {2}}{8}-\frac {1}{8}}\right ) \]
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