Optimal. Leaf size=198 \[ -\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\frac {\log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{4 \sqrt {1+\sqrt {2}}} \]
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Rubi [A]
time = 0.10, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {722, 1108,
648, 632, 210, 642} \begin {gather*} -\frac {1}{2} \sqrt {1+\sqrt {2}} \text {ArcTan}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {x+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\frac {1}{2} \sqrt {1+\sqrt {2}} \text {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 )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{4 \sqrt {1+\sqrt {2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 722
Rule 1108
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+x} \left (1+x^2\right )} \, dx &=2 \text {Subst}\left (\int \frac {1}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}-x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\text {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {1+\sqrt {2}}}\\ &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {2}}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {2}}-\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 )}{4 \sqrt {1+\sqrt {2}}}+\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 )}{4 \sqrt {1+\sqrt {2}}}\\ &=-\frac {\log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{4 \sqrt {1+\sqrt {2}}}-\frac {\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 )}{\sqrt {2}}-\frac {\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 )}{\sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 \sqrt {-1+\sqrt {2}}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )}{2 \sqrt {-1+\sqrt {2}}}-\frac {\log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{4 \sqrt {1+\sqrt {2}}}+\frac {\log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )}{4 \sqrt {1+\sqrt {2}}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 65, normalized size = 0.33 \begin {gather*} \sqrt {\frac {1}{2}+\frac {i}{2}} \tan ^{-1}\left (\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {1+x}\right )+\sqrt {\frac {1}{2}-\frac {i}{2}} \tan ^{-1}\left (\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.97, size = 269, normalized size = 1.36
method | result | size |
derivativedivides | \(-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x +\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x +\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(269\) |
default | \(-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x +\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{8}-\frac {\left (-2 \sqrt {2}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}+\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \ln \left (1+x +\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{8}+\frac {\left (2 \sqrt {2}-\frac {\left (-\sqrt {2+2 \sqrt {2}}\, \sqrt {2}+2 \sqrt {2+2 \sqrt {2}}\right ) \sqrt {2+2 \sqrt {2}}}{2}\right ) \arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{2 \sqrt {-2+2 \sqrt {2}}}\) | \(269\) |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+1\right ) \ln \left (-\frac {64 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+1\right ) \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{4} x +48 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+1\right ) x +40 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+1\right )-48 \sqrt {x +1}\, \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+9 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+1\right ) x +15 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}+1\right )+2 \sqrt {x +1}}{8 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} x +x -1}\right )}{2}+\RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {64 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{5} x -16 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3} x +24 \sqrt {x +1}\, \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2}-40 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{3}+\RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right ) x +7 \sqrt {x +1}+5 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )}{8 \RootOf \left (32 \textit {\_Z}^{4}+8 \textit {\_Z}^{2}+1\right )^{2} x +x +1}\right )\) | \(409\) |
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 [A]
time = 2.65, size = 269, normalized size = 1.36 \begin {gather*} \frac {1}{8} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 1\right )} \log \left (2 \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 1\right )} \log \left (-2 \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{4} \cdot 2^{\frac {3}{4}} \sqrt {2 \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} \sqrt {2 \, \sqrt {2} + 4} - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} - \sqrt {2} - 1\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{4} \cdot 2^{\frac {3}{4}} \sqrt {-2 \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 4 \, x + 4 \, \sqrt {2} + 4} \sqrt {2 \, \sqrt {2} + 4} - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + \sqrt {2} + 1\right ) \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 {1}{\sqrt {x + 1} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.80, size = 152, normalized size = 0.77 \begin {gather*} \frac {1}{2} \, \sqrt {\sqrt {2} + 1} \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 {\sqrt {2} + 1} \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 {\sqrt {2} - 1} \log \left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) - \frac {1}{4} \, \sqrt {\sqrt {2} - 1} \log \left (-2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.46, size = 226, normalized size = 1.14 \begin {gather*} \mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {x+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-8}-\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {x+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-8}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+2\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\right )-\mathrm {atanh}\left (\frac {16\,\sqrt {2}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {x+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+8}+\frac {16\,\sqrt {2}\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {x+1}}{128\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}+8}\right )\,\left (2\,\sqrt {-\frac {\sqrt {2}}{16}-\frac {1}{16}}-2\,\sqrt {\frac {\sqrt {2}}{16}-\frac {1}{16}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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