Optimal. Leaf size=205 \[ \frac {\log \left (x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\log \left (x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )}{2 \sqrt {2 \left (1+\sqrt {2}\right )}}-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {x+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {2 \sqrt {x+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {700, 1127, 1161, 618, 204, 1164, 628} \begin {gather*} \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 )}}-\sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \tan ^{-1}\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 )} \tan ^{-1}\left (\frac {2 \sqrt {x+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 700
Rule 1127
Rule 1161
Rule 1164
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x}}{1+x^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {\sqrt {2}-x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )+\operatorname {Subst}\left (\int \frac {\sqrt {2}+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {1}{2} \operatorname {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} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )+\frac {\operatorname {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 {\operatorname {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 )}}-\operatorname {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 )-\operatorname {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*}
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Mathematica [C] time = 0.03, size = 55, normalized size = 0.27 \begin {gather*} i \sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1+i}}\right )-i \sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-i}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [C] time = 0.11, size = 57, normalized size = 0.28 \begin {gather*} \sqrt {1-i} \tan ^{-1}\left (\sqrt {-\frac {1}{2}-\frac {i}{2}} \sqrt {x+1}\right )+\sqrt {1+i} \tan ^{-1}\left (\sqrt {-\frac {1}{2}+\frac {i}{2}} \sqrt {x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 260, normalized size = 1.27 \begin {gather*} \frac {1}{8} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} \log \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + x + \sqrt {2} + 1\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \sqrt {2 \, \sqrt {2} + 4} {\left (\sqrt {2} - 2\right )} \log \left (-\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + x + \sqrt {2} + 1\right ) - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + \frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 2 \, x + 2 \, \sqrt {2} + 2} \sqrt {2 \, \sqrt {2} + 4} - \sqrt {2} - 1\right ) - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} + 4} \arctan \left (-\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + \frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {-2^{\frac {3}{4}} \sqrt {x + 1} \sqrt {2 \, \sqrt {2} + 4} + 2 \, x + 2 \, \sqrt {2} + 2} \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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giac [A] time = 0.94, size = 160, normalized size = 0.78 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.70, size = 336, normalized size = 1.64 \begin {gather*} \frac {\sqrt {2}\, \left (2+2 \sqrt {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 (2+2 \sqrt {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 {\sqrt {2}\, \left (2+2 \sqrt {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 (2+2 \sqrt {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 {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (x +1+\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (x +1+\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \sqrt {2}\, \ln \left (x +1+\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (x +1+\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4} \end {gather*}
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*} \int \frac {\sqrt {x + 1}}{x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 109, normalized size = 0.53 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.92, size = 31, normalized size = 0.15 \begin {gather*} 2 \operatorname {RootSum} {\left (128 t^{4} + 16 t^{2} + 1, \left (t \mapsto t \log {\left (64 t^{3} + 4 t + \sqrt {x + 1} \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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