Optimal. Leaf size=172 \[ -\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}} \]
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Rubi [A] time = 0.137031, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1169, 634, 618, 204, 628} \[ -\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (x^2-\sqrt{2-\sqrt{2}} x+1\right )+\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (x^2+\sqrt{2-\sqrt{2}} x+1\right )-\frac{\tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}}+\frac{\tan ^{-1}\left (\frac{2 x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}\right )}{2 \sqrt{2-\sqrt{2}}} \]
Antiderivative was successfully verified.
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Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{2}+x^2}{1+\sqrt{2} x^2+x^4} \, dx &=\frac{\int \frac{\sqrt{2 \left (2-\sqrt{2}\right )}-\left (-1+\sqrt{2}\right ) x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx}{2 \sqrt{2-\sqrt{2}}}+\frac{\int \frac{\sqrt{2 \left (2-\sqrt{2}\right )}+\left (-1+\sqrt{2}\right ) x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx}{2 \sqrt{2-\sqrt{2}}}\\ &=\frac{\left (1-\sqrt{2}\right ) \int \frac{-\sqrt{2-\sqrt{2}}+2 x}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx}{4 \sqrt{2-\sqrt{2}}}+\frac{\left (-1+\sqrt{2}\right ) \int \frac{\sqrt{2-\sqrt{2}}+2 x}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx}{4 \sqrt{2-\sqrt{2}}}+\frac{1}{4} \sqrt{3+2 \sqrt{2}} \int \frac{1}{1-\sqrt{2-\sqrt{2}} x+x^2} \, dx+\frac{1}{4} \sqrt{3+2 \sqrt{2}} \int \frac{1}{1+\sqrt{2-\sqrt{2}} x+x^2} \, dx\\ &=-\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (1-\sqrt{2-\sqrt{2}} x+x^2\right )+\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (1+\sqrt{2-\sqrt{2}} x+x^2\right )-\frac{1}{2} \sqrt{3+2 \sqrt{2}} \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,-\sqrt{2-\sqrt{2}}+2 x\right )-\frac{1}{2} \sqrt{3+2 \sqrt{2}} \operatorname{Subst}\left (\int \frac{1}{-2-\sqrt{2}-x^2} \, dx,x,\sqrt{2-\sqrt{2}}+2 x\right )\\ &=-\frac{1}{2} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}-2 x}{\sqrt{2+\sqrt{2}}}\right )+\frac{1}{2} \sqrt{\frac{1}{2} \left (2+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2-\sqrt{2}}+2 x}{\sqrt{2+\sqrt{2}}}\right )-\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (1-\sqrt{2-\sqrt{2}} x+x^2\right )+\frac{1}{4} \sqrt{1-\frac{1}{\sqrt{2}}} \log \left (1+\sqrt{2-\sqrt{2}} x+x^2\right )\\ \end{align*}
Mathematica [C] time = 0.0346015, size = 53, normalized size = 0.31 \[ \frac{\sqrt{1-i} \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1-i}}\right )+\sqrt{1+i} \tan ^{-1}\left (\frac{\sqrt [4]{2} x}{\sqrt{1+i}}\right )}{2^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.151, size = 199, normalized size = 1.2 \begin{align*} -{\frac{\sqrt{2}\sqrt{2-\sqrt{2}}\ln \left ( 1+{x}^{2}-x\sqrt{2-\sqrt{2}} \right ) }{8}}+{\frac{\sqrt{2}}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) }+{\frac{1}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x-\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) }+{\frac{\sqrt{2}\sqrt{2-\sqrt{2}}\ln \left ( 1+{x}^{2}+x\sqrt{2-\sqrt{2}} \right ) }{8}}+{\frac{\sqrt{2}}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) }+{\frac{1}{2\,\sqrt{2+\sqrt{2}}}\arctan \left ({\frac{2\,x+\sqrt{2-\sqrt{2}}}{\sqrt{2+\sqrt{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + \sqrt{2}}{x^{4} + \sqrt{2} x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.6733, size = 387, normalized size = 2.25 \begin{align*} \frac{1}{4} \, \sqrt{\left (i - 1\right ) \, \sqrt{2}} \log \left (x + \frac{1}{2} \, \sqrt{2} \sqrt{\left (i - 1\right ) \, \sqrt{2}}\right ) - \frac{1}{4} \, \sqrt{\left (i - 1\right ) \, \sqrt{2}} \log \left (x - \frac{1}{2} \, \sqrt{2} \sqrt{\left (i - 1\right ) \, \sqrt{2}}\right ) + \frac{1}{4} \, \sqrt{-\left (i + 1\right ) \, \sqrt{2}} \log \left (x + \frac{1}{2} \, \sqrt{2} \sqrt{-\left (i + 1\right ) \, \sqrt{2}}\right ) - \frac{1}{4} \, \sqrt{-\left (i + 1\right ) \, \sqrt{2}} \log \left (x - \frac{1}{2} \, \sqrt{2} \sqrt{-\left (i + 1\right ) \, \sqrt{2}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: PolynomialError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} + \sqrt{2}}{x^{4} + \sqrt{2} x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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