Optimal. Leaf size=35 \[ \frac {\tan ^{-1}\left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1162, 617, 204} \[ \frac {\tan ^{-1}\left (\sqrt {2} x+1\right )}{\sqrt {2}}-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 1162
Rubi steps
\begin {align*} \int \frac {1+x^2}{1+x^4} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {2} x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {2} x+x^2} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} x\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} x\right )}{\sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt {2} x\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} x\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 30, normalized size = 0.86 \[ \frac {\tan ^{-1}\left (\sqrt {2} x+1\right )-\tan ^{-1}\left (1-\sqrt {2} x\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 29, normalized size = 0.83 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (x^{3} + x\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 39, normalized size = 1.11 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 88, normalized size = 2.51 \[ \frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x -1\right )}{2}+\frac {\sqrt {2}\, \arctan \left (\sqrt {2}\, x +1\right )}{2}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}-\sqrt {2}\, x +1}{x^{2}+\sqrt {2}\, x +1}\right )}{8}+\frac {\sqrt {2}\, \ln \left (\frac {x^{2}+\sqrt {2}\, x +1}{x^{2}-\sqrt {2}\, x +1}\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.42, size = 39, normalized size = 1.11 \[ \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x + \sqrt {2}\right )}\right ) + \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (2 \, x - \sqrt {2}\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 29, normalized size = 0.83 \[ \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,x^3}{2}+\frac {\sqrt {2}\,x}{2}\right )+\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 39, normalized size = 1.11 \[ \frac {\sqrt {2} \left (2 \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )} + 2 \operatorname {atan}{\left (\frac {\sqrt {2} x^{3}}{2} + \frac {\sqrt {2} x}{2} \right )}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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