Optimal. Leaf size=26 \[ \frac {x^3}{3}-2 x+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1154, 203} \[ \frac {x^3}{3}-2 x+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 203
Rule 1154
Rubi steps
\begin {align*} \int \frac {1+x^4}{2+x^2} \, dx &=\int \left (-2+x^2+\frac {5}{2+x^2}\right ) \, dx\\ &=-2 x+\frac {x^3}{3}+5 \int \frac {1}{2+x^2} \, dx\\ &=-2 x+\frac {x^3}{3}+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ \frac {x^3}{3}-2 x+\frac {5 \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 21, normalized size = 0.81 \[ \frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 21, normalized size = 0.81 \[ \frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 22, normalized size = 0.85 \[ \frac {x^{3}}{3}-2 x +\frac {5 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x}{2}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.22, size = 21, normalized size = 0.81 \[ \frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 2 \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 21, normalized size = 0.81 \[ \frac {5\,\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{2}-2\,x+\frac {x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.09, size = 26, normalized size = 1.00 \[ \frac {x^{3}}{3} - 2 x + \frac {5 \sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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