Optimal. Leaf size=19 \[ \frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {28, 199, 203} \begin {gather*} \frac {x}{2 \left (x^2+1\right )}+\frac {1}{2} \tan ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 199
Rule 203
Rubi steps
\begin {align*} \int \frac {1}{1+2 x^2+x^4} \, dx &=\int \frac {1}{\left (1+x^2\right )^2} \, dx\\ &=\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=\frac {x}{2 \left (1+x^2\right )}+\frac {1}{2} \tan ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 0.84 \begin {gather*} \frac {1}{2} \left (\frac {x}{x^2+1}+\tan ^{-1}(x)\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{1+2 x^2+x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 1.98, size = 19, normalized size = 1.00 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \arctan \relax (x) + x}{2 \, {\left (x^{2} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 16, normalized size = 0.84 \begin {gather*} \frac {x}{2 x^{2}+2}+\frac {\arctan \relax (x )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.06, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{2 \, {\left (x^{2} + 1\right )}} + \frac {1}{2} \, \arctan \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 16, normalized size = 0.84 \begin {gather*} \frac {\mathrm {atan}\relax (x)}{2}+\frac {x}{2\,\left (x^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 12, normalized size = 0.63 \begin {gather*} \frac {x}{2 x^{2} + 2} + \frac {\operatorname {atan}{\relax (x )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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