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 = 0.0031113, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {28, 199, 203} \[ \frac{x}{2 \left (x^2+1\right )}+\frac{1}{2} \tan ^{-1}(x) \]
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*}
Mathematica [A] time = 0.005355, size = 16, normalized size = 0.84 \[ \frac{1}{2} \left (\frac{x}{x^2+1}+\tan ^{-1}(x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 16, normalized size = 0.8 \begin{align*}{\frac{x}{2\,{x}^{2}+2}}+{\frac{\arctan \left ( x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48803, size = 20, normalized size = 1.05 \begin{align*} \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48237, size = 55, normalized size = 2.89 \begin{align*} \frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{2 \,{\left (x^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.098102, size = 12, normalized size = 0.63 \begin{align*} \frac{x}{2 x^{2} + 2} + \frac{\operatorname{atan}{\left (x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12473, size = 20, normalized size = 1.05 \begin{align*} \frac{x}{2 \,{\left (x^{2} + 1\right )}} + \frac{1}{2} \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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