Optimal. Leaf size=19 \[ -\frac{3 x}{2 \left (x^2+1\right )}-\frac{1}{2} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0040069, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {385, 203} \[ -\frac{3 x}{2 \left (x^2+1\right )}-\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 385
Rule 203
Rubi steps
\begin{align*} \int \frac{-2+x^2}{\left (1+x^2\right )^2} \, dx &=-\frac{3 x}{2 \left (1+x^2\right )}-\frac{1}{2} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{3 x}{2 \left (1+x^2\right )}-\frac{1}{2} \tan ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0071582, size = 19, normalized size = 1. \[ -\frac{3 x}{2 \left (x^2+1\right )}-\frac{1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 16, normalized size = 0.8 \begin{align*} -{\frac{3\,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 = 1.52571, size = 20, normalized size = 1.05 \begin{align*} -\frac{3 \, 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 = 1.24542, size = 59, normalized size = 3.11 \begin{align*} -\frac{{\left (x^{2} + 1\right )} \arctan \left (x\right ) + 3 \, 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 = 0.098991, size = 15, normalized size = 0.79 \begin{align*} - \frac{3 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 = 1.11447, size = 20, normalized size = 1.05 \begin{align*} -\frac{3 \, 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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