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 = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 203
Rule 385
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*}
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Mathematica [A] time = 0.01, size = 19, normalized size = 1.00 \[ -\frac {3 x}{2 \left (x^2+1\right )}-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 21, normalized size = 1.11 \[ -\frac {{\left (x^{2} + 1\right )} \arctan \relax (x) + 3 \, x}{2 \, {\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 15, normalized size = 0.79 \[ -\frac {3 \, x}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{2} \, \arctan \relax (x) \]
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 \[ -\frac {3 x}{2 \left (x^{2}+1\right )}-\frac {\arctan \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.34, size = 15, normalized size = 0.79 \[ -\frac {3 \, x}{2 \, {\left (x^{2} + 1\right )}} - \frac {1}{2} \, \arctan \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 17, normalized size = 0.89 \[ -\frac {\mathrm {atan}\relax (x)}{2}-\frac {3\,x}{2\,\left (x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 15, normalized size = 0.79 \[ - \frac {3 x}{2 x^{2} + 2} - \frac {\operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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