Optimal. Leaf size=6 \[ x-2 \tan ^{-1}(x) \]
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Rubi [A] time = 0.0034794, antiderivative size = 6, 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 = {388, 203} \[ x-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 388
Rule 203
Rubi steps
\begin{align*} \int \frac{-1+x^2}{1+x^2} \, dx &=x-2 \int \frac{1}{1+x^2} \, dx\\ &=x-2 \tan ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0036768, size = 6, normalized size = 1. \[ x-2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 7, normalized size = 1.2 \begin{align*} x-2\,\arctan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60279, size = 8, normalized size = 1.33 \begin{align*} x - 2 \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44712, size = 23, normalized size = 3.83 \begin{align*} x - 2 \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.079618, size = 5, normalized size = 0.83 \begin{align*} x - 2 \operatorname{atan}{\left (x \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15304, size = 8, normalized size = 1.33 \begin{align*} x - 2 \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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