Optimal. Leaf size=14 \[ \frac{x}{x^2+1}+2 \tan ^{-1}(x) \]
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Rubi [A] time = 0.0040127, antiderivative size = 14, 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} \[ \frac{x}{x^2+1}+2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 385
Rule 203
Rubi steps
\begin{align*} \int \frac{3+x^2}{\left (1+x^2\right )^2} \, dx &=\frac{x}{1+x^2}+2 \int \frac{1}{1+x^2} \, dx\\ &=\frac{x}{1+x^2}+2 \tan ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0059902, size = 14, normalized size = 1. \[ \frac{x}{x^2+1}+2 \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 15, normalized size = 1.1 \begin{align*}{\frac{x}{{x}^{2}+1}}+2\,\arctan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49805, size = 19, normalized size = 1.36 \begin{align*} \frac{x}{x^{2} + 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 = 1.30389, size = 53, normalized size = 3.79 \begin{align*} \frac{2 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) + x}{x^{2} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.092744, size = 10, normalized size = 0.71 \begin{align*} \frac{x}{x^{2} + 1} + 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 = 1.64378, size = 19, normalized size = 1.36 \begin{align*} \frac{x}{x^{2} + 1} + 2 \, \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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