Optimal. Leaf size=14 \[ \tan ^{-1}(x)-\frac{1}{2 \left (x^2+1\right )} \]
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Rubi [A] time = 0.0109896, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {1814, 12, 203} \[ \tan ^{-1}(x)-\frac{1}{2 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 12
Rule 203
Rubi steps
\begin{align*} \int \frac{1+x+x^2}{\left (1+x^2\right )^2} \, dx &=-\frac{1}{2 \left (1+x^2\right )}-\frac{1}{2} \int -\frac{2}{1+x^2} \, dx\\ &=-\frac{1}{2 \left (1+x^2\right )}+\int \frac{1}{1+x^2} \, dx\\ &=-\frac{1}{2 \left (1+x^2\right )}+\tan ^{-1}(x)\\ \end{align*}
Mathematica [A] time = 0.0067723, size = 14, normalized size = 1. \[ \tan ^{-1}(x)-\frac{1}{2 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}+2}}+\arctan \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.4843, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 1\right )}} + \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.980056, size = 58, normalized size = 4.14 \begin{align*} \frac{2 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) - 1}{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 = 0.104805, size = 10, normalized size = 0.71 \begin{align*} \operatorname{atan}{\left (x \right )} - \frac{1}{2 x^{2} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16207, size = 16, normalized size = 1.14 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 1\right )}} + \arctan \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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