Optimal. Leaf size=25 \[ -\frac{1}{10} \log \left (x^2+1\right )+\frac{1}{5} \log (x+2)+\frac{2}{5} \tan ^{-1}(x) \]
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Rubi [A] time = 0.0096625, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {706, 31, 635, 203, 260} \[ -\frac{1}{10} \log \left (x^2+1\right )+\frac{1}{5} \log (x+2)+\frac{2}{5} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 706
Rule 31
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{1}{(2+x) \left (1+x^2\right )} \, dx &=\frac{1}{5} \int \frac{1}{2+x} \, dx+\frac{1}{5} \int \frac{2-x}{1+x^2} \, dx\\ &=\frac{1}{5} \log (2+x)-\frac{1}{5} \int \frac{x}{1+x^2} \, dx+\frac{2}{5} \int \frac{1}{1+x^2} \, dx\\ &=\frac{2}{5} \tan ^{-1}(x)+\frac{1}{5} \log (2+x)-\frac{1}{10} \log \left (1+x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0059501, size = 25, normalized size = 1. \[ -\frac{1}{10} \log \left (x^2+1\right )+\frac{1}{5} \log (x+2)+\frac{2}{5} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 20, normalized size = 0.8 \begin{align*}{\frac{2\,\arctan \left ( x \right ) }{5}}+{\frac{\ln \left ( 2+x \right ) }{5}}-{\frac{\ln \left ({x}^{2}+1 \right ) }{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.60552, size = 26, normalized size = 1.04 \begin{align*} \frac{2}{5} \, \arctan \left (x\right ) - \frac{1}{10} \, \log \left (x^{2} + 1\right ) + \frac{1}{5} \, \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36182, size = 70, normalized size = 2.8 \begin{align*} \frac{2}{5} \, \arctan \left (x\right ) - \frac{1}{10} \, \log \left (x^{2} + 1\right ) + \frac{1}{5} \, \log \left (x + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11747, size = 20, normalized size = 0.8 \begin{align*} \frac{\log{\left (x + 2 \right )}}{5} - \frac{\log{\left (x^{2} + 1 \right )}}{10} + \frac{2 \operatorname{atan}{\left (x \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15265, size = 27, normalized size = 1.08 \begin{align*} \frac{2}{5} \, \arctan \left (x\right ) - \frac{1}{10} \, \log \left (x^{2} + 1\right ) + \frac{1}{5} \, \log \left ({\left | x + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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