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 = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 31
Rule 203
Rule 260
Rule 635
Rule 706
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 1.00 \[ -\frac {1}{10} \log \left (x^2+1\right )+\frac {1}{5} \log (x+2)+\frac {2}{5} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 19, normalized size = 0.76 \[ \frac {2}{5} \, \arctan \relax (x) - \frac {1}{10} \, \log \left (x^{2} + 1\right ) + \frac {1}{5} \, \log \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 20, normalized size = 0.80 \[ \frac {2}{5} \, \arctan \relax (x) - \frac {1}{10} \, \log \left (x^{2} + 1\right ) + \frac {1}{5} \, \log \left ({\left | x + 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 20, normalized size = 0.80 \[ \frac {2 \arctan \relax (x )}{5}+\frac {\ln \left (x +2\right )}{5}-\frac {\ln \left (x^{2}+1\right )}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 19, normalized size = 0.76 \[ \frac {2}{5} \, \arctan \relax (x) - \frac {1}{10} \, \log \left (x^{2} + 1\right ) + \frac {1}{5} \, \log \left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 25, normalized size = 1.00 \[ \frac {\ln \left (x+2\right )}{5}+\ln \left (x-\mathrm {i}\right )\,\left (-\frac {1}{10}-\frac {1}{5}{}\mathrm {i}\right )+\ln \left (x+1{}\mathrm {i}\right )\,\left (-\frac {1}{10}+\frac {1}{5}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.80 \[ \frac {\log {\left (x + 2 \right )}}{5} - \frac {\log {\left (x^{2} + 1 \right )}}{10} + \frac {2 \operatorname {atan}{\relax (x )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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