Optimal. Leaf size=10 \[ \frac{2}{x+1}+\log (x) \]
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Rubi [A] time = 0.0186148, antiderivative size = 10, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1594, 27, 894} \[ \frac{2}{x+1}+\log (x) \]
Antiderivative was successfully verified.
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Rule 1594
Rule 27
Rule 894
Rubi steps
\begin{align*} \int \frac{1+x^2}{x+2 x^2+x^3} \, dx &=\int \frac{1+x^2}{x \left (1+2 x+x^2\right )} \, dx\\ &=\int \frac{1+x^2}{x (1+x)^2} \, dx\\ &=\int \left (\frac{1}{x}-\frac{2}{(1+x)^2}\right ) \, dx\\ &=\frac{2}{1+x}+\log (x)\\ \end{align*}
Mathematica [A] time = 0.0048352, size = 10, normalized size = 1. \[ \frac{2}{x+1}+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 11, normalized size = 1.1 \begin{align*} 2\, \left ( 1+x \right ) ^{-1}+\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09606, size = 14, normalized size = 1.4 \begin{align*} \frac{2}{x + 1} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1844, size = 41, normalized size = 4.1 \begin{align*} \frac{{\left (x + 1\right )} \log \left (x\right ) + 2}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.081395, size = 7, normalized size = 0.7 \begin{align*} \log{\left (x \right )} + \frac{2}{x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14539, size = 15, normalized size = 1.5 \begin{align*} \frac{2}{x + 1} + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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