Optimal. Leaf size=12 \[ x+4 \log (x)-2 \log (x+1) \]
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Rubi [A] time = 0.0176414, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1593, 893} \[ x+4 \log (x)-2 \log (x+1) \]
Antiderivative was successfully verified.
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Rule 1593
Rule 893
Rubi steps
\begin{align*} \int \frac{4+3 x+x^2}{x+x^2} \, dx &=\int \frac{4+3 x+x^2}{x (1+x)} \, dx\\ &=\int \left (1+\frac{4}{x}-\frac{2}{1+x}\right ) \, dx\\ &=x+4 \log (x)-2 \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0032719, size = 12, normalized size = 1. \[ x+4 \log (x)-2 \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 13, normalized size = 1.1 \begin{align*} x+4\,\ln \left ( x \right ) -2\,\ln \left ( 1+x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00843, size = 16, normalized size = 1.33 \begin{align*} x - 2 \, \log \left (x + 1\right ) + 4 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44783, size = 39, normalized size = 3.25 \begin{align*} x - 2 \, \log \left (x + 1\right ) + 4 \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.092957, size = 12, normalized size = 1. \begin{align*} x + 4 \log{\left (x \right )} - 2 \log{\left (x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22549, size = 19, normalized size = 1.58 \begin{align*} x - 2 \, \log \left ({\left | x + 1 \right |}\right ) + 4 \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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