Optimal. Leaf size=14 \[ \log (x+1)-\frac{1}{2 (x+1)^2} \]
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Rubi [A] time = 0.0071718, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {683} \[ \log (x+1)-\frac{1}{2 (x+1)^2} \]
Antiderivative was successfully verified.
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Rule 683
Rubi steps
\begin{align*} \int \frac{2+2 x+x^2}{(1+x)^3} \, dx &=\int \left (\frac{1}{(1+x)^3}+\frac{1}{1+x}\right ) \, dx\\ &=-\frac{1}{2 (1+x)^2}+\log (1+x)\\ \end{align*}
Mathematica [A] time = 0.0050636, size = 14, normalized size = 1. \[ \log (x+1)-\frac{1}{2 (x+1)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 13, normalized size = 0.9 \begin{align*} -{\frac{1}{2\, \left ( 1+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 = 0.990921, size = 23, normalized size = 1.64 \begin{align*} -\frac{1}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.81557, size = 76, normalized size = 5.43 \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) - 1}{2 \,{\left (x^{2} + 2 \, x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.101164, size = 15, normalized size = 1.07 \begin{align*} \log{\left (x + 1 \right )} - \frac{1}{2 x^{2} + 4 x + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09139, size = 18, normalized size = 1.29 \begin{align*} -\frac{1}{2 \,{\left (x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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