Optimal. Leaf size=19 \[ \frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1) \]
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Rubi [A] time = 0.0110944, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {698} \[ \frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1) \]
Antiderivative was successfully verified.
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Rule 698
Rubi steps
\begin{align*} \int \frac{3+3 x+2 x^2}{(1+x)^3} \, dx &=\int \left (\frac{2}{(1+x)^3}-\frac{1}{(1+x)^2}+\frac{2}{1+x}\right ) \, dx\\ &=-\frac{1}{(1+x)^2}+\frac{1}{1+x}+2 \log (1+x)\\ \end{align*}
Mathematica [A] time = 0.007844, size = 19, normalized size = 1. \[ \frac{1}{x+1}-\frac{1}{(x+1)^2}+2 \log (x+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 20, normalized size = 1.1 \begin{align*} - \left ( 1+x \right ) ^{-2}+ \left ( 1+x \right ) ^{-1}+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 = 1.01557, size = 26, normalized size = 1.37 \begin{align*} \frac{x}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65808, size = 70, normalized size = 3.68 \begin{align*} \frac{2 \,{\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + x}{x^{2} + 2 \, x + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.091706, size = 15, normalized size = 0.79 \begin{align*} \frac{x}{x^{2} + 2 x + 1} + 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 = 1.07939, size = 20, normalized size = 1.05 \begin{align*} \frac{x}{{\left (x + 1\right )}^{2}} + 2 \, \log \left ({\left | x + 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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