Optimal. Leaf size=27 \[ \frac{x+1}{2 \left (1-(x+1)^2\right )}+\frac{1}{2} \tanh ^{-1}(x+1) \]
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Rubi [A] time = 0.0072031, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {247, 199, 206} \[ \frac{x+1}{2 \left (1-(x+1)^2\right )}+\frac{1}{2} \tanh ^{-1}(x+1) \]
Antiderivative was successfully verified.
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Rule 247
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\left (1-(1+x)^2\right )^2} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right )^2} \, dx,x,1+x\right )\\ &=\frac{1+x}{2 \left (1-(1+x)^2\right )}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,1+x\right )\\ &=\frac{1+x}{2 \left (1-(1+x)^2\right )}+\frac{1}{2} \tanh ^{-1}(1+x)\\ \end{align*}
Mathematica [A] time = 0.0162995, size = 26, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 (x+1)}{x (x+2)}-\log (x)+\log (x+2)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 24, normalized size = 0.9 \begin{align*} -{\frac{1}{4\,x}}-{\frac{\ln \left ( x \right ) }{4}}-{\frac{1}{8+4\,x}}+{\frac{\ln \left ( 2+x \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16811, size = 34, normalized size = 1.26 \begin{align*} -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \, \log \left (x + 2\right ) - \frac{1}{4} \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67117, size = 99, normalized size = 3.67 \begin{align*} \frac{{\left (x^{2} + 2 \, x\right )} \log \left (x + 2\right ) -{\left (x^{2} + 2 \, x\right )} \log \left (x\right ) - 2 \, x - 2}{4 \,{\left (x^{2} + 2 \, x\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.10968, size = 22, normalized size = 0.81 \begin{align*} - \frac{x + 1}{2 x^{2} + 4 x} - \frac{\log{\left (x \right )}}{4} + \frac{\log{\left (x + 2 \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11523, size = 36, normalized size = 1.33 \begin{align*} -\frac{x + 1}{2 \,{\left (x^{2} + 2 \, x\right )}} + \frac{1}{4} \, \log \left ({\left | x + 2 \right |}\right ) - \frac{1}{4} \, \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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