Optimal. Leaf size=4 \[ \tanh ^{-1}(x+1) \]
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Rubi [A] time = 0.00, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {247, 206} \begin {gather*} \tanh ^{-1}(x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 247
Rubi steps
\begin {align*} \int \frac {1}{1-(1+x)^2} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,1+x\right )\\ &=\tanh ^{-1}(1+x)\\ \end {align*}
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Mathematica [B] time = 0.00, size = 15, normalized size = 3.75 \begin {gather*} \frac {1}{2} \log (x+2)-\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{1-(1+x)^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.28, size = 11, normalized size = 2.75 \begin {gather*} \frac {1}{2} \, \log \left (x + 2\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 13, normalized size = 3.25 \begin {gather*} \frac {1}{2} \, \log \left ({\left | x + 2 \right |}\right ) - \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 12, normalized size = 3.00 \begin {gather*} -\frac {\ln \relax (x )}{2}+\frac {\ln \left (x +2\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.57, size = 11, normalized size = 2.75 \begin {gather*} \frac {1}{2} \, \log \left (x + 2\right ) - \frac {1}{2} \, \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 4, normalized size = 1.00 \begin {gather*} \mathrm {atanh}\left (x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 10, normalized size = 2.50 \begin {gather*} - \frac {\log {\relax (x )}}{2} + \frac {\log {\left (x + 2 \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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