Optimal. Leaf size=14 \[ -\frac {1}{2 (1+x)^2}+\log (1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, 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 = {697}
\begin {gather*} \log (x+1)-\frac {1}{2 (x+1)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
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*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{2 (1+x)^2}+\log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 13, normalized size = 0.93
method | result | size |
default | \(-\frac {1}{2 \left (x +1\right )^{2}}+\ln \left (x +1\right )\) | \(13\) |
norman | \(-\frac {1}{2 \left (x +1\right )^{2}}+\ln \left (x +1\right )\) | \(13\) |
risch | \(-\frac {1}{2 \left (x +1\right )^{2}}+\ln \left (x +1\right )\) | \(13\) |
meijerg | \(\frac {x \left (2+x \right )}{\left (x +1\right )^{2}}-\frac {x \left (6+9 x \right )}{6 \left (x +1\right )^{2}}+\ln \left (x +1\right )+\frac {x^{2}}{\left (x +1\right )^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 17, normalized size = 1.21 \begin {gather*} -\frac {1}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} + \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (12) = 24\).
time = 2.85, size = 28, normalized size = 2.00 \begin {gather*} \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) - 1}{2 \, {\left (x^{2} + 2 \, x + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 1.07 \begin {gather*} \log {\left (x + 1 \right )} - \frac {1}{2 x^{2} + 4 x + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{2 \, {\left (x + 1\right )}^{2}} + \log \left ({\left | x + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 12, normalized size = 0.86 \begin {gather*} \ln \left (x+1\right )-\frac {1}{2\,{\left (x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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