Optimal. Leaf size=21 \[ -\frac {1}{2 (1+x)^2}+\frac {2}{1+x}+\log (1+x) \]
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Rubi [A]
time = 0.00, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {45}
\begin {gather*} \frac {2}{x+1}-\frac {1}{2 (x+1)^2}+\log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^2}{(1+x)^3} \, dx &=\int \left (\frac {1}{(1+x)^3}-\frac {2}{(1+x)^2}+\frac {1}{1+x}\right ) \, dx\\ &=-\frac {1}{2 (1+x)^2}+\frac {2}{1+x}+\log (1+x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 1.00 \begin {gather*} -\frac {1}{2 (1+x)^2}+\frac {2}{1+x}+\log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.93, size = 29, normalized size = 1.38 \begin {gather*} \frac {\frac {3}{2}+2 x+\text {Log}\left [1+x\right ] \left (1+2 x+x^2\right )}{1+2 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 20, normalized size = 0.95
method | result | size |
norman | \(\frac {2 x +\frac {3}{2}}{\left (1+x \right )^{2}}+\ln \left (1+x \right )\) | \(17\) |
risch | \(\frac {2 x +\frac {3}{2}}{\left (1+x \right )^{2}}+\ln \left (1+x \right )\) | \(17\) |
meijerg | \(-\frac {x \left (9 x +6\right )}{6 \left (1+x \right )^{2}}+\ln \left (1+x \right )\) | \(19\) |
default | \(-\frac {1}{2 \left (1+x \right )^{2}}+\frac {2}{1+x}+\ln \left (1+x \right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 22, normalized size = 1.05 \begin {gather*} \frac {4 \, x + 3}{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 [A]
time = 0.34, size = 31, normalized size = 1.48 \begin {gather*} \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + 4 \, x + 3}{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 = 0.04, size = 19, normalized size = 0.90 \begin {gather*} \frac {4 x + 3}{2 x^{2} + 4 x + 2} + \log {\left (x + 1 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 21, normalized size = 1.00 \begin {gather*} \frac {\frac {1}{2} \left (4 x+3\right )}{\left (x+1\right )^{2}}+\ln \left |x+1\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 21, normalized size = 1.00 \begin {gather*} \ln \left (x+1\right )+\frac {2\,x+\frac {3}{2}}{x^2+2\,x+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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