Optimal. Leaf size=19 \[ -\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \]
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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 = {712}
\begin {gather*} \frac {1}{x+1}-\frac {1}{(x+1)^2}+2 \log (x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 712
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*}
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Mathematica [A]
time = 0.01, size = 19, normalized size = 1.00 \begin {gather*} -\frac {1}{(1+x)^2}+\frac {1}{1+x}+2 \log (1+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 20, normalized size = 1.05
method | result | size |
norman | \(\frac {x}{\left (x +1\right )^{2}}+2 \ln \left (x +1\right )\) | \(15\) |
risch | \(\frac {x}{\left (x +1\right )^{2}}+2 \ln \left (x +1\right )\) | \(15\) |
default | \(-\frac {1}{\left (x +1\right )^{2}}+\frac {1}{x +1}+2 \ln \left (x +1\right )\) | \(20\) |
meijerg | \(\frac {3 x \left (2+x \right )}{2 \left (x +1\right )^{2}}-\frac {x \left (6+9 x \right )}{3 \left (x +1\right )^{2}}+2 \ln \left (x +1\right )+\frac {3 x^{2}}{2 \left (x +1\right )^{2}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 19, normalized size = 1.00 \begin {gather*} \frac {x}{x^{2} + 2 \, x + 1} + 2 \, \log \left (x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.47, size = 27, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left (x^{2} + 2 \, x + 1\right )} \log \left (x + 1\right ) + x}{x^{2} + 2 \, x + 1} \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 = 0.79 \begin {gather*} \frac {x}{x^{2} + 2 x + 1} + 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 = 1.21, size = 15, normalized size = 0.79 \begin {gather*} \frac {x}{{\left (x + 1\right )}^{2}} + 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 = 14, normalized size = 0.74 \begin {gather*} 2\,\ln \left (x+1\right )+\frac {x}{{\left (x+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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