Optimal. Leaf size=24 \[ \frac {x^2}{4+x+x \left (3+x-\log \left (\frac {x}{1+x}\right )\right )} \]
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Rubi [F]
time = 0.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {8 x+13 x^2+4 x^3+\left (-x^2-x^3\right ) \log \left (\frac {x}{1+x}\right )}{16+48 x+56 x^2+32 x^3+9 x^4+x^5+\left (-8 x-16 x^2-10 x^3-2 x^4\right ) \log \left (\frac {x}{1+x}\right )+\left (x^2+x^3\right ) \log ^2\left (\frac {x}{1+x}\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (8+13 x+4 x^2-x (1+x) \log \left (\frac {x}{1+x}\right )\right )}{(1+x) \left ((2+x)^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx\\ &=\int \left (-\frac {x \left (-4-5 x+x^2+x^3\right )}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}+\frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )}\right ) \, dx\\ &=-\int \frac {x \left (-4-5 x+x^2+x^3\right )}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx+\int \frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )} \, dx\\ &=\int \frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )} \, dx-\int \left (\frac {1}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}-\frac {5 x}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}+\frac {x^3}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}-\frac {1}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2}\right ) \, dx\\ &=5 \int \frac {x}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx-\int \frac {1}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx-\int \frac {x^3}{\left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx+\int \frac {1}{(1+x) \left (4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )\right )^2} \, dx+\int \frac {x}{4+4 x+x^2-x \log \left (\frac {x}{1+x}\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 23, normalized size = 0.96 \begin {gather*} \frac {x^2}{(2+x)^2-x \log \left (\frac {x}{1+x}\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs.
\(2(24)=48\).
time = 3.01, size = 77, normalized size = 3.21
method | result | size |
norman | \(\frac {x^{2}}{x^{2}-x \ln \left (\frac {x}{x +1}\right )+4 x +4}\) | \(26\) |
risch | \(\frac {x^{2}}{x^{2}-x \ln \left (\frac {x}{x +1}\right )+4 x +4}\) | \(26\) |
derivativedivides | \(\frac {\left (1-\frac {1}{x +1}\right )^{2}}{\left (1-\frac {1}{x +1}\right )^{2} \ln \left (1-\frac {1}{x +1}\right )+\left (1-\frac {1}{x +1}\right )^{2}-\left (1-\frac {1}{x +1}\right ) \ln \left (1-\frac {1}{x +1}\right )+\frac {4}{x +1}}\) | \(77\) |
default | \(\frac {\left (1-\frac {1}{x +1}\right )^{2}}{\left (1-\frac {1}{x +1}\right )^{2} \ln \left (1-\frac {1}{x +1}\right )+\left (1-\frac {1}{x +1}\right )^{2}-\left (1-\frac {1}{x +1}\right ) \ln \left (1-\frac {1}{x +1}\right )+\frac {4}{x +1}}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{2} + x \log \left (x + 1\right ) - x \log \left (x\right ) + 4 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 25, normalized size = 1.04 \begin {gather*} \frac {x^{2}}{x^{2} - x \log \left (\frac {x}{x + 1}\right ) + 4 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 20, normalized size = 0.83 \begin {gather*} - \frac {x^{2}}{- x^{2} + x \log {\left (\frac {x}{x + 1} \right )} - 4 x - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (24) = 48\).
time = 0.43, size = 65, normalized size = 2.71 \begin {gather*} -\frac {x^{2}}{{\left (x + 1\right )}^{2} {\left (\frac {x \log \left (\frac {x}{x + 1}\right )}{x + 1} - \frac {x^{2} \log \left (\frac {x}{x + 1}\right )}{{\left (x + 1\right )}^{2}} + \frac {4 \, x}{x + 1} - \frac {x^{2}}{{\left (x + 1\right )}^{2}} - 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {8\,x+13\,x^2+4\,x^3-\ln \left (\frac {x}{x+1}\right )\,\left (x^3+x^2\right )}{48\,x+{\ln \left (\frac {x}{x+1}\right )}^2\,\left (x^3+x^2\right )-\ln \left (\frac {x}{x+1}\right )\,\left (2\,x^4+10\,x^3+16\,x^2+8\,x\right )+56\,x^2+32\,x^3+9\,x^4+x^5+16} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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