Optimal. Leaf size=27 \[ 4+2 x-x^2-\log (4)-\frac {4}{x (x+4 \log (x))} \]
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Rubi [F] time = 0.65, 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 {16+8 x+2 x^4-2 x^5+\left (16+16 x^3-16 x^4\right ) \log (x)+\left (32 x^2-32 x^3\right ) \log ^2(x)}{x^4+8 x^3 \log (x)+16 x^2 \log ^2(x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16+8 x+2 x^4-2 x^5+\left (16+16 x^3-16 x^4\right ) \log (x)+\left (32 x^2-32 x^3\right ) \log ^2(x)}{x^2 (x+4 \log (x))^2} \, dx\\ &=\int \left (-2 (-1+x)+\frac {4 (4+x)}{x^2 (x+4 \log (x))^2}+\frac {4}{x^2 (x+4 \log (x))}\right ) \, dx\\ &=-(1-x)^2+4 \int \frac {4+x}{x^2 (x+4 \log (x))^2} \, dx+4 \int \frac {1}{x^2 (x+4 \log (x))} \, dx\\ &=-(1-x)^2+4 \int \frac {1}{x^2 (x+4 \log (x))} \, dx+4 \int \left (\frac {4}{x^2 (x+4 \log (x))^2}+\frac {1}{x (x+4 \log (x))^2}\right ) \, dx\\ &=-(1-x)^2+4 \int \frac {1}{x (x+4 \log (x))^2} \, dx+4 \int \frac {1}{x^2 (x+4 \log (x))} \, dx+16 \int \frac {1}{x^2 (x+4 \log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 26, normalized size = 0.96 \begin {gather*} -2 \left (-x+\frac {x^2}{2}+\frac {2}{x (x+4 \log (x))}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 36, normalized size = 1.33 \begin {gather*} -\frac {x^{4} - 2 \, x^{3} + 4 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x) + 4}{x^{2} + 4 \, x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 22, normalized size = 0.81 \begin {gather*} -x^{2} + 2 \, x - \frac {4}{x^{2} + 4 \, x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 23, normalized size = 0.85
method | result | size |
risch | \(-x^{2}+2 x -\frac {4}{x \left (4 \ln \relax (x )+x \right )}\) | \(23\) |
norman | \(\frac {-4+8 x^{2} \ln \relax (x )+2 x^{3}-x^{4}-4 x^{3} \ln \relax (x )}{x \left (4 \ln \relax (x )+x \right )}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 36, normalized size = 1.33 \begin {gather*} -\frac {x^{4} - 2 \, x^{3} + 4 \, {\left (x^{3} - 2 \, x^{2}\right )} \log \relax (x) + 4}{x^{2} + 4 \, x \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 22, normalized size = 0.81 \begin {gather*} 2\,x-\frac {4}{x\,\left (x+4\,\ln \relax (x)\right )}-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 17, normalized size = 0.63 \begin {gather*} - x^{2} + 2 x - \frac {4}{x^{2} + 4 x \log {\relax (x )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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