Optimal. Leaf size=22 \[ 3+x-x^2-\frac {2}{5+\frac {x^2}{\log (x)}} \]
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Rubi [F] time = 0.42, 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 {-2 x+x^4-2 x^5+\left (4 x+10 x^2-20 x^3\right ) \log (x)+(25-50 x) \log ^2(x)}{x^4+10 x^2 \log (x)+25 \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 {-2 x+x^4-2 x^5+\left (4 x+10 x^2-20 x^3\right ) \log (x)+(25-50 x) \log ^2(x)}{\left (x^2+5 \log (x)\right )^2} \, dx\\ &=\int \left (1-2 x-\frac {2 x \left (5+2 x^2\right )}{5 \left (x^2+5 \log (x)\right )^2}+\frac {4 x}{5 \left (x^2+5 \log (x)\right )}\right ) \, dx\\ &=x-x^2-\frac {2}{5} \int \frac {x \left (5+2 x^2\right )}{\left (x^2+5 \log (x)\right )^2} \, dx+\frac {4}{5} \int \frac {x}{x^2+5 \log (x)} \, dx\\ &=x-x^2-\frac {2}{5} \int \left (\frac {5 x}{\left (x^2+5 \log (x)\right )^2}+\frac {2 x^3}{\left (x^2+5 \log (x)\right )^2}\right ) \, dx+\frac {4}{5} \int \frac {x}{x^2+5 \log (x)} \, dx\\ &=x-x^2-\frac {4}{5} \int \frac {x^3}{\left (x^2+5 \log (x)\right )^2} \, dx+\frac {4}{5} \int \frac {x}{x^2+5 \log (x)} \, dx-2 \int \frac {x}{\left (x^2+5 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.11, size = 24, normalized size = 1.09 \begin {gather*} x-x^2+\frac {2 x^2}{5 \left (x^2+5 \log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 39, normalized size = 1.77 \begin {gather*} -\frac {5 \, x^{4} - 5 \, x^{3} - 2 \, x^{2} + 25 \, {\left (x^{2} - x\right )} \log \relax (x)}{5 \, {\left (x^{2} + 5 \, \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 22, normalized size = 1.00 \begin {gather*} -x^{2} + x + \frac {2 \, x^{2}}{5 \, {\left (x^{2} + 5 \, \log \relax (x)\right )}} \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 = 1.05
method | result | size |
risch | \(-x^{2}+x +\frac {2 x^{2}}{5 \left (x^{2}+5 \ln \relax (x )\right )}\) | \(23\) |
norman | \(\frac {x^{3}-2 \ln \relax (x )-x^{4}+5 x \ln \relax (x )-5 x^{2} \ln \relax (x )}{x^{2}+5 \ln \relax (x )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 39, normalized size = 1.77 \begin {gather*} -\frac {5 \, x^{4} - 5 \, x^{3} - 2 \, x^{2} + 25 \, {\left (x^{2} - x\right )} \log \relax (x)}{5 \, {\left (x^{2} + 5 \, \log \relax (x)\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.38, size = 24, normalized size = 1.09 \begin {gather*} x+\frac {2\,x^2}{5\,\left (5\,\ln \relax (x)+x^2\right )}-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.11, size = 19, normalized size = 0.86 \begin {gather*} - x^{2} + \frac {2 x^{2}}{5 x^{2} + 25 \log {\relax (x )}} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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