Optimal. Leaf size=26 \[ 9+e^3-\frac {20}{2 x-x^2}+\frac {x^2}{\log (x)} \]
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Rubi [A] time = 0.35, antiderivative size = 21, normalized size of antiderivative = 0.81, number of steps used = 10, number of rules used = 7, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.106, Rules used = {1594, 27, 6688, 74, 2306, 2309, 2178} \begin {gather*} \frac {x^2}{\log (x)}-\frac {20}{(2-x) x} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 74
Rule 1594
Rule 2178
Rule 2306
Rule 2309
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x^3+4 x^4-x^5+\left (8 x^3-8 x^4+2 x^5\right ) \log (x)+(40-40 x) \log ^2(x)}{x^2 \left (4-4 x+x^2\right ) \log ^2(x)} \, dx\\ &=\int \frac {-4 x^3+4 x^4-x^5+\left (8 x^3-8 x^4+2 x^5\right ) \log (x)+(40-40 x) \log ^2(x)}{(-2+x)^2 x^2 \log ^2(x)} \, dx\\ &=\int \left (-\frac {40 (-1+x)}{(-2+x)^2 x^2}-\frac {x}{\log ^2(x)}+\frac {2 x}{\log (x)}\right ) \, dx\\ &=2 \int \frac {x}{\log (x)} \, dx-40 \int \frac {-1+x}{(-2+x)^2 x^2} \, dx-\int \frac {x}{\log ^2(x)} \, dx\\ &=-\frac {20}{(2-x) x}+\frac {x^2}{\log (x)}-2 \int \frac {x}{\log (x)} \, dx+2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {20}{(2-x) x}+2 \text {Ei}(2 \log (x))+\frac {x^2}{\log (x)}-2 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )\\ &=-\frac {20}{(2-x) x}+\frac {x^2}{\log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 21, normalized size = 0.81 \begin {gather*} \frac {10}{-2+x}-\frac {10}{x}+\frac {x^2}{\log (x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 27, normalized size = 1.04 \begin {gather*} \frac {x^{4} - 2 \, x^{3} + 20 \, \log \relax (x)}{{\left (x^{2} - 2 \, x\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 21, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{\log \relax (x)} + \frac {10}{x - 2} - \frac {10}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 20, normalized size = 0.77
method | result | size |
risch | \(\frac {20}{\left (x -2\right ) x}+\frac {x^{2}}{\ln \relax (x )}\) | \(20\) |
default | \(\frac {x^{2}}{\ln \relax (x )}+\frac {10}{x -2}-\frac {10}{x}\) | \(22\) |
norman | \(\frac {x^{4}-2 x^{3}+20 \ln \relax (x )}{x \left (x -2\right ) \ln \relax (x )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 27, normalized size = 1.04 \begin {gather*} \frac {x^{4} - 2 \, x^{3} + 20 \, \log \relax (x)}{{\left (x^{2} - 2 \, x\right )} \log \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.37, size = 19, normalized size = 0.73 \begin {gather*} \frac {20}{x\,\left (x-2\right )}+\frac {x^2}{\ln \relax (x)} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 14, normalized size = 0.54 \begin {gather*} \frac {x^{2}}{\log {\relax (x )}} + \frac {20}{x^{2} - 2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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