Optimal. Leaf size=26 \[ -3+x+\frac {64 e^{\frac {2 x^2}{e^4 (-2+x)}}}{(x+\log (x))^2} \]
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Rubi [B] time = 4.56, antiderivative size = 90, normalized size of antiderivative = 3.46, number of steps used = 12, number of rules used = 4, integrand size = 245, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6688, 12, 6742, 2288} \begin {gather*} \frac {64 e^{-\frac {2 x^2}{e^4 (2-x)}-4} \left ((4-x) x^3+(4-x) x^2 \log (x)\right )}{(2-x)^2 \left (\frac {x^2}{e^4 (2-x)^2}+\frac {2 x}{e^4 (2-x)}\right ) x (x+\log (x))^3}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2288
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {128 e^{\frac {2 x^2}{e^4 (-2+x)}} (-4+x) x^3+e^4 (-2+x)^2 x^4-128 e^{4+\frac {2 x^2}{e^4 (-2+x)}} (-2+x)^2 (1+x)+x^2 \left (128 e^{\frac {2 x^2}{e^4 (-2+x)}} (-4+x)+3 e^4 (-2+x)^2 x\right ) \log (x)+3 e^4 (-2+x)^2 x^2 \log ^2(x)+e^4 (-2+x)^2 x \log ^3(x)}{e^4 (2-x)^2 x (x+\log (x))^3} \, dx\\ &=\frac {\int \frac {128 e^{\frac {2 x^2}{e^4 (-2+x)}} (-4+x) x^3+e^4 (-2+x)^2 x^4-128 e^{4+\frac {2 x^2}{e^4 (-2+x)}} (-2+x)^2 (1+x)+x^2 \left (128 e^{\frac {2 x^2}{e^4 (-2+x)}} (-4+x)+3 e^4 (-2+x)^2 x\right ) \log (x)+3 e^4 (-2+x)^2 x^2 \log ^2(x)+e^4 (-2+x)^2 x \log ^3(x)}{(2-x)^2 x (x+\log (x))^3} \, dx}{e^4}\\ &=\frac {\int \left (\frac {e^4 x^3}{(x+\log (x))^3}+\frac {3 e^4 x^2 \log (x)}{(x+\log (x))^3}+\frac {3 e^4 x \log ^2(x)}{(x+\log (x))^3}+\frac {e^4 \log ^3(x)}{(x+\log (x))^3}+\frac {128 e^{\frac {2 x^2}{e^4 (-2+x)}} \left (-4 e^4+3 e^4 x^2-4 \left (1+\frac {e^4}{4}\right ) x^3+x^4-4 x^2 \log (x)+x^3 \log (x)\right )}{(2-x)^2 x (x+\log (x))^3}\right ) \, dx}{e^4}\\ &=3 \int \frac {x^2 \log (x)}{(x+\log (x))^3} \, dx+3 \int \frac {x \log ^2(x)}{(x+\log (x))^3} \, dx+\frac {128 \int \frac {e^{\frac {2 x^2}{e^4 (-2+x)}} \left (-4 e^4+3 e^4 x^2-4 \left (1+\frac {e^4}{4}\right ) x^3+x^4-4 x^2 \log (x)+x^3 \log (x)\right )}{(2-x)^2 x (x+\log (x))^3} \, dx}{e^4}+\int \frac {x^3}{(x+\log (x))^3} \, dx+\int \frac {\log ^3(x)}{(x+\log (x))^3} \, dx\\ &=3 \int \left (-\frac {x^3}{(x+\log (x))^3}+\frac {x^2}{(x+\log (x))^2}\right ) \, dx+3 \int \left (\frac {x^3}{(x+\log (x))^3}-\frac {2 x^2}{(x+\log (x))^2}+\frac {x}{x+\log (x)}\right ) \, dx+\frac {128 \int \frac {e^{\frac {2 x^2}{e^4 (-2+x)}} \left ((-4+x) x^3-e^4 (-2+x)^2 (1+x)+(-4+x) x^2 \log (x)\right )}{(2-x)^2 x (x+\log (x))^3} \, dx}{e^4}+\int \frac {x^3}{(x+\log (x))^3} \, dx+\int \left (1-\frac {x^3}{(x+\log (x))^3}+\frac {3 x^2}{(x+\log (x))^2}-\frac {3 x}{x+\log (x)}\right ) \, dx\\ &=x+\frac {64 e^{-4-\frac {2 x^2}{e^4 (2-x)}} \left ((4-x) x^3+(4-x) x^2 \log (x)\right )}{(2-x)^2 x \left (\frac {2 x}{e^4 (2-x)}+\frac {x^2}{e^4 (2-x)^2}\right ) (x+\log (x))^3}+2 \left (3 \int \frac {x^2}{(x+\log (x))^2} \, dx\right )-6 \int \frac {x^2}{(x+\log (x))^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 25, normalized size = 0.96 \begin {gather*} x+\frac {64 e^{\frac {2 x^2}{e^4 (-2+x)}}}{(x+\log (x))^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 48, normalized size = 1.85 \begin {gather*} \frac {x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + 64 \, e^{\left (\frac {2 \, x^{2} e^{\left (-4\right )}}{x - 2}\right )}}{x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.40, size = 52, normalized size = 2.00 \begin {gather*} \frac {x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + 64 \, e^{\left (\frac {2 \, x^{2}}{x e^{4} - 2 \, e^{4}}\right )}}{x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 24, normalized size = 0.92
method | result | size |
risch | \(x +\frac {64 \,{\mathrm e}^{\frac {2 x^{2} {\mathrm e}^{-4}}{x -2}}}{\left (x +\ln \relax (x )\right )^{2}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.43, size = 59, normalized size = 2.27 \begin {gather*} \frac {x^{3} + 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + 64 \, e^{\left (2 \, x e^{\left (-4\right )} + \frac {8}{x e^{4} - 2 \, e^{4}} + 4 \, e^{\left (-4\right )}\right )}}{x^{2} + 2 \, x \log \relax (x) + \log \relax (x)^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.93, size = 23, normalized size = 0.88 \begin {gather*} x+\frac {64\,{\mathrm {e}}^{\frac {2\,x^2\,{\mathrm {e}}^{-4}}{x-2}}}{{\left (x+\ln \relax (x)\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.75, size = 31, normalized size = 1.19 \begin {gather*} x + \frac {64 e^{\frac {2 x^{2}}{\left (x - 2\right ) e^{4}}}}{x^{2} + 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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