Optimal. Leaf size=24 \[ \left (4-\frac {2 e^{-4/x} \log ^2(x)}{5 (4+x)}\right )^2 \]
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Rubi [B] time = 6.13, antiderivative size = 57, normalized size of antiderivative = 2.38, number of steps used = 6, number of rules used = 7, integrand size = 107, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {6688, 12, 6742, 2228, 2178, 2288, 2554} \begin {gather*} \frac {4 e^{-8/x} \log ^3(x) (x \log (x)+4 \log (x))}{25 (x+4)^3}-\frac {16 e^{-4/x} \log (x) (x \log (x)+4 \log (x))}{5 (x+4)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2228
Rule 2288
Rule 2554
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{-8/x} \log (x) \left (2 x (4+x)-\left (-16-4 x+x^2\right ) \log (x)\right ) \left (-10 e^{4/x} (4+x)+\log ^2(x)\right )}{25 x^2 (4+x)^3} \, dx\\ &=\frac {8}{25} \int \frac {e^{-8/x} \log (x) \left (2 x (4+x)-\left (-16-4 x+x^2\right ) \log (x)\right ) \left (-10 e^{4/x} (4+x)+\log ^2(x)\right )}{x^2 (4+x)^3} \, dx\\ &=\frac {8}{25} \int \left (\frac {10 e^{-4/x} \log (x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^2}-\frac {e^{-8/x} \log ^3(x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^3}\right ) \, dx\\ &=-\left (\frac {8}{25} \int \frac {e^{-8/x} \log ^3(x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^3} \, dx\right )+\frac {16}{5} \int \frac {e^{-4/x} \log (x) \left (-8 x-2 x^2-16 \log (x)-4 x \log (x)+x^2 \log (x)\right )}{x^2 (4+x)^2} \, dx\\ &=-\frac {16 e^{-4/x} \log (x) (4 \log (x)+x \log (x))}{5 (4+x)^2}+\frac {4 e^{-8/x} \log ^3(x) (4 \log (x)+x \log (x))}{25 (4+x)^3}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.24, size = 39, normalized size = 1.62 \begin {gather*} -\frac {4 e^{-8/x} \left (20 e^{4/x} (4+x) \log ^2(x)-\log ^4(x)\right )}{25 (4+x)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 40, normalized size = 1.67 \begin {gather*} -\frac {4 \, {\left (20 \, {\left (x + 4\right )} e^{\frac {4}{x}} \log \relax (x)^{2} - \log \relax (x)^{4}\right )} e^{\left (-\frac {8}{x}\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 36, normalized size = 1.50
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{-\frac {8}{x}} \ln \relax (x )^{4}}{25 \left (4+x \right )^{2}}-\frac {16 \,{\mathrm e}^{-\frac {4}{x}} \ln \relax (x )^{2}}{5 \left (4+x \right )}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 39, normalized size = 1.62 \begin {gather*} \frac {4 \, {\left (e^{\left (-\frac {8}{x}\right )} \log \relax (x)^{4} - 20 \, {\left (x + 4\right )} e^{\left (-\frac {4}{x}\right )} \log \relax (x)^{2}\right )}}{25 \, {\left (x^{2} + 8 \, x + 16\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 {{\mathrm {e}}^{-\frac {8}{x}}\,\left (\left (-8\,x^2+32\,x+128\right )\,{\ln \relax (x)}^4+\left (16\,x^2+64\,x\right )\,{\ln \relax (x)}^3-{\mathrm {e}}^{4/x}\,\left (-80\,x^3+2560\,x+5120\right )\,{\ln \relax (x)}^2-{\mathrm {e}}^{4/x}\,\left (160\,x^3+1280\,x^2+2560\,x\right )\,\ln \relax (x)\right )}{25\,x^5+300\,x^4+1200\,x^3+1600\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.46, size = 66, normalized size = 2.75 \begin {gather*} \frac {\left (20 x \log {\relax (x )}^{4} + 80 \log {\relax (x )}^{4}\right ) e^{- \frac {8}{x}} + \left (- 400 x^{2} \log {\relax (x )}^{2} - 3200 x \log {\relax (x )}^{2} - 6400 \log {\relax (x )}^{2}\right ) e^{- \frac {4}{x}}}{125 x^{3} + 1500 x^{2} + 6000 x + 8000} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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