Optimal. Leaf size=29 \[ \frac {x}{16-x+\frac {1}{3} \left (-\frac {4 e^{-e^x+x}}{x}+x\right )} \]
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Rubi [F] time = 11.39, 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 {36 e^{2 e^x} x^2+e^{e^x} \left (-3 e^{2 x} x^2+e^x \left (-6 x+3 x^2\right )\right )}{4 e^{2 x}+e^{e^x+x} \left (-96 x+4 x^2\right )+e^{2 e^x} \left (576 x^2-48 x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^{e^x} x \left (e^x (-2+x)+12 e^{e^x} x-e^{2 x} x\right )}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ &=3 \int \frac {e^{e^x} x \left (e^x (-2+x)+12 e^{e^x} x-e^{2 x} x\right )}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ &=3 \int \left (-\frac {1}{4} e^{e^x} x^2+\frac {e^{e^x} x \left (-2+x-24 e^{e^x} x^2+e^{e^x} x^3\right )}{2 \left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )}-\frac {e^{2 e^x} x^2 \left (48-52 x+2 x^2+576 e^{e^x} x^2-48 e^{e^x} x^3+e^{e^x} x^4\right )}{4 \left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{2 e^x} x^2 \left (48-52 x+2 x^2+576 e^{e^x} x^2-48 e^{e^x} x^3+e^{e^x} x^4\right )}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+\frac {3}{2} \int \frac {e^{e^x} x \left (-2+x-24 e^{e^x} x^2+e^{e^x} x^3\right )}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \left (\frac {48 e^{2 e^x} x^2}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}-\frac {52 e^{2 e^x} x^3}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}+\frac {2 e^{2 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}+\frac {576 e^{3 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}-\frac {48 e^{3 e^x} x^5}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}+\frac {e^{3 e^x} x^6}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2}\right ) \, dx+\frac {3}{2} \int \frac {e^{e^x} x \left (-2+x-24 e^{e^x} x^2+e^{e^x} x^3\right )}{2 e^x+e^{e^x} (-24+x) x} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{3 e^x} x^6}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx-\frac {3}{2} \int \frac {e^{2 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+\frac {3}{2} \int \left (-\frac {2 e^{e^x} x}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}+\frac {e^{e^x} x^2}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}-\frac {24 e^{2 e^x} x^3}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}+\frac {e^{2 e^x} x^4}{2 e^x-24 e^{e^x} x+e^{e^x} x^2}\right ) \, dx-36 \int \frac {e^{2 e^x} x^2}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+36 \int \frac {e^{3 e^x} x^5}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx+39 \int \frac {e^{2 e^x} x^3}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx-432 \int \frac {e^{3 e^x} x^4}{\left (2 e^x-24 e^{e^x} x+e^{e^x} x^2\right )^2} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{3 e^x} x^6}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-\frac {3}{2} \int \frac {e^{2 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+\frac {3}{2} \int \frac {e^{e^x} x^2}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx+\frac {3}{2} \int \frac {e^{2 e^x} x^4}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx-3 \int \frac {e^{e^x} x}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx-36 \int \frac {e^{2 e^x} x^2}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+36 \int \frac {e^{3 e^x} x^5}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-36 \int \frac {e^{2 e^x} x^3}{2 e^x-24 e^{e^x} x+e^{e^x} x^2} \, dx+39 \int \frac {e^{2 e^x} x^3}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-432 \int \frac {e^{3 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ &=-\left (\frac {3}{4} \int e^{e^x} x^2 \, dx\right )-\frac {3}{4} \int \frac {e^{3 e^x} x^6}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-\frac {3}{2} \int \frac {e^{2 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+\frac {3}{2} \int \frac {e^{e^x} x^2}{2 e^x+e^{e^x} (-24+x) x} \, dx+\frac {3}{2} \int \frac {e^{2 e^x} x^4}{2 e^x+e^{e^x} (-24+x) x} \, dx-3 \int \frac {e^{e^x} x}{2 e^x+e^{e^x} (-24+x) x} \, dx-36 \int \frac {e^{2 e^x} x^2}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx+36 \int \frac {e^{3 e^x} x^5}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-36 \int \frac {e^{2 e^x} x^3}{2 e^x+e^{e^x} (-24+x) x} \, dx+39 \int \frac {e^{2 e^x} x^3}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx-432 \int \frac {e^{3 e^x} x^4}{\left (2 e^x+e^{e^x} (-24+x) x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 4.39, size = 32, normalized size = 1.10 \begin {gather*} \frac {3 \left (e^x-12 e^{e^x} x\right )}{2 e^x+e^{e^x} (-24+x) x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 39, normalized size = 1.34 \begin {gather*} -\frac {3 \, {\left (12 \, x e^{\left (x + e^{x}\right )} - e^{\left (2 \, x\right )}\right )}}{{\left (x^{2} - 24 \, x\right )} e^{\left (x + e^{x}\right )} + 2 \, e^{\left (2 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 33, normalized size = 1.14 \begin {gather*} -\frac {3 \, {\left (12 \, x e^{\left (e^{x}\right )} - e^{x}\right )}}{x^{2} e^{\left (e^{x}\right )} - 24 \, x e^{\left (e^{x}\right )} + 2 \, e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 39, normalized size = 1.34
method | result | size |
risch | \(-\frac {36}{x -24}+\frac {3 x \,{\mathrm e}^{x}}{\left (x -24\right ) \left ({\mathrm e}^{{\mathrm e}^{x}} x^{2}-24 x \,{\mathrm e}^{{\mathrm e}^{x}}+2 \,{\mathrm e}^{x}\right )}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 31, normalized size = 1.07 \begin {gather*} -\frac {3 \, {\left (12 \, x e^{\left (e^{x}\right )} - e^{x}\right )}}{{\left (x^{2} - 24 \, x\right )} e^{\left (e^{x}\right )} + 2 \, e^{x}} \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 = -0.03 \begin {gather*} -\int \frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (3\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (6\,x-3\,x^2\right )\right )-36\,x^2\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}}{4\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{2\,{\mathrm {e}}^x}\,\left (x^4-48\,x^3+576\,x^2\right )-{\mathrm {e}}^{x+{\mathrm {e}}^x}\,\left (96\,x-4\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 39, normalized size = 1.34 \begin {gather*} \frac {3 x e^{x}}{2 x e^{x} + \left (x^{3} - 48 x^{2} + 576 x\right ) e^{e^{x}} - 48 e^{x}} - \frac {36}{x - 24} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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