Optimal. Leaf size=27 \[ \frac {x}{-4+\frac {e^{-10+2 e^x}}{6 x (1+x)^2}} \]
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Rubi [F]
time = 9.66, 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 {e^{20-4 e^x} \left (-144 x^2-576 x^3-864 x^4-576 x^5-144 x^6\right )+e^{10-2 e^x} \left (12 x+36 x^2+24 x^3+e^x \left (-12 x^2-24 x^3-12 x^4\right )\right )}{1+e^{10-2 e^x} \left (-48 x-96 x^2-48 x^3\right )+e^{20-4 e^x} \left (576 x^2+2304 x^3+3456 x^4+2304 x^5+576 x^6\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 {12 e^{10} x (1+x) \left (-e^{2 e^x+x} x (1+x)-12 e^{10} x (1+x)^3+e^{2 e^x} (1+2 x)\right )}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=\left (12 e^{10}\right ) \int \frac {x (1+x) \left (-e^{2 e^x+x} x (1+x)-12 e^{10} x (1+x)^3+e^{2 e^x} (1+2 x)\right )}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=\left (12 e^{10}\right ) \int \left (-\frac {e^{2 e^x+x} x^2 (1+x)^2}{\left (e^{2 e^x}-24 e^{10} x-48 e^{10} x^2-24 e^{10} x^3\right )^2}-\frac {x (1+x) \left (-e^{2 e^x}+12 e^{10} x-2 e^{2 e^x} x+36 e^{10} x^2+36 e^{10} x^3+12 e^{10} x^4\right )}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}\right ) \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2 (1+x)^2}{\left (e^{2 e^x}-24 e^{10} x-48 e^{10} x^2-24 e^{10} x^3\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {x (1+x) \left (-e^{2 e^x}+12 e^{10} x-2 e^{2 e^x} x+36 e^{10} x^2+36 e^{10} x^3+12 e^{10} x^4\right )}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2 (1+x)^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {x (1+x) \left (12 e^{10} x (1+x)^3-e^{2 e^x} (1+2 x)\right )}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \left (\frac {e^{2 e^x+x} x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {2 e^{2 e^x+x} x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {e^{2 e^x+x} x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}\right ) \, dx\right )-\left (12 e^{10}\right ) \int \left (-\frac {12 e^{10} x^2 (1+x)^3 (1+3 x)}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {x \left (1+3 x+2 x^2\right )}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}\right ) \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {x \left (1+3 x+2 x^2\right )}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (144 e^{20}\right ) \int \frac {x^2 (1+x)^3 (1+3 x)}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {x \left (-1-3 x-2 x^2\right )}{e^{2 e^x}-24 e^{10} x (1+x)^2} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (144 e^{20}\right ) \int \frac {x^2 (1+x)^3 (1+3 x)}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \left (\frac {x}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}+\frac {3 x^2}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}+\frac {2 x^3}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3}\right ) \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (144 e^{20}\right ) \int \left (\frac {x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {6 x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {12 x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {10 x^5}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}+\frac {3 x^6}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2}\right ) \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {x}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (24 e^{10}\right ) \int \frac {x^3}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (36 e^{10}\right ) \int \frac {x^2}{-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3} \, dx+\left (144 e^{20}\right ) \int \frac {x^2}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (432 e^{20}\right ) \int \frac {x^6}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (864 e^{20}\right ) \int \frac {x^3}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (1440 e^{20}\right ) \int \frac {x^5}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx+\left (1728 e^{20}\right ) \int \frac {x^4}{\left (-e^{2 e^x}+24 e^{10} x+48 e^{10} x^2+24 e^{10} x^3\right )^2} \, dx\\ &=-\left (\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\right )-\left (12 e^{10}\right ) \int \frac {e^{2 e^x+x} x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (12 e^{10}\right ) \int \frac {x}{-e^{2 e^x}+24 e^{10} x (1+x)^2} \, dx-\left (24 e^{10}\right ) \int \frac {e^{2 e^x+x} x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx-\left (24 e^{10}\right ) \int \frac {x^3}{-e^{2 e^x}+24 e^{10} x (1+x)^2} \, dx-\left (36 e^{10}\right ) \int \frac {x^2}{-e^{2 e^x}+24 e^{10} x (1+x)^2} \, dx+\left (144 e^{20}\right ) \int \frac {x^2}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (432 e^{20}\right ) \int \frac {x^6}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (864 e^{20}\right ) \int \frac {x^3}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (1440 e^{20}\right ) \int \frac {x^5}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx+\left (1728 e^{20}\right ) \int \frac {x^4}{\left (e^{2 e^x}-24 e^{10} x (1+x)^2\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.31, size = 36, normalized size = 1.33 \begin {gather*} -\frac {12 e^{10} x^2 (1+x)^2}{-2 e^{2 e^x}+48 e^{10} x (1+x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 46, normalized size = 1.70
method | result | size |
risch | \(-\frac {x}{4}-\frac {x}{4 \left (24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{3}+48 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x^{2}+24 \,{\mathrm e}^{-2 \,{\mathrm e}^{x}+10} x -1\right )}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (23) = 46\).
time = 0.32, size = 51, normalized size = 1.89 \begin {gather*} -\frac {6 \, {\left (x^{4} e^{10} + 2 \, x^{3} e^{10} + x^{2} e^{10}\right )}}{24 \, x^{3} e^{10} + 48 \, x^{2} e^{10} + 24 \, x e^{10} - e^{\left (2 \, e^{x}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 44, normalized size = 1.63 \begin {gather*} -\frac {6 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (-2 \, e^{x} + 10\right )}}{24 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{\left (-2 \, e^{x} + 10\right )} - 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.18, size = 29, normalized size = 1.07 \begin {gather*} - \frac {x}{4} - \frac {x}{\left (96 x^{3} + 192 x^{2} + 96 x\right ) e^{10 - 2 e^{x}} - 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs.
\(2 (23) = 46\).
time = 0.44, size = 51, normalized size = 1.89 \begin {gather*} -\frac {6 \, {\left (x^{4} e^{10} + 2 \, x^{3} e^{10} + x^{2} e^{10}\right )}}{24 \, x^{3} e^{10} + 48 \, x^{2} e^{10} + 24 \, x e^{10} - e^{\left (2 \, e^{x}\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}}^{10-2\,{\mathrm {e}}^x}\,\left (12\,x-{\mathrm {e}}^x\,\left (12\,x^4+24\,x^3+12\,x^2\right )+36\,x^2+24\,x^3\right )-{\mathrm {e}}^{20-4\,{\mathrm {e}}^x}\,\left (144\,x^6+576\,x^5+864\,x^4+576\,x^3+144\,x^2\right )}{{\mathrm {e}}^{20-4\,{\mathrm {e}}^x}\,\left (576\,x^6+2304\,x^5+3456\,x^4+2304\,x^3+576\,x^2\right )-{\mathrm {e}}^{10-2\,{\mathrm {e}}^x}\,\left (48\,x^3+96\,x^2+48\,x\right )+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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