Optimal. Leaf size=27 \[ \frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \]
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Rubi [F] time = 2.57, 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^{\frac {8}{9} e^{-2 x} x^2} \left (-72 e^{2+2 x}+e^2 \left (64 x^2-64 x^3\right )+e^{2 x} \left (-144 e^{2+2 x}+e^2 \left (64 x-64 x^2\right )\right )\right )}{9 e^{8 x}+27 e^{6 x} x+27 e^{4 x} x^2+9 e^{2 x} x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (-18 e^{4 x}-8 (-1+x) x^2-e^{2 x} \left (9-8 x+8 x^2\right )\right )}{9 \left (e^{2 x}+x\right )^3} \, dx\\ &=\frac {8}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (-18 e^{4 x}-8 (-1+x) x^2-e^{2 x} \left (9-8 x+8 x^2\right )\right )}{\left (e^{2 x}+x\right )^3} \, dx\\ &=\frac {8}{9} \int \left (-\frac {18 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x}-\frac {9 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x (-1+2 x)}{\left (e^{2 x}+x\right )^3}-\frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (9-44 x+8 x^2\right )}{\left (e^{2 x}+x\right )^2}\right ) \, dx\\ &=-\left (\frac {8}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} \left (9-44 x+8 x^2\right )}{\left (e^{2 x}+x\right )^2} \, dx\right )-8 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x (-1+2 x)}{\left (e^{2 x}+x\right )^3} \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x} \, dx\\ &=-\left (\frac {8}{9} \int \left (\frac {9 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2}-\frac {44 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^2}+\frac {8 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^2}\right ) \, dx\right )-8 \int \left (-\frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^3}+\frac {2 e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^3}\right ) \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x} \, dx\\ &=-\left (\frac {64}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^2} \, dx\right )+8 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^3} \, dx-8 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x^2}{\left (e^{2 x}+x\right )^3} \, dx-16 \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2}}{e^{2 x}+x} \, dx+\frac {352}{9} \int \frac {e^{2-2 x+\frac {8}{9} e^{-2 x} x^2} x}{\left (e^{2 x}+x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.69, size = 27, normalized size = 1.00 \begin {gather*} \frac {4 e^{2+\frac {8}{9} e^{-2 x} x^2}}{\left (e^{2 x}+x\right )^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 38, normalized size = 1.41 \begin {gather*} \frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 6\right )}}{x^{2} e^{4} + 2 \, x e^{\left (2 \, x + 4\right )} + e^{\left (4 \, x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {8 \, {\left (8 \, {\left (x^{3} - x^{2}\right )} e^{2} + 2 \, {\left (4 \, {\left (x^{2} - x\right )} e^{2} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (2 \, x\right )} + 9 \, e^{\left (2 \, x + 2\right )}\right )} e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )}\right )}}{9 \, {\left (x^{3} e^{\left (2 \, x\right )} + 3 \, x^{2} e^{\left (4 \, x\right )} + 3 \, x e^{\left (6 \, x\right )} + e^{\left (8 \, x\right )}\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 23, normalized size = 0.85
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{2+\frac {8 x^{2} {\mathrm e}^{-2 x}}{9}}}{\left ({\mathrm e}^{2 x}+x \right )^{2}}\) | \(23\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 31, normalized size = 1.15 \begin {gather*} \frac {4 \, e^{\left (\frac {8}{9} \, x^{2} e^{\left (-2 \, x\right )} + 2\right )}}{x^{2} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.61, size = 31, normalized size = 1.15 \begin {gather*} \frac {4\,{\mathrm {e}}^2\,{\mathrm {e}}^{\frac {8\,x^2\,{\mathrm {e}}^{-2\,x}}{9}}}{{\mathrm {e}}^{4\,x}+2\,x\,{\mathrm {e}}^{2\,x}+x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 34, normalized size = 1.26 \begin {gather*} \frac {4 e^{2} e^{\frac {8 x^{2} e^{- 2 x}}{9}}}{x^{2} + 2 x e^{2 x} + e^{4 x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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