Optimal. Leaf size=23 \[ e^{-1+\frac {40}{-\frac {5 e^x}{18}+\frac {x^2}{16}}} \]
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Rubi [F]
time = 3.78, 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 {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (230400 e^x-103680 x\right )}{1600 e^{2 x}-720 e^x x^2+81 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {11520 e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (20 e^x-9 x\right )}{\left (40 e^x-9 x^2\right )^2} \, dx\\ &=11520 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} \left (20 e^x-9 x\right )}{\left (40 e^x-9 x^2\right )^2} \, dx\\ &=11520 \int \left (\frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{2 \left (40 e^x-9 x^2\right )}+\frac {9 e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} (-2+x) x}{2 \left (-40 e^x+9 x^2\right )^2}\right ) \, dx\\ &=5760 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{40 e^x-9 x^2} \, dx+51840 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} (-2+x) x}{\left (-40 e^x+9 x^2\right )^2} \, dx\\ &=5760 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{40 e^x-9 x^2} \, dx+51840 \int \left (-\frac {2 e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x}{\left (-40 e^x+9 x^2\right )^2}+\frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x^2}{\left (-40 e^x+9 x^2\right )^2}\right ) \, dx\\ &=5760 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}}}{40 e^x-9 x^2} \, dx+51840 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x^2}{\left (-40 e^x+9 x^2\right )^2} \, dx-103680 \int \frac {e^{\frac {-5760-40 e^x+9 x^2}{40 e^x-9 x^2}} x}{\left (-40 e^x+9 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.13, size = 19, normalized size = 0.83 \begin {gather*} e^{-1-\frac {5760}{40 e^x-9 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 27, normalized size = 1.17
method | result | size |
risch | \({\mathrm e}^{-\frac {-9 x^{2}+40 \,{\mathrm e}^{x}+5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}\) | \(27\) |
norman | \(\frac {9 x^{2} {\mathrm e}^{\frac {-40 \,{\mathrm e}^{x}+9 x^{2}-5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}-40 \,{\mathrm e}^{x} {\mathrm e}^{\frac {-40 \,{\mathrm e}^{x}+9 x^{2}-5760}{40 \,{\mathrm e}^{x}-9 x^{2}}}}{9 x^{2}-40 \,{\mathrm e}^{x}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.49, size = 17, normalized size = 0.74 \begin {gather*} e^{\left (\frac {5760}{9 \, x^{2} - 40 \, e^{x}} - 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 26, normalized size = 1.13 \begin {gather*} e^{\left (-\frac {9 \, x^{2} - 40 \, e^{x} - 5760}{9 \, x^{2} - 40 \, e^{x}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 22, normalized size = 0.96 \begin {gather*} e^{\frac {9 x^{2} - 40 e^{x} - 5760}{- 9 x^{2} + 40 e^{x}}} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.63, size = 51, normalized size = 2.22 \begin {gather*} {\mathrm {e}}^{\frac {9\,x^2}{40\,{\mathrm {e}}^x-9\,x^2}}\,{\mathrm {e}}^{-\frac {40\,{\mathrm {e}}^x}{40\,{\mathrm {e}}^x-9\,x^2}}\,{\mathrm {e}}^{-\frac {5760}{40\,{\mathrm {e}}^x-9\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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