Optimal. Leaf size=30 \[ \frac {4 e^{-6+\frac {2}{e^{-4+e^5-x}+x}}}{x (5+x)} \]
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Rubi [F]
time = 34.62, 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 {\exp \left (\frac {2 \left (1-3 e^{-4+e^5-x}-3 x\right )}{e^{-4+e^5-x}+x}\right ) \left (e^{-8+2 e^5-2 x} (-20-8 x)-40 x-28 x^2-8 e^{-4+e^5-x} x^2-8 x^3\right )}{25 x^4+10 x^5+x^6+e^{-8+2 e^5-2 x} \left (25 x^2+10 x^3+x^4\right )+e^{-4+e^5-x} \left (50 x^3+20 x^4+2 x^5\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 {4 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-2 e^{4+e^5+x} x^2-e^{2 e^5} (5+2 x)-e^{8+2 x} x \left (10+7 x+2 x^2\right )\right )}{x^2 (5+x)^2 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx\\ &=4 \int \frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-2 e^{4+e^5+x} x^2-e^{2 e^5} (5+2 x)-e^{8+2 x} x \left (10+7 x+2 x^2\right )\right )}{x^2 (5+x)^2 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx\\ &=4 \int \left (-\frac {2 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (1+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )^2}+\frac {2 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (2+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )}+\frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-10-7 x-2 x^2\right )}{x^3 (5+x)^2}\right ) \, dx\\ &=4 \int \frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) \left (-10-7 x-2 x^2\right )}{x^3 (5+x)^2} \, dx-8 \int \frac {\exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (1+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )^2} \, dx+8 \int \frac {\exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right ) (2+x)}{x^3 (5+x) \left (e^{e^5}+e^{4+x} x\right )} \, dx\\ &=4 \int \left (-\frac {2 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 x^3}-\frac {3 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{25 x^2}-\frac {2 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 x}+\frac {\exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 (5+x)^2}+\frac {2 \exp \left (\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 (5+x)}\right ) \, dx-8 \int \left (\frac {\exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 x^3 \left (e^{e^5}+e^{4+x} x\right )^2}+\frac {4 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{25 x^2 \left (e^{e^5}+e^{4+x} x\right )^2}-\frac {4 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 x \left (e^{e^5}+e^{4+x} x\right )^2}+\frac {4 \exp \left (2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 (5+x) \left (e^{e^5}+e^{4+x} x\right )^2}\right ) \, dx+8 \int \left (\frac {2 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{5 x^3 \left (e^{e^5}+e^{4+x} x\right )}+\frac {3 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{25 x^2 \left (e^{e^5}+e^{4+x} x\right )}-\frac {3 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 x \left (e^{e^5}+e^{4+x} x\right )}+\frac {3 \exp \left (e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}\right )}{125 (5+x) \left (e^{e^5}+e^{4+x} x\right )}\right ) \, dx\\ &=-\left (\frac {8}{125} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x} \, dx\right )+\frac {8}{125} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{5+x} \, dx-\frac {24}{125} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x \left (e^{e^5}+e^{4+x} x\right )} \, dx+\frac {24}{125} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{(5+x) \left (e^{e^5}+e^{4+x} x\right )} \, dx+\frac {32}{125} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x \left (e^{e^5}+e^{4+x} x\right )^2} \, dx-\frac {32}{125} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{(5+x) \left (e^{e^5}+e^{4+x} x\right )^2} \, dx-\frac {12}{25} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^2} \, dx+\frac {4}{5} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{(5+x)^2} \, dx+\frac {24}{25} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^2 \left (e^{e^5}+e^{4+x} x\right )} \, dx-\frac {32}{25} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^2 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx-\frac {8}{5} \int \frac {e^{\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^3} \, dx-\frac {8}{5} \int \frac {e^{2 e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^3 \left (e^{e^5}+e^{4+x} x\right )^2} \, dx+\frac {16}{5} \int \frac {e^{e^5+\frac {-6 e^{e^5}+e^{4+x} (2-6 x)}{e^{e^5}+e^{4+x} x}}}{x^3 \left (e^{e^5}+e^{4+x} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.22, size = 36, normalized size = 1.20 \begin {gather*} \frac {4 e^{-6+\frac {2 e^{4+x}}{e^{e^5}+e^{4+x} x}}}{x (5+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 41, normalized size = 1.37
method | result | size |
risch | \(\frac {4 \,{\mathrm e}^{-\frac {2 \left (3 \,{\mathrm e}^{{\mathrm e}^{5}-x -4}+3 x -1\right )}{{\mathrm e}^{{\mathrm e}^{5}-x -4}+x}}}{\left (5+x \right ) x}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.46, size = 35, normalized size = 1.17 \begin {gather*} \frac {4 \, e^{\left (\frac {2 \, e^{\left (x + 4\right )}}{x e^{\left (x + 4\right )} + e^{\left (e^{5}\right )}}\right )}}{x^{2} e^{6} + 5 \, x e^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 41, normalized size = 1.37 \begin {gather*} \frac {4 \, e^{\left (-\frac {2 \, {\left (3 \, x + 3 \, e^{\left (-x + e^{5} - 4\right )} - 1\right )}}{x + e^{\left (-x + e^{5} - 4\right )}}\right )}}{x^{2} + 5 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 34, normalized size = 1.13 \begin {gather*} \frac {4 e^{\frac {2 \left (- 3 x - 3 e^{- x - 4 + e^{5}} + 1\right )}{x + e^{- x - 4 + e^{5}}}}}{x^{2} + 5 x} \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. 104 vs.
\(2 (27) = 54\).
time = 0.98, size = 104, normalized size = 3.47 \begin {gather*} \frac {4 \, e^{\left (-\frac {2 \, {\left (x - e^{5} + e^{\left (-x + e^{5} - 4\right )} + 4\right )}}{{\left (x - e^{5} + 4\right )} e^{5} - 4 \, x + e^{10} - 4 \, e^{5} + e^{\left (-x + e^{5} + 1\right )} - 4 \, e^{\left (-x + e^{5} - 4\right )}} - \frac {2 \, {\left (3 \, e^{5} - 13\right )}}{e^{5} - 4}\right )}}{{\left (x - e^{5} + 4\right )}^{2} + 2 \, {\left (x - e^{5} + 4\right )} e^{5} - 3 \, x + e^{10} - 16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.81, size = 72, normalized size = 2.40 \begin {gather*} \frac {4\,{\mathrm {e}}^{-\frac {6\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}{x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}}\,{\mathrm {e}}^{-\frac {6\,x}{x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}}\,{\mathrm {e}}^{\frac {2}{x+{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^5}}}}{x^2+5\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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