Optimal. Leaf size=30 \[ \frac {1}{\frac {2}{-5+\frac {e^{e^{-x} x (-2+2 x)}}{x}}+x^2} \]
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Rubi [F]
time = 29.29, 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 {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 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 {e^{-e^{-x} \left (-4+e^x\right ) x} \left (-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )\right )}{x^2 \left (2 e^{2 e^{-x} x}+e^{2 e^{-x} x^2} x-5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ &=\int \left (-\frac {2}{x^3}+\frac {2 e^{-2 e^{-x} x-e^{-x} \left (-4+e^x\right ) x} \left (-3 e^x+2 x-6 x^2+2 x^3\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )}-\frac {2 e^{-e^{-x} \left (-4+e^x\right ) x} \left (2 e^x-4 x+12 x^2+5 e^x x^2+6 x^3-30 x^4+10 x^5\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}\right ) \, dx\\ &=\frac {1}{x^2}+2 \int \frac {e^{-2 e^{-x} x-e^{-x} \left (-4+e^x\right ) x} \left (-3 e^x+2 x-6 x^2+2 x^3\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )} \, dx-2 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} \left (2 e^x-4 x+12 x^2+5 e^x x^2+6 x^3-30 x^4+10 x^5\right )}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ &=\frac {1}{x^2}+2 \int \frac {e^{-x} \left (3 e^x-2 x \left (1-3 x+x^2\right )\right )}{x^3 \left (2+e^{2 e^{-x} (-1+x) x} x-5 x^2\right )} \, dx-2 \int \left (\frac {6 e^{-e^{-x} \left (-4+e^x\right ) x}}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {2 e^{x-e^{-x} \left (-4+e^x\right ) x}}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}-\frac {4 e^{-e^{-x} \left (-4+e^x\right ) x}}{x^2 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {12 e^{-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {5 e^{x-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}-\frac {30 e^{-e^{-x} \left (-4+e^x\right ) x} x}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}+\frac {10 e^{-e^{-x} \left (-4+e^x\right ) x} x^2}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2}\right ) \, dx\\ &=\frac {1}{x^2}+2 \int \left (-\frac {2 e^{-x}}{2+e^{2 e^{-x} (-1+x) x} x-5 x^2}-\frac {3}{x^3 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )}+\frac {2 e^{-x}}{x^2 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )}-\frac {6 e^{-x}}{x \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )}\right ) \, dx-4 \int \frac {e^{x-e^{-x} \left (-4+e^x\right ) x}}{x^3 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx+8 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x^2 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-10 \int \frac {e^{x-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-12 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-20 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x^2}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-24 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx+60 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ &=\frac {1}{x^2}-4 \int \frac {e^{-x}}{2+e^{2 e^{-x} (-1+x) x} x-5 x^2} \, dx+4 \int \frac {e^{-x}}{x^2 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )} \, dx-4 \int \frac {e^{4 e^{-x} x}}{x^3 \left (e^{2 e^{-x} x^2} x+e^{2 e^{-x} x} \left (2-5 x^2\right )\right )^2} \, dx-6 \int \frac {1}{x^3 \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )} \, dx+8 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x^2 \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-10 \int \frac {e^{4 e^{-x} x}}{x \left (e^{2 e^{-x} x^2} x+e^{2 e^{-x} x} \left (2-5 x^2\right )\right )^2} \, dx-12 \int \frac {e^{-x}}{x \left (-2-e^{2 e^{-x} (-1+x) x} x+5 x^2\right )} \, dx-12 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-20 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x^2}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx-24 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x}}{x \left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx+60 \int \frac {e^{-e^{-x} \left (-4+e^x\right ) x} x}{\left (-2 e^{2 e^{-x} x}-e^{2 e^{-x} x^2} x+5 e^{2 e^{-x} x} x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [F]
time = 123.01, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-2 e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x-50 e^x x^3+e^{e^{-x} \left (-2 x+2 x^2\right )} \left (-4 x+12 x^2-4 x^3+e^x \left (-2+20 x^2\right )\right )}{e^{x+2 e^{-x} \left (-2 x+2 x^2\right )} x^4+e^{x+e^{-x} \left (-2 x+2 x^2\right )} \left (4 x^3-10 x^5\right )+e^x \left (4 x^2-20 x^4+25 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.09, size = 33, normalized size = 1.10
method | result | size |
risch | \(\frac {1}{x^{2}}+\frac {2}{x^{2} \left (5 x^{2}-{\mathrm e}^{2 x \left (x -1\right ) {\mathrm e}^{-x}} x -2\right )}\) | \(33\) |
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. 62 vs.
\(2 (27) = 54\).
time = 0.33, size = 62, normalized size = 2.07 \begin {gather*} -\frac {5 \, x e^{\left (2 \, x e^{\left (-x\right )}\right )} - e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )}}{x^{2} e^{\left (2 \, x^{2} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 68 vs.
\(2 (27) = 54\).
time = 0.37, size = 68, normalized size = 2.27 \begin {gather*} -\frac {5 \, x e^{x} - e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )}}{x^{2} e^{\left ({\left (2 \, x^{2} + x e^{x} - 2 \, x\right )} e^{\left (-x\right )}\right )} - {\left (5 \, x^{3} - 2 \, x\right )} e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 32, normalized size = 1.07 \begin {gather*} - \frac {2}{- 5 x^{4} + x^{3} e^{\left (2 x^{2} - 2 x\right ) e^{- x}} + 2 x^{2}} + \frac {1}{x^{2}} \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. 2753 vs.
\(2 (27) = 54\).
time = 0.57, size = 2753, normalized size = 91.77 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.75, size = 41, normalized size = 1.37 \begin {gather*} \frac {1}{x^2}-\frac {2}{2\,x^2-5\,x^4+x^3\,{\mathrm {e}}^{-2\,x\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{-x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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