Optimal. Leaf size=27 \[ e^{\frac {\left (-2+x+x^2\right )^2}{\left (1-e^{e^8 x}\right )^2 x}} \]
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Rubi [F]
time = 26.59, 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 {4-4 x-3 x^2+2 x^3+x^4}{x-2 e^{e^8 x} x+e^{2 e^8 x} x}\right ) \left (4+3 x^2-4 x^3-3 x^4+e^{e^8 x} \left (-4-3 x^2+4 x^3+3 x^4+e^8 \left (-8 x+8 x^2+6 x^3-4 x^4-2 x^5\right )\right )\right )}{-x^2+3 e^{e^8 x} x^2-3 e^{2 e^8 x} x^2+e^{3 e^8 x} x^2} \, 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^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (2-x-x^2\right ) \left (-2-x-3 x^2-2 e^{8+e^8 x} x \left (-2+x+x^2\right )+e^{e^8 x} \left (2+x+3 x^2\right )\right )}{\left (1-e^{e^8 x}\right )^3 x^2} \, dx\\ &=\int \left (-\frac {2 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^3 x}+\frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-4-8 e^8 x-3 \left (1-\frac {8 e^8}{3}\right ) x^2+4 \left (1+\frac {3 e^8}{2}\right ) x^3+3 \left (1-\frac {4 e^8}{3}\right ) x^4-2 e^8 x^5\right )}{\left (1-e^{e^8 x}\right )^2 x^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^3 x} \, dx\right )+\int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-4-8 e^8 x-3 \left (1-\frac {8 e^8}{3}\right ) x^2+4 \left (1+\frac {3 e^8}{2}\right ) x^3+3 \left (1-\frac {4 e^8}{3}\right ) x^4-2 e^8 x^5\right )}{\left (1-e^{e^8 x}\right )^2 x^2} \, dx\\ &=-\left (2 \int \left (-\frac {4 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3}+\frac {4 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3 x}-\frac {3 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^3}+\frac {2 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^3}+\frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^3}\right ) \, dx\right )+\int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-4-3 x^2+4 x^3+3 x^4-2 e^8 x \left (-2+x+x^2\right )^2\right )}{\left (1-e^{e^8 x}\right )^2 x^2} \, dx\\ &=-\left (2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^3} \, dx\right )-4 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^3} \, dx+6 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^3} \, dx+8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3} \, dx-8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3 x} \, dx+\int \left (-\frac {3 e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (1-\frac {8 e^8}{3}\right )}{\left (-1+e^{e^8 x}\right )^2}-\frac {4 e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x^2}-\frac {8 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x}+\frac {2 e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (2+3 e^8\right ) x}{\left (-1+e^{e^8 x}\right )^2}-\frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \left (-3+4 e^8\right ) x^2}{\left (-1+e^{e^8 x}\right )^2}-\frac {2 e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^3} \, dx\right )-2 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^3}{\left (-1+e^{e^8 x}\right )^2} \, dx-4 \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x^2} \, dx-4 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^3} \, dx+6 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^3} \, dx+8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3} \, dx-8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^3 x} \, dx-8 \int \frac {e^{8+\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2 x} \, dx-\left (3-8 e^8\right ) \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}}}{\left (-1+e^{e^8 x}\right )^2} \, dx+\left (3-4 e^8\right ) \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x^2}{\left (-1+e^{e^8 x}\right )^2} \, dx+\left (2 \left (2+3 e^8\right )\right ) \int \frac {e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} x}{\left (-1+e^{e^8 x}\right )^2} \, dx\\ \end {aligned} 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Mathematica [A]
time = 0.09, size = 25, normalized size = 0.93 \begin {gather*} e^{\frac {\left (-2+x+x^2\right )^2}{\left (-1+e^{e^8 x}\right )^2 x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 33, normalized size = 1.22
method | result | size |
risch | \({\mathrm e}^{\frac {\left (2+x \right )^{2} \left (x -1\right )^{2}}{x \left ({\mathrm e}^{2 x \,{\mathrm e}^{8}}-2 \,{\mathrm e}^{x \,{\mathrm e}^{8}}+1\right )}}\) | \(33\) |
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. 106 vs.
\(2 (22) = 44\).
time = 0.43, size = 106, normalized size = 3.93 \begin {gather*} e^{\left (\frac {x^{3}}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} + \frac {2 \, x^{2}}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} - \frac {3 \, x}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1} + \frac {4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x} - \frac {4}{e^{\left (2 \, x e^{8}\right )} - 2 \, e^{\left (x e^{8}\right )} + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 40, normalized size = 1.48 \begin {gather*} e^{\left (\frac {x^{4} + 2 \, x^{3} - 3 \, x^{2} - 4 \, x + 4}{x e^{\left (2 \, x e^{8}\right )} - 2 \, x e^{\left (x e^{8}\right )} + x}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (20) = 40\).
time = 0.46, size = 41, normalized size = 1.52 \begin {gather*} e^{\frac {x^{4} + 2 x^{3} - 3 x^{2} - 4 x + 4}{x e^{2 x e^{8}} - 2 x e^{x e^{8}} + x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.22, size = 110, normalized size = 4.07 \begin {gather*} {\mathrm {e}}^{-\frac {4}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{-\frac {3\,x}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{\frac {4}{x-2\,x\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}}}\,{\mathrm {e}}^{\frac {x^3}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}}\,{\mathrm {e}}^{\frac {2\,x^2}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^8}-2\,{\mathrm {e}}^{x\,{\mathrm {e}}^8}+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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