Optimal. Leaf size=27 \[ e^{-x+\left (\sqrt [676]{e}+\log \left (-4+\frac {4-x}{x}+x\right )\right )^2} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(27)=54\).
time = 1.74, antiderivative size = 146, normalized size of antiderivative = 5.41, number of steps
used = 3, number of rules used = 3, integrand size = 109, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {1608, 2306,
2326} \begin {gather*} \frac {\left (\frac {x^2-5 x+4}{x}\right )^{2 \sqrt [676]{e}} e^{\log ^2\left (\frac {x^2-5 x+4}{x}\right )-x+\sqrt [338]{e}} \left (x^3-5 x^2+2 \left (4-x^2\right ) \log \left (\frac {x^2-5 x+4}{x}\right )+4 x\right )}{x \left (x^2-5 x+4\right ) \left (\frac {2 x \left (\frac {x^2-5 x+4}{x^2}+\frac {5-2 x}{x}\right ) \log \left (\frac {x^2-5 x+4}{x}\right )}{x^2-5 x+4}+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 1608
Rule 2306
Rule 2326
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\sqrt [338]{e}-x+2 \sqrt [676]{e} \log \left (\frac {4-5 x+x^2}{x}\right )+\log ^2\left (\frac {4-5 x+x^2}{x}\right )\right ) \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{x \left (4-5 x+x^2\right )} \, dx\\ &=\int \frac {e^{\sqrt [338]{e}-x+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (\frac {4-5 x+x^2}{x}\right )^{2 \sqrt [676]{e}} \left (-4 x+5 x^2-x^3+\sqrt [676]{e} \left (-8+2 x^2\right )+\left (-8+2 x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{x \left (4-5 x+x^2\right )} \, dx\\ &=\frac {e^{\sqrt [338]{e}-x+\log ^2\left (\frac {4-5 x+x^2}{x}\right )} \left (\frac {4-5 x+x^2}{x}\right )^{2 \sqrt [676]{e}} \left (4 x-5 x^2+x^3+2 \left (4-x^2\right ) \log \left (\frac {4-5 x+x^2}{x}\right )\right )}{x \left (4-5 x+x^2\right ) \left (1+\frac {2 x \left (\frac {5-2 x}{x}+\frac {4-5 x+x^2}{x^2}\right ) \log \left (\frac {4-5 x+x^2}{x}\right )}{4-5 x+x^2}\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.19, size = 39, normalized size = 1.44 \begin {gather*} e^{\sqrt [338]{e}-x+\log ^2\left (-5+\frac {4}{x}+x\right )} \left (-5+\frac {4}{x}+x\right )^{2 \sqrt [676]{e}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 41, normalized size = 1.52
method | result | size |
risch | \(\left (\frac {x^{2}-5 x +4}{x}\right )^{2 \,{\mathrm e}^{\frac {1}{676}}} {\mathrm e}^{\ln \left (\frac {x^{2}-5 x +4}{x}\right )^{2}+{\mathrm e}^{\frac {1}{338}}-x}\) | \(41\) |
norman | \({\mathrm e}^{\ln \left (\frac {x^{2}-5 x +4}{x}\right )^{2}+2 \,{\mathrm e}^{\frac {1}{676}} \ln \left (\frac {x^{2}-5 x +4}{x}\right )+{\mathrm e}^{\frac {1}{338}}-x}\) | \(42\) |
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. 71 vs.
\(2 (22) = 44\).
time = 0.47, size = 71, normalized size = 2.63 \begin {gather*} e^{\left (2 \, e^{\frac {1}{676}} \log \left (x - 1\right ) + \log \left (x - 1\right )^{2} + 2 \, e^{\frac {1}{676}} \log \left (x - 4\right ) + 2 \, \log \left (x - 1\right ) \log \left (x - 4\right ) + \log \left (x - 4\right )^{2} - 2 \, e^{\frac {1}{676}} \log \left (x\right ) - 2 \, \log \left (x - 1\right ) \log \left (x\right ) - 2 \, \log \left (x - 4\right ) \log \left (x\right ) + \log \left (x\right )^{2} - x + e^{\frac {1}{338}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 39, normalized size = 1.44 \begin {gather*} e^{\left (2 \, e^{\frac {1}{676}} \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right ) + \log \left (\frac {x^{2} - 5 \, x + 4}{x}\right )^{2} - x + e^{\frac {1}{338}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 31, normalized size = 1.15 \begin {gather*} e^{\left (2 \, e^{\frac {1}{676}} \log \left (x + \frac {4}{x} - 5\right ) + \log \left (x + \frac {4}{x} - 5\right )^{2} - x + e^{\frac {1}{338}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.87, size = 37, normalized size = 1.37 \begin {gather*} {\mathrm {e}}^{-x}\,{\mathrm {e}}^{{\ln \left (\frac {x^2-5\,x+4}{x}\right )}^2}\,{\mathrm {e}}^{{\mathrm {e}}^{1/338}}\,{\left (x+\frac {4}{x}-5\right )}^{2\,{\mathrm {e}}^{1/676}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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