Optimal. Leaf size=33 \[ \frac {-4 x+\log \left (e^{e^{x^2 (-5+\log (x))}}+\frac {6}{x}\right )}{x-x^2} \]
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Rubi [F]
time = 15.31, 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 {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{6 x^2-12 x^3+6 x^4+e^{e^{-5 x^2+x^2 \log (x)}} \left (x^3-2 x^4+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 {-6+6 x-24 x^2+e^{e^{-5 x^2+x^2 \log (x)}} \left (-4 x^3+e^{-5 x^2+x^2 \log (x)} \left (-9 x^3+9 x^4+\left (2 x^3-2 x^4\right ) \log (x)\right )\right )+\left (-6+12 x+e^{e^{-5 x^2+x^2 \log (x)}} \left (-x+2 x^2\right )\right ) \log \left (\frac {6+e^{e^{-5 x^2+x^2 \log (x)}} x}{x}\right )}{(1-x)^2 x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\\ &=\int \left (-\frac {24}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {6}{(-1+x)^2 x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {6}{(-1+x)^2 x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {4 e^{e^{-5 x^2} x^{x^2}} x}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {(-1+2 x) \log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x)^2 x^2}-\frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{e^{-5 x^2} x^{x^2}} x}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )-6 \int \frac {1}{(-1+x)^2 x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{(-1+x)^2 x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+\int \frac {(-1+2 x) \log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x)^2 x^2} \, dx-\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\\ &=\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}-4 \int \left (\frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx+6 \int \left (\frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx-6 \int \left (\frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {2}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {2}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-\int \frac {e^{-5 x^2} \left (-6 e^{5 x^2}-9 e^{e^{-5 x^2} x^{x^2}} x^{3+x^2}+2 e^{e^{-5 x^2} x^{x^2}} x^{3+x^2} \log (x)\right )}{(1-x) x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-\int \left (-\frac {9 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {2 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx\\ &=\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}-2 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+9 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-\int \left (\frac {6}{(-1+x) x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx\\ &=\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}+2 \int \frac {\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx}{x} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+9 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-(2 \log (x)) \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} (-9+2 \log (x))}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\\ &=\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}+2 \int \frac {\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx}{x} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-6 \int \left (\frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {1}{x^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}-\frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx+9 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-(2 \log (x)) \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+\int \left (-\frac {9 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}+\frac {2 e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )}\right ) \, dx\\ &=\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}+2 \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2} \log (x)}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx+2 \int \frac {\int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx}{x} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-2 \left (6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+2 \left (6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-(2 \log (x)) \int \frac {e^{-5 x^2+e^{-5 x^2} x^{x^2}} x^{1+x^2}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\\ &=\frac {\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(1-x) x}-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-4 \int \frac {e^{e^{-5 x^2} x^{x^2}}}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-2 \left (6 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+2 \left (6 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\right )+12 \int \frac {1}{(-1+x) \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-12 \int \frac {1}{x \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx-24 \int \frac {1}{(-1+x)^2 \left (6+e^{e^{-5 x^2} x^{x^2}} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.32, size = 37, normalized size = 1.12 \begin {gather*} \frac {4 x-\log \left (e^{e^{-5 x^2} x^{x^2}}+\frac {6}{x}\right )}{(-1+x) x} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 0.75, size = 222, normalized size = 6.73
method | result | size |
risch | \(-\frac {\ln \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )}{x \left (x -1\right )}+\frac {-i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )\right ) \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )}{x}\right ) \mathrm {csgn}\left (\frac {i}{x}\right )+i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )}{x}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i \left (x \,{\mathrm e}^{x^{x^{2}} {\mathrm e}^{-5 x^{2}}}+6\right )}{x}\right )^{2} \mathrm {csgn}\left (\frac {i}{x}\right )+8 x +2 \ln \left (x \right )}{2 x \left (x -1\right )}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 37, normalized size = 1.12 \begin {gather*} \frac {4 \, x - \log \left (x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6\right ) + \log \left (x\right )}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 39, normalized size = 1.18 \begin {gather*} \frac {4 \, x - \log \left (\frac {x e^{\left (e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6}{x}\right )}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 15.79, size = 31, normalized size = 0.94 \begin {gather*} - \frac {\log {\left (\frac {x e^{e^{x^{2} \log {\left (x \right )} - 5 x^{2}}} + 6}{x} \right )}}{x^{2} - x} + \frac {4}{x - 1} \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. 78 vs.
\(2 (33) = 66\).
time = 1.06, size = 78, normalized size = 2.36 \begin {gather*} \frac {4 \, x - \log \left ({\left (x e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2} + e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} + 6 \, e^{\left (x^{2} \log \left (x\right ) - 5 \, x^{2}\right )}\right )} e^{\left (-x^{2} \log \left (x\right ) + 5 \, x^{2}\right )}\right ) + \log \left (x\right )}{x^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.85, size = 37, normalized size = 1.12 \begin {gather*} \frac {4\,x-\ln \left (\frac {x\,{\mathrm {e}}^{x^{x^2}\,{\mathrm {e}}^{-5\,x^2}}+6}{x}\right )}{x\,\left (x-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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