Optimal. Leaf size=30 \[ \frac {5}{6+e^{4 e^{e^{3/x}}} \left (-1+e^{2/x}+x\right )} \]
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Rubi [F]
time = 36.19, 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 {e^{4 e^{e^{3/x}}} \left (10 e^{2/x}-5 x^2+e^{e^{3/x}+\frac {3}{x}} \left (-60+60 e^{2/x}+60 x\right )\right )}{36 x^2+e^{4 e^{e^{3/x}}} \left (-12 x^2+12 e^{2/x} x^2+12 x^3\right )+e^{8 e^{e^{3/x}}} \left (x^2+e^{4/x} x^2-2 x^3+x^4+e^{2/x} \left (-2 x^2+2 x^3\right )\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 {5 e^{4 e^{e^{3/x}}} \left (12 e^{e^{3/x}+\frac {5}{x}}+2 e^{2/x}+12 e^{e^{3/x}+\frac {3}{x}} (-1+x)-x^2\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right )^2 x^2} \, dx\\ &=5 \int \frac {e^{4 e^{e^{3/x}}} \left (12 e^{e^{3/x}+\frac {5}{x}}+2 e^{2/x}+12 e^{e^{3/x}+\frac {3}{x}} (-1+x)-x^2\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right )^2 x^2} \, dx\\ &=5 \int \left (\frac {12 e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2}-\frac {2 e^{-4 e^{e^{3/x}}} \left (-e^{4 e^{e^{3/x}}}+72 e^{e^{3/x}+\frac {1}{x}}-6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}-\frac {e^{-4 e^{e^{3/x}}} \left (12 e^{4 e^{e^{3/x}}}-2 e^{8 e^{e^{3/x}}}-432 e^{e^{3/x}+\frac {1}{x}}+72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+2 e^{8 e^{e^{3/x}}} x-72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x+e^{8 e^{e^{3/x}}} x^2\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}\right ) \, dx\\ &=-\left (5 \int \frac {e^{-4 e^{e^{3/x}}} \left (12 e^{4 e^{e^{3/x}}}-2 e^{8 e^{e^{3/x}}}-432 e^{e^{3/x}+\frac {1}{x}}+72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+2 e^{8 e^{e^{3/x}}} x-72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x+e^{8 e^{e^{3/x}}} x^2\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx\right )-10 \int \frac {e^{-4 e^{e^{3/x}}} \left (-e^{4 e^{e^{3/x}}}+72 e^{e^{3/x}+\frac {1}{x}}-6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}+6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} x\right )}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx+60 \int \frac {e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2} \, dx\\ &=-\left (5 \int \frac {e^{-4 e^{e^{3/x}}} \left (12 e^{4 e^{e^{3/x}}}-432 e^{e^{3/x}+\frac {1}{x}}-72 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} (-1+x)+e^{8 e^{e^{3/x}}} \left (-2+2 x+x^2\right )\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right )^2 x^2} \, dx\right )-10 \int \frac {e^{-4 e^{e^{3/x}}} \left (-e^{4 e^{e^{3/x}}}+72 e^{e^{3/x}+\frac {1}{x}}+6 e^{4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}} (-1+x)\right )}{\left (6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)\right ) x^2} \, dx-60 \text {Subst}\left (\int e^{-4 e^{e^{3 x}}+e^{3 x}+x} \, dx,x,\frac {1}{x}\right )\\ &=-\left (5 \int \left (\frac {e^{4 e^{e^{3/x}}}}{\left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}+\frac {12}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}-\frac {2 e^{4 e^{e^{3/x}}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}+\frac {72 e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}-\frac {432 e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}+\frac {2 e^{4 e^{e^{3/x}}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}-\frac {72 e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2}\right ) \, dx\right )-10 \int \left (-\frac {1}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}-\frac {6 e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}+\frac {72 e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}+\frac {6 e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )}\right ) \, dx-60 \text {Subst}\left (\int e^{-4 e^{x^3}+x^3} \, dx,x,e^{\frac {1}{x}}\right )\\ &=-\left (5 \int \frac {e^{4 e^{e^{3/x}}}}{\left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx\right )+10 \int \frac {e^{4 e^{e^{3/x}}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx-10 \int \frac {e^{4 e^{e^{3/x}}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx+10 \int \frac {1}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx-60 \int \frac {1}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx+60 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx-60 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx-60 \text {Subst}\left (\int e^{-4 e^{x^3}+x^3} \, dx,x,e^{\frac {1}{x}}\right )-360 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx+360 \int \frac {e^{e^{3/x}+\frac {1}{x}}}{x \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx-720 \int \frac {e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )} \, dx+2160 \int \frac {e^{-4 e^{e^{3/x}}+e^{3/x}+\frac {1}{x}}}{x^2 \left (6-e^{4 e^{e^{3/x}}}+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.15, size = 42, normalized size = 1.40 \begin {gather*} \frac {5}{6+e^{4 e^{e^{3/x}}+\frac {2}{x}}+e^{4 e^{e^{3/x}}} (-1+x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.49, size = 49, normalized size = 1.63
method | result | size |
risch | \(\frac {5}{{\mathrm e}^{\frac {4 \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x}}} x +2}{x}}+{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x}}}} x -{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{\frac {3}{x}}}}+6}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 26, normalized size = 0.87 \begin {gather*} \frac {5}{{\left (x + e^{\frac {2}{x}} - 1\right )} e^{\left (4 \, e^{\left (e^{\frac {3}{x}}\right )}\right )} + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 40, normalized size = 1.33 \begin {gather*} \frac {5}{{\left (x + e^{\frac {2}{x}} - 1\right )} e^{\left (4 \, e^{\left (\frac {x e^{\frac {3}{x}} + 3}{x} - \frac {3}{x}\right )}\right )} + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 20, normalized size = 0.67 \begin {gather*} \frac {5}{\left (x + e^{\frac {2}{x}} - 1\right ) e^{4 e^{e^{\frac {3}{x}}}} + 6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{3/x}}}\,\left (10\,{\mathrm {e}}^{2/x}-5\,x^2+{\mathrm {e}}^{{\mathrm {e}}^{3/x}}\,{\mathrm {e}}^{3/x}\,\left (60\,x+60\,{\mathrm {e}}^{2/x}-60\right )\right )}{{\mathrm {e}}^{8\,{\mathrm {e}}^{{\mathrm {e}}^{3/x}}}\,\left (x^2\,{\mathrm {e}}^{4/x}-{\mathrm {e}}^{2/x}\,\left (2\,x^2-2\,x^3\right )+x^2-2\,x^3+x^4\right )+{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{3/x}}}\,\left (12\,x^2\,{\mathrm {e}}^{2/x}-12\,x^2+12\,x^3\right )+36\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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