Integrand size = 150, antiderivative size = 26 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \]
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Time = 0.81 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {6820, 12, 6818} \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (2 x+e^{3 x}-2 e^{e^{-e^{4 x}} x}\right )^2 \]
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Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int 2 e^{-e^{4 x}} \left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right ) \left (2 e^{e^{4 x}}-2 e^{e^{-e^{4 x}} x}+3 e^{e^{4 x}+3 x}+8 e^{\left (4+e^{-e^{4 x}}\right ) x} x\right ) \, dx \\ & = 2 \int e^{-e^{4 x}} \left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right ) \left (2 e^{e^{4 x}}-2 e^{e^{-e^{4 x}} x}+3 e^{e^{4 x}+3 x}+8 e^{\left (4+e^{-e^{4 x}}\right ) x} x\right ) \, dx \\ & = \left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\left (e^{3 x}-2 e^{e^{-e^{4 x}} x}+2 x\right )^2 \]
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Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(25)=50\).
Time = 0.37 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.04
method | result | size |
risch | \({\mathrm e}^{6 x}+4 x \,{\mathrm e}^{3 x}+4 x^{2}+4 \,{\mathrm e}^{2 \,{\mathrm e}^{-{\mathrm e}^{4 x}} x}+2 \left (-2 \,{\mathrm e}^{3 x}-4 x \right ) {\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{4 x}} x}\) | \(53\) |
parallelrisch | \(4 x^{2}+4 x \,{\mathrm e}^{3 x}-4 \,{\mathrm e}^{\left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}} x +{\mathrm e}^{6 x}-2 \,{\mathrm e}^{3 x} {\mathrm e}^{\left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}}+{\mathrm e}^{2 \left (\ln \left (2\right ) {\mathrm e}^{{\mathrm e}^{4 x}}+x \right ) {\mathrm e}^{-{\mathrm e}^{4 x}}}\) | \(88\) |
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Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.54 \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=4 \, x^{2} - 2 \, {\left (2 \, x + e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} + 4 \, x e^{\left (3 \, x\right )} + e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} + e^{\left (6 \, x\right )} \]
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Timed out. \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\text {Timed out} \]
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\[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\int { -2 \, {\left ({\left (4 \, x e^{\left (4 \, x\right )} - 1\right )} e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (4 \, {\left (2 \, x^{2} + x e^{\left (3 \, x\right )}\right )} e^{\left (4 \, x\right )} - {\left (3 \, e^{\left (3 \, x\right )} + 2\right )} e^{\left (e^{\left (4 \, x\right )}\right )} - 2 \, x - e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (3 \, x\right )} + 4 \, x + 3 \, e^{\left (6 \, x\right )}\right )} e^{\left (e^{\left (4 \, x\right )}\right )}\right )} e^{\left (-e^{\left (4 \, x\right )}\right )} \,d x } \]
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\[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=\int { -2 \, {\left ({\left (4 \, x e^{\left (4 \, x\right )} - 1\right )} e^{\left (2 \, {\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (4 \, {\left (2 \, x^{2} + x e^{\left (3 \, x\right )}\right )} e^{\left (4 \, x\right )} - {\left (3 \, e^{\left (3 \, x\right )} + 2\right )} e^{\left (e^{\left (4 \, x\right )}\right )} - 2 \, x - e^{\left (3 \, x\right )}\right )} e^{\left ({\left (e^{\left (e^{\left (4 \, x\right )}\right )} \log \left (2\right ) + x\right )} e^{\left (-e^{\left (4 \, x\right )}\right )}\right )} - {\left (2 \, {\left (3 \, x + 1\right )} e^{\left (3 \, x\right )} + 4 \, x + 3 \, e^{\left (6 \, x\right )}\right )} e^{\left (e^{\left (4 \, x\right )}\right )}\right )} e^{\left (-e^{\left (4 \, x\right )}\right )} \,d x } \]
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Timed out. \[ \int e^{-e^{4 x}} \left (e^{2 e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (2-8 e^{4 x} x\right )+e^{e^{4 x}} \left (6 e^{6 x}+8 x+e^{3 x} (4+12 x)\right )+e^{e^{-e^{4 x}} \left (x+e^{e^{4 x}} \log (2)\right )} \left (-2 e^{3 x}+e^{e^{4 x}} \left (-4-6 e^{3 x}\right )-4 x+e^{4 x} \left (8 e^{3 x} x+16 x^2\right )\right )\right ) \, dx=-\int {\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left ({\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\ln \left (2\right )\right )}\,\left (4\,x+2\,{\mathrm {e}}^{3\,x}-{\mathrm {e}}^{4\,x}\,\left (8\,x\,{\mathrm {e}}^{3\,x}+16\,x^2\right )+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (6\,{\mathrm {e}}^{3\,x}+4\right )\right )-{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\left (8\,x+6\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{3\,x}\,\left (12\,x+4\right )\right )+{\mathrm {e}}^{2\,{\mathrm {e}}^{-{\mathrm {e}}^{4\,x}}\,\left (x+{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}\,\ln \left (2\right )\right )}\,\left (8\,x\,{\mathrm {e}}^{4\,x}-2\right )\right ) \,d x \]
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