Integrand size = 167, antiderivative size = 23 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{x^2 \left (1-5 e^{-x}+x+e^4 x\right )^2} \]
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\[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=\int \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (50 \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x-50 \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x^2+10 \exp \left (-x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (-2-\left (2+3 e^4\right ) x+\left (1+e^4\right ) x^2\right )+2 \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (1+3 \left (1+e^4\right ) x+2 \left (1+e^4\right )^2 x^2\right )\right ) \, dx \\ & = 2 \int \exp \left (e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (1+3 \left (1+e^4\right ) x+2 \left (1+e^4\right )^2 x^2\right ) \, dx+10 \int \exp \left (-x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \left (-2-\left (2+3 e^4\right ) x+\left (1+e^4\right ) x^2\right ) \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )\right ) x^2 \, dx \\ & = 2 \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x \left (1+3 \left (1+e^4\right ) x+2 \left (1+e^4\right )^2 x^2\right ) \, dx+10 \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \left (-2-\left (2+3 e^4\right ) x+\left (1+e^4\right ) x^2\right ) \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx \\ & = 2 \int \left (\exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x+3 \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) \left (1+e^4\right ) x^2+2 \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) \left (1+e^4\right )^2 x^3\right ) \, dx+10 \int \left (-2 \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x-\exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) \left (2+3 e^4\right ) x^2+\exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) \left (1+e^4\right ) x^3\right ) \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx \\ & = 2 \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x \, dx-20 \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx+50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x \, dx-50 \int \exp \left (-2 x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx+\left (6 \left (1+e^4\right )\right ) \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x^2 \, dx+\left (10 \left (1+e^4\right )\right ) \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^3 \, dx+\left (4 \left (1+e^4\right )^2\right ) \int \exp \left (e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2\right ) x^3 \, dx-\left (10 \left (2+3 e^4\right )\right ) \int \exp \left (-x+e^{-2 x} \left (-5+e^x\right )^2 x^2+2 e^{-x} \left (1+e^4\right ) \left (-5+e^x\right ) x^3+\left (1+e^4\right )^2 x^4\right ) x^2 \, dx \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{e^{-2 x} x^2 \left (-5+e^{4+x} x+e^x (1+x)\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(20)=40\).
Time = 1.57 (sec) , antiderivative size = 73, normalized size of antiderivative = 3.17
method | result | size |
norman | \({\mathrm e}^{\left (\left (x^{4} {\mathrm e}^{8}+\left (2 x^{4}+2 x^{3}\right ) {\mathrm e}^{4}+x^{4}+2 x^{3}+x^{2}\right ) {\mathrm e}^{2 x}+\left (-10 x^{3} {\mathrm e}^{4}-10 x^{3}-10 x^{2}\right ) {\mathrm e}^{x}+25 x^{2}\right ) {\mathrm e}^{-2 x}}\) | \(73\) |
parallelrisch | \({\mathrm e}^{\left (\left (x^{4} {\mathrm e}^{8}+\left (2 x^{4}+2 x^{3}\right ) {\mathrm e}^{4}+x^{4}+2 x^{3}+x^{2}\right ) {\mathrm e}^{2 x}+\left (-10 x^{3} {\mathrm e}^{4}-10 x^{3}-10 x^{2}\right ) {\mathrm e}^{x}+25 x^{2}\right ) {\mathrm e}^{-2 x}}\) | \(73\) |
risch | \({\mathrm e}^{-x^{2} \left (-2 x^{2} {\mathrm e}^{4+2 x}-x^{2} {\mathrm e}^{2 x +8}+10 x \,{\mathrm e}^{4+x}-2 x \,{\mathrm e}^{4+2 x}-{\mathrm e}^{2 x} x^{2}+10 \,{\mathrm e}^{x} x -2 x \,{\mathrm e}^{2 x}+10 \,{\mathrm e}^{x}-{\mathrm e}^{2 x}-25\right ) {\mathrm e}^{-2 x}}\) | \(82\) |
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Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (20) = 40\).
Time = 0.26 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{\left ({\left (25 \, x^{2} + {\left (x^{4} e^{8} + x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{4} + x^{3}\right )} e^{4} - 2 \, x\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{3} e^{4} + x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (-2 \, x\right )} + 2 \, x\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (19) = 38\).
Time = 0.30 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{\left (25 x^{2} + \left (- 10 x^{3} e^{4} - 10 x^{3} - 10 x^{2}\right ) e^{x} + \left (x^{4} + x^{4} e^{8} + 2 x^{3} + x^{2} + \left (2 x^{4} + 2 x^{3}\right ) e^{4}\right ) e^{2 x}\right ) e^{- 2 x}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (20) = 40\).
Time = 0.62 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.09 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=e^{\left (x^{4} e^{8} + 2 \, x^{4} e^{4} + x^{4} + 2 \, x^{3} e^{4} - 10 \, x^{3} e^{\left (-x\right )} - 10 \, x^{3} e^{\left (-x + 4\right )} + 2 \, x^{3} - 10 \, x^{2} e^{\left (-x\right )} + 25 \, x^{2} e^{\left (-2 \, x\right )} + x^{2}\right )} \]
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\[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx=\int { -2 \, {\left (25 \, x^{2} - {\left (2 \, x^{3} e^{8} + 2 \, x^{3} + 3 \, x^{2} + {\left (4 \, x^{3} + 3 \, x^{2}\right )} e^{4} + x\right )} e^{\left (2 \, x\right )} - 5 \, {\left (x^{3} - 2 \, x^{2} + {\left (x^{3} - 3 \, x^{2}\right )} e^{4} - 2 \, x\right )} e^{x} - 25 \, x\right )} e^{\left ({\left (25 \, x^{2} + {\left (x^{4} e^{8} + x^{4} + 2 \, x^{3} + x^{2} + 2 \, {\left (x^{4} + x^{3}\right )} e^{4}\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{3} e^{4} + x^{3} + x^{2}\right )} e^{x}\right )} e^{\left (-2 \, x\right )} - 2 \, x\right )} \,d x } \]
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Time = 8.42 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.48 \[ \int e^{-2 x+e^{-2 x} \left (25 x^2+e^x \left (-10 x^2-10 x^3-10 e^4 x^3\right )+e^{2 x} \left (x^2+2 x^3+x^4+e^8 x^4+e^4 \left (2 x^3+2 x^4\right )\right )\right )} \left (50 x-50 x^2+e^{2 x} \left (2 x+6 x^2+4 x^3+4 e^8 x^3+e^4 \left (6 x^2+8 x^3\right )\right )+e^x \left (-20 x-20 x^2+10 x^3+e^4 \left (-30 x^2+10 x^3\right )\right )\right ) \, dx={\mathrm {e}}^{2\,x^3\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x^4\,{\mathrm {e}}^4}\,{\mathrm {e}}^{x^4\,{\mathrm {e}}^8}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x^3}\,{\mathrm {e}}^{-10\,x^2\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{-10\,x^3\,{\mathrm {e}}^{-x}}\,{\mathrm {e}}^{25\,x^2\,{\mathrm {e}}^{-2\,x}} \]
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