Optimal. Leaf size=30 \[ x+\frac {4}{3+\frac {3 e x}{\left (e^{\frac {x}{e^5}}-x\right ) (2+x)}} \]
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Rubi [F] time = 3.37, 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^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-16 x^2-6 x^3\right )\right )+e^{\frac {x}{e^5}} \left (e \left (8 x+4 x^2\right )+e^5 \left (-24 x-24 x^2-6 x^3+e \left (-8+12 x+6 x^2\right )\right )\right )}{e^{5+\frac {2 x}{e^5}} \left (12+12 x+3 x^2\right )+e^{5+\frac {x}{e^5}} \left (-24 x-24 x^2-6 x^3+e \left (12 x+6 x^2\right )\right )+e^5 \left (12 x^2+3 e^2 x^2+12 x^3+3 x^4+e \left (-12 x^2-6 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 {3 e^6 x^2+4 e^{\frac {x}{e^5}} x (2+x)+3 e^{4+\frac {2 x}{e^5}} (2+x)^2-6 e^{4+\frac {x}{e^5}} x (2+x)^2+3 e^4 x^2 (2+x)^2-2 e^5 x^2 (8+3 x)+2 e^{5+\frac {x}{e^5}} \left (-4+6 x+3 x^2\right )}{3 e^4 \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx\\ &=\frac {\int \frac {3 e^6 x^2+4 e^{\frac {x}{e^5}} x (2+x)+3 e^{4+\frac {2 x}{e^5}} (2+x)^2-6 e^{4+\frac {x}{e^5}} x (2+x)^2+3 e^4 x^2 (2+x)^2-2 e^5 x^2 (8+3 x)+2 e^{5+\frac {x}{e^5}} \left (-4+6 x+3 x^2\right )}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}\\ &=\frac {\int \left (3 e^4+\frac {4 \left (-2 e^5+2 x+x^2\right )}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )}+\frac {4 x \left (-2 (2-e) e^5+2 \left (2-e-2 e^5\right ) x+\left (4-e-e^5\right ) x^2+x^3\right )}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}\right ) \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {-2 e^5+2 x+x^2}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )} \, dx}{3 e^4}+\frac {4 \int \frac {x \left (-2 (2-e) e^5+2 \left (2-e-2 e^5\right ) x+\left (4-e-e^5\right ) x^2+x^3\right )}{(2+x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {x \left (-2 (2-e) e^5+2 \left (2-e-2 e^5\right ) x+\left (4-e-e^5\right ) x^2+x^3\right )}{(2+x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}+\frac {4 \int \frac {-2 e^5+2 x+x^2}{(2+x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )} \, dx}{3 e^4}\\ &=x+\frac {4 \int \left (\frac {2 e^6}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}+\frac {4 e^6}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}-\frac {2 e^5 x}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}+\frac {\left (2-e-e^5\right ) x^2}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}+\frac {x^3}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2}\right ) \, dx}{3 e^4}+\frac {4 \int \left (\frac {2 e^5}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )}+\frac {x}{2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2}\right ) \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {x^3}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx}{3 e^4}+\frac {4 \int \frac {x}{2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2} \, dx}{3 e^4}-\frac {1}{3} (8 e) \int \frac {x}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx+\frac {1}{3} (8 e) \int \frac {1}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )} \, dx+\frac {1}{3} \left (8 e^2\right ) \int \frac {1}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx+\frac {1}{3} \left (16 e^2\right ) \int \frac {1}{(-2-x) \left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx+\frac {\left (4 \left (2-e-e^5\right )\right ) \int \frac {x^2}{\left (2 e^{\frac {x}{e^5}}-2 \left (1-\frac {e}{2}\right ) x+e^{\frac {x}{e^5}} x-x^2\right )^2} \, dx}{3 e^4}\\ &=x+\frac {4 \int \frac {x^3}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}+\frac {4 \int \frac {x}{e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)} \, dx}{3 e^4}-\frac {1}{3} (8 e) \int \frac {x}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx+\frac {1}{3} (8 e) \int \frac {1}{(-2-x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )} \, dx+\frac {1}{3} \left (8 e^2\right ) \int \frac {1}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx+\frac {1}{3} \left (16 e^2\right ) \int \frac {1}{(-2-x) \left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx+\frac {\left (4 \left (2-e-e^5\right )\right ) \int \frac {x^2}{\left (e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)\right )^2} \, dx}{3 e^4}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.12, size = 33, normalized size = 1.10 \begin {gather*} \frac {1}{3} x \left (3-\frac {4 e}{e x+e^{\frac {x}{e^5}} (2+x)-x (2+x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.82, size = 79, normalized size = 2.63 \begin {gather*} \frac {{\left (3 \, x^{2} - 4 \, x\right )} e^{6} - 3 \, {\left (x^{3} + 2 \, x^{2}\right )} e^{5} + 3 \, {\left (x^{2} + 2 \, x\right )} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}}{3 \, {\left (x e^{6} - {\left (x^{2} + 2 \, x\right )} e^{5} + {\left (x + 2\right )} e^{\left ({\left (x + 5 \, e^{5}\right )} e^{\left (-5\right )}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 551, normalized size = 18.37 \begin {gather*} \frac {3 \, x^{6} e^{4} - 3 \, x^{5} e^{9} - 6 \, x^{5} e^{5} + 18 \, x^{5} e^{4} - 3 \, x^{5} e^{\left (x e^{\left (-5\right )} + 4\right )} + 3 \, x^{4} e^{10} - 18 \, x^{4} e^{9} + 3 \, x^{4} e^{6} - 20 \, x^{4} e^{5} + 36 \, x^{4} e^{4} + 3 \, x^{4} e^{\left (x e^{\left (-5\right )} + 9\right )} + 3 \, x^{4} e^{\left (x e^{\left (-5\right )} + 5\right )} - 18 \, x^{4} e^{\left (x e^{\left (-5\right )} + 4\right )} + 14 \, x^{3} e^{10} - 36 \, x^{3} e^{9} + 2 \, x^{3} e^{6} - 8 \, x^{3} e^{5} + 24 \, x^{3} e^{4} + 18 \, x^{3} e^{\left (x e^{\left (-5\right )} + 9\right )} + 12 \, x^{3} e^{\left (x e^{\left (-5\right )} + 5\right )} - 36 \, x^{3} e^{\left (x e^{\left (-5\right )} + 4\right )} - 6 \, x^{2} e^{11} + 8 \, x^{2} e^{10} - 24 \, x^{2} e^{9} - 8 \, x^{2} e^{6} + 16 \, x^{2} e^{5} - 6 \, x^{2} e^{\left (x e^{\left (-5\right )} + 10\right )} + 36 \, x^{2} e^{\left (x e^{\left (-5\right )} + 9\right )} + 12 \, x^{2} e^{\left (x e^{\left (-5\right )} + 5\right )} - 24 \, x^{2} e^{\left (x e^{\left (-5\right )} + 4\right )} + 8 \, x e^{11} - 16 \, x e^{10} - 12 \, x e^{\left (x e^{\left (-5\right )} + 10\right )} + 24 \, x e^{\left (x e^{\left (-5\right )} + 9\right )}}{3 \, {\left (x^{5} e^{4} - x^{4} e^{9} - 2 \, x^{4} e^{5} + 6 \, x^{4} e^{4} - x^{4} e^{\left (x e^{\left (-5\right )} + 4\right )} + x^{3} e^{10} - 6 \, x^{3} e^{9} + x^{3} e^{6} - 8 \, x^{3} e^{5} + 12 \, x^{3} e^{4} + x^{3} e^{\left (x e^{\left (-5\right )} + 9\right )} + x^{3} e^{\left (x e^{\left (-5\right )} + 5\right )} - 6 \, x^{3} e^{\left (x e^{\left (-5\right )} + 4\right )} + 6 \, x^{2} e^{10} - 12 \, x^{2} e^{9} + 2 \, x^{2} e^{6} - 8 \, x^{2} e^{5} + 8 \, x^{2} e^{4} + 6 \, x^{2} e^{\left (x e^{\left (-5\right )} + 9\right )} + 4 \, x^{2} e^{\left (x e^{\left (-5\right )} + 5\right )} - 12 \, x^{2} e^{\left (x e^{\left (-5\right )} + 4\right )} - 2 \, x e^{11} + 8 \, x e^{10} - 8 \, x e^{9} - 2 \, x e^{\left (x e^{\left (-5\right )} + 10\right )} + 12 \, x e^{\left (x e^{\left (-5\right )} + 9\right )} + 4 \, x e^{\left (x e^{\left (-5\right )} + 5\right )} - 8 \, x e^{\left (x e^{\left (-5\right )} + 4\right )} - 4 \, e^{\left (x e^{\left (-5\right )} + 10\right )} + 8 \, e^{\left (x e^{\left (-5\right )} + 9\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 37, normalized size = 1.23
method | result | size |
risch | \(x -\frac {4 \,{\mathrm e} x}{3 \left (x \,{\mathrm e}-x^{2}+x \,{\mathrm e}^{x \,{\mathrm e}^{-5}}-2 x +2 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}\right )}\) | \(37\) |
norman | \(\frac {\left (2 \,{\mathrm e}-4\right ) {\mathrm e}^{x \,{\mathrm e}^{-5}}+\left ({\mathrm e}^{2}-\frac {16 \,{\mathrm e}}{3}+4\right ) x +{\mathrm e}^{x \,{\mathrm e}^{-5}} x^{2}+{\mathrm e}^{x \,{\mathrm e}^{-5}} {\mathrm e} x -x^{3}}{x \,{\mathrm e}-x^{2}+x \,{\mathrm e}^{x \,{\mathrm e}^{-5}}-2 x +2 \,{\mathrm e}^{x \,{\mathrm e}^{-5}}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 59, normalized size = 1.97 \begin {gather*} \frac {3 \, x^{3} - 3 \, x^{2} {\left (e - 2\right )} + 4 \, x e - 3 \, {\left (x^{2} + 2 \, x\right )} e^{\left (x e^{\left (-5\right )}\right )}}{3 \, {\left (x^{2} - x {\left (e - 2\right )} - {\left (x + 2\right )} e^{\left (x e^{\left (-5\right )}\right )}\right )}} \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}}^{x\,{\mathrm {e}}^{-5}}\,\left (\mathrm {e}\,\left (4\,x^2+8\,x\right )-{\mathrm {e}}^5\,\left (24\,x-\mathrm {e}\,\left (6\,x^2+12\,x-8\right )+24\,x^2+6\,x^3\right )\right )+{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-5}+5}\,\left (3\,x^2+12\,x+12\right )+{\mathrm {e}}^5\,\left (3\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (6\,x^3+16\,x^2\right )+12\,x^2+12\,x^3+3\,x^4\right )}{{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-5}+5}\,\left (3\,x^2+12\,x+12\right )-{\mathrm {e}}^{x\,{\mathrm {e}}^{-5}+5}\,\left (24\,x-\mathrm {e}\,\left (6\,x^2+12\,x\right )+24\,x^2+6\,x^3\right )+{\mathrm {e}}^5\,\left (3\,x^2\,{\mathrm {e}}^2-\mathrm {e}\,\left (6\,x^3+12\,x^2\right )+12\,x^2+12\,x^3+3\,x^4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 34, normalized size = 1.13 \begin {gather*} x - \frac {4 e x}{- 3 x^{2} - 6 x + 3 e x + \left (3 x + 6\right ) e^{\frac {x}{e^{5}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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