Optimal. Leaf size=31 \[ \frac {e^x}{2-e^{-e^{(5-x)^2}+x}}+\frac {x}{2} \]
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Rubi [F] time = 4.06, 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^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{8 e^{2 e^{25-10 x+x^2}}+2 e^{2 x}-8 e^{e^{25-10 x+x^2}+x}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{2 \left (2 e^{e^{(-5+x)^2}}-e^x\right )^2} \, dx\\ &=\frac {1}{2} \int \frac {e^{2 x}+e^{2 e^{25-10 x+x^2}} \left (4+4 e^x\right )+e^{e^{25-10 x+x^2}} \left (-4 e^x+e^{25-8 x+x^2} (20-4 x)\right )}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (\frac {4 e^{2 e^{(-5+x)^2}}+e^{2 x}-4 e^{e^{(-5+x)^2}+x}+4 e^{2 e^{(-5+x)^2}+x}}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2}-\frac {4 e^{25+e^{(-5+x)^2}-8 x+x^2} (-5+x)}{\left (-2 e^{e^{(-5+x)^2}}+e^x\right )^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {4 e^{2 e^{(-5+x)^2}}+e^{2 x}-4 e^{e^{(-5+x)^2}+x}+4 e^{2 e^{(-5+x)^2}+x}}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2} \, dx-2 \int \frac {e^{25+e^{(-5+x)^2}-8 x+x^2} (-5+x)}{\left (-2 e^{e^{(-5+x)^2}}+e^x\right )^2} \, dx\\ &=\frac {1}{2} \int \left (1+\frac {8 e^{3 e^{25-10 x+x^2}}}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2}+\frac {4 e^{2 e^{25-10 x+x^2}}}{-2 e^{e^{(-5+x)^2}}+e^x}\right ) \, dx-2 \int \left (-\frac {5 e^{25+e^{(-5+x)^2}-8 x+x^2}}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2}+\frac {e^{25+e^{(-5+x)^2}-8 x+x^2} x}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2}\right ) \, dx\\ &=\frac {x}{2}+2 \int \frac {e^{2 e^{25-10 x+x^2}}}{-2 e^{e^{(-5+x)^2}}+e^x} \, dx-2 \int \frac {e^{25+e^{(-5+x)^2}-8 x+x^2} x}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2} \, dx+4 \int \frac {e^{3 e^{25-10 x+x^2}}}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2} \, dx+10 \int \frac {e^{25+e^{(-5+x)^2}-8 x+x^2}}{\left (2 e^{e^{(-5+x)^2}}-e^x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.22, size = 35, normalized size = 1.13 \begin {gather*} \frac {1}{2} \left (e^x+\frac {e^x}{-1+2 e^{e^{25-10 x+x^2}-x}}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 71, normalized size = 2.29 \begin {gather*} \frac {2 \, {\left (x + e^{x}\right )} e^{\left ({\left (x e^{\left (2 \, x\right )} + e^{\left (x^{2} - 8 \, x + 25\right )}\right )} e^{\left (-2 \, x\right )}\right )} - x e^{\left (2 \, x\right )}}{2 \, {\left (2 \, e^{\left ({\left (x e^{\left (2 \, x\right )} + e^{\left (x^{2} - 8 \, x + 25\right )}\right )} e^{\left (-2 \, x\right )}\right )} - e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 936, normalized size = 30.19 result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 29, normalized size = 0.94
method | result | size |
risch | \(\frac {x}{2}+\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{2 x}}{2 \left ({\mathrm e}^{x}-2 \,{\mathrm e}^{{\mathrm e}^{\left (x -5\right )^{2}}}\right )}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 40, normalized size = 1.29 \begin {gather*} \frac {x e^{x} - 2 \, {\left (x + e^{x}\right )} e^{\left (e^{\left (x^{2} - 10 \, x + 25\right )}\right )}}{2 \, {\left (e^{x} - 2 \, e^{\left (e^{\left (x^{2} - 10 \, x + 25\right )}\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 87, normalized size = 2.81 \begin {gather*} \frac {x}{2}+\frac {{\mathrm {e}}^x}{2}+\frac {{\mathrm {e}}^{3\,x}+10\,{\mathrm {e}}^{x^2-7\,x+25}-2\,x\,{\mathrm {e}}^{x^2-7\,x+25}}{2\,\left (2\,{\mathrm {e}}^{{\mathrm {e}}^{-10\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{25}}-{\mathrm {e}}^x\right )\,\left (10\,{\mathrm {e}}^{x^2-9\,x+25}+{\mathrm {e}}^x-2\,x\,{\mathrm {e}}^{x^2-9\,x+25}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.27, size = 31, normalized size = 1.00 \begin {gather*} \frac {x}{2} + \frac {e^{x}}{2} + \frac {e^{2 x}}{- 2 e^{x} + 4 e^{e^{x^{2} - 10 x + 25}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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