Optimal. Leaf size=28 \[ 1+\frac {4}{5+\frac {1}{2} \left (e^8-x\right )+\left (e^x-x\right )^2} \]
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Rubi [A] time = 0.34, antiderivative size = 30, normalized size of antiderivative = 1.07, number of steps used = 3, number of rules used = 3, integrand size = 121, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {6688, 12, 6686} \begin {gather*} \frac {8}{2 x^2-4 e^x x-x+2 e^{2 x}+e^8+10} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 6686
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 \left (1-4 e^{2 x}-4 x+4 e^x (1+x)\right )}{\left (2 e^{2 x}+10 \left (1+\frac {e^8}{10}\right )-x-4 e^x x+2 x^2\right )^2} \, dx\\ &=8 \int \frac {1-4 e^{2 x}-4 x+4 e^x (1+x)}{\left (2 e^{2 x}+10 \left (1+\frac {e^8}{10}\right )-x-4 e^x x+2 x^2\right )^2} \, dx\\ &=\frac {8}{10+e^8+2 e^{2 x}-x-4 e^x x+2 x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.03, size = 30, normalized size = 1.07 \begin {gather*} \frac {8}{10+e^8+2 e^{2 x}-x-4 e^x x+2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 27, normalized size = 0.96 \begin {gather*} \frac {8}{2 \, x^{2} - 4 \, x e^{x} - x + e^{8} + 2 \, e^{\left (2 \, x\right )} + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.81, size = 27, normalized size = 0.96 \begin {gather*} \frac {16}{2 \, x^{2} - 4 \, x e^{x} - x + e^{8} + 2 \, e^{\left (2 \, x\right )} + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.56, size = 28, normalized size = 1.00
method | result | size |
norman | \(\frac {8}{2 \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x +2 x^{2}+{\mathrm e}^{8}-x +10}\) | \(28\) |
risch | \(\frac {8}{2 \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x +2 x^{2}+{\mathrm e}^{8}-x +10}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 27, normalized size = 0.96 \begin {gather*} \frac {8}{2 \, x^{2} - 4 \, x e^{x} - x + e^{8} + 2 \, e^{\left (2 \, x\right )} + 10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.71, size = 71, normalized size = 2.54 \begin {gather*} -\frac {\frac {16\,x^2}{{\mathrm {e}}^8+10}+\frac {16\,{\mathrm {e}}^{2\,x}}{{\mathrm {e}}^8+10}-\frac {8\,x}{{\mathrm {e}}^8+10}-\frac {32\,x\,{\mathrm {e}}^x}{{\mathrm {e}}^8+10}}{2\,{\mathrm {e}}^{2\,x}-x+{\mathrm {e}}^8-4\,x\,{\mathrm {e}}^x+2\,x^2+10} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 26, normalized size = 0.93 \begin {gather*} \frac {8}{2 x^{2} - 4 x e^{x} - x + 2 e^{2 x} + 10 + e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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