Optimal. Leaf size=24 \[ 1+x+\frac {x}{e^4}-e^5 x-\frac {4}{3+e^x+x} \]
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Rubi [F] time = 0.55, 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 {9+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\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 {e^{2 x} \left (1+e^4-e^9\right )+2 e^x \left (1-e^9\right ) (3+x)+\left (1-e^9\right ) (3+x)^2+2 e^{4+x} (5+x)+e^4 \left (13+6 x+x^2\right )}{e^4 \left (3+e^x+x\right )^2} \, dx\\ &=\frac {\int \frac {e^{2 x} \left (1+e^4-e^9\right )+2 e^x \left (1-e^9\right ) (3+x)+\left (1-e^9\right ) (3+x)^2+2 e^{4+x} (5+x)+e^4 \left (13+6 x+x^2\right )}{\left (3+e^x+x\right )^2} \, dx}{e^4}\\ &=\frac {\int \left (1+e^4-e^9-\frac {4 e^4 (2+x)}{\left (3+e^x+x\right )^2}+\frac {4 e^4}{3+e^x+x}\right ) \, dx}{e^4}\\ &=\frac {\left (1+e^4-e^9\right ) x}{e^4}-4 \int \frac {2+x}{\left (3+e^x+x\right )^2} \, dx+4 \int \frac {1}{3+e^x+x} \, dx\\ &=\frac {\left (1+e^4-e^9\right ) x}{e^4}+4 \int \frac {1}{3+e^x+x} \, dx-4 \int \left (\frac {2}{\left (3+e^x+x\right )^2}+\frac {x}{\left (3+e^x+x\right )^2}\right ) \, dx\\ &=\frac {\left (1+e^4-e^9\right ) x}{e^4}-4 \int \frac {x}{\left (3+e^x+x\right )^2} \, dx+4 \int \frac {1}{3+e^x+x} \, dx-8 \int \frac {1}{\left (3+e^x+x\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.05, size = 30, normalized size = 1.25 \begin {gather*} \frac {\left (1+e^4-e^9\right ) x-\frac {4 e^4}{3+e^x+x}}{e^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.49, size = 67, normalized size = 2.79 \begin {gather*} -\frac {{\left (x^{2} + 3 \, x\right )} e^{13} - {\left (x^{2} + 3 \, x - 4\right )} e^{8} - {\left (x^{2} + 3 \, x\right )} e^{4} + {\left (x e^{9} - x e^{4} - x\right )} e^{\left (x + 4\right )}}{{\left (x + 3\right )} e^{8} + e^{\left (x + 8\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 71, normalized size = 2.96 \begin {gather*} -\frac {x^{2} e^{9} - x^{2} e^{4} - x^{2} + 3 \, x e^{9} - 3 \, x e^{4} + x e^{\left (x + 9\right )} - x e^{\left (x + 4\right )} - x e^{x} - 3 \, x + 4 \, e^{4}}{x e^{4} + 3 \, e^{4} + e^{\left (x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 28, normalized size = 1.17
method | result | size |
risch | \({\mathrm e}^{4} {\mathrm e}^{-4} x -x \,{\mathrm e}^{-4} {\mathrm e}^{9}+x \,{\mathrm e}^{-4}-\frac {4}{{\mathrm e}^{x}+3+x}\) | \(28\) |
norman | \(\frac {3 \,{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) {\mathrm e}^{x}-{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x^{2}-{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x \,{\mathrm e}^{x}+{\mathrm e}^{-4} \left (9 \,{\mathrm e}^{4} {\mathrm e}^{5}-13 \,{\mathrm e}^{4}-9\right )}{{\mathrm e}^{x}+3+x}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 57, normalized size = 2.38 \begin {gather*} -\frac {x^{2} {\left (e^{9} - e^{4} - 1\right )} + x {\left (e^{9} - e^{4} - 1\right )} e^{x} + 3 \, x {\left (e^{9} - e^{4} - 1\right )} + 4 \, e^{4}}{x e^{4} + 3 \, e^{4} + e^{\left (x + 4\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 30, normalized size = 1.25 \begin {gather*} x+x\,{\mathrm {e}}^{-4}-x\,{\mathrm {e}}^5-\frac {4\,{\mathrm {e}}^4}{{\mathrm {e}}^{x+4}+3\,{\mathrm {e}}^4+x\,{\mathrm {e}}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} \frac {x \left (- e^{9} + 1 + e^{4}\right )}{e^{4}} - \frac {4}{x + e^{x} + 3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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