Optimal. Leaf size=32 \[ 1+\frac {e^x x}{3+e^{-2 x+2 \left (1-e+e^4\right ) x} x^2} \]
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Rubi [F] time = 3.08, 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^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )}{9+6 e^{-2 e x+2 e^4 x} x^2+e^{-4 e x+4 e^4 x} x^4} \, 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^{4 e x} \left (e^x (3+3 x)+e^{x-2 e x+2 e^4 x} \left (-x^2+x^3+2 e x^3-2 e^4 x^3\right )\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx\\ &=\int \left (\frac {6 e^{x+4 e x} \left (1-e \left (1-e^3\right ) x\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2}+\frac {e^{(1-2 e) x+4 e x} \left (-1+\left (1+2 e-2 e^4\right ) x\right )}{3 e^{2 e x}+e^{2 e^4 x} x^2}\right ) \, dx\\ &=6 \int \frac {e^{x+4 e x} \left (1-e \left (1-e^3\right ) x\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx+\int \frac {e^{(1-2 e) x+4 e x} \left (-1+\left (1+2 e-2 e^4\right ) x\right )}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx\\ &=6 \int \frac {e^{(1+4 e) x} \left (1-e \left (1-e^3\right ) x\right )}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx+\int \frac {e^{(1+2 e) x} \left (-1+\left (1+2 e-2 e^4\right ) x\right )}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx\\ &=6 \int \left (\frac {e^{(1+4 e) x}}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2}+\frac {e^{1+(1+4 e) x} \left (-1+e^3\right ) x}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2}\right ) \, dx+\int \left (-\frac {e^{(1+2 e) x}}{3 e^{2 e x}+e^{2 e^4 x} x^2}-\frac {e^{(1+2 e) x} \left (-1-2 e+2 e^4\right ) x}{3 e^{2 e x}+e^{2 e^4 x} x^2}\right ) \, dx\\ &=6 \int \frac {e^{(1+4 e) x}}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx-\left (6 \left (1-e^3\right )\right ) \int \frac {e^{1+(1+4 e) x} x}{\left (3 e^{2 e x}+e^{2 e^4 x} x^2\right )^2} \, dx+\left (1+2 e-2 e^4\right ) \int \frac {e^{(1+2 e) x} x}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx-\int \frac {e^{(1+2 e) x}}{3 e^{2 e x}+e^{2 e^4 x} x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 34, normalized size = 1.06 \begin {gather*} \frac {e^{(1+2 e) x} x}{3 e^{2 e x}+e^{2 e^4 x} x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 48, normalized size = 1.50 \begin {gather*} \frac {x e^{\left (2 \, x e^{4} - 2 \, x e + x\right )}}{x^{2} e^{\left (4 \, x e^{4} - 4 \, x e\right )} + 3 \, e^{\left (2 \, x e^{4} - 2 \, x e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {{\left (2 \, x^{3} e^{4} - 2 \, x^{3} e - x^{3} + x^{2}\right )} e^{\left (2 \, x e^{4} - 2 \, x e + x\right )} - 3 \, {\left (x + 1\right )} e^{x}}{x^{4} e^{\left (4 \, x e^{4} - 4 \, x e\right )} + 6 \, x^{2} e^{\left (2 \, x e^{4} - 2 \, x e\right )} + 9}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 24, normalized size = 0.75
method | result | size |
risch | \(\frac {{\mathrm e}^{x} x}{x^{2} {\mathrm e}^{-2 x \left (-{\mathrm e}^{4}+{\mathrm e}\right )}+3}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 31, normalized size = 0.97 \begin {gather*} \frac {x e^{\left (2 \, x e + x\right )}}{x^{2} e^{\left (2 \, x e^{4}\right )} + 3 \, e^{\left (2 \, x e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 61, normalized size = 1.91 \begin {gather*} \frac {x^2\,{\mathrm {e}}^x-x^3\,\left ({\mathrm {e}}^{x+1}-{\mathrm {e}}^{x+4}\right )}{\left (x^2\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^4-2\,x\,\mathrm {e}}+3\right )\,\left (x-x^2\,\mathrm {e}+x^2\,{\mathrm {e}}^4\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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