Optimal. Leaf size=27 \[ e^{-5+\frac {x (2+x) (-2+x (1-x-\log (3)))}{(1+x)^2}} \]
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Rubi [F] time = 4.16, 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 {\exp \left (\frac {-5-14 x-5 x^2-x^3-x^4+\left (-2 x^2-x^3\right ) \log (3)}{1+2 x+x^2}\right ) \left (-4+4 x-3 x^2-5 x^3-2 x^4+\left (-4 x-3 x^2-x^3\right ) \log (3)\right )}{1+3 x+3 x^2+x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {-5-14 x-x^4-x^3 (1+\log (3))-x^2 (5+\log (9))}{1+2 x+x^2}\right ) \left (-4-2 x^4+4 x (1-\log (3))-3 x^2 (1+\log (3))-x^3 (5+\log (3))\right )}{1+3 x+3 x^2+x^3} \, dx\\ &=\int \frac {\exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) \left (-4-2 x^4+4 x (1-\log (3))-3 x^2 (1+\log (3))-x^3 (5+\log (3))\right )}{(1+x)^3} \, dx\\ &=\int \left (-2 \exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) x+\exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) (1-\log (3))+\frac {\exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) (3-\log (3))}{(1+x)^2}+\frac {\exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) (-8+\log (9))}{(1+x)^3}\right ) \, dx\\ &=-\left (2 \int \exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) x \, dx\right )+(1-\log (3)) \int \exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right ) \, dx+(3-\log (3)) \int \frac {\exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right )}{(1+x)^2} \, dx+(-8+\log (9)) \int \frac {\exp \left (-\frac {5+14 x+x^4+x^3 (1+\log (3))+x^2 (5+\log (9))}{(1+x)^2}\right )}{(1+x)^3} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 7.86, size = 41, normalized size = 1.52 \begin {gather*} 3^{-\frac {x^2 (2+x)}{(1+x)^2}} e^{-\frac {5+14 x+5 x^2+x^3+x^4}{(1+x)^2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 41, normalized size = 1.52 \begin {gather*} e^{\left (-\frac {x^{4} + x^{3} + 5 \, x^{2} + {\left (x^{3} + 2 \, x^{2}\right )} \log \relax (3) + 14 \, x + 5}{x^{2} + 2 \, x + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 106, normalized size = 3.93 \begin {gather*} e^{\left (-\frac {x^{4}}{x^{2} + 2 \, x + 1} - \frac {x^{3} \log \relax (3)}{x^{2} + 2 \, x + 1} - \frac {x^{3}}{x^{2} + 2 \, x + 1} - \frac {2 \, x^{2} \log \relax (3)}{x^{2} + 2 \, x + 1} - \frac {5 \, x^{2}}{x^{2} + 2 \, x + 1} - \frac {14 \, x}{x^{2} + 2 \, x + 1} - \frac {5}{x^{2} + 2 \, x + 1}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 38, normalized size = 1.41
method | result | size |
risch | \({\mathrm e}^{-\frac {x^{3} \ln \relax (3)+x^{4}+2 x^{2} \ln \relax (3)+x^{3}+5 x^{2}+14 x +5}{\left (x +1\right )^{2}}}\) | \(38\) |
gosper | \({\mathrm e}^{-\frac {x^{3} \ln \relax (3)+x^{4}+2 x^{2} \ln \relax (3)+x^{3}+5 x^{2}+14 x +5}{x^{2}+2 x +1}}\) | \(43\) |
norman | \(\frac {x^{2} {\mathrm e}^{\frac {\left (-x^{3}-2 x^{2}\right ) \ln \relax (3)-x^{4}-x^{3}-5 x^{2}-14 x -5}{x^{2}+2 x +1}}+2 x \,{\mathrm e}^{\frac {\left (-x^{3}-2 x^{2}\right ) \ln \relax (3)-x^{4}-x^{3}-5 x^{2}-14 x -5}{x^{2}+2 x +1}}+{\mathrm e}^{\frac {\left (-x^{3}-2 x^{2}\right ) \ln \relax (3)-x^{4}-x^{3}-5 x^{2}-14 x -5}{x^{2}+2 x +1}}}{\left (x +1\right )^{2}}\) | \(153\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.69, size = 55, normalized size = 2.04 \begin {gather*} e^{\left (-x^{2} - x \log \relax (3) + x - \frac {\log \relax (3)}{x^{2} + 2 \, x + 1} + \frac {\log \relax (3)}{x + 1} + \frac {4}{x^{2} + 2 \, x + 1} - \frac {3}{x + 1} - 6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 98, normalized size = 3.63 \begin {gather*} {\left (\frac {1}{3}\right )}^{\frac {x^3+2\,x^2}{x^2+2\,x+1}}\,{\mathrm {e}}^{-\frac {x^3}{x^2+2\,x+1}}\,{\mathrm {e}}^{-\frac {x^4}{x^2+2\,x+1}}\,{\mathrm {e}}^{-\frac {5\,x^2}{x^2+2\,x+1}}\,{\mathrm {e}}^{-\frac {5}{x^2+2\,x+1}}\,{\mathrm {e}}^{-\frac {14\,x}{x^2+2\,x+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.51, size = 39, normalized size = 1.44 \begin {gather*} e^{\frac {- x^{4} - x^{3} - 5 x^{2} - 14 x + \left (- x^{3} - 2 x^{2}\right ) \log {\relax (3 )} - 5}{x^{2} + 2 x + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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