Optimal. Leaf size=17 \[ x-\frac {x^5}{\left (3-e^2+x\right )^2} \]
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Rubi [B] time = 0.17, antiderivative size = 74, normalized size of antiderivative = 4.35, number of steps used = 2, number of rules used = 1, integrand size = 100, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.010, Rules used = {2074} \begin {gather*} -x^3+2 \left (3-e^2\right ) x^2-\left (26-18 e^2+3 e^4\right ) x-\frac {5 \left (3-e^2\right )^4}{x-e^2+3}+\frac {\left (3-e^2\right )^5}{\left (x-e^2+3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-26 \left (1+\frac {3}{26} e^2 \left (-6+e^2\right )\right )-\frac {2 \left (-3+e^2\right )^5}{\left (-3+e^2-x\right )^3}+\frac {5 \left (-3+e^2\right )^4}{\left (-3+e^2-x\right )^2}-4 \left (-3+e^2\right ) x-3 x^2\right ) \, dx\\ &=-\left (\left (26-18 e^2+3 e^4\right ) x\right )+2 \left (3-e^2\right ) x^2-x^3+\frac {\left (3-e^2\right )^5}{\left (3-e^2+x\right )^2}-\frac {5 \left (3-e^2\right )^4}{3-e^2+x}\\ \end {aligned} \end {gather*}
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Mathematica [B] time = 0.04, size = 88, normalized size = 5.18 \begin {gather*} -\frac {\left (-3+e^2\right )^5}{\left (-3+e^2-x\right )^2}+\frac {5 \left (-3+e^2\right )^4}{-3+e^2-x}-\left (-89+60 e^2-10 e^4\right ) \left (-3+e^2-x\right )-5 \left (-3+e^2\right ) \left (-3+e^2-x\right )^2+\left (-3+e^2-x\right )^3 \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 91, normalized size = 5.35 \begin {gather*} -\frac {x^{5} - x^{3} + 102 \, x^{2} + 4 \, {\left (2 \, x + 15\right )} e^{8} - 4 \, {\left (x^{2} + 24 \, x + 90\right )} e^{6} + {\left (36 \, x^{2} + 431 \, x + 1080\right )} e^{4} - 2 \, {\left (53 \, x^{2} + 429 \, x + 810\right )} e^{2} + 639 \, x - 4 \, e^{10} + 972}{x^{2} - 2 \, {\left (x + 3\right )} e^{2} + 6 \, x + e^{4} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 0.10, size = 51, normalized size = 3.00
method | result | size |
norman | \(\frac {-x^{5}-54+x^{3}+\left (18 \,{\mathrm e}^{2}-3 \,{\mathrm e}^{4}-27\right ) x +2 \,{\mathrm e}^{6}-18 \,{\mathrm e}^{4}+54 \,{\mathrm e}^{2}}{\left ({\mathrm e}^{2}-3-x \right )^{2}}\) | \(51\) |
gosper | \(\frac {-x^{5}+2 \,{\mathrm e}^{6}-3 x \,{\mathrm e}^{4}+x^{3}-18 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +54 \,{\mathrm e}^{2}-27 x -54}{{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}-6 \,{\mathrm e}^{2}+6 x +9}\) | \(66\) |
risch | \(-3 x \,{\mathrm e}^{4}-2 x^{2} {\mathrm e}^{2}-x^{3}+18 \,{\mathrm e}^{2} x +6 x^{2}-26 x +\frac {\left (60 \,{\mathrm e}^{6}-5 \,{\mathrm e}^{8}-270 \,{\mathrm e}^{4}+540 \,{\mathrm e}^{2}-405\right ) x +4 \,{\mathrm e}^{10}-60 \,{\mathrm e}^{8}+360 \,{\mathrm e}^{6}-1080 \,{\mathrm e}^{4}+1620 \,{\mathrm e}^{2}-972}{{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}-6 \,{\mathrm e}^{2}+6 x +9}\) | \(96\) |
default | \(-3 x \,{\mathrm e}^{4}-2 x^{2} {\mathrm e}^{2}-x^{3}+18 \,{\mathrm e}^{2} x +6 x^{2}-26 x -\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{2}+9\right ) \textit {\_Z}^{2}+\left (-18 \,{\mathrm e}^{2}+3 \,{\mathrm e}^{4}+27\right ) \textit {\_Z} -27 \,{\mathrm e}^{2}+9 \,{\mathrm e}^{4}-{\mathrm e}^{6}+27\right )}{\sum }\frac {\left (540 \,{\mathrm e}^{2} \textit {\_R} -270 \textit {\_R} \,{\mathrm e}^{4}-5 \textit {\_R} \,{\mathrm e}^{8}+60 \textit {\_R} \,{\mathrm e}^{6}+1215 \,{\mathrm e}^{2}-810 \,{\mathrm e}^{4}-45 \,{\mathrm e}^{8}+3 \,{\mathrm e}^{10}+270 \,{\mathrm e}^{6}-405 \textit {\_R} -729\right ) \ln \left (x -\textit {\_R} \right )}{9+{\mathrm e}^{4}-2 \,{\mathrm e}^{2} \textit {\_R} +\textit {\_R}^{2}-6 \,{\mathrm e}^{2}+6 \textit {\_R}}\right )}{3}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.36, size = 91, normalized size = 5.35 \begin {gather*} -x^{3} - 2 \, x^{2} {\left (e^{2} - 3\right )} - x {\left (3 \, e^{4} - 18 \, e^{2} + 26\right )} - \frac {5 \, x {\left (e^{8} - 12 \, e^{6} + 54 \, e^{4} - 108 \, e^{2} + 81\right )} - 4 \, e^{10} + 60 \, e^{8} - 360 \, e^{6} + 1080 \, e^{4} - 1620 \, e^{2} + 972}{x^{2} - 2 \, x {\left (e^{2} - 3\right )} + e^{4} - 6 \, e^{2} + 9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.33, size = 109, normalized size = 6.41 \begin {gather*} x\,\left (9\,{\left ({\mathrm {e}}^2-3\right )}^2-\left (3\,{\mathrm {e}}^2-9\right )\,\left (4\,{\mathrm {e}}^2-12\right )+1\right )-x^2\,\left (2\,{\mathrm {e}}^2-6\right )-\frac {1080\,{\mathrm {e}}^4-1620\,{\mathrm {e}}^2-360\,{\mathrm {e}}^6+60\,{\mathrm {e}}^8-4\,{\mathrm {e}}^{10}+x\,\left (270\,{\mathrm {e}}^4-540\,{\mathrm {e}}^2-60\,{\mathrm {e}}^6+5\,{\mathrm {e}}^8+405\right )+972}{x^2+\left (6-2\,{\mathrm {e}}^2\right )\,x-6\,{\mathrm {e}}^2+{\mathrm {e}}^4+9}-x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.77, size = 100, normalized size = 5.88 \begin {gather*} - x^{3} - x^{2} \left (-6 + 2 e^{2}\right ) - x \left (- 18 e^{2} + 26 + 3 e^{4}\right ) - \frac {x \left (- 60 e^{6} - 540 e^{2} + 405 + 270 e^{4} + 5 e^{8}\right ) - 360 e^{6} - 4 e^{10} - 1620 e^{2} + 972 + 1080 e^{4} + 60 e^{8}}{x^{2} + x \left (6 - 2 e^{2}\right ) - 6 e^{2} + 9 + e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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