Optimal. Leaf size=32 \[ 2+x+\left (x+\frac {2}{6+e^{\frac {3}{e^{4/3}}}-x+\frac {x^2}{2}}\right )^2 \]
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Rubi [A] time = 0.56, antiderivative size = 54, normalized size of antiderivative = 1.69, number of steps used = 8, number of rules used = 5, integrand size = 202, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2074, 618, 204, 629, 638} \begin {gather*} x^2+\frac {8 x}{x^2-2 x+2 \left (6+e^{\frac {3}{e^{4/3}}}\right )}+\frac {16}{\left (x^2-2 x+2 \left (6+e^{\frac {3}{e^{4/3}}}\right )\right )^2}+x \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 629
Rule 638
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+2 x+\frac {8}{-2 \left (6+e^{\frac {3}{e^{4/3}}}\right )+2 x-x^2}+\frac {64 (1-x)}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^3}+\frac {16 \left (12+2 e^{\frac {3}{e^{4/3}}}-x\right )}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}\right ) \, dx\\ &=x+x^2+8 \int \frac {1}{-2 \left (6+e^{\frac {3}{e^{4/3}}}\right )+2 x-x^2} \, dx+16 \int \frac {12+2 e^{\frac {3}{e^{4/3}}}-x}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2} \, dx+64 \int \frac {1-x}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^3} \, dx\\ &=x+x^2+\frac {16}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}+\frac {8 x}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2}+8 \int \frac {1}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2} \, dx-16 \operatorname {Subst}\left (\int \frac {1}{-4 \left (11+2 e^{\frac {3}{e^{4/3}}}\right )-x^2} \, dx,x,2-2 x\right )\\ &=x+x^2+\frac {16}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}+\frac {8 x}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2}+\frac {8 \tan ^{-1}\left (\frac {1-x}{\sqrt {11+2 e^{\frac {3}{e^{4/3}}}}}\right )}{\sqrt {11+2 e^{\frac {3}{e^{4/3}}}}}-16 \operatorname {Subst}\left (\int \frac {1}{-4 \left (11+2 e^{\frac {3}{e^{4/3}}}\right )-x^2} \, dx,x,-2+2 x\right )\\ &=x+x^2+\frac {16}{\left (2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2\right )^2}+\frac {8 x}{2 \left (6+e^{\frac {3}{e^{4/3}}}\right )-2 x+x^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 52, normalized size = 1.62 \begin {gather*} x+x^2+\frac {16}{\left (12+2 e^{\frac {3}{e^{4/3}}}-2 x+x^2\right )^2}+\frac {8 x}{12+2 e^{\frac {3}{e^{4/3}}}-2 x+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 107, normalized size = 3.34 \begin {gather*} \frac {x^{6} - 3 \, x^{5} + 24 \, x^{4} - 12 \, x^{3} + 80 \, x^{2} + 4 \, {\left (x^{2} + x\right )} e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 4 \, {\left (x^{4} - x^{3} + 10 \, x^{2} + 16 \, x\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 240 \, x + 16}{x^{4} - 4 \, x^{3} + 28 \, x^{2} + 4 \, {\left (x^{2} - 2 \, x + 12\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} - 48 \, x + 4 \, e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 144} \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 {2 \, x^{7} - 11 \, x^{6} + 90 \, x^{5} - 264 \, x^{4} + 1016 \, x^{3} - 1152 \, x^{2} + 8 \, {\left (2 \, x + 1\right )} e^{\left (9 \, e^{\left (-\frac {4}{3}\right )}\right )} + 4 \, {\left (6 \, x^{3} - 9 \, x^{2} + 66 \, x + 44\right )} e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 2 \, {\left (6 \, x^{5} - 21 \, x^{4} + 156 \, x^{3} - 204 \, x^{2} + 704 \, x + 624\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 2336 \, x + 2944}{x^{6} - 6 \, x^{5} + 48 \, x^{4} - 152 \, x^{3} + 576 \, x^{2} + 12 \, {\left (x^{2} - 2 \, x + 12\right )} e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 6 \, {\left (x^{4} - 4 \, x^{3} + 28 \, x^{2} - 48 \, x + 144\right )} e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} - 864 \, x + 8 \, e^{\left (9 \, e^{\left (-\frac {4}{3}\right )}\right )} + 1728}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.17, size = 81, normalized size = 2.53
method | result | size |
risch | \(x^{2}+x +\frac {2 x^{3}-4 x^{2}+\left (4 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}+24\right ) x +4}{{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}+{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}} x^{2}-2 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}} x +\frac {x^{4}}{4}+12 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-x^{3}+7 x^{2}-12 x +36}\) | \(81\) |
norman | \(\frac {x^{6}+\left (12 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}+84\right ) x^{3}+\left (-12 \,{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}-168 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-592\right ) x^{2}+\left (36 \,{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}+448 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}+1392\right ) x -3 x^{5}-16 \,{\mathrm e}^{9 \,{\mathrm e}^{-\frac {4}{3}}}-288 \,{\mathrm e}^{6 \,{\mathrm e}^{-\frac {4}{3}}}-1728 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-3440}{\left (x^{2}+2 \,{\mathrm e}^{3 \,{\mathrm e}^{-\frac {4}{3}}}-2 x +12\right )^{2}}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.35, size = 75, normalized size = 2.34 \begin {gather*} x^{2} + x + \frac {8 \, {\left (x^{3} - 2 \, x^{2} + 2 \, x {\left (e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 6\right )} + 2\right )}}{x^{4} - 4 \, x^{3} + 4 \, x^{2} {\left (e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 7\right )} - 8 \, x {\left (e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 6\right )} + 4 \, e^{\left (6 \, e^{\left (-\frac {4}{3}\right )}\right )} + 48 \, e^{\left (3 \, e^{\left (-\frac {4}{3}\right )}\right )} + 144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.25, size = 94, normalized size = 2.94 \begin {gather*} x+\frac {8\,x^3+\left (48\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+4\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-\frac {4}{3}}}-{\left (2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+12\right )}^2+128\right )\,x^2+\left (64\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+4\,{\mathrm {e}}^{6\,{\mathrm {e}}^{-\frac {4}{3}}}-{\left (2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+12\right )}^2+240\right )\,x+16}{{\left (x^2-2\,x+2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{-\frac {4}{3}}}+12\right )}^2}+x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.44, size = 90, normalized size = 2.81 \begin {gather*} x^{2} + x + \frac {8 x^{3} - 16 x^{2} + x \left (16 e^{\frac {3}{e^{\frac {4}{3}}}} + 96\right ) + 16}{x^{4} - 4 x^{3} + x^{2} \left (4 e^{\frac {3}{e^{\frac {4}{3}}}} + 28\right ) + x \left (-48 - 8 e^{\frac {3}{e^{\frac {4}{3}}}}\right ) + 4 e^{\frac {6}{e^{\frac {4}{3}}}} + 48 e^{\frac {3}{e^{\frac {4}{3}}}} + 144} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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