Integrand size = 398, antiderivative size = 34 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^x}{\log \left (-x+\log \left (x+\frac {1}{9} \left (2+\frac {3}{-1-2 x+x^2}\right )^2\right )\right )} \]
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Time = 0.87 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.005, Rules used = {6820, 2326} \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^x}{\log \left (\log \left (\frac {9 x^5-32 x^4+2 x^3+56 x^2+x+1}{9 \left (-x^2+2 x+1\right )^2}\right )-x\right )} \]
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Rule 2326
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7-\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \left (x-\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )\right )}{\left (1+3 x+57 x^2+113 x^3-84 x^4-57 x^5+50 x^6-9 x^7\right ) \left (x-\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )} \, dx \\ & = \frac {e^x}{\log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (1+2 x-x^2\right )^2}\right )\right )} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^x}{\log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9 \left (-1-2 x+x^2\right )^2}\right )\right )} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.04 (sec) , antiderivative size = 257, normalized size of antiderivative = 7.56
\[\frac {{\mathrm e}^{x}}{\ln \left (\ln \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )-2 \ln \left (x^{2}-2 x -1\right )+\frac {i \pi \,\operatorname {csgn}\left (i \left (x^{2}-2 x -1\right )^{2}\right ) {\left (-\operatorname {csgn}\left (i \left (x^{2}-2 x -1\right )^{2}\right )+\operatorname {csgn}\left (i \left (x^{2}-2 x -1\right )\right )\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right )+\operatorname {csgn}\left (i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{5}-\frac {32}{9} x^{4}+\frac {2}{9} x^{3}+\frac {56}{9} x^{2}+\frac {1}{9} x +\frac {1}{9}\right )}{\left (x^{2}-2 x -1\right )^{2}}\right )+\operatorname {csgn}\left (\frac {i}{\left (x^{2}-2 x -1\right )^{2}}\right )\right )}{2}-x \right )}\]
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Time = 0.25 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.65 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^{x}}{\log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (24) = 48\).
Time = 4.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^{x}}{\log {\left (- x + \log {\left (\frac {9 x^{5} - 32 x^{4} + 2 x^{3} + 56 x^{2} + x + 1}{9 x^{4} - 36 x^{3} + 18 x^{2} + 36 x + 9} \right )} \right )}} \]
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Time = 0.50 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {e^{x}}{\log \left (-x - 2 \, \log \left (3\right ) + \log \left (9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1\right ) - 2 \, \log \left (x^{2} - 2 \, x - 1\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 262 vs. \(2 (31) = 62\).
Time = 7.03 (sec) , antiderivative size = 262, normalized size of antiderivative = 7.71 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {x e^{x} - e^{x} \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )}{x \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right ) - \log \left (9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1\right ) \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right ) + \log \left (9 \, x^{4} - 36 \, x^{3} + 18 \, x^{2} + 36 \, x + 9\right ) \log \left (-x + \log \left (\frac {9 \, x^{5} - 32 \, x^{4} + 2 \, x^{3} + 56 \, x^{2} + x + 1}{9 \, {\left (x^{4} - 4 \, x^{3} + 2 \, x^{2} + 4 \, x + 1\right )}}\right )\right )} \]
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Time = 12.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.68 \[ \int \frac {e^x \left (-4+111 x-48 x^2-125 x^3+3 x^4+111 x^5-59 x^6+9 x^7\right )+\left (e^x \left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8\right )+e^x \left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log \left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )}{\left (x+3 x^2+57 x^3+113 x^4-84 x^5-57 x^6+50 x^7-9 x^8+\left (-1-3 x-57 x^2-113 x^3+84 x^4+57 x^5-50 x^6+9 x^7\right ) \log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right ) \log ^2\left (-x+\log \left (\frac {1+x+56 x^2+2 x^3-32 x^4+9 x^5}{9+36 x+18 x^2-36 x^3+9 x^4}\right )\right )} \, dx=\frac {{\mathrm {e}}^x}{\ln \left (\ln \left (\frac {9\,x^5-32\,x^4+2\,x^3+56\,x^2+x+1}{9\,x^4-36\,x^3+18\,x^2+36\,x+9}\right )-x\right )} \]
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