Integrand size = 257, antiderivative size = 22 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=11+x^2-\frac {3-2 x}{(x-e (1+x))^4} \]
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Leaf count is larger than twice the leaf count of optimal. \(51\) vs. \(2(22)=44\).
Time = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.32, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.004, Rules used = {2099} \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=x^2-\frac {2}{(1-e) (e-(1-e) x)^3}-\frac {3-5 e}{(1-e) (e-(1-e) x)^4} \]
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Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (2 x+\frac {4 (-3+5 e)}{(e-(1-e) x)^5}-\frac {6}{(e-(1-e) x)^4}\right ) \, dx \\ & = x^2-\frac {3-5 e}{(1-e) (e-(1-e) x)^4}-\frac {2}{(1-e) (e-(1-e) x)^3} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=-2 x+(1+x)^2+\frac {3-5 e}{(-1+e) (e-x+e x)^4}+\frac {2}{(-1+e) (e-x+e x)^3} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(23)=46\).
Time = 0.59 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.77
method | result | size |
risch | \(x^{2}+\frac {-3+2 x}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\) | \(105\) |
norman | \(\frac {\left ({\mathrm e}^{4}-4 \,{\mathrm e}^{3}+6 \,{\mathrm e}^{2}-4 \,{\mathrm e}+1\right ) x^{6}+{\mathrm e}^{-2} \left ({\mathrm e}^{6}+18 \,{\mathrm e}^{2}-36 \,{\mathrm e}+18\right ) x^{2}+\left (6 \,{\mathrm e}^{8}-12 \,{\mathrm e}^{7}+6 \,{\mathrm e}^{6}+3+3 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{3}+18 \,{\mathrm e}^{2}-12 \,{\mathrm e}\right ) {\mathrm e}^{-4} x^{4}+2 \,{\mathrm e}^{-1} \left (7 \,{\mathrm e}-6\right ) x +4 \left ({\mathrm e}^{7}-{\mathrm e}^{6}+3 \,{\mathrm e}^{3}-9 \,{\mathrm e}^{2}+9 \,{\mathrm e}-3\right ) {\mathrm e}^{-3} x^{3}+4 \,{\mathrm e} \left ({\mathrm e}^{3}-3 \,{\mathrm e}^{2}+3 \,{\mathrm e}-1\right ) x^{5}}{\left (x \,{\mathrm e}+{\mathrm e}-x \right )^{4}}\) | \(188\) |
default | \(x^{2}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left ({\mathrm e}^{5}-10 \,{\mathrm e}^{2}+5 \,{\mathrm e}-5 \,{\mathrm e}^{4}+10 \,{\mathrm e}^{3}-1\right ) \textit {\_Z}^{5}+\left (5 \,{\mathrm e}^{5}-20 \,{\mathrm e}^{2}+5 \,{\mathrm e}-20 \,{\mathrm e}^{4}+30 \,{\mathrm e}^{3}\right ) \textit {\_Z}^{4}+\left (10 \,{\mathrm e}^{5}-10 \,{\mathrm e}^{2}-30 \,{\mathrm e}^{4}+30 \,{\mathrm e}^{3}\right ) \textit {\_Z}^{3}+\left (10 \,{\mathrm e}^{5}-20 \,{\mathrm e}^{4}+10 \,{\mathrm e}^{3}\right ) \textit {\_Z}^{2}+\left (5 \,{\mathrm e}^{5}-5 \,{\mathrm e}^{4}\right ) \textit {\_Z} +{\mathrm e}^{5}\right )}{\sum }\frac {\left (-6+3 \left (1-{\mathrm e}\right ) \textit {\_R} +7 \,{\mathrm e}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{4} {\mathrm e}^{5}+4 \textit {\_R}^{3} {\mathrm e}^{5}-5 \textit {\_R}^{4} {\mathrm e}^{4}+6 \textit {\_R}^{2} {\mathrm e}^{5}-16 \textit {\_R}^{3} {\mathrm e}^{4}+10 \textit {\_R}^{4} {\mathrm e}^{3}+4 \textit {\_R} \,{\mathrm e}^{5}-18 \textit {\_R}^{2} {\mathrm e}^{4}+24 \textit {\_R}^{3} {\mathrm e}^{3}-10 \textit {\_R}^{4} {\mathrm e}^{2}+{\mathrm e}^{5}-8 \textit {\_R} \,{\mathrm e}^{4}+18 \textit {\_R}^{2} {\mathrm e}^{3}-16 \textit {\_R}^{3} {\mathrm e}^{2}+5 \textit {\_R}^{4} {\mathrm e}-{\mathrm e}^{4}+4 \textit {\_R} \,{\mathrm e}^{3}-6 \textit {\_R}^{2} {\mathrm e}^{2}+4 \textit {\_R}^{3} {\mathrm e}-\textit {\_R}^{4}}\right )}{5}\) | \(260\) |
gosper | \(\frac {x \left (6 \,{\mathrm e}^{6} x^{5}+12 \,{\mathrm e}^{6} x^{4}+6 x^{3} {\mathrm e}^{8}+6 x^{3} {\mathrm e}^{6}+x \,{\mathrm e}^{8}+3 x^{3} {\mathrm e}^{4}+12 x^{2} {\mathrm e}^{4}-4 x^{5} {\mathrm e}^{5}+18 x^{3} {\mathrm e}^{2}-36 x \,{\mathrm e}^{3}-12 x^{2} {\mathrm e}-12 x^{3} {\mathrm e}-12 x^{3} {\mathrm e}^{3}-36 x^{2} {\mathrm e}^{3}+36 x^{2} {\mathrm e}^{2}-4 x^{4} {\mathrm e}^{5}+4 x^{4} {\mathrm e}^{8}-4 x^{2} {\mathrm e}^{7}+x^{5} {\mathrm e}^{4}+18 x \,{\mathrm e}^{4}+14 \,{\mathrm e}^{4}-12 \,{\mathrm e}^{3}+3 x^{3}+18 \,{\mathrm e}^{2} x +4 x^{2} {\mathrm e}^{8}+{\mathrm e}^{8} x^{5}-4 \,{\mathrm e}^{7} x^{5}-12 \,{\mathrm e}^{7} x^{3}-12 x^{4} {\mathrm e}^{7}\right ) {\mathrm e}^{-4}}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\) | \(362\) |
parallelrisch | \(\frac {\left (6 \,{\mathrm e}^{6} x^{6}+12 \,{\mathrm e}^{6} x^{5}+6 \,{\mathrm e}^{6} x^{4}+4 x^{3} {\mathrm e}^{8}+12 x^{3} {\mathrm e}^{4}+18 x^{2} {\mathrm e}^{4}-4 x^{5} {\mathrm e}^{5}+36 x^{3} {\mathrm e}^{2}-12 x^{4} {\mathrm e}^{3}-12 x \,{\mathrm e}^{3}-12 x^{3} {\mathrm e}-36 x^{3} {\mathrm e}^{3}-36 x^{2} {\mathrm e}^{3}-12 x^{4} {\mathrm e}+18 x^{2} {\mathrm e}^{2}+18 x^{4} {\mathrm e}^{2}+6 x^{4} {\mathrm e}^{8}-4 x^{6} {\mathrm e}^{5}+3 x^{4} {\mathrm e}^{4}+x^{6} {\mathrm e}^{4}+14 x \,{\mathrm e}^{4}+3 x^{4}+x^{2} {\mathrm e}^{8}+{\mathrm e}^{8} x^{6}+4 \,{\mathrm e}^{8} x^{5}-12 \,{\mathrm e}^{7} x^{5}-4 \,{\mathrm e}^{7} x^{3}-12 x^{4} {\mathrm e}^{7}-4 \,{\mathrm e}^{7} x^{6}\right ) {\mathrm e}^{-4}}{x^{4} {\mathrm e}^{4}+4 x^{3} {\mathrm e}^{4}-4 x^{4} {\mathrm e}^{3}+6 x^{2} {\mathrm e}^{4}-12 x^{3} {\mathrm e}^{3}+6 x^{4} {\mathrm e}^{2}+4 x \,{\mathrm e}^{4}-12 x^{2} {\mathrm e}^{3}+12 x^{3} {\mathrm e}^{2}-4 x^{4} {\mathrm e}+{\mathrm e}^{4}-4 x \,{\mathrm e}^{3}+6 x^{2} {\mathrm e}^{2}-4 x^{3} {\mathrm e}+x^{4}}\) | \(371\) |
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (23) = 46\).
Time = 0.24 (sec) , antiderivative size = 155, normalized size of antiderivative = 7.05 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {x^{6} + {\left (x^{6} + 4 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + x^{2}\right )} e^{4} - 4 \, {\left (x^{6} + 3 \, x^{5} + 3 \, x^{4} + x^{3}\right )} e^{3} + 6 \, {\left (x^{6} + 2 \, x^{5} + x^{4}\right )} e^{2} - 4 \, {\left (x^{6} + x^{5}\right )} e + 2 \, x - 3}{x^{4} + {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 4 \, x + 1\right )} e^{4} - 4 \, {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + x\right )} e^{3} + 6 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{2} - 4 \, {\left (x^{4} + x^{3}\right )} e} \]
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Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (19) = 38\).
Time = 2.70 (sec) , antiderivative size = 88, normalized size of antiderivative = 4.00 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=x^{2} + \frac {2 x - 3}{x^{4} \left (- 4 e^{3} - 4 e + 1 + 6 e^{2} + e^{4}\right ) + x^{3} \left (- 12 e^{3} - 4 e + 12 e^{2} + 4 e^{4}\right ) + x^{2} \left (- 12 e^{3} + 6 e^{2} + 6 e^{4}\right ) + x \left (- 4 e^{3} + 4 e^{4}\right ) + e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (23) = 46\).
Time = 0.20 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.59 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=x^{2} + \frac {2 \, x - 3}{x^{4} {\left (e^{4} - 4 \, e^{3} + 6 \, e^{2} - 4 \, e + 1\right )} + 4 \, x^{3} {\left (e^{4} - 3 \, e^{3} + 3 \, e^{2} - e\right )} + 6 \, x^{2} {\left (e^{4} - 2 \, e^{3} + e^{2}\right )} + 4 \, x {\left (e^{4} - e^{3}\right )} + e^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 6.14 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {x^{2} e^{10} - 10 \, x^{2} e^{9} + 45 \, x^{2} e^{8} - 120 \, x^{2} e^{7} + 210 \, x^{2} e^{6} - 252 \, x^{2} e^{5} + 210 \, x^{2} e^{4} - 120 \, x^{2} e^{3} + 45 \, x^{2} e^{2} - 10 \, x^{2} e + x^{2}}{e^{10} - 10 \, e^{9} + 45 \, e^{8} - 120 \, e^{7} + 210 \, e^{6} - 252 \, e^{5} + 210 \, e^{4} - 120 \, e^{3} + 45 \, e^{2} - 10 \, e + 1} + \frac {2 \, x - 3}{{\left (x e - x + e\right )}^{4}} \]
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Time = 12.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 2.18 \[ \int \frac {-12+6 x-2 x^6+e^2 \left (-20 x^4-40 x^5-20 x^6\right )+e^4 \left (-10 x^2-40 x^3-60 x^4-40 x^5-10 x^6\right )+e^5 \left (2 x+10 x^2+20 x^3+20 x^4+10 x^5+2 x^6\right )+e \left (14-6 x+10 x^5+10 x^6\right )+e^3 \left (20 x^3+60 x^4+60 x^5+20 x^6\right )}{-x^5+e^2 \left (-10 x^3-20 x^4-10 x^5\right )+e^4 \left (-5 x-20 x^2-30 x^3-20 x^4-5 x^5\right )+e^5 \left (1+5 x+10 x^2+10 x^3+5 x^4+x^5\right )+e \left (5 x^4+5 x^5\right )+e^3 \left (10 x^2+30 x^3+30 x^4+10 x^5\right )} \, dx=\frac {2}{{\left (\mathrm {e}+x\,\left (\mathrm {e}-1\right )\right )}^3\,\left (\mathrm {e}-1\right )}+x^2-\frac {5\,\mathrm {e}-3}{{\left (\mathrm {e}+x\,\left (\mathrm {e}-1\right )\right )}^4\,\left (\mathrm {e}-1\right )} \]
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