Integrand size = 210, antiderivative size = 24 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^2}{2 x+(1+x)^2 \left (-4+e^3+x\right )^2} \]
[Out]
\[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \left (\left (3-e^3\right ) \left (4-e^3\right )^2+2 \left (55-42 e^3+11 e^6-e^9\right ) x+\left (42-34 e^3+10 e^6-e^9\right ) x^2-\left (17-8 e^3+e^6\right ) x^3\right )}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}+\frac {2 \left (-3+e^3-x\right )}{\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4}\right ) \, dx \\ & = 2 \int \frac {\left (3-e^3\right ) \left (4-e^3\right )^2+2 \left (55-42 e^3+11 e^6-e^9\right ) x+\left (42-34 e^3+10 e^6-e^9\right ) x^2-\left (17-8 e^3+e^6\right ) x^3}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+2 \int \frac {-3+e^3-x}{\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4} \, dx \\ & = \frac {17-8 e^3+e^6}{2 \left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )}+\frac {1}{2} \int \frac {2 \left (317-303 e^3+108 e^6-17 e^9+e^{12}\right )+2 \left (237-244 e^3+94 e^6-16 e^9+e^{12}\right ) x-2 \left (69-55 e^3+13 e^6-e^9\right ) x^2}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+2 \int \left (\frac {3 \left (1-\frac {e^3}{3}\right )}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4}+\frac {x}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4}\right ) \, dx \\ & = \frac {17-8 e^3+e^6}{2 \left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )}+\frac {1}{2} \int \left (\frac {2 \left (317-303 e^3+108 e^6-17 e^9+e^{12}\right )}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}+\frac {2 \left (237-244 e^3+94 e^6-16 e^9+e^{12}\right ) x}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}+\frac {2 \left (-69+55 e^3-13 e^6+e^9\right ) x^2}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2}\right ) \, dx+2 \int \frac {x}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx+\left (2 \left (3-e^3\right )\right ) \int \frac {1}{-\left (-4+e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx \\ & = \frac {17-8 e^3+e^6}{2 \left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )}+2 \int \frac {x}{-\left (4-e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx+\left (2 \left (3-e^3\right )\right ) \int \frac {1}{-\left (4-e^3\right )^2-2 \left (13-7 e^3+e^6\right ) x-\left (1-4 e^3+e^6\right ) x^2+2 \left (3-e^3\right ) x^3-x^4} \, dx+\left (-69+55 e^3-13 e^6+e^9\right ) \int \frac {x^2}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+\left (317-303 e^3+108 e^6-17 e^9+e^{12}\right ) \int \frac {1}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx+\left (237-244 e^3+94 e^6-16 e^9+e^{12}\right ) \int \frac {x}{\left (\left (-4+e^3\right )^2+2 \left (13-7 e^3+e^6\right ) x+\left (1-4 e^3+e^6\right ) x^2-2 \left (3-e^3\right ) x^3+x^4\right )^2} \, dx \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.83 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^2}{16+26 x+x^2-6 x^3+x^4+e^6 (1+x)^2+2 e^3 (-4+x) (1+x)^2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs. \(2(23)=46\).
Time = 0.67 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.46
method | result | size |
risch | \(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\) | \(59\) |
gosper | \(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\) | \(65\) |
norman | \(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\) | \(65\) |
parallelrisch | \(\frac {x^{2}}{x^{2} {\mathrm e}^{6}+2 x^{3} {\mathrm e}^{3}+x^{4}+2 x \,{\mathrm e}^{6}-4 x^{2} {\mathrm e}^{3}-6 x^{3}+{\mathrm e}^{6}-14 x \,{\mathrm e}^{3}+x^{2}-8 \,{\mathrm e}^{3}+26 x +16}\) | \(65\) |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}+\left (4 \,{\mathrm e}^{3}-12\right ) \textit {\_Z}^{7}+\left (6 \,{\mathrm e}^{6}-32 \,{\mathrm e}^{3}+38\right ) \textit {\_Z}^{6}+\left (4 \,{\mathrm e}^{9}-24 \,{\mathrm e}^{6}+24 \,{\mathrm e}^{3}+40\right ) \textit {\_Z}^{5}+\left ({\mathrm e}^{12}-60 \,{\mathrm e}^{6}+248 \,{\mathrm e}^{3}-279\right ) \textit {\_Z}^{4}+\left (-40 \,{\mathrm e}^{9}+4 \,{\mathrm e}^{12}+124 \,{\mathrm e}^{6}-76 \,{\mathrm e}^{3}-140\right ) \textit {\_Z}^{3}+\left (-80 \,{\mathrm e}^{9}+6 \,{\mathrm e}^{12}+398 \,{\mathrm e}^{6}-872 \,{\mathrm e}^{3}+708\right ) \textit {\_Z}^{2}+\left (-60 \,{\mathrm e}^{9}+4 \,{\mathrm e}^{12}+340 \,{\mathrm e}^{6}-864 \,{\mathrm e}^{3}+832\right ) \textit {\_Z} +256+{\mathrm e}^{12}-16 \,{\mathrm e}^{9}+96 \,{\mathrm e}^{6}-256 \,{\mathrm e}^{3}\right )}{\sum }\frac {\left (-\textit {\_R}^{5}+\left (3-{\mathrm e}^{3}\right ) \textit {\_R}^{4}+\left ({\mathrm e}^{6}-7 \,{\mathrm e}^{3}+13\right ) \textit {\_R}^{2}+\left (16+{\mathrm e}^{6}-8 \,{\mathrm e}^{3}\right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{208+9 \textit {\_R}^{5} {\mathrm e}^{6}+7 \textit {\_R}^{6} {\mathrm e}^{3}-30 \textit {\_R}^{4} {\mathrm e}^{6}-48 \textit {\_R}^{5} {\mathrm e}^{3}-60 \textit {\_R}^{3} {\mathrm e}^{6}+30 \textit {\_R}^{4} {\mathrm e}^{3}+248 \textit {\_R}^{3} {\mathrm e}^{3}-57 \textit {\_R}^{2} {\mathrm e}^{3}-436 \textit {\_R} \,{\mathrm e}^{3}-279 \textit {\_R}^{3}-105 \textit {\_R}^{2}+2 \textit {\_R}^{7}-21 \textit {\_R}^{6}+57 \textit {\_R}^{5}+50 \textit {\_R}^{4}+5 \textit {\_R}^{4} {\mathrm e}^{9}+{\mathrm e}^{12} \textit {\_R}^{3}+3 \,{\mathrm e}^{12} \textit {\_R}^{2}+354 \textit {\_R} +{\mathrm e}^{12}-15 \,{\mathrm e}^{9}+85 \,{\mathrm e}^{6}-216 \,{\mathrm e}^{3}+3 \textit {\_R} \,{\mathrm e}^{12}-30 \textit {\_R}^{2} {\mathrm e}^{9}-40 \textit {\_R} \,{\mathrm e}^{9}+93 \textit {\_R}^{2} {\mathrm e}^{6}+199 \textit {\_R} \,{\mathrm e}^{6}}\right )}{2}\) | \(354\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^{2}}{x^{4} - 6 \, x^{3} + x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{6} + 2 \, {\left (x^{3} - 2 \, x^{2} - 7 \, x - 4\right )} e^{3} + 26 \, x + 16} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (19) = 38\).
Time = 3.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^{2}}{x^{4} + x^{3} \left (-6 + 2 e^{3}\right ) + x^{2} \left (- 4 e^{3} + 1 + e^{6}\right ) + x \left (- 14 e^{3} + 26 + 2 e^{6}\right ) - 8 e^{3} + 16 + e^{6}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.04 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^{2}}{x^{4} + 2 \, x^{3} {\left (e^{3} - 3\right )} + x^{2} {\left (e^{6} - 4 \, e^{3} + 1\right )} + 2 \, x {\left (e^{6} - 7 \, e^{3} + 13\right )} + e^{6} - 8 \, e^{3} + 16} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.42 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^{2}}{x^{4} + 2 \, x^{3} e^{3} - 6 \, x^{3} + x^{2} e^{6} - 4 \, x^{2} e^{3} + x^{2} + 2 \, x e^{6} - 14 \, x e^{3} + 26 \, x + e^{6} - 8 \, e^{3} + 16} \]
[In]
[Out]
Time = 12.63 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {32 x+26 x^2+6 x^4-2 x^5+e^6 \left (2 x+2 x^2\right )+e^3 \left (-16 x-14 x^2-2 x^4\right )}{256+832 x+708 x^2-140 x^3-279 x^4+40 x^5+38 x^6-12 x^7+x^8+e^{12} \left (1+4 x+6 x^2+4 x^3+x^4\right )+e^9 \left (-16-60 x-80 x^2-40 x^3+4 x^5\right )+e^6 \left (96+340 x+398 x^2+124 x^3-60 x^4-24 x^5+6 x^6\right )+e^3 \left (-256-864 x-872 x^2-76 x^3+248 x^4+24 x^5-32 x^6+4 x^7\right )} \, dx=\frac {x^2}{x^4+\left (2\,{\mathrm {e}}^3-6\right )\,x^3+\left ({\mathrm {e}}^6-4\,{\mathrm {e}}^3+1\right )\,x^2+\left (2\,{\mathrm {e}}^6-14\,{\mathrm {e}}^3+26\right )\,x-8\,{\mathrm {e}}^3+{\mathrm {e}}^6+16} \]
[In]
[Out]