Integrand size = 119, antiderivative size = 18 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=e^{2+\frac {x}{\left (1+\frac {1}{x}+x\right )^2}}-x \] Output:
exp(2+x/(1+1/x+x)^2)-x
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=e^{2+\frac {1}{\left (1+x+x^2\right )^2}+\frac {-1+x}{1+x+x^2}}-x \] Input:
Integrate[(-1 - 3*x - 6*x^2 - 7*x^3 - 6*x^4 - 3*x^5 - x^6 + E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + 2*x + 3*x^2 + 2*x^3 + x^4))*(3*x^2 + x^3 - x^4 ))/(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6),x]
Output:
E^(2 + (1 + x + x^2)^(-2) + (-1 + x)/(1 + x + x^2)) - x
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1}{x^6+3 x^5+6 x^4+7 x^3+6 x^2+3 x+1} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 i \left (\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1\right )}{3 \sqrt {3} \left (-2 x+i \sqrt {3}-1\right )}+\frac {4 i \left (\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1\right )}{3 \sqrt {3} \left (2 x+i \sqrt {3}+1\right )}-\frac {4 \left (\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1\right )}{3 \left (-2 x+i \sqrt {3}-1\right )^2}-\frac {4 \left (\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1\right )}{3 \left (2 x+i \sqrt {3}+1\right )^2}-\frac {8 i \left (\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1\right )}{3 \sqrt {3} \left (-2 x+i \sqrt {3}-1\right )^3}-\frac {8 i \left (\left (-x^4+x^3+3 x^2\right ) \exp \left (\frac {2 x^4+5 x^3+6 x^2+4 x+2}{x^4+2 x^3+3 x^2+2 x+1}\right )-x^6-3 x^5-6 x^4-7 x^3-6 x^2-3 x-1\right )}{3 \sqrt {3} \left (2 x+i \sqrt {3}+1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-x^6-3 x^5-\left (e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}+6\right ) x^4+\left (e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}-7\right ) x^3+3 \left (e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}-2\right ) x^2-3 x-1}{\left (x^2+x+1\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {6 x^2}{\left (x^2+x+1\right )^3}-\frac {3 x}{\left (x^2+x+1\right )^3}-\frac {1}{\left (x^2+x+1\right )^3}-\frac {x^6}{\left (x^2+x+1\right )^3}-\frac {3 x^5}{\left (x^2+x+1\right )^3}-\frac {6 x^4}{\left (x^2+x+1\right )^3}-\frac {7 x^3}{\left (x^2+x+1\right )^3}+\frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}} \left (-x^2+x+3\right ) x^2}{\left (x^2+x+1\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(x+2) x^5}{6 \left (x^2+x+1\right )^2}+\frac {(x+2) x^4}{2 \left (x^2+x+1\right )^2}+\frac {(2 x+3) x^3}{3 \left (x^2+x+1\right )}+\frac {(x+2) x^3}{\left (x^2+x+1\right )^2}-\frac {7 (2 x+1) x^3}{6 \left (x^2+x+1\right )^2}+\frac {(3 x+5) x^2}{2 \left (x^2+x+1\right )}-\frac {5 x^2}{6}+\frac {5 (x+2) x}{6 \left (x^2+x+1\right )}+\frac {(x+2) x}{\left (x^2+x+1\right )^2}-\frac {10 x}{3}-\frac {16}{9} \left (3+i \sqrt {3}\right ) \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (-2 x+i \sqrt {3}-1\right )^3}dx+\frac {16 i \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (-2 x+i \sqrt {3}-1\right )^3}dx}{3 \sqrt {3}}-\frac {8}{9} \left (3-i \sqrt {3}\right ) \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx+2 \left (1-i \sqrt {3}\right ) \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx-\frac {4}{3} \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (-2 x+i \sqrt {3}-1\right )^2}dx+\frac {16}{9} \left (3-i \sqrt {3}\right ) \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (2 x+i \sqrt {3}+1\right )^3}dx+\frac {16 i \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (2 x+i \sqrt {3}+1\right )^3}dx}{3 \sqrt {3}}-\frac {8}{9} \left (3+i \sqrt {3}\right ) \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (2 x+i \sqrt {3}+1\right )^2}dx+2 \left (1+i \sqrt {3}\right ) \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (2 x+i \sqrt {3}+1\right )^2}dx-\frac {4}{3} \int \frac {e^{\frac {2 x^4+5 x^3+6 x^2+4 x+2}{\left (x^2+x+1\right )^2}}}{\left (2 x+i \sqrt {3}+1\right )^2}dx-\frac {2 (x+1)}{x^2+x+1}+\frac {2 x+1}{6 \left (x^2+x+1\right )}+\frac {x+2}{2 \left (x^2+x+1\right )^2}-\frac {2 x+1}{6 \left (x^2+x+1\right )^2}\) |
Input:
Int[(-1 - 3*x - 6*x^2 - 7*x^3 - 6*x^4 - 3*x^5 - x^6 + E^((2 + 4*x + 6*x^2 + 5*x^3 + 2*x^4)/(1 + 2*x + 3*x^2 + 2*x^3 + x^4))*(3*x^2 + x^3 - x^4))/(1 + 3*x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6),x]
Output:
$Aborted
Time = 1.95 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.94
| method | result | size |
| risch | \(-x +{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{\left (x^{2}+x +1\right )^{2}}}\) | \(35\) |
| parallelrisch | \(-x +{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+6\) | \(48\) |
| parts | \(-x +\frac {x^{4} {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+2 x \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+3 \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} x^{2}+2 \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} x^{3}+{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}}{\left (x^{2}+x +1\right )^{2}}\) | \(242\) |
| norman | \(\frac {x^{4} {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+x^{3}+3 x +4 x^{2}-x^{5}+2 x \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+3 \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} x^{2}+2 \,{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} x^{3}+2+{\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}}{\left (x^{2}+x +1\right )^{2}}\) | \(255\) |
| orering | \(\frac {\left (-5+x \right ) \left (\left (-x^{4}+x^{3}+3 x^{2}\right ) {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}-x^{6}-3 x^{5}-6 x^{4}-7 x^{3}-6 x^{2}-3 x -1\right )}{x^{6}+3 x^{5}+6 x^{4}+7 x^{3}+6 x^{2}+3 x +1}+\frac {\left (x^{6}+4 x^{5}+9 x^{3}+21 x^{2}+3 x +1\right ) \left (x^{2}+x +1\right )^{3} \left (\frac {\left (-4 x^{3}+3 x^{2}+6 x \right ) {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}+\left (-x^{4}+x^{3}+3 x^{2}\right ) \left (\frac {8 x^{3}+15 x^{2}+12 x +4}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}-\frac {\left (2 x^{4}+5 x^{3}+6 x^{2}+4 x +2\right ) \left (4 x^{3}+6 x^{2}+6 x +2\right )}{\left (x^{4}+2 x^{3}+3 x^{2}+2 x +1\right )^{2}}\right ) {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}-6 x^{5}-15 x^{4}-24 x^{3}-21 x^{2}-12 x -3}{x^{6}+3 x^{5}+6 x^{4}+7 x^{3}+6 x^{2}+3 x +1}-\frac {\left (\left (-x^{4}+x^{3}+3 x^{2}\right ) {\mathrm e}^{\frac {2 x^{4}+5 x^{3}+6 x^{2}+4 x +2}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}}-x^{6}-3 x^{5}-6 x^{4}-7 x^{3}-6 x^{2}-3 x -1\right ) \left (6 x^{5}+15 x^{4}+24 x^{3}+21 x^{2}+12 x +3\right )}{\left (x^{6}+3 x^{5}+6 x^{4}+7 x^{3}+6 x^{2}+3 x +1\right )^{2}}\right )}{x \left (2 x^{8}+x^{7}-20 x^{6}-45 x^{5}-42 x^{4}-15 x^{3}+2 x^{2}+12 x +6\right )}\) | \(607\) |
Input:
int(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2+2*x+1 ))-x^6-3*x^5-6*x^4-7*x^3-6*x^2-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1), x,method=_RETURNVERBOSE)
Output:
-x+exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^2+x+1)^2)
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (17) = 34\).
Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.56 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=-x + e^{\left (\frac {2 \, x^{4} + 5 \, x^{3} + 6 \, x^{2} + 4 \, x + 2}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right )} \] Input:
integrate(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2 +2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3 *x+1),x, algorithm="fricas")
Output:
-x + e^((2*x^4 + 5*x^3 + 6*x^2 + 4*x + 2)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (14) = 28\).
Time = 0.10 (sec) , antiderivative size = 41, normalized size of antiderivative = 2.28 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=- x + e^{\frac {2 x^{4} + 5 x^{3} + 6 x^{2} + 4 x + 2}{x^{4} + 2 x^{3} + 3 x^{2} + 2 x + 1}} \] Input:
integrate(((-x**4+x**3+3*x**2)*exp((2*x**4+5*x**3+6*x**2+4*x+2)/(x**4+2*x* *3+3*x**2+2*x+1))-x**6-3*x**5-6*x**4-7*x**3-6*x**2-3*x-1)/(x**6+3*x**5+6*x **4+7*x**3+6*x**2+3*x+1),x)
Output:
-x + exp((2*x**4 + 5*x**3 + 6*x**2 + 4*x + 2)/(x**4 + 2*x**3 + 3*x**2 + 2* x + 1))
Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (17) = 34\).
Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 16.94 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=-x + \frac {8 \, x^{3} + 18 \, x^{2} + 16 \, x + 9}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} - \frac {8 \, x^{3} + 9 \, x^{2} + 8 \, x + 2}{2 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} - \frac {4 \, x^{3} + 6 \, x^{2} + 8 \, x + 3}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} + \frac {7 \, {\left (2 \, x^{3} + 6 \, x^{2} + 4 \, x + 3\right )}}{6 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} + \frac {2 \, x^{3} + 3 \, x^{2} + 4 \, x + 3}{2 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )}} - \frac {2 \, x^{3} + 3 \, x^{2} + 2 \, x + 2}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {2 \, x^{3} - 3 \, x^{2} - 2 \, x - 3}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + e^{\left (\frac {x}{x^{2} + x + 1} + \frac {1}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} - \frac {1}{x^{2} + x + 1} + 2\right )} \] Input:
integrate(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2 +2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3 *x+1),x, algorithm="maxima")
Output:
-x + 1/6*(8*x^3 + 18*x^2 + 16*x + 9)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) - 1/2 *(8*x^3 + 9*x^2 + 8*x + 2)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) - 1/6*(4*x^3 + 6*x^2 + 8*x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 7/6*(2*x^3 + 6*x^2 + 4* x + 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 1/2*(2*x^3 + 3*x^2 + 4*x + 3)/(x^ 4 + 2*x^3 + 3*x^2 + 2*x + 1) - (2*x^3 + 3*x^2 + 2*x + 2)/(x^4 + 2*x^3 + 3* x^2 + 2*x + 1) + (2*x^3 - 3*x^2 - 2*x - 3)/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + e^(x/(x^2 + x + 1) + 1/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) - 1/(x^2 + x + 1 ) + 2)
Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (17) = 34\).
Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 7.00 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=-x + e^{\left (\frac {2 \, x^{4}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {5 \, x^{3}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {6 \, x^{2}}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {4 \, x}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1} + \frac {2}{x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1}\right )} \] Input:
integrate(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2 +2*x+1))-x^6-3*x^5-6*x^4-7*x^3-6*x^2-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3 *x+1),x, algorithm="giac")
Output:
-x + e^(2*x^4/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 5*x^3/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 6*x^2/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1) + 4*x/(x^4 + 2*x^3 + 3 *x^2 + 2*x + 1) + 2/(x^4 + 2*x^3 + 3*x^2 + 2*x + 1))
Time = 2.97 (sec) , antiderivative size = 130, normalized size of antiderivative = 7.22 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx={\mathrm {e}}^{\frac {4\,x}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {2\,x^4}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {6\,x^2}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {5\,x^3}{x^4+2\,x^3+3\,x^2+2\,x+1}}\,{\mathrm {e}}^{\frac {2}{x^4+2\,x^3+3\,x^2+2\,x+1}}-x \] Input:
int(-(3*x - exp((4*x + 6*x^2 + 5*x^3 + 2*x^4 + 2)/(2*x + 3*x^2 + 2*x^3 + x ^4 + 1))*(3*x^2 + x^3 - x^4) + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6 + 1)/(3 *x + 6*x^2 + 7*x^3 + 6*x^4 + 3*x^5 + x^6 + 1),x)
Output:
exp((4*x)/(2*x + 3*x^2 + 2*x^3 + x^4 + 1))*exp((2*x^4)/(2*x + 3*x^2 + 2*x^ 3 + x^4 + 1))*exp((6*x^2)/(2*x + 3*x^2 + 2*x^3 + x^4 + 1))*exp((5*x^3)/(2* x + 3*x^2 + 2*x^3 + x^4 + 1))*exp(2/(2*x + 3*x^2 + 2*x^3 + x^4 + 1)) - x
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.89 \[ \int \frac {-1-3 x-6 x^2-7 x^3-6 x^4-3 x^5-x^6+e^{\frac {2+4 x+6 x^2+5 x^3+2 x^4}{1+2 x+3 x^2+2 x^3+x^4}} \left (3 x^2+x^3-x^4\right )}{1+3 x+6 x^2+7 x^3+6 x^4+3 x^5+x^6} \, dx=e^{\frac {x^{3}}{x^{4}+2 x^{3}+3 x^{2}+2 x +1}} e^{2}-x \] Input:
int(((-x^4+x^3+3*x^2)*exp((2*x^4+5*x^3+6*x^2+4*x+2)/(x^4+2*x^3+3*x^2+2*x+1 ))-x^6-3*x^5-6*x^4-7*x^3-6*x^2-3*x-1)/(x^6+3*x^5+6*x^4+7*x^3+6*x^2+3*x+1), x)
Output:
e**(x**3/(x**4 + 2*x**3 + 3*x**2 + 2*x + 1))*e**2 - x