Integrand size = 277, antiderivative size = 29 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=x-\frac {4 x^2}{\left (\frac {e^{\frac {x}{5-e}} (-4+x)}{x}+x\right )^2} \] Output:
x-4*x^2/((-4+x)/x*exp(x/(5-exp(1)))+x)^2
Timed out. \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=\text {\$Aborted} \] Input:
Integrate[(-5*x^6 + E*x^6 + (320 - 240*x + 60*x^2 - 5*x^3 + E*(-64 + 48*x - 12*x^2 + x^3))/E^((3*x)/(-5 + E)) + (-240*x^2 + 120*x^3 - 15*x^4 + E*(48 *x^2 - 24*x^3 + 3*x^4))/E^((2*x)/(-5 + E)) + (-320*x^3 + 132*x^4 - 23*x^5 + E*(64*x^3 - 20*x^4 + 3*x^5))/E^(x/(-5 + E)))/(-5*x^6 + E*x^6 + (320 - 24 0*x + 60*x^2 - 5*x^3 + E*(-64 + 48*x - 12*x^2 + x^3))/E^((3*x)/(-5 + E)) + (-240*x^2 + 120*x^3 - 15*x^4 + E*(48*x^2 - 24*x^3 + 3*x^4))/E^((2*x)/(-5 + E)) + (60*x^4 - 15*x^5 + E*(-12*x^4 + 3*x^5))/E^(x/(-5 + E))),x]
Output:
$Aborted
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.
| \(\displaystyle \int \frac {e x^6-5 x^6+e^{-\frac {3 x}{e-5}} \left (-5 x^3+60 x^2+e \left (x^3-12 x^2+48 x-64\right )-240 x+320\right )+e^{-\frac {x}{e-5}} \left (-23 x^5+132 x^4-320 x^3+e \left (3 x^5-20 x^4+64 x^3\right )\right )+e^{-\frac {2 x}{e-5}} \left (-15 x^4+120 x^3-240 x^2+e \left (3 x^4-24 x^3+48 x^2\right )\right )}{e x^6-5 x^6+e^{-\frac {x}{e-5}} \left (-15 x^5+60 x^4+e \left (3 x^5-12 x^4\right )\right )+e^{-\frac {3 x}{e-5}} \left (-5 x^3+60 x^2+e \left (x^3-12 x^2+48 x-64\right )-240 x+320\right )+e^{-\frac {2 x}{e-5}} \left (-15 x^4+120 x^3-240 x^2+e \left (3 x^4-24 x^3+48 x^2\right )\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {(e-5) x^6+e^{-\frac {3 x}{e-5}} \left (-5 x^3+60 x^2+e \left (x^3-12 x^2+48 x-64\right )-240 x+320\right )+e^{-\frac {x}{e-5}} \left (-23 x^5+132 x^4-320 x^3+e \left (3 x^5-20 x^4+64 x^3\right )\right )+e^{-\frac {2 x}{e-5}} \left (-15 x^4+120 x^3-240 x^2+e \left (3 x^4-24 x^3+48 x^2\right )\right )}{e x^6-5 x^6+e^{-\frac {x}{e-5}} \left (-15 x^5+60 x^4+e \left (3 x^5-12 x^4\right )\right )+e^{-\frac {3 x}{e-5}} \left (-5 x^3+60 x^2+e \left (x^3-12 x^2+48 x-64\right )-240 x+320\right )+e^{-\frac {2 x}{e-5}} \left (-15 x^4+120 x^3-240 x^2+e \left (3 x^4-24 x^3+48 x^2\right )\right )}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {(e-5) x^6+e^{-\frac {3 x}{e-5}} \left (-5 x^3+60 x^2+e \left (x^3-12 x^2+48 x-64\right )-240 x+320\right )+e^{-\frac {x}{e-5}} \left (-23 x^5+132 x^4-320 x^3+e \left (3 x^5-20 x^4+64 x^3\right )\right )+e^{-\frac {2 x}{e-5}} \left (-15 x^4+120 x^3-240 x^2+e \left (3 x^4-24 x^3+48 x^2\right )\right )}{(e-5) x^6+e^{-\frac {x}{e-5}} \left (-15 x^5+60 x^4+e \left (3 x^5-12 x^4\right )\right )+e^{-\frac {3 x}{e-5}} \left (-5 x^3+60 x^2+e \left (x^3-12 x^2+48 x-64\right )-240 x+320\right )+e^{-\frac {2 x}{e-5}} \left (-15 x^4+120 x^3-240 x^2+e \left (3 x^4-24 x^3+48 x^2\right )\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-5 \left (1-\frac {e}{5}\right ) e^{\frac {3 x}{e-5}} x^6-15 \left (1-\frac {e}{5}\right ) e^{\frac {x}{e-5}} (x-4)^2 x^2+e^{\frac {2 x}{e-5}+1} \left (3 x^2-20 x+64\right ) x^3-e^{\frac {2 x}{e-5}} \left (23 x^2-132 x+320\right ) x^3-5 \left (1-\frac {e}{5}\right ) (x-4)^3}{(5-e) \left (-e^{\frac {x}{e-5}} x^2-x+4\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-\left ((5-e) e^{-\frac {3 x}{5-e}} x^6\right )+e^{1-\frac {2 x}{5-e}} \left (3 x^2-20 x+64\right ) x^3-e^{-\frac {2 x}{5-e}} \left (23 x^2-132 x+320\right ) x^3-3 (5-e) e^{-\frac {x}{5-e}} (4-x)^2 x^2+(5-e) (4-x)^3}{\left (-e^{-\frac {x}{5-e}} x^2-x+4\right )^3}dx}{5-e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (\frac {8 e^{\frac {3 x}{-5+e}} \left (-x^2+(9-e) x-8 (5-e)\right ) x^5}{(4-x) \left (-e^{\frac {x}{-5+e}} x^2-x+4\right )^3}+\frac {8 e^{\frac {2 x}{-5+e}} \left (-x^2+(9-e) x-8 (5-e)\right ) x^3}{(4-x) \left (-e^{\frac {x}{-5+e}} x^2-x+4\right )^2}+5 \left (1-\frac {e}{5}\right )\right )dx}{5-e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-8192 (5-e) \int \frac {e^{\frac {3 x}{-5+e}}}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx-32768 (5-e) \int \frac {e^{\frac {3 x}{-5+e}}}{(x-4) \left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx-2048 (5-e) \int \frac {e^{\frac {3 x}{-5+e}} x}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx-512 (5-e) \int \frac {e^{\frac {3 x}{-5+e}} x^2}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx+512 (5-e) \int \frac {e^{\frac {2 x}{-5+e}}}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^2}dx+2048 (5-e) \int \frac {e^{\frac {2 x}{-5+e}}}{(x-4) \left (e^{\frac {x}{-5+e}} x^2+x-4\right )^2}dx+128 (5-e) \int \frac {e^{\frac {2 x}{-5+e}} x}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^2}dx+32 (5-e) \int \frac {e^{\frac {2 x}{-5+e}} x^2}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^2}dx-8 \int \frac {e^{\frac {3 x}{-5+e}} x^6}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx+8 (5-e) \int \frac {e^{\frac {3 x}{-5+e}} x^5}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx-32 (5-e) \int \frac {e^{\frac {3 x}{-5+e}} x^4}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx+8 \int \frac {e^{\frac {2 x}{-5+e}} x^4}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^2}dx-128 (5-e) \int \frac {e^{\frac {3 x}{-5+e}} x^3}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^3}dx-8 (5-e) \int \frac {e^{\frac {2 x}{-5+e}} x^3}{\left (e^{\frac {x}{-5+e}} x^2+x-4\right )^2}dx+(5-e) x}{5-e}\) |
Input:
Int[(-5*x^6 + E*x^6 + (320 - 240*x + 60*x^2 - 5*x^3 + E*(-64 + 48*x - 12*x ^2 + x^3))/E^((3*x)/(-5 + E)) + (-240*x^2 + 120*x^3 - 15*x^4 + E*(48*x^2 - 24*x^3 + 3*x^4))/E^((2*x)/(-5 + E)) + (-320*x^3 + 132*x^4 - 23*x^5 + E*(6 4*x^3 - 20*x^4 + 3*x^5))/E^(x/(-5 + E)))/(-5*x^6 + E*x^6 + (320 - 240*x + 60*x^2 - 5*x^3 + E*(-64 + 48*x - 12*x^2 + x^3))/E^((3*x)/(-5 + E)) + (-240 *x^2 + 120*x^3 - 15*x^4 + E*(48*x^2 - 24*x^3 + 3*x^4))/E^((2*x)/(-5 + E)) + (60*x^4 - 15*x^5 + E*(-12*x^4 + 3*x^5))/E^(x/(-5 + E))),x]
Output:
$Aborted
Time = 9.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31
| method | result | size |
| risch | \(x -\frac {4 x^{4}}{\left ({\mathrm e}^{-\frac {x}{{\mathrm e}-5}} x +x^{2}-4 \,{\mathrm e}^{-\frac {x}{{\mathrm e}-5}}\right )^{2}}\) | \(38\) |
| norman | \(\frac {x^{5}+64 \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}+{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}} x^{3}-32 x^{2} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}-16 x \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}-4 \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}} x^{2}+2 x^{4} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}}{\left ({\mathrm e}^{-\frac {x}{{\mathrm e}-5}} x +x^{2}-4 \,{\mathrm e}^{-\frac {x}{{\mathrm e}-5}}\right )^{2}}\) | \(128\) |
| parallelrisch | \(\frac {x^{5} {\mathrm e}+2 \,{\mathrm e} \,{\mathrm e}^{-\frac {x}{{\mathrm e}-5}} x^{4}+{\mathrm e} \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}} x^{3}+4 x^{4} {\mathrm e}+8 \,{\mathrm e} \,{\mathrm e}^{-\frac {x}{{\mathrm e}-5}} x^{3}-5 x^{5}-10 x^{4} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}-5 \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}} x^{3}-64 \,{\mathrm e} \,{\mathrm e}^{-\frac {x}{{\mathrm e}-5}} x^{2}-48 \,{\mathrm e} \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}} x -20 x^{4}-40 x^{3} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}+128 \,{\mathrm e} \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}+320 x^{2} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}+240 x \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}-640 \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}}{\left ({\mathrm e}-5\right ) \left (x^{4}+2 x^{3} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}+{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}} x^{2}-8 x^{2} {\mathrm e}^{-\frac {x}{{\mathrm e}-5}}-8 x \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}+16 \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}-5}}\right )}\) | \(306\) |
Input:
int((((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+320)*exp(-x/(exp(1)-5 ))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x^2)*exp(-x/(exp(1)- 5))^2+((3*x^5-20*x^4+64*x^3)*exp(1)-23*x^5+132*x^4-320*x^3)*exp(-x/(exp(1) -5))+x^6*exp(1)-5*x^6)/(((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+32 0)*exp(-x/(exp(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x ^2)*exp(-x/(exp(1)-5))^2+((3*x^5-12*x^4)*exp(1)-15*x^5+60*x^4)*exp(-x/(exp (1)-5))+x^6*exp(1)-5*x^6),x,method=_RETURNVERBOSE)
Output:
x-4*x^4/(exp(-x/(exp(1)-5))*x+x^2-4*exp(-x/(exp(1)-5)))^2
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (28) = 56\).
Time = 0.10 (sec) , antiderivative size = 100, normalized size of antiderivative = 3.45 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=\frac {x^{5} - 4 \, x^{4} + 2 \, {\left (x^{4} - 4 \, x^{3}\right )} e^{\left (-\frac {x}{e - 5}\right )} + {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} e^{\left (-\frac {2 \, x}{e - 5}\right )}}{x^{4} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (-\frac {x}{e - 5}\right )} + {\left (x^{2} - 8 \, x + 16\right )} e^{\left (-\frac {2 \, x}{e - 5}\right )}} \] Input:
integrate((((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+320)*exp(-x/(ex p(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x^2)*exp(-x/(e xp(1)-5))^2+((3*x^5-20*x^4+64*x^3)*exp(1)-23*x^5+132*x^4-320*x^3)*exp(-x/( exp(1)-5))+x^6*exp(1)-5*x^6)/(((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-24 0*x+320)*exp(-x/(exp(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3 -240*x^2)*exp(-x/(exp(1)-5))^2+((3*x^5-12*x^4)*exp(1)-15*x^5+60*x^4)*exp(- x/(exp(1)-5))+x^6*exp(1)-5*x^6),x, algorithm="fricas")
Output:
(x^5 - 4*x^4 + 2*(x^4 - 4*x^3)*e^(-x/(e - 5)) + (x^3 - 8*x^2 + 16*x)*e^(-2 *x/(e - 5)))/(x^4 + 2*(x^3 - 4*x^2)*e^(-x/(e - 5)) + (x^2 - 8*x + 16)*e^(- 2*x/(e - 5)))
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=- \frac {4 x^{4}}{x^{4} + \left (2 x^{3} - 8 x^{2}\right ) e^{- \frac {x}{-5 + e}} + \left (x^{2} - 8 x + 16\right ) e^{- \frac {2 x}{-5 + e}}} + x \] Input:
integrate((((x**3-12*x**2+48*x-64)*exp(1)-5*x**3+60*x**2-240*x+320)*exp(-x /(exp(1)-5))**3+((3*x**4-24*x**3+48*x**2)*exp(1)-15*x**4+120*x**3-240*x**2 )*exp(-x/(exp(1)-5))**2+((3*x**5-20*x**4+64*x**3)*exp(1)-23*x**5+132*x**4- 320*x**3)*exp(-x/(exp(1)-5))+x**6*exp(1)-5*x**6)/(((x**3-12*x**2+48*x-64)* exp(1)-5*x**3+60*x**2-240*x+320)*exp(-x/(exp(1)-5))**3+((3*x**4-24*x**3+48 *x**2)*exp(1)-15*x**4+120*x**3-240*x**2)*exp(-x/(exp(1)-5))**2+((3*x**5-12 *x**4)*exp(1)-15*x**5+60*x**4)*exp(-x/(exp(1)-5))+x**6*exp(1)-5*x**6),x)
Output:
-4*x**4/(x**4 + (2*x**3 - 8*x**2)*exp(-x/(-5 + E)) + (x**2 - 8*x + 16)*exp (-2*x/(-5 + E))) + x
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (28) = 56\).
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.17 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=\frac {x^{5} e^{\left (\frac {2 \, x}{e - 5}\right )} + x^{3} - 4 \, x^{2} + 2 \, {\left (x^{4} - 16 \, x^{2}\right )} e^{\left (\frac {x}{e - 5}\right )} - 16 \, x + 64}{x^{4} e^{\left (\frac {2 \, x}{e - 5}\right )} + x^{2} + 2 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (\frac {x}{e - 5}\right )} - 8 \, x + 16} \] Input:
integrate((((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+320)*exp(-x/(ex p(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x^2)*exp(-x/(e xp(1)-5))^2+((3*x^5-20*x^4+64*x^3)*exp(1)-23*x^5+132*x^4-320*x^3)*exp(-x/( exp(1)-5))+x^6*exp(1)-5*x^6)/(((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-24 0*x+320)*exp(-x/(exp(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3 -240*x^2)*exp(-x/(exp(1)-5))^2+((3*x^5-12*x^4)*exp(1)-15*x^5+60*x^4)*exp(- x/(exp(1)-5))+x^6*exp(1)-5*x^6),x, algorithm="maxima")
Output:
(x^5*e^(2*x/(e - 5)) + x^3 - 4*x^2 + 2*(x^4 - 16*x^2)*e^(x/(e - 5)) - 16*x + 64)/(x^4*e^(2*x/(e - 5)) + x^2 + 2*(x^3 - 4*x^2)*e^(x/(e - 5)) - 8*x + 16)
Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (28) = 56\).
Time = 3.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.72 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=\frac {x^{5} e^{\left (\frac {2 \, x}{e - 5}\right )} + 2 \, x^{4} e^{\left (\frac {x}{e - 5}\right )} + x^{3} - 32 \, x^{2} e^{\left (\frac {x}{e - 5}\right )} - 4 \, x^{2} - 16 \, x + 64}{x^{4} e^{\left (\frac {2 \, x}{e - 5}\right )} + 2 \, x^{3} e^{\left (\frac {x}{e - 5}\right )} - 8 \, x^{2} e^{\left (\frac {x}{e - 5}\right )} + x^{2} - 8 \, x + 16} \] Input:
integrate((((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+320)*exp(-x/(ex p(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x^2)*exp(-x/(e xp(1)-5))^2+((3*x^5-20*x^4+64*x^3)*exp(1)-23*x^5+132*x^4-320*x^3)*exp(-x/( exp(1)-5))+x^6*exp(1)-5*x^6)/(((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-24 0*x+320)*exp(-x/(exp(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3 -240*x^2)*exp(-x/(exp(1)-5))^2+((3*x^5-12*x^4)*exp(1)-15*x^5+60*x^4)*exp(- x/(exp(1)-5))+x^6*exp(1)-5*x^6),x, algorithm="giac")
Output:
(x^5*e^(2*x/(e - 5)) + 2*x^4*e^(x/(e - 5)) + x^3 - 32*x^2*e^(x/(e - 5)) - 4*x^2 - 16*x + 64)/(x^4*e^(2*x/(e - 5)) + 2*x^3*e^(x/(e - 5)) - 8*x^2*e^(x /(e - 5)) + x^2 - 8*x + 16)
Time = 3.08 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.03 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=\frac {x\,\left (x-4\right )\,\left (x+x^3\,{\mathrm {e}}^{\frac {2\,x}{\mathrm {e}-5}}-4\right )+2\,x^3\,{\mathrm {e}}^{\frac {x}{\mathrm {e}-5}}\,\left (x-4\right )}{{\left (x+x^2\,{\mathrm {e}}^{\frac {x}{\mathrm {e}-5}}-4\right )}^2} \] Input:
int((exp(-(3*x)/(exp(1) - 5))*(exp(1)*(48*x - 12*x^2 + x^3 - 64) - 240*x + 60*x^2 - 5*x^3 + 320) + exp(-(2*x)/(exp(1) - 5))*(exp(1)*(48*x^2 - 24*x^3 + 3*x^4) - 240*x^2 + 120*x^3 - 15*x^4) + exp(-x/(exp(1) - 5))*(exp(1)*(64 *x^3 - 20*x^4 + 3*x^5) - 320*x^3 + 132*x^4 - 23*x^5) + x^6*exp(1) - 5*x^6) /(exp(-(3*x)/(exp(1) - 5))*(exp(1)*(48*x - 12*x^2 + x^3 - 64) - 240*x + 60 *x^2 - 5*x^3 + 320) - exp(-x/(exp(1) - 5))*(exp(1)*(12*x^4 - 3*x^5) - 60*x ^4 + 15*x^5) + exp(-(2*x)/(exp(1) - 5))*(exp(1)*(48*x^2 - 24*x^3 + 3*x^4) - 240*x^2 + 120*x^3 - 15*x^4) + x^6*exp(1) - 5*x^6),x)
Output:
(x*(x - 4)*(x + x^3*exp((2*x)/(exp(1) - 5)) - 4) + 2*x^3*exp(x/(exp(1) - 5 ))*(x - 4))/(x + x^2*exp(x/(exp(1) - 5)) - 4)^2
Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.10 \[ \int \frac {-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (-320 x^3+132 x^4-23 x^5+e \left (64 x^3-20 x^4+3 x^5\right )\right )}{-5 x^6+e x^6+e^{-\frac {3 x}{-5+e}} \left (320-240 x+60 x^2-5 x^3+e \left (-64+48 x-12 x^2+x^3\right )\right )+e^{-\frac {2 x}{-5+e}} \left (-240 x^2+120 x^3-15 x^4+e \left (48 x^2-24 x^3+3 x^4\right )\right )+e^{-\frac {x}{-5+e}} \left (60 x^4-15 x^5+e \left (-12 x^4+3 x^5\right )\right )} \, dx=\frac {x \left (e^{\frac {2 x}{e -5}} x^{4}-4 e^{\frac {2 x}{e -5}} x^{3}+2 e^{\frac {x}{e -5}} x^{3}-8 e^{\frac {x}{e -5}} x^{2}+x^{2}-8 x +16\right )}{e^{\frac {2 x}{e -5}} x^{4}+2 e^{\frac {x}{e -5}} x^{3}-8 e^{\frac {x}{e -5}} x^{2}+x^{2}-8 x +16} \] Input:
int((((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+320)*exp(-x/(exp(1)-5 ))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x^2)*exp(-x/(exp(1)- 5))^2+((3*x^5-20*x^4+64*x^3)*exp(1)-23*x^5+132*x^4-320*x^3)*exp(-x/(exp(1) -5))+x^6*exp(1)-5*x^6)/(((x^3-12*x^2+48*x-64)*exp(1)-5*x^3+60*x^2-240*x+32 0)*exp(-x/(exp(1)-5))^3+((3*x^4-24*x^3+48*x^2)*exp(1)-15*x^4+120*x^3-240*x ^2)*exp(-x/(exp(1)-5))^2+((3*x^5-12*x^4)*exp(1)-15*x^5+60*x^4)*exp(-x/(exp (1)-5))+x^6*exp(1)-5*x^6),x)
Output:
(x*(e**((2*x)/(e - 5))*x**4 - 4*e**((2*x)/(e - 5))*x**3 + 2*e**(x/(e - 5)) *x**3 - 8*e**(x/(e - 5))*x**2 + x**2 - 8*x + 16))/(e**((2*x)/(e - 5))*x**4 + 2*e**(x/(e - 5))*x**3 - 8*e**(x/(e - 5))*x**2 + x**2 - 8*x + 16)