Integrand size = 135, antiderivative size = 32 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=e^{(-4+x) \left (x-\frac {3 \left (1-\frac {x}{e^3}+\frac {x}{8+x}\right )}{x}\right )}-x \] Output:
exp((x-3*(x/(x+8)+1-x/exp(3))/x)*(-4+x))-x
Time = 0.17 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=e^{3-\frac {3 \left (4+3 e^3\right )}{e^3}+\frac {12}{x}-\frac {\left (-3+4 e^3\right ) x}{e^3}+x^2+\frac {36}{8+x}}-x \] Input:
Integrate[(E^3*(-64*x^2 - 16*x^3 - x^4) + E^((-96*x + 12*x^2 + 3*x^3 + E^3 *(96 - 38*x^2 + 4*x^3 + x^4))/(E^3*(8*x + x^2)))*(192*x^2 + 48*x^3 + 3*x^4 + E^3*(-768 - 192*x - 304*x^2 + 64*x^3 + 28*x^4 + 2*x^5)))/(E^3*(64*x^2 + 16*x^3 + x^4)),x]
Output:
E^(3 - (3*(4 + 3*E^3))/E^3 + 12/x - ((-3 + 4*E^3)*x)/E^3 + x^2 + 36/(8 + x )) - 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.
| \(\displaystyle \int \frac {\left (3 x^4+48 x^3+192 x^2+e^3 \left (2 x^5+28 x^4+64 x^3-304 x^2-192 x-768\right )\right ) \exp \left (\frac {3 x^3+12 x^2+e^3 \left (x^4+4 x^3-38 x^2+96\right )-96 x}{e^3 \left (x^2+8 x\right )}\right )+e^3 \left (-x^4-16 x^3-64 x^2\right )}{e^3 \left (x^4+16 x^3+64 x^2\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int -\frac {e^3 \left (x^4+16 x^3+64 x^2\right )-\exp \left (-\frac {-3 x^3-12 x^2+96 x-e^3 \left (x^4+4 x^3-38 x^2+96\right )}{e^3 \left (x^2+8 x\right )}\right ) \left (3 x^4+48 x^3+192 x^2-2 e^3 \left (-x^5-14 x^4-32 x^3+152 x^2+96 x+384\right )\right )}{x^4+16 x^3+64 x^2}dx}{e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {e^3 \left (x^4+16 x^3+64 x^2\right )-\exp \left (-\frac {-3 x^3-12 x^2+96 x-e^3 \left (x^4+4 x^3-38 x^2+96\right )}{e^3 \left (x^2+8 x\right )}\right ) \left (3 x^4+48 x^3+192 x^2-2 e^3 \left (-x^5-14 x^4-32 x^3+152 x^2+96 x+384\right )\right )}{x^4+16 x^3+64 x^2}dx}{e^3}\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle -\frac {\int \frac {e^3 \left (x^4+16 x^3+64 x^2\right )-\exp \left (-\frac {-3 x^3-12 x^2+96 x-e^3 \left (x^4+4 x^3-38 x^2+96\right )}{e^3 \left (x^2+8 x\right )}\right ) \left (3 x^4+48 x^3+192 x^2-2 e^3 \left (-x^5-14 x^4-32 x^3+152 x^2+96 x+384\right )\right )}{x^2 \left (x^2+16 x+64\right )}dx}{e^3}\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle -\frac {\int \frac {e^3 \left (x^4+16 x^3+64 x^2\right )-\exp \left (-\frac {-3 x^3-12 x^2+96 x-e^3 \left (x^4+4 x^3-38 x^2+96\right )}{e^3 \left (x^2+8 x\right )}\right ) \left (3 x^4+48 x^3+192 x^2-2 e^3 \left (-x^5-14 x^4-32 x^3+152 x^2+96 x+384\right )\right )}{x^2 (x+8)^2}dx}{e^3}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\int \left (\frac {\exp \left (\frac {(x-4) \left (e^3 x^3+\left (3+8 e^3\right ) x^2+6 \left (4-e^3\right ) x-24 e^3\right )}{e^3 x (x+8)}\right ) \left (-2 e^3 x^5-3 \left (1+\frac {28 e^3}{3}\right ) x^4-48 \left (1+\frac {4 e^3}{3}\right ) x^3-192 \left (1-\frac {19 e^3}{12}\right ) x^2+192 e^3 x+768 e^3\right )}{x^2 (x+8)^2}+e^3\right )dx}{e^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\left (3-4 e^3\right ) \int \exp \left (\frac {(x-4) \left (e^3 x^3+\left (3+8 e^3\right ) x^2+6 \left (4-e^3\right ) x-24 e^3\right )}{e^3 x (x+8)}\right )dx+12 \int \frac {\exp \left (\frac {(x-4) \left (e^3 x^3+\left (3+8 e^3\right ) x^2+6 \left (4-e^3\right ) x-24 e^3\right )}{e^3 x (x+8)}+3\right )}{x^2}dx-2 \int \exp \left (\frac {(x-4) \left (e^3 x^3+\left (3+8 e^3\right ) x^2+6 \left (4-e^3\right ) x-24 e^3\right )}{e^3 x (x+8)}+3\right ) xdx+36 \int \frac {\exp \left (\frac {(x-4) \left (e^3 x^3+\left (3+8 e^3\right ) x^2+6 \left (4-e^3\right ) x-24 e^3\right )}{e^3 x (x+8)}+3\right )}{(x+8)^2}dx+e^3 x}{e^3}\) |
Input:
Int[(E^3*(-64*x^2 - 16*x^3 - x^4) + E^((-96*x + 12*x^2 + 3*x^3 + E^3*(96 - 38*x^2 + 4*x^3 + x^4))/(E^3*(8*x + x^2)))*(192*x^2 + 48*x^3 + 3*x^4 + E^3 *(-768 - 192*x - 304*x^2 + 64*x^3 + 28*x^4 + 2*x^5)))/(E^3*(64*x^2 + 16*x^ 3 + x^4)),x]
Output:
$Aborted
Time = 1.81 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.59
| method | result | size |
| risch | \(-x +{\mathrm e}^{\frac {\left (x -4\right ) \left (x^{3} {\mathrm e}^{3}+8 x^{2} {\mathrm e}^{3}-6 x \,{\mathrm e}^{3}+3 x^{2}-24 \,{\mathrm e}^{3}+24 x \right ) {\mathrm e}^{-3}}{x \left (x +8\right )}}\) | \(51\) |
| parallelrisch | \({\mathrm e}^{-3} \left (-x \,{\mathrm e}^{3}+{\mathrm e}^{3} {\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x \left (x +8\right )}}+32 \,{\mathrm e}^{3}\right )\) | \(65\) |
| parts | \(-x +\frac {x^{2} {\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}+8 x \,{\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}}{x \left (x +8\right )}\) | \(116\) |
| norman | \(\frac {x^{2} {\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}+64 x -x^{3}+8 x \,{\mathrm e}^{\frac {\left (\left (x^{4}+4 x^{3}-38 x^{2}+96\right ) {\mathrm e}^{3}+3 x^{3}+12 x^{2}-96 x \right ) {\mathrm e}^{-3}}{x^{2}+8 x}}}{x \left (x +8\right )}\) | \(120\) |
Input:
int((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+192*x^2) *exp(((x^4+4*x^3-38*x^2+96)*exp(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp(3))+(- x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x,method=_RETURNVERB OSE)
Output:
-x+exp((x-4)*(x^3*exp(3)+8*x^2*exp(3)-6*x*exp(3)+3*x^2-24*exp(3)+24*x)*exp (-3)/x/(x+8))
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=-x + e^{\left (\frac {{\left (3 \, x^{3} + 12 \, x^{2} + {\left (x^{4} + 4 \, x^{3} - 38 \, x^{2} + 96\right )} e^{3} - 96 \, x\right )} e^{\left (-3\right )}}{x^{2} + 8 \, x}\right )} \] Input:
integrate((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+19 2*x^2)*exp(((x^4+4*x^3-38*x^2+96)*exp(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp( 3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x, algorithm=" fricas")
Output:
-x + e^((3*x^3 + 12*x^2 + (x^4 + 4*x^3 - 38*x^2 + 96)*e^3 - 96*x)*e^(-3)/( x^2 + 8*x))
Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=- x + e^{\frac {3 x^{3} + 12 x^{2} - 96 x + \left (x^{4} + 4 x^{3} - 38 x^{2} + 96\right ) e^{3}}{\left (x^{2} + 8 x\right ) e^{3}}} \] Input:
integrate((((2*x**5+28*x**4+64*x**3-304*x**2-192*x-768)*exp(3)+3*x**4+48*x **3+192*x**2)*exp(((x**4+4*x**3-38*x**2+96)*exp(3)+3*x**3+12*x**2-96*x)/(x **2+8*x)/exp(3))+(-x**4-16*x**3-64*x**2)*exp(3))/(x**4+16*x**3+64*x**2)/ex p(3),x)
Output:
-x + exp((3*x**3 + 12*x**2 - 96*x + (x**4 + 4*x**3 - 38*x**2 + 96)*exp(3)) *exp(-3)/(x**2 + 8*x))
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (30) = 60\).
Time = 0.37 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=-{\left ({\left (x - \frac {64}{x + 8} - 16 \, \log \left (x + 8\right )\right )} e^{3} + 16 \, {\left (\frac {8}{x + 8} + \log \left (x + 8\right )\right )} e^{3} - \frac {64 \, e^{3}}{x + 8} - e^{\left (x^{2} + 3 \, x e^{\left (-3\right )} - 4 \, x + \frac {36}{x + 8} + \frac {12}{x} - 12 \, e^{\left (-3\right )} - 3\right )}\right )} e^{\left (-3\right )} \] Input:
integrate((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+19 2*x^2)*exp(((x^4+4*x^3-38*x^2+96)*exp(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp( 3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x, algorithm=" maxima")
Output:
-((x - 64/(x + 8) - 16*log(x + 8))*e^3 + 16*(8/(x + 8) + log(x + 8))*e^3 - 64*e^3/(x + 8) - e^(x^2 + 3*x*e^(-3) - 4*x + 36/(x + 8) + 12/x - 12*e^(-3 ) - 3))*e^(-3)
Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (30) = 60\).
Time = 0.64 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.09 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=-{\left (x e^{3} - e^{\left (\frac {x^{4} e^{3} + 4 \, x^{3} e^{3} + 3 \, x^{3} - 38 \, x^{2} e^{3} + 12 \, x^{2} - 96 \, x + 96 \, e^{3}}{x^{2} e^{3} + 8 \, x e^{3}} + 3\right )}\right )} e^{\left (-3\right )} \] Input:
integrate((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+19 2*x^2)*exp(((x^4+4*x^3-38*x^2+96)*exp(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp( 3))+(-x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x, algorithm=" giac")
Output:
-(x*e^3 - e^((x^4*e^3 + 4*x^3*e^3 + 3*x^3 - 38*x^2*e^3 + 12*x^2 - 96*x + 9 6*e^3)/(x^2*e^3 + 8*x*e^3) + 3))*e^(-3)
Time = 0.76 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.44 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx={\mathrm {e}}^{\frac {x^4}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {4\,x^3}{x^2+8\,x}}\,{\mathrm {e}}^{-\frac {38\,x^2}{x^2+8\,x}}\,{\mathrm {e}}^{-\frac {96\,x\,{\mathrm {e}}^{-3}}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {96}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {3\,x^3\,{\mathrm {e}}^{-3}}{x^2+8\,x}}\,{\mathrm {e}}^{\frac {12\,x^2\,{\mathrm {e}}^{-3}}{x^2+8\,x}}-x \] Input:
int(-(exp(-3)*(exp(3)*(64*x^2 + 16*x^3 + x^4) - exp((exp(-3)*(exp(3)*(4*x^ 3 - 38*x^2 + x^4 + 96) - 96*x + 12*x^2 + 3*x^3))/(8*x + x^2))*(192*x^2 - e xp(3)*(192*x + 304*x^2 - 64*x^3 - 28*x^4 - 2*x^5 + 768) + 48*x^3 + 3*x^4)) )/(64*x^2 + 16*x^3 + x^4),x)
Output:
exp(x^4/(8*x + x^2))*exp((4*x^3)/(8*x + x^2))*exp(-(38*x^2)/(8*x + x^2))*e xp(-(96*x*exp(-3))/(8*x + x^2))*exp(96/(8*x + x^2))*exp((3*x^3*exp(-3))/(8 *x + x^2))*exp((12*x^2*exp(-3))/(8*x + x^2)) - x
Time = 21.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.16 \[ \int \frac {e^3 \left (-64 x^2-16 x^3-x^4\right )+e^{\frac {-96 x+12 x^2+3 x^3+e^3 \left (96-38 x^2+4 x^3+x^4\right )}{e^3 \left (8 x+x^2\right )}} \left (192 x^2+48 x^3+3 x^4+e^3 \left (-768-192 x-304 x^2+64 x^3+28 x^4+2 x^5\right )\right )}{e^3 \left (64 x^2+16 x^3+x^4\right )} \, dx=\frac {e^{\frac {e^{3} x^{4}+8 e^{3} x^{3}+48 e^{3} x +96 e^{3}+3 x^{3}+12 x^{2}}{e^{3} x^{2}+8 e^{3} x}}-e^{\frac {4 e^{3} x^{2}+32 e^{3} x +96}{e^{3} x +8 e^{3}}} e^{6} x}{e^{\frac {4 e^{3} x^{2}+32 e^{3} x +96}{e^{3} x +8 e^{3}}} e^{6}} \] Input:
int((((2*x^5+28*x^4+64*x^3-304*x^2-192*x-768)*exp(3)+3*x^4+48*x^3+192*x^2) *exp(((x^4+4*x^3-38*x^2+96)*exp(3)+3*x^3+12*x^2-96*x)/(x^2+8*x)/exp(3))+(- x^4-16*x^3-64*x^2)*exp(3))/(x^4+16*x^3+64*x^2)/exp(3),x)
Output:
(e**((e**3*x**4 + 8*e**3*x**3 + 48*e**3*x + 96*e**3 + 3*x**3 + 12*x**2)/(e **3*x**2 + 8*e**3*x)) - e**((4*e**3*x**2 + 32*e**3*x + 96)/(e**3*x + 8*e** 3))*e**6*x)/(e**((4*e**3*x**2 + 32*e**3*x + 96)/(e**3*x + 8*e**3))*e**6)