Integrand size = 131, antiderivative size = 28 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x}{\frac {3}{-2+x}+\frac {e^{8 e^x}}{x^2}}-\log (x) \] Output:
x/(3/(-2+x)+exp(8*exp(x))/x^2)-ln(x)
Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {(-2+x) x^3}{e^{8 e^x} (-2+x)+3 x^2}-\log (x) \] Input:
Integrate[(-9*x^4 - 6*x^5 + 6*x^6 + E^(16*E^x)*(-4 + 4*x - x^2) + E^(8*E^x )*(12*x^2 + 6*x^3 - 12*x^4 + 3*x^5 + E^x*(-32*x^4 + 32*x^5 - 8*x^6)))/(9*x ^5 + E^(16*E^x)*(4*x - 4*x^2 + x^3) + E^(8*E^x)*(-12*x^3 + 6*x^4)),x]
Output:
((-2 + x)*x^3)/(E^(8*E^x)*(-2 + x) + 3*x^2) - Log[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.
| \(\displaystyle \int \frac {6 x^6-6 x^5-9 x^4+e^{16 e^x} \left (-x^2+4 x-4\right )+e^{8 e^x} \left (3 x^5-12 x^4+6 x^3+12 x^2+e^x \left (-8 x^6+32 x^5-32 x^4\right )\right )}{9 x^5+e^{8 e^x} \left (6 x^4-12 x^3\right )+e^{16 e^x} \left (x^3-4 x^2+4 x\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {6 x^6-6 x^5-9 x^4+e^{16 e^x} \left (-x^2+4 x-4\right )+e^{8 e^x} \left (3 x^5-12 x^4+6 x^3+12 x^2+e^x \left (-8 x^6+32 x^5-32 x^4\right )\right )}{x \left (-3 x^2-e^{8 e^x} x+2 e^{8 e^x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 e^{8 e^x} x^2}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}+\frac {12 e^{8 e^x} x}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {e^{16 e^x} (x-2)^2}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2 x}+\frac {6 x^5}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}+\frac {3 e^{8 e^x} x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {6 x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {12 e^{8 e^x} x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {8 e^{x+8 e^x} (x-2)^2 x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}-\frac {9 x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \frac {e^{16 e^x}}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-4 \int \frac {e^{16 e^x}}{x \left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+12 \int \frac {e^{8 e^x} x}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-\int \frac {e^{16 e^x} x}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+6 \int \frac {e^{8 e^x} x^2}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+6 \int \frac {x^5}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-8 \int \frac {e^{x+8 e^x} x^5}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-6 \int \frac {x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+3 \int \frac {e^{8 e^x} x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx+32 \int \frac {e^{x+8 e^x} x^4}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-9 \int \frac {x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-12 \int \frac {e^{8 e^x} x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx-32 \int \frac {e^{x+8 e^x} x^3}{\left (3 x^2+e^{8 e^x} x-2 e^{8 e^x}\right )^2}dx\) |
Input:
Int[(-9*x^4 - 6*x^5 + 6*x^6 + E^(16*E^x)*(-4 + 4*x - x^2) + E^(8*E^x)*(12* x^2 + 6*x^3 - 12*x^4 + 3*x^5 + E^x*(-32*x^4 + 32*x^5 - 8*x^6)))/(9*x^5 + E ^(16*E^x)*(4*x - 4*x^2 + x^3) + E^(8*E^x)*(-12*x^3 + 6*x^4)),x]
Output:
$Aborted
Time = 0.43 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25
| method | result | size |
| risch | \(-\ln \left (x \right )+\frac {\left (-2+x \right ) x^{3}}{{\mathrm e}^{8 \,{\mathrm e}^{x}} x +3 x^{2}-2 \,{\mathrm e}^{8 \,{\mathrm e}^{x}}}\) | \(35\) |
| parallelrisch | \(-\frac {-3 x^{4}+9 x^{2} \ln \left (x \right )+3 \ln \left (x \right ) {\mathrm e}^{8 \,{\mathrm e}^{x}} x +6 x^{3}-6 \ln \left (x \right ) {\mathrm e}^{8 \,{\mathrm e}^{x}}}{3 \left ({\mathrm e}^{8 \,{\mathrm e}^{x}} x +3 x^{2}-2 \,{\mathrm e}^{8 \,{\mathrm e}^{x}}\right )}\) | \(62\) |
Input:
int(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12* x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8* exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x,method=_RETURNVERBOSE)
Output:
-ln(x)+(-2+x)*x^3/(exp(8*exp(x))*x+3*x^2-2*exp(8*exp(x)))
Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.64 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (x\right ) - {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )} \log \left (x\right )}{3 \, x^{2} + {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )}} \] Input:
integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x ^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)* exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm="fricas")
Output:
(x^4 - 2*x^3 - 3*x^2*log(x) - (x - 2)*e^(8*e^x)*log(x))/(3*x^2 + (x - 2)*e ^(8*e^x))
Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=- \log {\left (x \right )} + \frac {x^{4} - 2 x^{3}}{3 x^{2} + \left (x - 2\right ) e^{8 e^{x}}} \] Input:
integrate(((-x**2+4*x-4)*exp(8*exp(x))**2+((-8*x**6+32*x**5-32*x**4)*exp(x )+3*x**5-12*x**4+6*x**3+12*x**2)*exp(8*exp(x))+6*x**6-6*x**5-9*x**4)/((x** 3-4*x**2+4*x)*exp(8*exp(x))**2+(6*x**4-12*x**3)*exp(8*exp(x))+9*x**5),x)
Output:
-log(x) + (x**4 - 2*x**3)/(3*x**2 + (x - 2)*exp(8*exp(x)))
Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x^{4} - 2 \, x^{3}}{3 \, x^{2} + {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )}} - \log \left (x\right ) \] Input:
integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x ^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)* exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm="maxima")
Output:
(x^4 - 2*x^3)/(3*x^2 + (x - 2)*e^(8*e^x)) - log(x)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.07 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=\frac {x^{4} - 2 \, x^{3} - 3 \, x^{2} \log \left (x\right ) - x e^{\left (8 \, e^{x}\right )} \log \left (x\right ) + 2 \, e^{\left (8 \, e^{x}\right )} \log \left (x\right )}{3 \, x^{2} + x e^{\left (8 \, e^{x}\right )} - 2 \, e^{\left (8 \, e^{x}\right )}} \] Input:
integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x ^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)* exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm="giac")
Output:
(x^4 - 2*x^3 - 3*x^2*log(x) - x*e^(8*e^x)*log(x) + 2*e^(8*e^x)*log(x))/(3* x^2 + x*e^(8*e^x) - 2*e^(8*e^x))
Time = 3.56 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=-\frac {3\,x^2\,\ln \left (x\right )+2\,x^3-x^4-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\ln \left (x\right )+x\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\ln \left (x\right )}{x\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}+3\,x^2} \] Input:
int(-(exp(16*exp(x))*(x^2 - 4*x + 4) - exp(8*exp(x))*(12*x^2 - exp(x)*(32* x^4 - 32*x^5 + 8*x^6) + 6*x^3 - 12*x^4 + 3*x^5) + 9*x^4 + 6*x^5 - 6*x^6)/( exp(16*exp(x))*(4*x - 4*x^2 + x^3) - exp(8*exp(x))*(12*x^3 - 6*x^4) + 9*x^ 5),x)
Output:
-(3*x^2*log(x) + 2*x^3 - x^4 - 2*exp(8*exp(x))*log(x) + x*exp(8*exp(x))*lo g(x))/(x*exp(8*exp(x)) - 2*exp(8*exp(x)) + 3*x^2)
\[ \int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx=-4 \left (\int \frac {e^{16 e^{x}}}{e^{16 e^{x}} x^{3}-4 e^{16 e^{x}} x^{2}+4 e^{16 e^{x}} x +6 e^{8 e^{x}} x^{4}-12 e^{8 e^{x}} x^{3}+9 x^{5}}d x \right )+4 \left (\int \frac {e^{16 e^{x}}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )+6 \left (\int \frac {x^{5}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )-6 \left (\int \frac {x^{4}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )-9 \left (\int \frac {x^{3}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )-8 \left (\int \frac {e^{8 e^{x}+x} x^{5}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )+32 \left (\int \frac {e^{8 e^{x}+x} x^{4}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )-32 \left (\int \frac {e^{8 e^{x}+x} x^{3}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )-\left (\int \frac {e^{16 e^{x}} x}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )+3 \left (\int \frac {e^{8 e^{x}} x^{4}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )-12 \left (\int \frac {e^{8 e^{x}} x^{3}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )+6 \left (\int \frac {e^{8 e^{x}} x^{2}}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right )+12 \left (\int \frac {e^{8 e^{x}} x}{e^{16 e^{x}} x^{2}-4 e^{16 e^{x}} x +4 e^{16 e^{x}}+6 e^{8 e^{x}} x^{3}-12 e^{8 e^{x}} x^{2}+9 x^{4}}d x \right ) \] Input:
int(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12* x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8* exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x)
Output:
- 4*int(e**(16*e**x)/(e**(16*e**x)*x**3 - 4*e**(16*e**x)*x**2 + 4*e**(16* e**x)*x + 6*e**(8*e**x)*x**4 - 12*e**(8*e**x)*x**3 + 9*x**5),x) + 4*int(e* *(16*e**x)/(e**(16*e**x)*x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**( 8*e**x)*x**3 - 12*e**(8*e**x)*x**2 + 9*x**4),x) + 6*int(x**5/(e**(16*e**x) *x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e**x)*x**3 - 12*e**(8* e**x)*x**2 + 9*x**4),x) - 6*int(x**4/(e**(16*e**x)*x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e**x)*x**3 - 12*e**(8*e**x)*x**2 + 9*x**4),x) - 9*int(x**3/(e**(16*e**x)*x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e* *(8*e**x)*x**3 - 12*e**(8*e**x)*x**2 + 9*x**4),x) - 8*int((e**(8*e**x + x) *x**5)/(e**(16*e**x)*x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e* *x)*x**3 - 12*e**(8*e**x)*x**2 + 9*x**4),x) + 32*int((e**(8*e**x + x)*x**4 )/(e**(16*e**x)*x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e**x)*x **3 - 12*e**(8*e**x)*x**2 + 9*x**4),x) - 32*int((e**(8*e**x + x)*x**3)/(e* *(16*e**x)*x**2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e**x)*x**3 - 12*e**(8*e**x)*x**2 + 9*x**4),x) - int((e**(16*e**x)*x)/(e**(16*e**x)*x** 2 - 4*e**(16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e**x)*x**3 - 12*e**(8*e**x )*x**2 + 9*x**4),x) + 3*int((e**(8*e**x)*x**4)/(e**(16*e**x)*x**2 - 4*e**( 16*e**x)*x + 4*e**(16*e**x) + 6*e**(8*e**x)*x**3 - 12*e**(8*e**x)*x**2 + 9 *x**4),x) - 12*int((e**(8*e**x)*x**3)/(e**(16*e**x)*x**2 - 4*e**(16*e**x)* x + 4*e**(16*e**x) + 6*e**(8*e**x)*x**3 - 12*e**(8*e**x)*x**2 + 9*x**4)...