Integrand size = 105, antiderivative size = 28 \[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=e^{e^{e^{\frac {e^{5 \left (-3-x+5 x^2\right )}}{x}+x}}}+x \] Output:
exp(exp(exp(x+exp(25*x^2-5*x-15)/x)))+x
Time = 0.13 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=e^{e^{e^{\frac {e^{-15-5 x+25 x^2}}{x}+x}}}+x \] Input:
Integrate[(x^2 + E^(E^E^((E^(-15 - 5*x + 25*x^2) + x^2)/x) + E^((E^(-15 - 5*x + 25*x^2) + x^2)/x) + (E^(-15 - 5*x + 25*x^2) + x^2)/x)*(x^2 + E^(-15 - 5*x + 25*x^2)*(-1 - 5*x + 50*x^2)))/x^2,x]
Output:
E^E^E^(E^(-15 - 5*x + 25*x^2)/x + 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.
| \(\displaystyle \int \frac {\left (x^2+e^{25 x^2-5 x-15} \left (50 x^2-5 x-1\right )\right ) \exp \left (\frac {x^2+e^{25 x^2-5 x-15}}{x}+e^{e^{\frac {x^2+e^{25 x^2-5 x-15}}{x}}}+e^{\frac {x^2+e^{25 x^2-5 x-15}}{x}}\right )+x^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {(5 x-1) (10 x+1) \exp \left (25 x^2+e^{e^{\frac {e^{25 x^2-5 x-15}}{x}+x}}+e^{\frac {e^{25 x^2-5 x-15}}{x}+x}+\frac {e^{25 x^2-5 x-15}}{x}-4 x-15\right )}{x^2}+\exp \left (e^{e^{\frac {e^{25 x^2-5 x-15}}{x}+x}}+e^{\frac {e^{25 x^2-5 x-15}}{x}+x}+\frac {e^{25 x^2-5 x-15}}{x}+x\right )+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \exp \left (x+e^{e^{x+\frac {e^{25 x^2-5 x-15}}{x}}}+e^{x+\frac {e^{25 x^2-5 x-15}}{x}}+\frac {e^{25 x^2-5 x-15}}{x}\right )dx+50 \int \exp \left (25 x^2-4 x+e^{e^{x+\frac {e^{25 x^2-5 x-15}}{x}}}+e^{x+\frac {e^{25 x^2-5 x-15}}{x}}-15+\frac {e^{25 x^2-5 x-15}}{x}\right )dx-\int \frac {\exp \left (25 x^2-4 x+e^{e^{x+\frac {e^{25 x^2-5 x-15}}{x}}}+e^{x+\frac {e^{25 x^2-5 x-15}}{x}}-15+\frac {e^{25 x^2-5 x-15}}{x}\right )}{x^2}dx-5 \int \frac {\exp \left (25 x^2-4 x+e^{e^{x+\frac {e^{25 x^2-5 x-15}}{x}}}+e^{x+\frac {e^{25 x^2-5 x-15}}{x}}-15+\frac {e^{25 x^2-5 x-15}}{x}\right )}{x}dx+x\) |
Input:
Int[(x^2 + E^(E^E^((E^(-15 - 5*x + 25*x^2) + x^2)/x) + E^((E^(-15 - 5*x + 25*x^2) + x^2)/x) + (E^(-15 - 5*x + 25*x^2) + x^2)/x)*(x^2 + E^(-15 - 5*x + 25*x^2)*(-1 - 5*x + 50*x^2)))/x^2,x]
Output:
$Aborted
Time = 2.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89
| method | result | size |
| risch | \(x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{25 x^{2}-5 x -15}+x^{2}}{x}}}}\) | \(25\) |
| parallelrisch | \(x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {{\mathrm e}^{25 x^{2}-5 x -15}+x^{2}}{x}}}}\) | \(25\) |
Input:
int((((50*x^2-5*x-1)*exp(25*x^2-5*x-15)+x^2)*exp((exp(25*x^2-5*x-15)+x^2)/ x)*exp(exp((exp(25*x^2-5*x-15)+x^2)/x))*exp(exp(exp((exp(25*x^2-5*x-15)+x^ 2)/x)))+x^2)/x^2,x,method=_RETURNVERBOSE)
Output:
x+exp(exp(exp((exp(25*x^2-5*x-15)+x^2)/x)))
Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (22) = 44\).
Time = 0.08 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.50 \[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx={\left (x e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x} + e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )}\right )} + e^{\left (\frac {x^{2} + x e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )} + x e^{\left (e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )}\right )} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )}\right )} e^{\left (-\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x} - e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )}\right )} \] Input:
integrate((((50*x^2-5*x-1)*exp(25*x^2-5*x-15)+x^2)*exp((exp(25*x^2-5*x-15) +x^2)/x)*exp(exp((exp(25*x^2-5*x-15)+x^2)/x))*exp(exp(exp((exp(25*x^2-5*x- 15)+x^2)/x)))+x^2)/x^2,x, algorithm="fricas")
Output:
(x*e^((x^2 + e^(25*x^2 - 5*x - 15))/x + e^((x^2 + e^(25*x^2 - 5*x - 15))/x )) + e^((x^2 + x*e^((x^2 + e^(25*x^2 - 5*x - 15))/x) + x*e^(e^((x^2 + e^(2 5*x^2 - 5*x - 15))/x)) + e^(25*x^2 - 5*x - 15))/x))*e^(-(x^2 + e^(25*x^2 - 5*x - 15))/x - e^((x^2 + e^(25*x^2 - 5*x - 15))/x))
Time = 7.62 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=x + e^{e^{e^{\frac {x^{2} + e^{25 x^{2} - 5 x - 15}}{x}}}} \] Input:
integrate((((50*x**2-5*x-1)*exp(25*x**2-5*x-15)+x**2)*exp((exp(25*x**2-5*x -15)+x**2)/x)*exp(exp((exp(25*x**2-5*x-15)+x**2)/x))*exp(exp(exp((exp(25*x **2-5*x-15)+x**2)/x)))+x**2)/x**2,x)
Output:
x + exp(exp(exp((x**2 + exp(25*x**2 - 5*x - 15))/x)))
Time = 0.33 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=x + e^{\left (e^{\left (e^{\left (x + \frac {e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )}\right )}\right )} \] Input:
integrate((((50*x^2-5*x-1)*exp(25*x^2-5*x-15)+x^2)*exp((exp(25*x^2-5*x-15) +x^2)/x)*exp(exp((exp(25*x^2-5*x-15)+x^2)/x))*exp(exp(exp((exp(25*x^2-5*x- 15)+x^2)/x)))+x^2)/x^2,x, algorithm="maxima")
Output:
x + e^(e^(e^(x + e^(25*x^2 - 5*x - 15)/x)))
\[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=\int { \frac {x^{2} + {\left (x^{2} + {\left (50 \, x^{2} - 5 \, x - 1\right )} e^{\left (25 \, x^{2} - 5 \, x - 15\right )}\right )} e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x} + e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )} + e^{\left (e^{\left (\frac {x^{2} + e^{\left (25 \, x^{2} - 5 \, x - 15\right )}}{x}\right )}\right )}\right )}}{x^{2}} \,d x } \] Input:
integrate((((50*x^2-5*x-1)*exp(25*x^2-5*x-15)+x^2)*exp((exp(25*x^2-5*x-15) +x^2)/x)*exp(exp((exp(25*x^2-5*x-15)+x^2)/x))*exp(exp(exp((exp(25*x^2-5*x- 15)+x^2)/x)))+x^2)/x^2,x, algorithm="giac")
Output:
integrate((x^2 + (x^2 + (50*x^2 - 5*x - 1)*e^(25*x^2 - 5*x - 15))*e^((x^2 + e^(25*x^2 - 5*x - 15))/x + e^((x^2 + e^(25*x^2 - 5*x - 15))/x) + e^(e^(( x^2 + e^(25*x^2 - 5*x - 15))/x))))/x^2, x)
Time = 1.59 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=x+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{\frac {{\mathrm {e}}^{-5\,x}\,{\mathrm {e}}^{-15}\,{\mathrm {e}}^{25\,x^2}}{x}}\,{\mathrm {e}}^x}} \] Input:
int((x^2 - exp(exp((exp(25*x^2 - 5*x - 15) + x^2)/x))*exp(exp(exp((exp(25* x^2 - 5*x - 15) + x^2)/x)))*exp((exp(25*x^2 - 5*x - 15) + x^2)/x)*(exp(25* x^2 - 5*x - 15)*(5*x - 50*x^2 + 1) - x^2))/x^2,x)
Output:
x + exp(exp(exp((exp(-5*x)*exp(-15)*exp(25*x^2))/x)*exp(x)))
\[ \int \frac {x^2+e^{e^{e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}}+e^{\frac {e^{-15-5 x+25 x^2}+x^2}{x}}+\frac {e^{-15-5 x+25 x^2}+x^2}{x}} \left (x^2+e^{-15-5 x+25 x^2} \left (-1-5 x+50 x^2\right )\right )}{x^2} \, dx=\frac {\left (\int e^{\frac {e^{e^{\frac {e^{25 x^{2}}+e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}}+5 x} e^{15} x +e^{25 x^{2}}+e^{\frac {e^{25 x^{2}}+6 e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}} e^{15} x +e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}}d x \right ) e^{15}+50 \left (\int \frac {e^{\frac {e^{e^{\frac {e^{25 x^{2}}+e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}}+5 x} e^{15} x +e^{25 x^{2}}+e^{\frac {e^{25 x^{2}}+6 e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}} e^{15} x +25 e^{5 x} e^{15} x^{3}}{e^{5 x} e^{15} x}}}{e^{4 x}}d x \right )-\left (\int \frac {e^{\frac {e^{e^{\frac {e^{25 x^{2}}+e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}}+5 x} e^{15} x +e^{25 x^{2}}+e^{\frac {e^{25 x^{2}}+6 e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}} e^{15} x +25 e^{5 x} e^{15} x^{3}}{e^{5 x} e^{15} x}}}{e^{4 x} x^{2}}d x \right )-5 \left (\int \frac {e^{\frac {e^{e^{\frac {e^{25 x^{2}}+e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}}+5 x} e^{15} x +e^{25 x^{2}}+e^{\frac {e^{25 x^{2}}+6 e^{5 x} e^{15} x^{2}}{e^{5 x} e^{15} x}} e^{15} x +25 e^{5 x} e^{15} x^{3}}{e^{5 x} e^{15} x}}}{e^{4 x} x}d x \right )+e^{15} x}{e^{15}} \] Input:
int((((50*x^2-5*x-1)*exp(25*x^2-5*x-15)+x^2)*exp((exp(25*x^2-5*x-15)+x^2)/ x)*exp(exp((exp(25*x^2-5*x-15)+x^2)/x))*exp(exp(exp((exp(25*x^2-5*x-15)+x^ 2)/x)))+x^2)/x^2,x)
Output:
(int(e**((e**(e**((e**(25*x**2) + e**(5*x)*e**15*x**2)/(e**(5*x)*e**15*x)) + 5*x)*e**15*x + e**(25*x**2) + e**((e**(25*x**2) + 6*e**(5*x)*e**15*x**2 )/(e**(5*x)*e**15*x))*e**15*x + e**(5*x)*e**15*x**2)/(e**(5*x)*e**15*x)),x )*e**15 + 50*int(e**((e**(e**((e**(25*x**2) + e**(5*x)*e**15*x**2)/(e**(5* x)*e**15*x)) + 5*x)*e**15*x + e**(25*x**2) + e**((e**(25*x**2) + 6*e**(5*x )*e**15*x**2)/(e**(5*x)*e**15*x))*e**15*x + 25*e**(5*x)*e**15*x**3)/(e**(5 *x)*e**15*x))/e**(4*x),x) - int(e**((e**(e**((e**(25*x**2) + e**(5*x)*e**1 5*x**2)/(e**(5*x)*e**15*x)) + 5*x)*e**15*x + e**(25*x**2) + e**((e**(25*x* *2) + 6*e**(5*x)*e**15*x**2)/(e**(5*x)*e**15*x))*e**15*x + 25*e**(5*x)*e** 15*x**3)/(e**(5*x)*e**15*x))/(e**(4*x)*x**2),x) - 5*int(e**((e**(e**((e**( 25*x**2) + e**(5*x)*e**15*x**2)/(e**(5*x)*e**15*x)) + 5*x)*e**15*x + e**(2 5*x**2) + e**((e**(25*x**2) + 6*e**(5*x)*e**15*x**2)/(e**(5*x)*e**15*x))*e **15*x + 25*e**(5*x)*e**15*x**3)/(e**(5*x)*e**15*x))/(e**(4*x)*x),x) + e** 15*x)/e**15