Integrand size = 145, antiderivative size = 27 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x (3+x)}{-5+e^{\left (-e^{-5+3 e^{10}}+x\right )^2}+x} \] Output:
x/(exp((x-exp(3*exp(5)^2-5))^2)+x-5)*(3+x)
Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 6.21 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {e^{2 e^{-5+3 e^{10}} x} x (3+x)}{e^{e^{-10+6 e^{10}}+x^2}+e^{2 e^{-5+3 e^{10}} x} (-5+x)} \] Input:
Integrate[(-15 - 10*x + x^2 + E^(E^(-10 + 6*E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*(3 + 2*x - 6*x^2 - 2*x^3 + E^(-5 + 3*E^10)*(6*x + 2*x^2)))/(25 + E^(2 *E^(-10 + 6*E^10) - 4*E^(-5 + 3*E^10)*x + 2*x^2) - 10*x + x^2 + E^(E^(-10 + 6*E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*(-10 + 2*x)),x]
Output:
(E^(2*E^(-5 + 3*E^10)*x)*x*(3 + x))/(E^(E^(-10 + 6*E^10) + x^2) + E^(2*E^( -5 + 3*E^10)*x)*(-5 + 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.
| \(\displaystyle \int \frac {x^2+e^{x^2-2 e^{3 e^{10}-5} x+e^{6 e^{10}-10}} \left (-2 x^3-6 x^2+e^{3 e^{10}-5} \left (2 x^2+6 x\right )+2 x+3\right )-10 x-15}{\exp \left (2 x^2-4 e^{3 e^{10}-5} x+2 e^{6 e^{10}-10}\right )+x^2+e^{x^2-2 e^{3 e^{10}-5} x+e^{6 e^{10}-10}} (2 x-10)-10 x+25} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{4 e^{3 e^{10}-5} x} \left (x^2+e^{x^2-2 e^{3 e^{10}-5} x+e^{6 e^{10}-10}} \left (-2 x^3-6 x^2+e^{3 e^{10}-5} \left (2 x^2+6 x\right )+2 x+3\right )-10 x-15\right )}{\left (-e^{x^2+e^{6 e^{10}-10}}-e^{2 e^{3 e^{10}-5} x} x+5 e^{2 e^{3 e^{10}-5} x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{4 e^{3 e^{10}-5} x-5} x (x+3) \left (2 e^5 x^2-2 \left (5 e^5+e^{3 e^{10}}\right ) x+10 e^{3 e^{10}}-e^5\right )}{\left (-e^{x^2+e^{6 e^{10}-10}}-e^{2 e^{3 e^{10}-5} x} x+5 e^{2 e^{3 e^{10}-5} x}\right )^2}+\frac {e^{2 e^{3 e^{10}-5} x-5} \left (2 e^5 x^3+2 \left (3 e^5-e^{3 e^{10}}\right ) x^2-2 \left (e^5+3 e^{3 e^{10}}\right ) x-3 e^5\right )}{-e^{x^2+e^{6 e^{10}-10}}-e^{2 e^{3 e^{10}-5} x} x+5 e^{2 e^{3 e^{10}-5} x}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {x^3}{x+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-5}dx-2 \left (3-e^{3 e^{10}-5}\right ) \int \frac {x^2}{x+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-5}dx-3 \left (e^5-10 e^{3 e^{10}}\right ) \int \frac {e^{4 e^{-5+3 e^{10}} x-5} x}{\left (e^{2 e^{-5+3 e^{10}} x} x-5 e^{2 e^{-5+3 e^{10}} x}+e^{x^2+e^{-10+6 e^{10}}}\right )^2}dx-\left (31 e^5-4 e^{3 e^{10}}\right ) \int \frac {e^{4 e^{-5+3 e^{10}} x-5} x^2}{\left (e^{2 e^{-5+3 e^{10}} x} x-5 e^{2 e^{-5+3 e^{10}} x}+e^{x^2+e^{-10+6 e^{10}}}\right )^2}dx+2 \int \frac {e^{4 e^{-5+3 e^{10}} x} x^4}{\left (e^{2 e^{-5+3 e^{10}} x} x-5 e^{2 e^{-5+3 e^{10}} x}+e^{x^2+e^{-10+6 e^{10}}}\right )^2}dx-2 \left (2 e^5+e^{3 e^{10}}\right ) \int \frac {e^{4 e^{-5+3 e^{10}} x-5} x^3}{\left (e^{2 e^{-5+3 e^{10}} x} x-5 e^{2 e^{-5+3 e^{10}} x}+e^{x^2+e^{-10+6 e^{10}}}\right )^2}dx+3 \int \frac {1}{x+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-5}dx+2 \left (1+3 e^{3 e^{10}-5}\right ) \int \frac {x}{x+e^{\frac {\left (e^{3 e^{10}}-e^5 x\right )^2}{e^{10}}}-5}dx\) |
Input:
Int[(-15 - 10*x + x^2 + E^(E^(-10 + 6*E^10) - 2*E^(-5 + 3*E^10)*x + x^2)*( 3 + 2*x - 6*x^2 - 2*x^3 + E^(-5 + 3*E^10)*(6*x + 2*x^2)))/(25 + E^(2*E^(-1 0 + 6*E^10) - 4*E^(-5 + 3*E^10)*x + 2*x^2) - 10*x + x^2 + E^(E^(-10 + 6*E^ 10) - 2*E^(-5 + 3*E^10)*x + x^2)*(-10 + 2*x)),x]
Output:
$Aborted
Time = 0.66 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22
| method | result | size |
| risch | \(\frac {\left (3+x \right ) x}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\) | \(33\) |
| parallelrisch | \(\frac {x^{2}+3 x}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\) | \(42\) |
| norman | \(\frac {x^{2}-3 \,{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}+15}{x +{\mathrm e}^{{\mathrm e}^{6 \,{\mathrm e}^{10}-10}-2 x \,{\mathrm e}^{3 \,{\mathrm e}^{10}-5}+x^{2}}-5}\) | \(70\) |
Input:
int((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp(5)^2- 5)^2-2*x*exp(3*exp(5)^2-5)+x^2)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^2-2*x* exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5) ^2-5)+x^2)+x^2-10*x+25),x,method=_RETURNVERBOSE)
Output:
(3+x)*x/(x+exp(exp(6*exp(10)-10)-2*x*exp(3*exp(10)-5)+x^2)-5)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.30 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^{2} + 3 \, x}{x + e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 5} \] Input:
integrate((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp (5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^ 2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3* exp(5)^2-5)+x^2)+x^2-10*x+25),x, algorithm="fricas")
Output:
(x^2 + 3*x)/(x + e^(x^2 - 2*x*e^(3*e^10 - 5) + e^(6*e^10 - 10)) - 5)
Time = 0.71 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.26 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^{2} + 3 x}{x + e^{x^{2} - 2 x e^{-5 + 3 e^{10}} + e^{-10 + 6 e^{10}}} - 5} \] Input:
integrate((((2*x**2+6*x)*exp(3*exp(5)**2-5)-2*x**3-6*x**2+2*x+3)*exp(exp(3 *exp(5)**2-5)**2-2*x*exp(3*exp(5)**2-5)+x**2)+x**2-10*x-15)/(exp(exp(3*exp (5)**2-5)**2-2*x*exp(3*exp(5)**2-5)+x**2)**2+(2*x-10)*exp(exp(3*exp(5)**2- 5)**2-2*x*exp(3*exp(5)**2-5)+x**2)+x**2-10*x+25),x)
Output:
(x**2 + 3*x)/(x + exp(x**2 - 2*x*exp(-5 + 3*exp(10)) + exp(-10 + 6*exp(10) )) - 5)
Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {{\left (x^{2} - 5 \, x + 40\right )} e^{\left (2 \, x e^{\left (3 \, e^{10} - 5\right )}\right )} - 8 \, e^{\left (x^{2} + e^{\left (6 \, e^{10} - 10\right )}\right )}}{{\left (x - 5\right )} e^{\left (2 \, x e^{\left (3 \, e^{10} - 5\right )}\right )} + e^{\left (x^{2} + e^{\left (6 \, e^{10} - 10\right )}\right )}} \] Input:
integrate((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp (5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^ 2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3* exp(5)^2-5)+x^2)+x^2-10*x+25),x, algorithm="maxima")
Output:
((x^2 - 5*x + 40)*e^(2*x*e^(3*e^10 - 5)) - 8*e^(x^2 + e^(6*e^10 - 10)))/(( x - 5)*e^(2*x*e^(3*e^10 - 5)) + e^(x^2 + e^(6*e^10 - 10)))
\[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\int { \frac {x^{2} - {\left (2 \, x^{3} + 6 \, x^{2} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{\left (3 \, e^{10} - 5\right )} - 2 \, x - 3\right )} e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 10 \, x - 15}{x^{2} + 2 \, {\left (x - 5\right )} e^{\left (x^{2} - 2 \, x e^{\left (3 \, e^{10} - 5\right )} + e^{\left (6 \, e^{10} - 10\right )}\right )} - 10 \, x + e^{\left (2 \, x^{2} - 4 \, x e^{\left (3 \, e^{10} - 5\right )} + 2 \, e^{\left (6 \, e^{10} - 10\right )}\right )} + 25} \,d x } \] Input:
integrate((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp (5)^2-5)^2-2*x*exp(3*exp(5)^2-5)+x^2)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^ 2-2*x*exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3* exp(5)^2-5)+x^2)+x^2-10*x+25),x, algorithm="giac")
Output:
integrate((x^2 - (2*x^3 + 6*x^2 - 2*(x^2 + 3*x)*e^(3*e^10 - 5) - 2*x - 3)* e^(x^2 - 2*x*e^(3*e^10 - 5) + e^(6*e^10 - 10)) - 10*x - 15)/(x^2 + 2*(x - 5)*e^(x^2 - 2*x*e^(3*e^10 - 5) + e^(6*e^10 - 10)) - 10*x + e^(2*x^2 - 4*x* e^(3*e^10 - 5) + 2*e^(6*e^10 - 10)) + 25), x)
Time = 39.32 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {x^2+3\,x}{x+{\mathrm {e}}^{x^2-2\,{\mathrm {e}}^{3\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-5}\,x+{\mathrm {e}}^{6\,{\mathrm {e}}^{10}}\,{\mathrm {e}}^{-10}}-5} \] Input:
int(-(10*x - x^2 - exp(exp(6*exp(10) - 10) - 2*x*exp(3*exp(10) - 5) + x^2) *(2*x + exp(3*exp(10) - 5)*(6*x + 2*x^2) - 6*x^2 - 2*x^3 + 3) + 15)/(exp(2 *exp(6*exp(10) - 10) - 4*x*exp(3*exp(10) - 5) + 2*x^2) - 10*x + exp(exp(6* exp(10) - 10) - 2*x*exp(3*exp(10) - 5) + x^2)*(2*x - 10) + x^2 + 25),x)
Output:
(3*x + x^2)/(x + exp(x^2 + exp(6*exp(10))*exp(-10) - 2*x*exp(3*exp(10))*ex p(-5)) - 5)
Time = 1.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 4.41 \[ \int \frac {-15-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} \left (3+2 x-6 x^2-2 x^3+e^{-5+3 e^{10}} \left (6 x+2 x^2\right )\right )}{25+e^{2 e^{-10+6 e^{10}}-4 e^{-5+3 e^{10}} x+2 x^2}-10 x+x^2+e^{e^{-10+6 e^{10}}-2 e^{-5+3 e^{10}} x+x^2} (-10+2 x)} \, dx=\frac {-3 e^{\frac {e^{6 e^{10}}+e^{10} x^{2}}{e^{10}}}+e^{\frac {2 e^{3 e^{10}} x}{e^{5}}} x^{2}+15 e^{\frac {2 e^{3 e^{10}} x}{e^{5}}}}{e^{\frac {e^{6 e^{10}}+e^{10} x^{2}}{e^{10}}}+e^{\frac {2 e^{3 e^{10}} x}{e^{5}}} x -5 e^{\frac {2 e^{3 e^{10}} x}{e^{5}}}} \] Input:
int((((2*x^2+6*x)*exp(3*exp(5)^2-5)-2*x^3-6*x^2+2*x+3)*exp(exp(3*exp(5)^2- 5)^2-2*x*exp(3*exp(5)^2-5)+x^2)+x^2-10*x-15)/(exp(exp(3*exp(5)^2-5)^2-2*x* exp(3*exp(5)^2-5)+x^2)^2+(2*x-10)*exp(exp(3*exp(5)^2-5)^2-2*x*exp(3*exp(5) ^2-5)+x^2)+x^2-10*x+25),x)
Output:
( - 3*e**((e**(6*e**10) + e**10*x**2)/e**10) + e**((2*e**(3*e**10)*x)/e**5 )*x**2 + 15*e**((2*e**(3*e**10)*x)/e**5))/(e**((e**(6*e**10) + e**10*x**2) /e**10) + e**((2*e**(3*e**10)*x)/e**5)*x - 5*e**((2*e**(3*e**10)*x)/e**5))