Integrand size = 117, antiderivative size = 33 \[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx=e^{5+e^{x \left (-1+\frac {6}{-5+x}-x^2-\left (-x+x^2\right )^2\right )}} \]
Time = 0.36 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx=e^{5+e^{-\frac {x \left (-11+x-10 x^2+12 x^3-7 x^4+x^5\right )}{-5+x}}} \]
Integrate[(E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x) ) + (11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x))*(-55 + 10*x - 1 51*x^2 + 260*x^3 - 211*x^4 + 58*x^5 - 5*x^6))/(25 - 10*x + x^2),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.
\(\displaystyle \int \frac {\left (-5 x^6+58 x^5-211 x^4+260 x^3-151 x^2+10 x-55\right ) \exp \left (\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{x-5}+e^{\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{x-5}}+5\right )}{x^2-10 x+25} \, dx\) |
\(\Big \downarrow \) 7277 |
\(\displaystyle 4 \int -\frac {\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) \left (5 x^6-58 x^5+211 x^4-260 x^3+151 x^2-10 x+55\right )}{4 (5-x)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) \left (5 x^6-58 x^5+211 x^4-260 x^3+151 x^2-10 x+55\right )}{(5-x)^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\int \left (5 \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) x^4-8 \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) x^3+6 \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) x^2+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right )+\frac {30 \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right )}{(x-5)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right )dx-30 \int \frac {\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right )}{(x-5)^2}dx-6 \int \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) x^2dx+8 \int \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) x^3dx-5 \int \exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}+\exp \left (-\frac {-x^6+7 x^5-12 x^4+10 x^3-x^2+11 x}{5-x}\right )+5\right ) x^4dx\) |
Int[(E^(5 + E^((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x)) + (1 1*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(-5 + x))*(-55 + 10*x - 151*x^2 + 260*x^3 - 211*x^4 + 58*x^5 - 5*x^6))/(25 - 10*x + x^2),x]
3.17.1.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.69 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03
method | result | size |
risch | \({\mathrm e}^{{\mathrm e}^{-\frac {x \left (x^{5}-7 x^{4}+12 x^{3}-10 x^{2}+x -11\right )}{-5+x}}+5}\) | \(34\) |
parallelrisch | \({\mathrm e}^{{\mathrm e}^{\frac {-x^{6}+7 x^{5}-12 x^{4}+10 x^{3}-x^{2}+11 x}{-5+x}}+5}\) | \(40\) |
norman | \(\frac {x \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{6}+7 x^{5}-12 x^{4}+10 x^{3}-x^{2}+11 x}{-5+x}}+5}-5 \,{\mathrm e}^{{\mathrm e}^{\frac {-x^{6}+7 x^{5}-12 x^{4}+10 x^{3}-x^{2}+11 x}{-5+x}}+5}}{-5+x}\) | \(90\) |
int((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5-12*x^4 +10*x^3-x^2+11*x)/(-5+x))*exp(exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x)/(-5+ x))+5)/(x^2-10*x+25),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (28) = 56\).
Time = 0.25 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.15 \[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx=e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - {\left (x - 5\right )} e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5}\right )} - 16 \, x + 25}{x - 5} + \frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5}\right )} \]
integrate((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5- 12*x^4+10*x^3-x^2+11*x)/(-5+x))*exp(exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x )/(-5+x))+5)/(x^2-10*x+25),x, algorithm=\
e^(-(x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - (x - 5)*e^(-(x^6 - 7*x^5 + 12*x ^4 - 10*x^3 + x^2 - 11*x)/(x - 5)) - 16*x + 25)/(x - 5) + (x^6 - 7*x^5 + 1 2*x^4 - 10*x^3 + x^2 - 11*x)/(x - 5))
Time = 0.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx=e^{e^{\frac {- x^{6} + 7 x^{5} - 12 x^{4} + 10 x^{3} - x^{2} + 11 x}{x - 5}} + 5} \]
integrate((-5*x**6+58*x**5-211*x**4+260*x**3-151*x**2+10*x-55)*exp((-x**6+ 7*x**5-12*x**4+10*x**3-x**2+11*x)/(-5+x))*exp(exp((-x**6+7*x**5-12*x**4+10 *x**3-x**2+11*x)/(-5+x))+5)/(x**2-10*x+25),x)
Time = 0.89 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx=e^{\left (e^{\left (-x^{5} + 2 \, x^{4} - 2 \, x^{3} - x + \frac {30}{x - 5} + 6\right )} + 5\right )} \]
integrate((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5- 12*x^4+10*x^3-x^2+11*x)/(-5+x))*exp(exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x )/(-5+x))+5)/(x^2-10*x+25),x, algorithm=\
\[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx=\int { -\frac {{\left (5 \, x^{6} - 58 \, x^{5} + 211 \, x^{4} - 260 \, x^{3} + 151 \, x^{2} - 10 \, x + 55\right )} e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5} + e^{\left (-\frac {x^{6} - 7 \, x^{5} + 12 \, x^{4} - 10 \, x^{3} + x^{2} - 11 \, x}{x - 5}\right )} + 5\right )}}{x^{2} - 10 \, x + 25} \,d x } \]
integrate((-5*x^6+58*x^5-211*x^4+260*x^3-151*x^2+10*x-55)*exp((-x^6+7*x^5- 12*x^4+10*x^3-x^2+11*x)/(-5+x))*exp(exp((-x^6+7*x^5-12*x^4+10*x^3-x^2+11*x )/(-5+x))+5)/(x^2-10*x+25),x, algorithm=\
integrate(-(5*x^6 - 58*x^5 + 211*x^4 - 260*x^3 + 151*x^2 - 10*x + 55)*e^(- (x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - 11*x)/(x - 5) + e^(-(x^6 - 7*x^5 + 12*x^4 - 10*x^3 + x^2 - 11*x)/(x - 5)) + 5)/(x^2 - 10*x + 25), x)
Time = 8.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09 \[ \int \frac {e^{5+e^{\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}}+\frac {11 x-x^2+10 x^3-12 x^4+7 x^5-x^6}{-5+x}} \left (-55+10 x-151 x^2+260 x^3-211 x^4+58 x^5-5 x^6\right )}{25-10 x+x^2} \, dx={\mathrm {e}}^{{\mathrm {e}}^{\frac {11\,x}{x-5}}\,{\mathrm {e}}^{-\frac {x^2}{x-5}}\,{\mathrm {e}}^{-\frac {x^6}{x-5}}\,{\mathrm {e}}^{\frac {7\,x^5}{x-5}}\,{\mathrm {e}}^{\frac {10\,x^3}{x-5}}\,{\mathrm {e}}^{-\frac {12\,x^4}{x-5}}}\,{\mathrm {e}}^5 \]
int(-(exp(exp((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(x - 5)) + 5)*e xp((11*x - x^2 + 10*x^3 - 12*x^4 + 7*x^5 - x^6)/(x - 5))*(151*x^2 - 10*x - 260*x^3 + 211*x^4 - 58*x^5 + 5*x^6 + 55))/(x^2 - 10*x + 25),x)