Integrand size = 150, antiderivative size = 30 \[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=e^{e^{e^{-x^2} x}} x+\left (4-\frac {5}{x}+x\right )^{5 x/3} \]
Time = 1.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=e^{e^{e^{-x^2} x}} x+\left (4-\frac {5}{x}+x\right )^{5 x/3} \]
Integrate[(E^E^(x/E^x^2)*(E^x^2*(-15 + 12*x + 3*x^2) + E^(x/E^x^2)*(-15*x + 12*x^2 + 33*x^3 - 24*x^4 - 6*x^5)) + ((-5 + 4*x + x^2)/x)^((5*x)/3)*(E^x ^2*(25 + 5*x^2) + E^x^2*(-25 + 20*x + 5*x^2)*Log[(-5 + 4*x + x^2)/x]))/(E^ x^2*(-15 + 12*x + 3*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.
\(\displaystyle \int \frac {e^{-x^2} \left (\left (\frac {x^2+4 x-5}{x}\right )^{5 x/3} \left (e^{x^2} \left (5 x^2+25\right )+e^{x^2} \left (5 x^2+20 x-25\right ) \log \left (\frac {x^2+4 x-5}{x}\right )\right )+e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (3 x^2+12 x-15\right )+e^{e^{-x^2} x} \left (-6 x^5-24 x^4+33 x^3+12 x^2-15 x\right )\right )\right )}{3 x^2+12 x-15} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {5 \left (x-\frac {5}{x}+4\right )^{5 x/3} \left (x^2+x^2 \log \left (x-\frac {5}{x}+4\right )+4 x \log \left (x-\frac {5}{x}+4\right )-5 \log \left (x-\frac {5}{x}+4\right )+5\right )}{3 (x-1) (x+5)}+e^{e^{e^{-x^2} x}-x^2} \left (e^{e^{-x^2} x} x+e^{x^2}-2 e^{e^{-x^2} x} x^3\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int e^{e^{e^{-x^2} x}}dx+\int e^{-x^2+e^{-x^2} x+e^{e^{-x^2} x}} xdx-2 \int e^{-x^2+e^{-x^2} x+e^{e^{-x^2} x}} x^3dx+\frac {5}{3} \int \left (x+4-\frac {5}{x}\right )^{5 x/3}dx+\frac {5}{3} \int \frac {\left (x+4-\frac {5}{x}\right )^{5 x/3}}{x-1}dx-\frac {25}{3} \int \frac {\left (x+4-\frac {5}{x}\right )^{5 x/3}}{x+5}dx-\frac {25}{3} \int \frac {\left (x+4-\frac {5}{x}\right )^{\frac {5 x}{3}-1}}{x}dx-\frac {5}{3} \int x \left (x+4-\frac {5}{x}\right )^{\frac {5 x}{3}-1}dx+\left (x-\frac {5}{x}+4\right )^{5 x/3}\) |
Int[(E^E^(x/E^x^2)*(E^x^2*(-15 + 12*x + 3*x^2) + E^(x/E^x^2)*(-15*x + 12*x ^2 + 33*x^3 - 24*x^4 - 6*x^5)) + ((-5 + 4*x + x^2)/x)^((5*x)/3)*(E^x^2*(25 + 5*x^2) + E^x^2*(-25 + 20*x + 5*x^2)*Log[(-5 + 4*x + x^2)/x]))/(E^x^2*(- 15 + 12*x + 3*x^2)),x]
3.11.97.3.1 Defintions of rubi rules used
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.60
\[x \,{\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{-x^{2}}}}+x^{-\frac {5 x}{3}} \left (x^{2}+4 x -5\right )^{\frac {5 x}{3}} {\mathrm e}^{-\frac {5 i \pi \,\operatorname {csgn}\left (\frac {i \left (x^{2}+4 x -5\right )}{x}\right ) x \left (-\operatorname {csgn}\left (\frac {i \left (x^{2}+4 x -5\right )}{x}\right )+\operatorname {csgn}\left (i \left (x^{2}+4 x -5\right )\right )\right ) \left (-\operatorname {csgn}\left (\frac {i \left (x^{2}+4 x -5\right )}{x}\right )+\operatorname {csgn}\left (\frac {i}{x}\right )\right )}{6}}\]
int((((-6*x^5-24*x^4+33*x^3+12*x^2-15*x)*exp(x/exp(x^2))+(3*x^2+12*x-15)*e xp(x^2))*exp(exp(x/exp(x^2)))+((5*x^2+20*x-25)*exp(x^2)*ln((x^2+4*x-5)/x)+ (5*x^2+25)*exp(x^2))*exp(5/3*x*ln((x^2+4*x-5)/x)))/(3*x^2+12*x-15)/exp(x^2 ),x)
x*exp(exp(x*exp(-x^2)))+x^(-5/3*x)*(x^2+4*x-5)^(5/3*x)*exp(-5/6*I*Pi*csgn( I/x*(x^2+4*x-5))*x*(-csgn(I/x*(x^2+4*x-5))+csgn(I*(x^2+4*x-5)))*(-csgn(I/x *(x^2+4*x-5))+csgn(I/x)))
Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=x e^{\left (e^{\left (x e^{\left (-x^{2}\right )}\right )}\right )} + \left (\frac {x^{2} + 4 \, x - 5}{x}\right )^{\frac {5}{3} \, x} \]
integrate((((-6*x^5-24*x^4+33*x^3+12*x^2-15*x)*exp(x/exp(x^2))+(3*x^2+12*x -15)*exp(x^2))*exp(exp(x/exp(x^2)))+((5*x^2+20*x-25)*exp(x^2)*log((x^2+4*x -5)/x)+(5*x^2+25)*exp(x^2))*exp(5/3*x*log((x^2+4*x-5)/x)))/(3*x^2+12*x-15) /exp(x^2),x, algorithm=\
Timed out. \[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=\text {Timed out} \]
integrate((((-6*x**5-24*x**4+33*x**3+12*x**2-15*x)*exp(x/exp(x**2))+(3*x** 2+12*x-15)*exp(x**2))*exp(exp(x/exp(x**2)))+((5*x**2+20*x-25)*exp(x**2)*ln ((x**2+4*x-5)/x)+(5*x**2+25)*exp(x**2))*exp(5/3*x*ln((x**2+4*x-5)/x)))/(3* x**2+12*x-15)/exp(x**2),x)
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.43 \[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=\frac {x e^{\left (\frac {5}{3} \, x \log \left (x\right ) + e^{\left (x e^{\left (-x^{2}\right )}\right )}\right )} + e^{\left (\frac {5}{3} \, x \log \left (x + 5\right ) + \frac {5}{3} \, x \log \left (x - 1\right )\right )}}{x^{\frac {5}{3} \, x}} \]
integrate((((-6*x^5-24*x^4+33*x^3+12*x^2-15*x)*exp(x/exp(x^2))+(3*x^2+12*x -15)*exp(x^2))*exp(exp(x/exp(x^2)))+((5*x^2+20*x-25)*exp(x^2)*log((x^2+4*x -5)/x)+(5*x^2+25)*exp(x^2))*exp(5/3*x*log((x^2+4*x-5)/x)))/(3*x^2+12*x-15) /exp(x^2),x, algorithm=\
\[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=\int { \frac {{\left (5 \, {\left ({\left (x^{2} + 4 \, x - 5\right )} e^{\left (x^{2}\right )} \log \left (\frac {x^{2} + 4 \, x - 5}{x}\right ) + {\left (x^{2} + 5\right )} e^{\left (x^{2}\right )}\right )} \left (\frac {x^{2} + 4 \, x - 5}{x}\right )^{\frac {5}{3} \, x} + 3 \, {\left ({\left (x^{2} + 4 \, x - 5\right )} e^{\left (x^{2}\right )} - {\left (2 \, x^{5} + 8 \, x^{4} - 11 \, x^{3} - 4 \, x^{2} + 5 \, x\right )} e^{\left (x e^{\left (-x^{2}\right )}\right )}\right )} e^{\left (e^{\left (x e^{\left (-x^{2}\right )}\right )}\right )}\right )} e^{\left (-x^{2}\right )}}{3 \, {\left (x^{2} + 4 \, x - 5\right )}} \,d x } \]
integrate((((-6*x^5-24*x^4+33*x^3+12*x^2-15*x)*exp(x/exp(x^2))+(3*x^2+12*x -15)*exp(x^2))*exp(exp(x/exp(x^2)))+((5*x^2+20*x-25)*exp(x^2)*log((x^2+4*x -5)/x)+(5*x^2+25)*exp(x^2))*exp(5/3*x*log((x^2+4*x-5)/x)))/(3*x^2+12*x-15) /exp(x^2),x, algorithm=\
integrate(1/3*(5*((x^2 + 4*x - 5)*e^(x^2)*log((x^2 + 4*x - 5)/x) + (x^2 + 5)*e^(x^2))*((x^2 + 4*x - 5)/x)^(5/3*x) + 3*((x^2 + 4*x - 5)*e^(x^2) - (2* x^5 + 8*x^4 - 11*x^3 - 4*x^2 + 5*x)*e^(x*e^(-x^2)))*e^(e^(x*e^(-x^2))))*e^ (-x^2)/(x^2 + 4*x - 5), x)
Time = 9.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-x^2} \left (e^{e^{e^{-x^2} x}} \left (e^{x^2} \left (-15+12 x+3 x^2\right )+e^{e^{-x^2} x} \left (-15 x+12 x^2+33 x^3-24 x^4-6 x^5\right )\right )+\left (\frac {-5+4 x+x^2}{x}\right )^{5 x/3} \left (e^{x^2} \left (25+5 x^2\right )+e^{x^2} \left (-25+20 x+5 x^2\right ) \log \left (\frac {-5+4 x+x^2}{x}\right )\right )\right )}{-15+12 x+3 x^2} \, dx=x\,{\mathrm {e}}^{{\mathrm {e}}^{x\,{\mathrm {e}}^{-x^2}}}+{\left (x-\frac {5}{x}+4\right )}^{\frac {5\,x}{3}} \]