Integrand size = 152, antiderivative size = 25 \[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=1+e^{\left (4+e^x+\frac {10}{x^2}+x-x (4+x)\right )^2} x \]
Time = 0.16 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.20 \[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=e^{\frac {\left (-10-\left (4+e^x\right ) x^2+3 x^3+x^4\right )^2}{x^4}} x \]
Integrate[(E^((100 + 80*x^2 - 60*x^3 - 4*x^4 + E^(2*x)*x^4 - 24*x^5 + x^6 + 6*x^7 + x^8 + E^x*(20*x^2 + 8*x^4 - 6*x^5 - 2*x^6))/x^4)*(-400 - 160*x^2 + 60*x^3 + x^4 - 24*x^5 + 2*E^(2*x)*x^5 + 2*x^6 + 18*x^7 + 4*x^8 + E^x*(- 40*x^2 + 20*x^3 + 2*x^5 - 10*x^6 - 2*x^7)))/x^4,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.
\(\displaystyle \int \frac {\left (4 x^8+18 x^7+2 x^6+2 e^{2 x} x^5-24 x^5+x^4+60 x^3-160 x^2+e^x \left (-2 x^7-10 x^6+2 x^5+20 x^3-40 x^2\right )-400\right ) \exp \left (\frac {x^8+6 x^7+x^6-24 x^5+e^{2 x} x^4-4 x^4-60 x^3+80 x^2+e^x \left (-2 x^6-6 x^5+8 x^4+20 x^2\right )+100}{x^4}\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}} \left (4 x^8+18 x^7+2 x^6+2 e^{2 x} x^5-24 x^5+x^4+60 x^3-160 x^2+e^x \left (-2 x^7-10 x^6+2 x^5+20 x^3-40 x^2\right )-400\right )}{x^4}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (2 x \exp \left (\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+2 x\right )-\frac {2 e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+x} \left (x^5+5 x^4-x^3-10 x+20\right )}{x^2}+\frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}} \left (4 x^8+18 x^7+2 x^6-24 x^5+x^4+60 x^3-160 x^2-400\right )}{x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \int \exp \left (\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+2 x\right ) xdx+\int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}}dx-400 \int \frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}}}{x^4}dx-160 \int \frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}}}{x^2}dx-40 \int \frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+x}}{x^2}dx+60 \int \frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}}}{x}dx+20 \int \frac {e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+x}}{x}dx-24 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}} xdx+2 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+x} xdx+2 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}} x^2dx-10 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+x} x^2dx+18 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}} x^3dx-2 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}+x} x^3dx+4 \int e^{\frac {\left (x^4+3 x^3-e^x x^2-4 x^2-10\right )^2}{x^4}} x^4dx\) |
Int[(E^((100 + 80*x^2 - 60*x^3 - 4*x^4 + E^(2*x)*x^4 - 24*x^5 + x^6 + 6*x^ 7 + x^8 + E^x*(20*x^2 + 8*x^4 - 6*x^5 - 2*x^6))/x^4)*(-400 - 160*x^2 + 60* x^3 + x^4 - 24*x^5 + 2*E^(2*x)*x^5 + 2*x^6 + 18*x^7 + 4*x^8 + E^x*(-40*x^2 + 20*x^3 + 2*x^5 - 10*x^6 - 2*x^7)))/x^4,x]
3.12.32.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(72\) vs. \(2(23)=46\).
Time = 5.71 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.92
method | result | size |
parallelrisch | \(x \,{\mathrm e}^{\frac {{\mathrm e}^{2 x} x^{4}+\left (-2 x^{6}-6 x^{5}+8 x^{4}+20 x^{2}\right ) {\mathrm e}^{x}+x^{8}+6 x^{7}+x^{6}-24 x^{5}-4 x^{4}-60 x^{3}+80 x^{2}+100}{x^{4}}}\) | \(73\) |
risch | \(x \,{\mathrm e}^{-\frac {-x^{8}+2 x^{6} {\mathrm e}^{x}-6 x^{7}+6 x^{5} {\mathrm e}^{x}-x^{6}-8 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{2 x} x^{4}+24 x^{5}+4 x^{4}-20 \,{\mathrm e}^{x} x^{2}+60 x^{3}-80 x^{2}-100}{x^{4}}}\) | \(83\) |
int((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8+18*x^ 7+2*x^6-24*x^5+x^4+60*x^3-160*x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x^5+8*x ^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^4)/x^4,x ,method=_RETURNVERBOSE)
x*exp((exp(x)^2*x^4+(-2*x^6-6*x^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^ 5-4*x^4-60*x^3+80*x^2+100)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=x e^{\left (\frac {x^{8} + 6 \, x^{7} + x^{6} - 24 \, x^{5} + x^{4} e^{\left (2 \, x\right )} - 4 \, x^{4} - 60 \, x^{3} + 80 \, x^{2} - 2 \, {\left (x^{6} + 3 \, x^{5} - 4 \, x^{4} - 10 \, x^{2}\right )} e^{x} + 100}{x^{4}}\right )} \]
integrate((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8 +18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x ^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^4) /x^4,x, algorithm=\
x*e^((x^8 + 6*x^7 + x^6 - 24*x^5 + x^4*e^(2*x) - 4*x^4 - 60*x^3 + 80*x^2 - 2*(x^6 + 3*x^5 - 4*x^4 - 10*x^2)*e^x + 100)/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (22) = 44\).
Time = 1.61 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.84 \[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=x e^{\frac {x^{8} + 6 x^{7} + x^{6} - 24 x^{5} + x^{4} e^{2 x} - 4 x^{4} - 60 x^{3} + 80 x^{2} + \left (- 2 x^{6} - 6 x^{5} + 8 x^{4} + 20 x^{2}\right ) e^{x} + 100}{x^{4}}} \]
integrate((2*x**5*exp(x)**2+(-2*x**7-10*x**6+2*x**5+20*x**3-40*x**2)*exp(x )+4*x**8+18*x**7+2*x**6-24*x**5+x**4+60*x**3-160*x**2-400)*exp((exp(x)**2* x**4+(-2*x**6-6*x**5+8*x**4+20*x**2)*exp(x)+x**8+6*x**7+x**6-24*x**5-4*x** 4-60*x**3+80*x**2+100)/x**4)/x**4,x)
x*exp((x**8 + 6*x**7 + x**6 - 24*x**5 + x**4*exp(2*x) - 4*x**4 - 60*x**3 + 80*x**2 + (-2*x**6 - 6*x**5 + 8*x**4 + 20*x**2)*exp(x) + 100)/x**4)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (26) = 52\).
Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.44 \[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=x e^{\left (x^{4} + 6 \, x^{3} - 2 \, x^{2} e^{x} + x^{2} - 6 \, x e^{x} - 24 \, x - \frac {60}{x} + \frac {20 \, e^{x}}{x^{2}} + \frac {80}{x^{2}} + \frac {100}{x^{4}} + e^{\left (2 \, x\right )} + 8 \, e^{x} - 4\right )} \]
integrate((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8 +18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x ^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^4) /x^4,x, algorithm=\
x*e^(x^4 + 6*x^3 - 2*x^2*e^x + x^2 - 6*x*e^x - 24*x - 60/x + 20*e^x/x^2 + 80/x^2 + 100/x^4 + e^(2*x) + 8*e^x - 4)
\[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=\int { \frac {{\left (4 \, x^{8} + 18 \, x^{7} + 2 \, x^{6} + 2 \, x^{5} e^{\left (2 \, x\right )} - 24 \, x^{5} + x^{4} + 60 \, x^{3} - 160 \, x^{2} - 2 \, {\left (x^{7} + 5 \, x^{6} - x^{5} - 10 \, x^{3} + 20 \, x^{2}\right )} e^{x} - 400\right )} e^{\left (\frac {x^{8} + 6 \, x^{7} + x^{6} - 24 \, x^{5} + x^{4} e^{\left (2 \, x\right )} - 4 \, x^{4} - 60 \, x^{3} + 80 \, x^{2} - 2 \, {\left (x^{6} + 3 \, x^{5} - 4 \, x^{4} - 10 \, x^{2}\right )} e^{x} + 100}{x^{4}}\right )}}{x^{4}} \,d x } \]
integrate((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8 +18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x ^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^4) /x^4,x, algorithm=\
integrate((4*x^8 + 18*x^7 + 2*x^6 + 2*x^5*e^(2*x) - 24*x^5 + x^4 + 60*x^3 - 160*x^2 - 2*(x^7 + 5*x^6 - x^5 - 10*x^3 + 20*x^2)*e^x - 400)*e^((x^8 + 6 *x^7 + x^6 - 24*x^5 + x^4*e^(2*x) - 4*x^4 - 60*x^3 + 80*x^2 - 2*(x^6 + 3*x ^5 - 4*x^4 - 10*x^2)*e^x + 100)/x^4)/x^4, x)
Time = 9.90 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.88 \[ \int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x \left (20 x^2+8 x^4-6 x^5-2 x^6\right )}{x^4}} \left (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x \left (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7\right )\right )}{x^4} \, dx=x\,{\mathrm {e}}^{-6\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-24\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {20\,{\mathrm {e}}^x}{x^2}}\,{\mathrm {e}}^{6\,x^3}\,{\mathrm {e}}^{-\frac {60}{x}}\,{\mathrm {e}}^{\frac {80}{x^2}}\,{\mathrm {e}}^{\frac {100}{x^4}}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{8\,{\mathrm {e}}^x} \]
int((exp((x^4*exp(2*x) + exp(x)*(20*x^2 + 8*x^4 - 6*x^5 - 2*x^6) + 80*x^2 - 60*x^3 - 4*x^4 - 24*x^5 + x^6 + 6*x^7 + x^8 + 100)/x^4)*(2*x^5*exp(2*x) - exp(x)*(40*x^2 - 20*x^3 - 2*x^5 + 10*x^6 + 2*x^7) - 160*x^2 + 60*x^3 + x ^4 - 24*x^5 + 2*x^6 + 18*x^7 + 4*x^8 - 400))/x^4,x)