Integrand size = 90, antiderivative size = 22 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=e^{e^x+\frac {1024 e^{-5 (5+2 x)^4}}{x^5}} \] Output:
exp(1024/exp((5+2*x)^4)^5/x^5+exp(x))
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=e^{e^x+\frac {1024 e^{-5 (5+2 x)^4}}{x^5}} \] Input:
Integrate[(E^(E^x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4) )/x^5)*(E^x*x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4)*(-5 - 5000*x - 6000*x^2 - 2400*x^3 - 320*x^4))/x^5))/x,x]
Output:
E^(E^x + 1024/(E^(5*(5 + 2*x)^4)*x^5))
Time = 0.91 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {7257}
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.
\(\displaystyle \int \frac {\left (\frac {1024 e^{-80 x^4-800 x^3-3000 x^2-5000 x-3125} \left (-320 x^4-2400 x^3-6000 x^2-5000 x-5\right )}{x^5}+e^x x\right ) \exp \left (\frac {1024 e^{-80 x^4-800 x^3-3000 x^2-5000 x-3125}}{x^5}+e^x\right )}{x} \, dx\) |
\(\Big \downarrow \) 7257 |
\(\displaystyle \exp \left (\frac {1024 e^{-80 x^4-800 x^3-3000 x^2-5000 x-3125}}{x^5}+e^x\right )\) |
Input:
Int[(E^(E^x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4))/x^5) *(E^x*x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4)*(-5 - 500 0*x - 6000*x^2 - 2400*x^3 - 320*x^4))/x^5))/x,x]
Output:
E^(E^x + (1024*E^(-3125 - 5000*x - 3000*x^2 - 800*x^3 - 80*x^4))/x^5)
Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim p[q*(F^v/Log[F]), x] /; !FalseQ[q]] /; FreeQ[F, x]
Time = 182.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14
method | result | size |
risch | \({\mathrm e}^{\frac {x^{5} {\mathrm e}^{x}+1024 \,{\mathrm e}^{-5 \left (5+2 x \right )^{4}}}{x^{5}}}\) | \(25\) |
parallelrisch | \({\mathrm e}^{\frac {\left ({\mathrm e}^{x} x^{5} {\mathrm e}^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x +3125}+1024\right ) {\mathrm e}^{-80 x^{4}-800 x^{3}-3000 x^{2}-5000 x -3125}}{x^{5}}}\) | \(60\) |
Input:
int((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x^3+600 *x^2+1000*x+625)^5+exp(x)*x)*exp(1024/x^5/exp(16*x^4+160*x^3+600*x^2+1000* x+625)^5+exp(x))/x,x,method=_RETURNVERBOSE)
Output:
exp((x^5*exp(x)+1024*exp(-5*(5+2*x)^4))/x^5)
Time = 0.09 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=e^{\left (\frac {x^{5} e^{x} + 1024 \, e^{\left (-80 \, x^{4} - 800 \, x^{3} - 3000 \, x^{2} - 5000 \, x - 3125\right )}}{x^{5}}\right )} \] Input:
integrate((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x ^3+600*x^2+1000*x+625)^5+exp(x)*x)*exp(1024/x^5/exp(16*x^4+160*x^3+600*x^2 +1000*x+625)^5+exp(x))/x,x, algorithm="fricas")
Output:
e^((x^5*e^x + 1024*e^(-80*x^4 - 800*x^3 - 3000*x^2 - 5000*x - 3125))/x^5)
Time = 0.33 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=e^{e^{x} + \frac {1024 e^{- 80 x^{4} - 800 x^{3} - 3000 x^{2} - 5000 x - 3125}}{x^{5}}} \] Input:
integrate((1024*(-320*x**4-2400*x**3-6000*x**2-5000*x-5)/x**5/exp(16*x**4+ 160*x**3+600*x**2+1000*x+625)**5+exp(x)*x)*exp(1024/x**5/exp(16*x**4+160*x **3+600*x**2+1000*x+625)**5+exp(x))/x,x)
Output:
exp(exp(x) + 1024*exp(-80*x**4 - 800*x**3 - 3000*x**2 - 5000*x - 3125)/x** 5)
Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=e^{\left (\frac {1024 \, e^{\left (-80 \, x^{4} - 800 \, x^{3} - 3000 \, x^{2} - 5000 \, x - 3125\right )}}{x^{5}} + e^{x}\right )} \] Input:
integrate((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x ^3+600*x^2+1000*x+625)^5+exp(x)*x)*exp(1024/x^5/exp(16*x^4+160*x^3+600*x^2 +1000*x+625)^5+exp(x))/x,x, algorithm="maxima")
Output:
e^(1024*e^(-80*x^4 - 800*x^3 - 3000*x^2 - 5000*x - 3125)/x^5 + e^x)
Time = 0.21 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=e^{\left (\frac {1024 \, e^{\left (-80 \, x^{4} - 800 \, x^{3} - 3000 \, x^{2} - 5000 \, x - 3125\right )}}{x^{5}} + e^{x}\right )} \] Input:
integrate((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x ^3+600*x^2+1000*x+625)^5+exp(x)*x)*exp(1024/x^5/exp(16*x^4+160*x^3+600*x^2 +1000*x+625)^5+exp(x))/x,x, algorithm="giac")
Output:
e^(1024*e^(-80*x^4 - 800*x^3 - 3000*x^2 - 5000*x - 3125)/x^5 + e^x)
Time = 4.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx={\mathrm {e}}^{\frac {1024\,{\mathrm {e}}^{-5000\,x}\,{\mathrm {e}}^{-3125}\,{\mathrm {e}}^{-80\,x^4}\,{\mathrm {e}}^{-800\,x^3}\,{\mathrm {e}}^{-3000\,x^2}}{x^5}}\,{\mathrm {e}}^{{\mathrm {e}}^x} \] Input:
int((exp(exp(x) + (1024*exp(- 5000*x - 3000*x^2 - 800*x^3 - 80*x^4 - 3125) )/x^5)*(x*exp(x) - (exp(- 5000*x - 3000*x^2 - 800*x^3 - 80*x^4 - 3125)*(51 20000*x + 6144000*x^2 + 2457600*x^3 + 327680*x^4 + 5120))/x^5))/x,x)
Output:
exp((1024*exp(-5000*x)*exp(-3125)*exp(-80*x^4)*exp(-800*x^3)*exp(-3000*x^2 ))/x^5)*exp(exp(x))
\[ \int \frac {e^{e^x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4}}{x^5}} \left (e^x x+\frac {1024 e^{-3125-5000 x-3000 x^2-800 x^3-80 x^4} \left (-5-5000 x-6000 x^2-2400 x^3-320 x^4\right )}{x^5}\right )}{x} \, dx=\frac {\left (\int e^{\frac {e^{80 x^{4}+800 x^{3}+3000 x^{2}+5001 x} e^{3125} x^{5}+e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{6}+1024}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{5}}}d x \right ) e^{3125}-5120 \left (\int \frac {e^{\frac {e^{80 x^{4}+800 x^{3}+3000 x^{2}+5001 x} e^{3125} x^{5}+1024}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{5}}}}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} x^{6}}d x \right )-5120000 \left (\int \frac {e^{\frac {e^{80 x^{4}+800 x^{3}+3000 x^{2}+5001 x} e^{3125} x^{5}+1024}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{5}}}}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} x^{5}}d x \right )-6144000 \left (\int \frac {e^{\frac {e^{80 x^{4}+800 x^{3}+3000 x^{2}+5001 x} e^{3125} x^{5}+1024}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{5}}}}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} x^{4}}d x \right )-2457600 \left (\int \frac {e^{\frac {e^{80 x^{4}+800 x^{3}+3000 x^{2}+5001 x} e^{3125} x^{5}+1024}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{5}}}}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} x^{3}}d x \right )-327680 \left (\int \frac {e^{\frac {e^{80 x^{4}+800 x^{3}+3000 x^{2}+5001 x} e^{3125} x^{5}+1024}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} e^{3125} x^{5}}}}{e^{80 x^{4}+800 x^{3}+3000 x^{2}+5000 x} x^{2}}d x \right )}{e^{3125}} \] Input:
int((1024*(-320*x^4-2400*x^3-6000*x^2-5000*x-5)/x^5/exp(16*x^4+160*x^3+600 *x^2+1000*x+625)^5+exp(x)*x)*exp(1024/x^5/exp(16*x^4+160*x^3+600*x^2+1000* x+625)^5+exp(x))/x,x)
Output:
(int(e**((e**(80*x**4 + 800*x**3 + 3000*x**2 + 5001*x)*e**3125*x**5 + e**( 80*x**4 + 800*x**3 + 3000*x**2 + 5000*x)*e**3125*x**6 + 1024)/(e**(80*x**4 + 800*x**3 + 3000*x**2 + 5000*x)*e**3125*x**5)),x)*e**3125 - 5120*int(e** ((e**(80*x**4 + 800*x**3 + 3000*x**2 + 5001*x)*e**3125*x**5 + 1024)/(e**(8 0*x**4 + 800*x**3 + 3000*x**2 + 5000*x)*e**3125*x**5))/(e**(80*x**4 + 800* x**3 + 3000*x**2 + 5000*x)*x**6),x) - 5120000*int(e**((e**(80*x**4 + 800*x **3 + 3000*x**2 + 5001*x)*e**3125*x**5 + 1024)/(e**(80*x**4 + 800*x**3 + 3 000*x**2 + 5000*x)*e**3125*x**5))/(e**(80*x**4 + 800*x**3 + 3000*x**2 + 50 00*x)*x**5),x) - 6144000*int(e**((e**(80*x**4 + 800*x**3 + 3000*x**2 + 500 1*x)*e**3125*x**5 + 1024)/(e**(80*x**4 + 800*x**3 + 3000*x**2 + 5000*x)*e* *3125*x**5))/(e**(80*x**4 + 800*x**3 + 3000*x**2 + 5000*x)*x**4),x) - 2457 600*int(e**((e**(80*x**4 + 800*x**3 + 3000*x**2 + 5001*x)*e**3125*x**5 + 1 024)/(e**(80*x**4 + 800*x**3 + 3000*x**2 + 5000*x)*e**3125*x**5))/(e**(80* x**4 + 800*x**3 + 3000*x**2 + 5000*x)*x**3),x) - 327680*int(e**((e**(80*x* *4 + 800*x**3 + 3000*x**2 + 5001*x)*e**3125*x**5 + 1024)/(e**(80*x**4 + 80 0*x**3 + 3000*x**2 + 5000*x)*e**3125*x**5))/(e**(80*x**4 + 800*x**3 + 3000 *x**2 + 5000*x)*x**2),x))/e**3125