Integrand size = 177, antiderivative size = 33 \[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=\left (e^{-x+5 x \left (4-\left (8+x \log \left (e^{-x} x\right )\right )^2\right )}-x\right )^2 \] Output:
(exp(5*(4-(8+x*ln(x/exp(x)))^2)*x-x)-x)^2
Leaf count is larger than twice the leaf count of optimal. \(173\) vs. \(2(33)=66\).
Time = 0.28 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.24 \[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=e^{-2 x \left (301+80 x^2+10 x^4+5 x^2 \log ^2\left (e^{-x} x\right )\right )} x^{20 x^4} \left (e^{-x} x\right )^{-20 x^2 \left (8+x^2\right )} \left (e^{10 x^3 \left (8+x \left (x-\log (x)+\log \left (e^{-x} x\right )\right )\right )}-e^{x \left (301+5 x^4+5 x^2 \log ^2(x)+80 x \left (x-\log (x)+\log \left (e^{-x} x\right )\right )+5 x^2 \left (x-\log (x)+\log \left (e^{-x} x\right )\right )^2\right )} x^{1+80 x^2+10 x^3 \left (-\log (x)+\log \left (e^{-x} x\right )\right )}\right )^2 \] Input:
Integrate[2*x + E^(-602*x - 160*x^2*Log[x/E^x] - 10*x^3*Log[x/E^x]^2)*(-60 2 - 160*x + 160*x^2 + (-320*x - 20*x^2 + 20*x^3)*Log[x/E^x] - 30*x^2*Log[x /E^x]^2) + E^(-301*x - 80*x^2*Log[x/E^x] - 5*x^3*Log[x/E^x]^2)*(-2 + 602*x + 160*x^2 - 160*x^3 + (320*x^2 + 20*x^3 - 20*x^4)*Log[x/E^x] + 30*x^3*Log [x/E^x]^2),x]
Output:
(x^(20*x^4)*(E^(10*x^3*(8 + x*(x - Log[x] + Log[x/E^x]))) - E^(x*(301 + 5* x^4 + 5*x^2*Log[x]^2 + 80*x*(x - Log[x] + Log[x/E^x]) + 5*x^2*(x - Log[x] + Log[x/E^x])^2))*x^(1 + 80*x^2 + 10*x^3*(-Log[x] + Log[x/E^x])))^2)/(E^(2 *x*(301 + 80*x^2 + 10*x^4 + 5*x^2*Log[x/E^x]^2))*(x/E^x)^(20*x^2*(8 + x^2) ))
Leaf count is larger than twice the leaf count of optimal. \(208\) vs. \(2(33)=66\).
Time = 3.38 (sec) , antiderivative size = 208, normalized size of antiderivative = 6.30, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {2009}
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.
| \(\displaystyle \int \left (\left (160 x^2-30 x^2 \log ^2\left (e^{-x} x\right )+\left (20 x^3-20 x^2-320 x\right ) \log \left (e^{-x} x\right )-160 x-602\right ) \exp \left (-10 x^3 \log ^2\left (e^{-x} x\right )-160 x^2 \log \left (e^{-x} x\right )-602 x\right )+\left (-160 x^3+30 x^3 \log ^2\left (e^{-x} x\right )+160 x^2+\left (-20 x^4+20 x^3+320 x^2\right ) \log \left (e^{-x} x\right )+602 x-2\right ) \exp \left (-5 x^3 \log ^2\left (e^{-x} x\right )-80 x^2 \log \left (e^{-x} x\right )-301 x\right )+2 x\right ) \, dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^2+\left (e^{-x} x\right )^{-160 x^2} e^{-10 x^3 \log ^2\left (e^{-x} x\right )-602 x}-\frac {2 \left (e^{-x} x\right )^{-80 x^2} e^{-5 x^3 \log ^2\left (e^{-x} x\right )-301 x} \left (-80 x^3+15 x^3 \log ^2\left (e^{-x} x\right )+80 x^2+10 \left (-x^4+x^3+16 x^2\right ) \log \left (e^{-x} x\right )+301 x\right )}{15 x^2 \log ^2\left (e^{-x} x\right )+10 e^x \left (e^{-x}-e^{-x} x\right ) x^2 \log \left (e^{-x} x\right )+80 e^x \left (e^{-x}-e^{-x} x\right ) x+160 x \log \left (e^{-x} x\right )+301}\) |
Input:
Int[2*x + E^(-602*x - 160*x^2*Log[x/E^x] - 10*x^3*Log[x/E^x]^2)*(-602 - 16 0*x + 160*x^2 + (-320*x - 20*x^2 + 20*x^3)*Log[x/E^x] - 30*x^2*Log[x/E^x]^ 2) + E^(-301*x - 80*x^2*Log[x/E^x] - 5*x^3*Log[x/E^x]^2)*(-2 + 602*x + 160 *x^2 - 160*x^3 + (320*x^2 + 20*x^3 - 20*x^4)*Log[x/E^x] + 30*x^3*Log[x/E^x ]^2),x]
Output:
x^2 + E^(-602*x - 10*x^3*Log[x/E^x]^2)/(x/E^x)^(160*x^2) - (2*E^(-301*x - 5*x^3*Log[x/E^x]^2)*(301*x + 80*x^2 - 80*x^3 + 10*(16*x^2 + x^3 - x^4)*Log [x/E^x] + 15*x^3*Log[x/E^x]^2))/((x/E^x)^(80*x^2)*(301 + 80*E^x*x*(E^(-x) - x/E^x) + 160*x*Log[x/E^x] + 10*E^x*x^2*(E^(-x) - x/E^x)*Log[x/E^x] + 15* x^2*Log[x/E^x]^2))
Leaf count of result is larger than twice the leaf count of optimal. \(71\) vs. \(2(31)=62\).
Time = 1.23 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.18
| method | result | size |
| parallelrisch | \(x^{2}-2 \,{\mathrm e}^{-5 \ln \left (x \,{\mathrm e}^{-x}\right )^{2} x^{3}-80 \ln \left (x \,{\mathrm e}^{-x}\right ) x^{2}-301 x} x +{\mathrm e}^{-10 \ln \left (x \,{\mathrm e}^{-x}\right )^{2} x^{3}-160 \ln \left (x \,{\mathrm e}^{-x}\right ) x^{2}-602 x}\) | \(72\) |
| risch | \(\text {Expression too large to display}\) | \(1157\) |
Input:
int((-30*x^2*ln(x/exp(x))^2+(20*x^3-20*x^2-320*x)*ln(x/exp(x))+160*x^2-160 *x-602)*exp(-5*x^3*ln(x/exp(x))^2-80*x^2*ln(x/exp(x))-301*x)^2+(30*x^3*ln( x/exp(x))^2+(-20*x^4+20*x^3+320*x^2)*ln(x/exp(x))-160*x^3+160*x^2+602*x-2) *exp(-5*x^3*ln(x/exp(x))^2-80*x^2*ln(x/exp(x))-301*x)+2*x,x,method=_RETURN VERBOSE)
Output:
x^2-2*exp(-5*x^3*ln(x/exp(x))^2-80*x^2*ln(x/exp(x))-301*x)*x+exp(-5*x^3*ln (x/exp(x))^2-80*x^2*ln(x/exp(x))-301*x)^2
Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.09 \[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=x^{2} - 2 \, x e^{\left (-5 \, x^{3} \log \left (x e^{\left (-x\right )}\right )^{2} - 80 \, x^{2} \log \left (x e^{\left (-x\right )}\right ) - 301 \, x\right )} + e^{\left (-10 \, x^{3} \log \left (x e^{\left (-x\right )}\right )^{2} - 160 \, x^{2} \log \left (x e^{\left (-x\right )}\right ) - 602 \, x\right )} \] Input:
integrate((-30*x^2*log(x/exp(x))^2+(20*x^3-20*x^2-320*x)*log(x/exp(x))+160 *x^2-160*x-602)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)^2+( 30*x^3*log(x/exp(x))^2+(-20*x^4+20*x^3+320*x^2)*log(x/exp(x))-160*x^3+160* x^2+602*x-2)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)+2*x,x, algorithm="fricas")
Output:
x^2 - 2*x*e^(-5*x^3*log(x*e^(-x))^2 - 80*x^2*log(x*e^(-x)) - 301*x) + e^(- 10*x^3*log(x*e^(-x))^2 - 160*x^2*log(x*e^(-x)) - 602*x)
Timed out. \[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=\text {Timed out} \] Input:
integrate((-30*x**2*ln(x/exp(x))**2+(20*x**3-20*x**2-320*x)*ln(x/exp(x))+1 60*x**2-160*x-602)*exp(-5*x**3*ln(x/exp(x))**2-80*x**2*ln(x/exp(x))-301*x) **2+(30*x**3*ln(x/exp(x))**2+(-20*x**4+20*x**3+320*x**2)*ln(x/exp(x))-160* x**3+160*x**2+602*x-2)*exp(-5*x**3*ln(x/exp(x))**2-80*x**2*ln(x/exp(x))-30 1*x)+2*x,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (29) = 58\).
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.52 \[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=x^{2} - 2 \, x e^{\left (-5 \, x^{5} + 10 \, x^{4} \log \left (x\right ) - 5 \, x^{3} \log \left (x\right )^{2} + 80 \, x^{3} - 80 \, x^{2} \log \left (x\right ) - 301 \, x\right )} + e^{\left (-10 \, x^{5} + 20 \, x^{4} \log \left (x\right ) - 10 \, x^{3} \log \left (x\right )^{2} + 160 \, x^{3} - 160 \, x^{2} \log \left (x\right ) - 602 \, x\right )} \] Input:
integrate((-30*x^2*log(x/exp(x))^2+(20*x^3-20*x^2-320*x)*log(x/exp(x))+160 *x^2-160*x-602)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)^2+( 30*x^3*log(x/exp(x))^2+(-20*x^4+20*x^3+320*x^2)*log(x/exp(x))-160*x^3+160* x^2+602*x-2)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)+2*x,x, algorithm="maxima")
Output:
x^2 - 2*x*e^(-5*x^5 + 10*x^4*log(x) - 5*x^3*log(x)^2 + 80*x^3 - 80*x^2*log (x) - 301*x) + e^(-10*x^5 + 20*x^4*log(x) - 10*x^3*log(x)^2 + 160*x^3 - 16 0*x^2*log(x) - 602*x)
\[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=\int { 2 \, {\left (15 \, x^{3} \log \left (x e^{\left (-x\right )}\right )^{2} - 80 \, x^{3} + 80 \, x^{2} - 10 \, {\left (x^{4} - x^{3} - 16 \, x^{2}\right )} \log \left (x e^{\left (-x\right )}\right ) + 301 \, x - 1\right )} e^{\left (-5 \, x^{3} \log \left (x e^{\left (-x\right )}\right )^{2} - 80 \, x^{2} \log \left (x e^{\left (-x\right )}\right ) - 301 \, x\right )} - 2 \, {\left (15 \, x^{2} \log \left (x e^{\left (-x\right )}\right )^{2} - 80 \, x^{2} - 10 \, {\left (x^{3} - x^{2} - 16 \, x\right )} \log \left (x e^{\left (-x\right )}\right ) + 80 \, x + 301\right )} e^{\left (-10 \, x^{3} \log \left (x e^{\left (-x\right )}\right )^{2} - 160 \, x^{2} \log \left (x e^{\left (-x\right )}\right ) - 602 \, x\right )} + 2 \, x \,d x } \] Input:
integrate((-30*x^2*log(x/exp(x))^2+(20*x^3-20*x^2-320*x)*log(x/exp(x))+160 *x^2-160*x-602)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)^2+( 30*x^3*log(x/exp(x))^2+(-20*x^4+20*x^3+320*x^2)*log(x/exp(x))-160*x^3+160* x^2+602*x-2)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)+2*x,x, algorithm="giac")
Output:
integrate(2*(15*x^3*log(x*e^(-x))^2 - 80*x^3 + 80*x^2 - 10*(x^4 - x^3 - 16 *x^2)*log(x*e^(-x)) + 301*x - 1)*e^(-5*x^3*log(x*e^(-x))^2 - 80*x^2*log(x* e^(-x)) - 301*x) - 2*(15*x^2*log(x*e^(-x))^2 - 80*x^2 - 10*(x^3 - x^2 - 16 *x)*log(x*e^(-x)) + 80*x + 301)*e^(-10*x^3*log(x*e^(-x))^2 - 160*x^2*log(x *e^(-x)) - 602*x) + 2*x, x)
Time = 1.87 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.79 \[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=x^2+\frac {x^{20\,x^4}\,{\mathrm {e}}^{-602\,x}\,{\mathrm {e}}^{-10\,x^5}\,{\mathrm {e}}^{160\,x^3}\,{\mathrm {e}}^{-10\,x^3\,{\ln \left (x\right )}^2}}{x^{160\,x^2}}-\frac {2\,x\,x^{10\,x^4}\,{\mathrm {e}}^{-301\,x}\,{\mathrm {e}}^{-5\,x^5}\,{\mathrm {e}}^{80\,x^3}\,{\mathrm {e}}^{-5\,x^3\,{\ln \left (x\right )}^2}}{x^{80\,x^2}} \] Input:
int(2*x + exp(- 301*x - 5*x^3*log(x*exp(-x))^2 - 80*x^2*log(x*exp(-x)))*(6 02*x + 30*x^3*log(x*exp(-x))^2 + 160*x^2 - 160*x^3 + log(x*exp(-x))*(320*x ^2 + 20*x^3 - 20*x^4) - 2) - exp(- 602*x - 10*x^3*log(x*exp(-x))^2 - 160*x ^2*log(x*exp(-x)))*(160*x + 30*x^2*log(x*exp(-x))^2 + log(x*exp(-x))*(320* x + 20*x^2 - 20*x^3) - 160*x^2 + 602),x)
Output:
x^2 + (x^(20*x^4)*exp(-602*x)*exp(-10*x^5)*exp(160*x^3)*exp(-10*x^3*log(x) ^2))/x^(160*x^2) - (2*x*x^(10*x^4)*exp(-301*x)*exp(-5*x^5)*exp(80*x^3)*exp (-5*x^3*log(x)^2))/x^(80*x^2)
\[ \int \left (2 x+e^{-602 x-160 x^2 \log \left (e^{-x} x\right )-10 x^3 \log ^2\left (e^{-x} x\right )} \left (-602-160 x+160 x^2+\left (-320 x-20 x^2+20 x^3\right ) \log \left (e^{-x} x\right )-30 x^2 \log ^2\left (e^{-x} x\right )\right )+e^{-301 x-80 x^2 \log \left (e^{-x} x\right )-5 x^3 \log ^2\left (e^{-x} x\right )} \left (-2+602 x+160 x^2-160 x^3+\left (320 x^2+20 x^3-20 x^4\right ) \log \left (e^{-x} x\right )+30 x^3 \log ^2\left (e^{-x} x\right )\right )\right ) \, dx=\int \left (\left (-30 x^{2} \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )^{2}+\left (20 x^{3}-20 x^{2}-320 x \right ) \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )+160 x^{2}-160 x -602\right ) \left ({\mathrm e}^{-5 x^{3} \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )^{2}-80 x^{2} \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )-301 x}\right )^{2}+\left (30 x^{3} \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )^{2}+\left (-20 x^{4}+20 x^{3}+320 x^{2}\right ) \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )-160 x^{3}+160 x^{2}+602 x -2\right ) {\mathrm e}^{-5 x^{3} \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )^{2}-80 x^{2} \mathrm {log}\left (\frac {x}{{\mathrm e}^{x}}\right )-301 x}+2 x \right )d x \] Input:
int((-30*x^2*log(x/exp(x))^2+(20*x^3-20*x^2-320*x)*log(x/exp(x))+160*x^2-1 60*x-602)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)^2+(30*x^3 *log(x/exp(x))^2+(-20*x^4+20*x^3+320*x^2)*log(x/exp(x))-160*x^3+160*x^2+60 2*x-2)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)+2*x,x)
Output:
int((-30*x^2*log(x/exp(x))^2+(20*x^3-20*x^2-320*x)*log(x/exp(x))+160*x^2-1 60*x-602)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)^2+(30*x^3 *log(x/exp(x))^2+(-20*x^4+20*x^3+320*x^2)*log(x/exp(x))-160*x^3+160*x^2+60 2*x-2)*exp(-5*x^3*log(x/exp(x))^2-80*x^2*log(x/exp(x))-301*x)+2*x,x)