Integrand size = 320, antiderivative size = 30 \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=x \log (3) \log \left (e^{4+\frac {x}{20-e^{3-e^x}}}-x\right ) \]
Time = 3.96 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=x \log (3) \log \left (e^{\frac {1}{20} \left (80+x-\frac {e^3 x}{e^3-20 e^{e^x}}\right )}-x\right ) \]
Integrate[(-400*x*Log[3] - E^(6 - 2*E^x)*x*Log[3] + 40*E^(3 - E^x)*x*Log[3 ] + E^((-80 + 4*E^(3 - E^x) - x)/(-20 + E^(3 - E^x)))*(20*x*Log[3] + E^(3 - E^x)*(-(x*Log[3]) - E^x*x^2*Log[3])) + (-400*x*Log[3] - E^(6 - 2*E^x)*x* Log[3] + 40*E^(3 - E^x)*x*Log[3] + E^((-80 + 4*E^(3 - E^x) - x)/(-20 + E^( 3 - E^x)))*(400*Log[3] + E^(6 - 2*E^x)*Log[3] - 40*E^(3 - E^x)*Log[3]))*Lo g[E^((-80 + 4*E^(3 - E^x) - x)/(-20 + E^(3 - E^x))) - x])/(E^((-80 + 4*E^( 3 - E^x) - x)/(-20 + E^(3 - E^x)))*(400 + E^(6 - 2*E^x) - 40*E^(3 - E^x)) - 400*x - E^(6 - 2*E^x)*x + 40*E^(3 - E^x)*x),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.
| \(\displaystyle \int \frac {e^{\frac {-x+4 e^{3-e^x}-80}{e^{3-e^x}-20}} \left (e^{3-e^x} \left (-e^x x^2 \log (3)-x \log (3)\right )+20 x \log (3)\right )-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)-400 x \log (3)+\left (-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)-400 x \log (3)+e^{\frac {-x+4 e^{3-e^x}-80}{e^{3-e^x}-20}} \left (e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)+400 \log (3)\right )\right ) \log \left (e^{\frac {-x+4 e^{3-e^x}-80}{e^{3-e^x}-20}}-x\right )}{e^{\frac {-x+4 e^{3-e^x}-80}{e^{3-e^x}-20}} \left (e^{6-2 e^x}-40 e^{3-e^x}+400\right )-e^{6-2 e^x} x+40 e^{3-e^x} x-400 x} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {e^{e^x} (x+80)}{20 e^{e^x}-e^3}} \left (-e^{x+e^x+3} x^2+20 e^{2 e^x} x-e^{e^x+3} x-400 e^{2 e^x}+40 e^{e^x+3}-e^6\right ) \log (3)}{\left (20 e^{e^x}-e^3\right )^2 \left (e^{\frac {e^{e^x} (x+80)}{20 e^{e^x}-e^3}}-e^{\frac {4 e^3}{20 e^{e^x}-e^3}} x\right )}+\log (3) \left (\log \left (e^{\frac {e^{e^x} (x+80)-4 e^3}{20 e^{e^x}-e^3}}-x\right )+1\right )\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {e^{\frac {e^{e^x} (x+80)}{20 e^{e^x}-e^3}} \left (-e^{x+e^x+3} x^2+20 e^{2 e^x} x-e^{e^x+3} x-400 e^{2 e^x}+40 e^{e^x+3}-e^6\right ) \log (3)}{\left (20 e^{e^x}-e^3\right )^2 \left (e^{\frac {e^{e^x} (x+80)}{20 e^{e^x}-e^3}}-e^{\frac {4 e^3}{20 e^{e^x}-e^3}} x\right )}+\log (3) \left (\log \left (e^{\frac {e^{e^x} (x+80)-4 e^3}{20 e^{e^x}-e^3}}-x\right )+1\right )\right )dx\) |
Int[(-400*x*Log[3] - E^(6 - 2*E^x)*x*Log[3] + 40*E^(3 - E^x)*x*Log[3] + E^ ((-80 + 4*E^(3 - E^x) - x)/(-20 + E^(3 - E^x)))*(20*x*Log[3] + E^(3 - E^x) *(-(x*Log[3]) - E^x*x^2*Log[3])) + (-400*x*Log[3] - E^(6 - 2*E^x)*x*Log[3] + 40*E^(3 - E^x)*x*Log[3] + E^((-80 + 4*E^(3 - E^x) - x)/(-20 + E^(3 - E^ x)))*(400*Log[3] + E^(6 - 2*E^x)*Log[3] - 40*E^(3 - E^x)*Log[3]))*Log[E^(( -80 + 4*E^(3 - E^x) - x)/(-20 + E^(3 - E^x))) - x])/(E^((-80 + 4*E^(3 - E^ x) - x)/(-20 + E^(3 - E^x)))*(400 + E^(6 - 2*E^x) - 40*E^(3 - E^x)) - 400* x - E^(6 - 2*E^x)*x + 40*E^(3 - E^x)*x),x]
3.8.93.3.1 Defintions of rubi rules used
Time = 157.76 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20
| method | result | size |
| risch | \(\ln \left (3\right ) \ln \left ({\mathrm e}^{-\frac {-4 \,{\mathrm e}^{-{\mathrm e}^{x}+3}+x +80}{{\mathrm e}^{-{\mathrm e}^{x}+3}-20}}-x \right ) x\) | \(36\) |
| parallelrisch | \(\ln \left (3\right ) x \ln \left ({\mathrm e}^{\frac {4 \,{\mathrm e}^{-{\mathrm e}^{x}+3}-x -80}{{\mathrm e}^{-{\mathrm e}^{x}+3}-20}}-x \right )\) | \(37\) |
int((((ln(3)*exp(-exp(x)+3)^2-40*ln(3)*exp(-exp(x)+3)+400*ln(3))*exp((4*ex p(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*ln(3)*exp(-exp(x)+3)^2+40*x*ln(3 )*exp(-exp(x)+3)-400*x*ln(3))*ln(exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+ 3)-20))-x)+((-x^2*ln(3)*exp(x)-x*ln(3))*exp(-exp(x)+3)+20*x*ln(3))*exp((4* exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*ln(3)*exp(-exp(x)+3)^2+40*x*ln (3)*exp(-exp(x)+3)-400*x*ln(3))/((exp(-exp(x)+3)^2-40*exp(-exp(x)+3)+400)* exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*exp(-exp(x)+3)^2+40*x*e xp(-exp(x)+3)-400*x),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.17 \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=x \log \left (3\right ) \log \left (-x + e^{\left (-\frac {x - 4 \, e^{\left (-e^{x} + 3\right )} + 80}{e^{\left (-e^{x} + 3\right )} - 20}\right )}\right ) \]
integrate((((log(3)*exp(-exp(x)+3)^2-40*log(3)*exp(-exp(x)+3)+400*log(3))* exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*log(3)*exp(-exp(x)+3)^2 +40*x*log(3)*exp(-exp(x)+3)-400*x*log(3))*log(exp((4*exp(-exp(x)+3)-x-80)/ (exp(-exp(x)+3)-20))-x)+((-x^2*log(3)*exp(x)-x*log(3))*exp(-exp(x)+3)+20*x *log(3))*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*log(3)*exp(-ex p(x)+3)^2+40*x*log(3)*exp(-exp(x)+3)-400*x*log(3))/((exp(-exp(x)+3)^2-40*e xp(-exp(x)+3)+400)*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*exp( -exp(x)+3)^2+40*x*exp(-exp(x)+3)-400*x),x, algorithm=\
Timed out. \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=\text {Timed out} \]
integrate((((ln(3)*exp(-exp(x)+3)**2-40*ln(3)*exp(-exp(x)+3)+400*ln(3))*ex p((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*ln(3)*exp(-exp(x)+3)**2+4 0*x*ln(3)*exp(-exp(x)+3)-400*x*ln(3))*ln(exp((4*exp(-exp(x)+3)-x-80)/(exp( -exp(x)+3)-20))-x)+((-x**2*ln(3)*exp(x)-x*ln(3))*exp(-exp(x)+3)+20*x*ln(3) )*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*ln(3)*exp(-exp(x)+3)* *2+40*x*ln(3)*exp(-exp(x)+3)-400*x*ln(3))/((exp(-exp(x)+3)**2-40*exp(-exp( x)+3)+400)*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*exp(-exp(x)+ 3)**2+40*x*exp(-exp(x)+3)-400*x),x)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (26) = 52\).
Time = 0.63 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.93 \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=\frac {4 \, x e^{3} \log \left (3\right ) + {\left (x e^{3} \log \left (3\right ) - 20 \, x e^{\left (e^{x}\right )} \log \left (3\right )\right )} \log \left (-x e^{\left (-\frac {4 \, e^{3}}{e^{3} - 20 \, e^{\left (e^{x}\right )}}\right )} + e^{\left (-\frac {x e^{\left (e^{x}\right )}}{e^{3} - 20 \, e^{\left (e^{x}\right )}} - \frac {80 \, e^{\left (e^{x}\right )}}{e^{3} - 20 \, e^{\left (e^{x}\right )}}\right )}\right )}{e^{3} - 20 \, e^{\left (e^{x}\right )}} \]
integrate((((log(3)*exp(-exp(x)+3)^2-40*log(3)*exp(-exp(x)+3)+400*log(3))* exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*log(3)*exp(-exp(x)+3)^2 +40*x*log(3)*exp(-exp(x)+3)-400*x*log(3))*log(exp((4*exp(-exp(x)+3)-x-80)/ (exp(-exp(x)+3)-20))-x)+((-x^2*log(3)*exp(x)-x*log(3))*exp(-exp(x)+3)+20*x *log(3))*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*log(3)*exp(-ex p(x)+3)^2+40*x*log(3)*exp(-exp(x)+3)-400*x*log(3))/((exp(-exp(x)+3)^2-40*e xp(-exp(x)+3)+400)*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*exp( -exp(x)+3)^2+40*x*exp(-exp(x)+3)-400*x),x, algorithm=\
(4*x*e^3*log(3) + (x*e^3*log(3) - 20*x*e^(e^x)*log(3))*log(-x*e^(-4*e^3/(e ^3 - 20*e^(e^x))) + e^(-x*e^(e^x)/(e^3 - 20*e^(e^x)) - 80*e^(e^x)/(e^3 - 2 0*e^(e^x)))))/(e^3 - 20*e^(e^x))
Timed out. \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=\text {Timed out} \]
integrate((((log(3)*exp(-exp(x)+3)^2-40*log(3)*exp(-exp(x)+3)+400*log(3))* exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*log(3)*exp(-exp(x)+3)^2 +40*x*log(3)*exp(-exp(x)+3)-400*x*log(3))*log(exp((4*exp(-exp(x)+3)-x-80)/ (exp(-exp(x)+3)-20))-x)+((-x^2*log(3)*exp(x)-x*log(3))*exp(-exp(x)+3)+20*x *log(3))*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*log(3)*exp(-ex p(x)+3)^2+40*x*log(3)*exp(-exp(x)+3)-400*x*log(3))/((exp(-exp(x)+3)^2-40*e xp(-exp(x)+3)+400)*exp((4*exp(-exp(x)+3)-x-80)/(exp(-exp(x)+3)-20))-x*exp( -exp(x)+3)^2+40*x*exp(-exp(x)+3)-400*x),x, algorithm=\
Time = 12.74 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (20 x \log (3)+e^{3-e^x} \left (-x \log (3)-e^x x^2 \log (3)\right )\right )+\left (-400 x \log (3)-e^{6-2 e^x} x \log (3)+40 e^{3-e^x} x \log (3)+e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400 \log (3)+e^{6-2 e^x} \log (3)-40 e^{3-e^x} \log (3)\right )\right ) \log \left (e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}}-x\right )}{e^{\frac {-80+4 e^{3-e^x}-x}{-20+e^{3-e^x}}} \left (400+e^{6-2 e^x}-40 e^{3-e^x}\right )-400 x-e^{6-2 e^x} x+40 e^{3-e^x} x} \, dx=x\,\ln \left ({\mathrm {e}}^{-\frac {x}{{\mathrm {e}}^3\,{\mathrm {e}}^{-{\mathrm {e}}^x}-20}}\,{\mathrm {e}}^{-\frac {80}{{\mathrm {e}}^3\,{\mathrm {e}}^{-{\mathrm {e}}^x}-20}}\,{\mathrm {e}}^{\frac {4\,{\mathrm {e}}^3\,{\mathrm {e}}^{-{\mathrm {e}}^x}}{{\mathrm {e}}^3\,{\mathrm {e}}^{-{\mathrm {e}}^x}-20}}-x\right )\,\ln \left (3\right ) \]
int((400*x*log(3) - exp(-(x - 4*exp(3 - exp(x)) + 80)/(exp(3 - exp(x)) - 2 0))*(20*x*log(3) - exp(3 - exp(x))*(x*log(3) + x^2*exp(x)*log(3))) + log(e xp(-(x - 4*exp(3 - exp(x)) + 80)/(exp(3 - exp(x)) - 20)) - x)*(400*x*log(3 ) - exp(-(x - 4*exp(3 - exp(x)) + 80)/(exp(3 - exp(x)) - 20))*(400*log(3) - 40*exp(3 - exp(x))*log(3) + exp(6 - 2*exp(x))*log(3)) - 40*x*exp(3 - exp (x))*log(3) + x*exp(6 - 2*exp(x))*log(3)) - 40*x*exp(3 - exp(x))*log(3) + x*exp(6 - 2*exp(x))*log(3))/(400*x - 40*x*exp(3 - exp(x)) + x*exp(6 - 2*ex p(x)) - exp(-(x - 4*exp(3 - exp(x)) + 80)/(exp(3 - exp(x)) - 20))*(exp(6 - 2*exp(x)) - 40*exp(3 - exp(x)) + 400)),x)