Integrand size = 135, antiderivative size = 29 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\left (-x+\frac {10}{\left (x-e^{-x} x\right ) \log \left (\frac {5}{\log (625)}\right )}\right )^2 \] Output:
(5/ln(5/4/ln(5))/(1/2*x-1/2*x/exp(x))-x)^2
Time = 0.10 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {\frac {100 e^{2 x}}{\left (-1+e^x\right )^2 x^2}+\log \left (\frac {5}{\log (625)}\right ) \left (-\frac {20}{-1+e^x}+x^2 \log \left (\frac {5}{\log (625)}\right )\right )}{\log ^2\left (\frac {5}{\log (625)}\right )} \] Input:
Integrate[(-200*E^(3*x) + E^(2*x)*(200 - 200*x) + (-20*E^x*x^3 + 20*E^(2*x )*x^3)*Log[5/Log[625]] + (-2*x^4 + 6*E^x*x^4 - 6*E^(2*x)*x^4 + 2*E^(3*x)*x ^4)*Log[5/Log[625]]^2)/((-x^3 + 3*E^x*x^3 - 3*E^(2*x)*x^3 + E^(3*x)*x^3)*L og[5/Log[625]]^2),x]
Output:
((100*E^(2*x))/((-1 + E^x)^2*x^2) + Log[5/Log[625]]*(-20/(-1 + E^x) + x^2* Log[5/Log[625]]))/Log[5/Log[625]]^2
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.
| \(\displaystyle \int \frac {\left (6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4-2 x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )+\left (20 e^{2 x} x^3-20 e^x x^3\right ) \log \left (\frac {5}{\log (625)}\right )+e^{2 x} (200-200 x)-200 e^{3 x}}{\left (3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3-x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 \left (-100 e^{2 x} (1-x)+100 e^{3 x}+\left (-3 e^x x^4+3 e^{2 x} x^4-e^{3 x} x^4+x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )+10 \left (e^x x^3-e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )\right )}{-3 e^x x^3+3 e^{2 x} x^3-e^{3 x} x^3+x^3}dx}{\log ^2\left (\frac {5}{\log (625)}\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \int \frac {-100 e^{2 x} (1-x)+100 e^{3 x}+\left (-3 e^x x^4+3 e^{2 x} x^4-e^{3 x} x^4+x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )+10 \left (e^x x^3-e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )}{-3 e^x x^3+3 e^{2 x} x^3-e^{3 x} x^3+x^3}dx}{\log ^2\left (\frac {5}{\log (625)}\right )}\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {2 \int \left (-\frac {100 e^{2 x} (x-1)}{\left (-1+e^x\right )^3 x^3}-\frac {100 e^{3 x}}{\left (-1+e^x\right )^3 x^3}+x \log ^2\left (\frac {5}{\log (625)}\right )+\frac {10 e^x \log \left (\frac {5}{\log (625)}\right )}{\left (-1+e^x\right )^2}\right )dx}{\log ^2\left (\frac {5}{\log (625)}\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 \left (-100 \int \frac {1}{\left (-1+e^x\right )^2 x^3}dx-200 \int \frac {1}{\left (-1+e^x\right ) x^3}dx-100 \int \frac {1}{\left (-1+e^x\right )^3 x^2}dx-200 \int \frac {1}{\left (-1+e^x\right )^2 x^2}dx-100 \int \frac {1}{\left (-1+e^x\right ) x^2}dx+\frac {50}{x^2}+\frac {1}{2} x^2 \log ^2\left (\frac {5}{\log (625)}\right )+\frac {10 \log \left (\frac {5}{\log (625)}\right )}{1-e^x}\right )}{\log ^2\left (\frac {5}{\log (625)}\right )}\) |
Input:
Int[(-200*E^(3*x) + E^(2*x)*(200 - 200*x) + (-20*E^x*x^3 + 20*E^(2*x)*x^3) *Log[5/Log[625]] + (-2*x^4 + 6*E^x*x^4 - 6*E^(2*x)*x^4 + 2*E^(3*x)*x^4)*Lo g[5/Log[625]]^2)/((-x^3 + 3*E^x*x^3 - 3*E^(2*x)*x^3 + E^(3*x)*x^3)*Log[5/L og[625]]^2),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(102\) vs. \(2(30)=60\).
Time = 2.54 (sec) , antiderivative size = 103, normalized size of antiderivative = 3.55
| method | result | size |
| risch | \(x^{2}+\frac {100}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2} x^{2}}-\frac {20 \left (x^{2} \ln \left (5\right ) {\mathrm e}^{x}-\ln \left (\ln \left (5\right )\right ) {\mathrm e}^{x} x^{2}-2 x^{2} \ln \left (2\right ) {\mathrm e}^{x}-x^{2} \ln \left (5\right )+\ln \left (\ln \left (5\right )\right ) x^{2}+2 x^{2} \ln \left (2\right )-10 \,{\mathrm e}^{x}+5\right )}{\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right )^{2} x^{2} \left ({\mathrm e}^{x}-1\right )^{2}}\) | \(103\) |
| parallelrisch | \(\frac {\ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} {\mathrm e}^{2 x} x^{4}-2 \,{\mathrm e}^{x} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} x^{4}+\ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} x^{4}-20 \,{\mathrm e}^{x} x^{2} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )+20 x^{2} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )+100 \,{\mathrm e}^{2 x}}{\ln \left (\frac {5}{4 \ln \left (5\right )}\right )^{2} \left ({\mathrm e}^{2 x}-2 \,{\mathrm e}^{x}+1\right ) x^{2}}\) | \(105\) |
| norman | \(\frac {20 x^{2}+\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right ) x^{4}-20 \,{\mathrm e}^{x} x^{2}+\left (-2 \ln \left (5\right )+2 \ln \left (\ln \left (5\right )\right )+4 \ln \left (2\right )\right ) x^{4} {\mathrm e}^{x}+\left (\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right ) x^{4} {\mathrm e}^{2 x}+\frac {100 \,{\mathrm e}^{2 x}}{\ln \left (5\right )-2 \ln \left (2\right )-\ln \left (\ln \left (5\right )\right )}}{x^{2} \left ({\mathrm e}^{x}-1\right )^{2} \ln \left (\frac {5}{4 \ln \left (5\right )}\right )}\) | \(109\) |
Input:
int(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*ln(5/4/ln(5))^2+(2 0*exp(x)^2*x^3-20*exp(x)*x^3)*ln(5/4/ln(5))-200*exp(x)^3+(-200*x+200)*exp( x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/ln(5/4/ln(5))^2,x,met hod=_RETURNVERBOSE)
Output:
x^2+100/(ln(5)-2*ln(2)-ln(ln(5)))^2/x^2-20/(ln(5)-2*ln(2)-ln(ln(5)))^2/x^2 *(x^2*ln(5)*exp(x)-ln(ln(5))*exp(x)*x^2-2*x^2*ln(2)*exp(x)-x^2*ln(5)+ln(ln (5))*x^2+2*x^2*ln(2)-10*exp(x)+5)/(exp(x)-1)^2
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (27) = 54\).
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.03 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {{\left (x^{4} e^{\left (2 \, x\right )} - 2 \, x^{4} e^{x} + x^{4}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2} - 20 \, {\left (x^{2} e^{x} - x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right ) + 100 \, e^{\left (2 \, x\right )}}{{\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2}} \] Input:
integrate(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log( 5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3)*log(5/4/log(5))-200*exp(x)^3+(-200*x +200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/log (5))^2,x, algorithm="fricas")
Output:
((x^4*e^(2*x) - 2*x^4*e^x + x^4)*log(5/4/log(5))^2 - 20*(x^2*e^x - x^2)*lo g(5/4/log(5)) + 100*e^(2*x))/((x^2*e^(2*x) - 2*x^2*e^x + x^2)*log(5/4/log( 5))^2)
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (22) = 44\).
Time = 0.45 (sec) , antiderivative size = 386, normalized size of antiderivative = 13.31 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {x^{2} \left (- 4 \log {\left (2 \right )} \log {\left (5 \right )} - 2 \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + \log {\left (\log {\left (5 \right )} \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}^{2} + \log {\left (5 \right )}^{2}\right ) + \frac {100}{x^{2}}}{- 4 \log {\left (2 \right )} \log {\left (5 \right )} - 2 \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + \log {\left (\log {\left (5 \right )} \right )}^{2} + 4 \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 4 \log {\left (2 \right )}^{2} + \log {\left (5 \right )}^{2}} + \frac {- 640 x^{2} \log {\left (2 \right )} - 320 x^{2} \log {\left (\log {\left (5 \right )} \right )} + 320 x^{2} \log {\left (5 \right )} + \left (- 320 x^{2} \log {\left (5 \right )} + 320 x^{2} \log {\left (\log {\left (5 \right )} \right )} + 640 x^{2} \log {\left (2 \right )} + 3200\right ) e^{x} - 1600}{- 64 x^{2} \log {\left (2 \right )} \log {\left (5 \right )} - 32 x^{2} \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + 16 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 64 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 64 x^{2} \log {\left (2 \right )}^{2} + 16 x^{2} \log {\left (5 \right )}^{2} + \left (- 32 x^{2} \log {\left (5 \right )}^{2} - 128 x^{2} \log {\left (2 \right )}^{2} - 128 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} - 32 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 64 x^{2} \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + 128 x^{2} \log {\left (2 \right )} \log {\left (5 \right )}\right ) e^{x} + \left (- 64 x^{2} \log {\left (2 \right )} \log {\left (5 \right )} - 32 x^{2} \log {\left (5 \right )} \log {\left (\log {\left (5 \right )} \right )} + 16 x^{2} \log {\left (\log {\left (5 \right )} \right )}^{2} + 64 x^{2} \log {\left (2 \right )} \log {\left (\log {\left (5 \right )} \right )} + 64 x^{2} \log {\left (2 \right )}^{2} + 16 x^{2} \log {\left (5 \right )}^{2}\right ) e^{2 x}} \] Input:
integrate(((2*x**4*exp(x)**3-6*exp(x)**2*x**4+6*exp(x)*x**4-2*x**4)*ln(5/4 /ln(5))**2+(20*exp(x)**2*x**3-20*exp(x)*x**3)*ln(5/4/ln(5))-200*exp(x)**3+ (-200*x+200)*exp(x)**2)/(x**3*exp(x)**3-3*exp(x)**2*x**3+3*exp(x)*x**3-x** 3)/ln(5/4/ln(5))**2,x)
Output:
(x**2*(-4*log(2)*log(5) - 2*log(5)*log(log(5)) + log(log(5))**2 + 4*log(2) *log(log(5)) + 4*log(2)**2 + log(5)**2) + 100/x**2)/(-4*log(2)*log(5) - 2* log(5)*log(log(5)) + log(log(5))**2 + 4*log(2)*log(log(5)) + 4*log(2)**2 + log(5)**2) + (-640*x**2*log(2) - 320*x**2*log(log(5)) + 320*x**2*log(5) + (-320*x**2*log(5) + 320*x**2*log(log(5)) + 640*x**2*log(2) + 3200)*exp(x) - 1600)/(-64*x**2*log(2)*log(5) - 32*x**2*log(5)*log(log(5)) + 16*x**2*lo g(log(5))**2 + 64*x**2*log(2)*log(log(5)) + 64*x**2*log(2)**2 + 16*x**2*lo g(5)**2 + (-32*x**2*log(5)**2 - 128*x**2*log(2)**2 - 128*x**2*log(2)*log(l og(5)) - 32*x**2*log(log(5))**2 + 64*x**2*log(5)*log(log(5)) + 128*x**2*lo g(2)*log(5))*exp(x) + (-64*x**2*log(2)*log(5) - 32*x**2*log(5)*log(log(5)) + 16*x**2*log(log(5))**2 + 64*x**2*log(2)*log(log(5)) + 64*x**2*log(2)**2 + 16*x**2*log(5)**2)*exp(2*x))
Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (27) = 54\).
Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 6.72 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {{\left (\log \left (5\right )^{2} - 4 \, {\left (\log \left (5\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, \log \left (5\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} + 20 \, x^{2} {\left (\log \left (5\right ) - 2 \, \log \left (2\right ) - \log \left (\log \left (5\right )\right )\right )} + {\left ({\left (\log \left (5\right )^{2} - 4 \, {\left (\log \left (5\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, \log \left (5\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} + 100\right )} e^{\left (2 \, x\right )} - 2 \, {\left ({\left (\log \left (5\right )^{2} - 4 \, {\left (\log \left (5\right ) - \log \left (\log \left (5\right )\right )\right )} \log \left (2\right ) + 4 \, \log \left (2\right )^{2} - 2 \, \log \left (5\right ) \log \left (\log \left (5\right )\right ) + \log \left (\log \left (5\right )\right )^{2}\right )} x^{4} + 10 \, x^{2} {\left (\log \left (5\right ) - 2 \, \log \left (2\right ) - \log \left (\log \left (5\right )\right )\right )}\right )} e^{x}}{{\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2}} \] Input:
integrate(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log( 5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3)*log(5/4/log(5))-200*exp(x)^3+(-200*x +200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/log (5))^2,x, algorithm="maxima")
Output:
((log(5)^2 - 4*(log(5) - log(log(5)))*log(2) + 4*log(2)^2 - 2*log(5)*log(l og(5)) + log(log(5))^2)*x^4 + 20*x^2*(log(5) - 2*log(2) - log(log(5))) + ( (log(5)^2 - 4*(log(5) - log(log(5)))*log(2) + 4*log(2)^2 - 2*log(5)*log(lo g(5)) + log(log(5))^2)*x^4 + 100)*e^(2*x) - 2*((log(5)^2 - 4*(log(5) - log (log(5)))*log(2) + 4*log(2)^2 - 2*log(5)*log(log(5)) + log(log(5))^2)*x^4 + 10*x^2*(log(5) - 2*log(2) - log(log(5))))*e^x)/((x^2*e^(2*x) - 2*x^2*e^x + x^2)*log(5/4/log(5))^2)
Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (27) = 54\).
Time = 0.13 (sec) , antiderivative size = 189, normalized size of antiderivative = 6.52 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {x^{4} e^{\left (2 \, x\right )} \log \left (5\right )^{2} - 2 \, x^{4} e^{x} \log \left (5\right )^{2} - 2 \, x^{4} e^{\left (2 \, x\right )} \log \left (5\right ) \log \left (4 \, \log \left (5\right )\right ) + 4 \, x^{4} e^{x} \log \left (5\right ) \log \left (4 \, \log \left (5\right )\right ) + x^{4} e^{\left (2 \, x\right )} \log \left (4 \, \log \left (5\right )\right )^{2} - 2 \, x^{4} e^{x} \log \left (4 \, \log \left (5\right )\right )^{2} + x^{4} \log \left (5\right )^{2} - 2 \, x^{4} \log \left (5\right ) \log \left (4 \, \log \left (5\right )\right ) + x^{4} \log \left (4 \, \log \left (5\right )\right )^{2} - 20 \, x^{2} e^{x} \log \left (5\right ) + 20 \, x^{2} e^{x} \log \left (4 \, \log \left (5\right )\right ) + 20 \, x^{2} \log \left (5\right ) - 20 \, x^{2} \log \left (4 \, \log \left (5\right )\right ) + 100 \, e^{\left (2 \, x\right )}}{{\left (x^{2} e^{\left (2 \, x\right )} - 2 \, x^{2} e^{x} + x^{2}\right )} \log \left (\frac {5}{4 \, \log \left (5\right )}\right )^{2}} \] Input:
integrate(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log( 5))^2+(20*exp(x)^2*x^3-20*exp(x)*x^3)*log(5/4/log(5))-200*exp(x)^3+(-200*x +200)*exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/log (5))^2,x, algorithm="giac")
Output:
(x^4*e^(2*x)*log(5)^2 - 2*x^4*e^x*log(5)^2 - 2*x^4*e^(2*x)*log(5)*log(4*lo g(5)) + 4*x^4*e^x*log(5)*log(4*log(5)) + x^4*e^(2*x)*log(4*log(5))^2 - 2*x ^4*e^x*log(4*log(5))^2 + x^4*log(5)^2 - 2*x^4*log(5)*log(4*log(5)) + x^4*l og(4*log(5))^2 - 20*x^2*e^x*log(5) + 20*x^2*e^x*log(4*log(5)) + 20*x^2*log (5) - 20*x^2*log(4*log(5)) + 100*e^(2*x))/((x^2*e^(2*x) - 2*x^2*e^x + x^2) *log(5/4/log(5))^2)
Time = 3.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.69 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {100}{x^2\,{\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}^2}+x^2-\frac {x^3\,\ln \left (\frac {1099511627776\,{\ln \left (5\right )}^{20}}{95367431640625}\right )-200\,x+20\,x^3\,\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}{2\,x^3\,{\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}^2\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {20\,\left (10\,x-x^3\,\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )\right )}{x^3\,{\ln \left (\frac {5}{4\,\ln \left (5\right )}\right )}^2\,\left ({\mathrm {e}}^x-1\right )} \] Input:
int(-(200*exp(3*x) + exp(2*x)*(200*x - 200) + log(5/(4*log(5)))*(20*x^3*ex p(x) - 20*x^3*exp(2*x)) - log(5/(4*log(5)))^2*(6*x^4*exp(x) - 6*x^4*exp(2* x) + 2*x^4*exp(3*x) - 2*x^4))/(log(5/(4*log(5)))^2*(3*x^3*exp(x) - 3*x^3*e xp(2*x) + x^3*exp(3*x) - x^3)),x)
Output:
100/(x^2*log(5/(4*log(5)))^2) + x^2 - (x^3*log((1099511627776*log(5)^20)/9 5367431640625) - 200*x + 20*x^3*log(5/(4*log(5))))/(2*x^3*log(5/(4*log(5)) )^2*(exp(2*x) - 2*exp(x) + 1)) + (20*(10*x - x^3*log(5/(4*log(5)))))/(x^3* log(5/(4*log(5)))^2*(exp(x) - 1))
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 3.86 \[ \int \frac {-200 e^{3 x}+e^{2 x} (200-200 x)+\left (-20 e^x x^3+20 e^{2 x} x^3\right ) \log \left (\frac {5}{\log (625)}\right )+\left (-2 x^4+6 e^x x^4-6 e^{2 x} x^4+2 e^{3 x} x^4\right ) \log ^2\left (\frac {5}{\log (625)}\right )}{\left (-x^3+3 e^x x^3-3 e^{2 x} x^3+e^{3 x} x^3\right ) \log ^2\left (\frac {5}{\log (625)}\right )} \, dx=\frac {e^{2 x} \mathrm {log}\left (\frac {5}{4 \,\mathrm {log}\left (5\right )}\right )^{2} x^{4}-10 e^{2 x} \mathrm {log}\left (\frac {5}{4 \,\mathrm {log}\left (5\right )}\right ) x^{2}+100 e^{2 x}-2 e^{x} \mathrm {log}\left (\frac {5}{4 \,\mathrm {log}\left (5\right )}\right )^{2} x^{4}+\mathrm {log}\left (\frac {5}{4 \,\mathrm {log}\left (5\right )}\right )^{2} x^{4}+10 \,\mathrm {log}\left (\frac {5}{4 \,\mathrm {log}\left (5\right )}\right ) x^{2}}{\mathrm {log}\left (\frac {5}{4 \,\mathrm {log}\left (5\right )}\right )^{2} x^{2} \left (e^{2 x}-2 e^{x}+1\right )} \] Input:
int(((2*x^4*exp(x)^3-6*exp(x)^2*x^4+6*exp(x)*x^4-2*x^4)*log(5/4/log(5))^2+ (20*exp(x)^2*x^3-20*exp(x)*x^3)*log(5/4/log(5))-200*exp(x)^3+(-200*x+200)* exp(x)^2)/(x^3*exp(x)^3-3*exp(x)^2*x^3+3*exp(x)*x^3-x^3)/log(5/4/log(5))^2 ,x)
Output:
(e**(2*x)*log(5/(4*log(5)))**2*x**4 - 10*e**(2*x)*log(5/(4*log(5)))*x**2 + 100*e**(2*x) - 2*e**x*log(5/(4*log(5)))**2*x**4 + log(5/(4*log(5)))**2*x* *4 + 10*log(5/(4*log(5)))*x**2)/(log(5/(4*log(5)))**2*x**2*(e**(2*x) - 2*e **x + 1))