Integrand size = 187, antiderivative size = 32 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=-5 x+\left (-4+\frac {e^{2 x}}{5}\right )^2 \left (1+x+x^2+\frac {x}{\log (x)}\right )^2 \] Output:
(x+x/ln(x)+x^2+1)^2*(1/5*exp(x)^2-4)^2-5*x
Leaf count is larger than twice the leaf count of optimal. \(91\) vs. \(2(32)=64\).
Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.84 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=\frac {1}{25} \left (675 x+1200 x^2+800 x^3+400 x^4-40 e^{2 x} \left (1+x+x^2\right )^2+e^{4 x} \left (1+x+x^2\right )^2+\frac {\left (-20+e^{2 x}\right )^2 x^2}{\log ^2(x)}+\frac {2 \left (-20+e^{2 x}\right )^2 x \left (1+x+x^2\right )}{\log (x)}\right ) \] Input:
Integrate[(-800*x + 80*E^(2*x)*x - 2*E^(4*x)*x + (-800 + 80*E^(2*x) - 800* x^2 + E^(4*x)*(-2 + 2*x^2))*Log[x] + (800 + 1600*x + 2400*x^2 + E^(2*x)*(- 80 - 320*x - 400*x^2 - 160*x^3) + E^(4*x)*(2 + 12*x + 14*x^2 + 8*x^3))*Log [x]^2 + (675 + 2400*x + 2400*x^2 + 1600*x^3 + E^(2*x)*(-160 - 400*x - 480* x^2 - 320*x^3 - 80*x^4) + E^(4*x)*(6 + 14*x + 18*x^2 + 12*x^3 + 4*x^4))*Lo g[x]^3)/(25*Log[x]^3),x]
Output:
(675*x + 1200*x^2 + 800*x^3 + 400*x^4 - 40*E^(2*x)*(1 + x + x^2)^2 + E^(4* x)*(1 + x + x^2)^2 + ((-20 + E^(2*x))^2*x^2)/Log[x]^2 + (2*(-20 + E^(2*x)) ^2*x*(1 + x + x^2))/Log[x])/25
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.
| \(\displaystyle \int \frac {\left (-800 x^2+e^{4 x} \left (2 x^2-2\right )+80 e^{2 x}-800\right ) \log (x)+\left (2400 x^2+e^{2 x} \left (-160 x^3-400 x^2-320 x-80\right )+e^{4 x} \left (8 x^3+14 x^2+12 x+2\right )+1600 x+800\right ) \log ^2(x)+\left (1600 x^3+2400 x^2+e^{2 x} \left (-80 x^4-320 x^3-480 x^2-400 x-160\right )+e^{4 x} \left (4 x^4+12 x^3+18 x^2+14 x+6\right )+2400 x+675\right ) \log ^3(x)+80 e^{2 x} x-2 e^{4 x} x-800 x}{25 \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{25} \int -\frac {-\left (\left (1600 x^3+2400 x^2+2400 x-80 e^{2 x} \left (x^4+4 x^3+6 x^2+5 x+2\right )+2 e^{4 x} \left (2 x^4+6 x^3+9 x^2+7 x+3\right )+675\right ) \log ^3(x)\right )-2 \left (1200 x^2+800 x-40 e^{2 x} \left (2 x^3+5 x^2+4 x+1\right )+e^{4 x} \left (4 x^3+7 x^2+6 x+1\right )+400\right ) \log ^2(x)+2 \left (400 x^2-40 e^{2 x}+e^{4 x} \left (1-x^2\right )+400\right ) \log (x)-80 e^{2 x} x+2 e^{4 x} x+800 x}{\log ^3(x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{25} \int \frac {-\left (\left (1600 x^3+2400 x^2+2400 x-80 e^{2 x} \left (x^4+4 x^3+6 x^2+5 x+2\right )+2 e^{4 x} \left (2 x^4+6 x^3+9 x^2+7 x+3\right )+675\right ) \log ^3(x)\right )-2 \left (1200 x^2+800 x-40 e^{2 x} \left (2 x^3+5 x^2+4 x+1\right )+e^{4 x} \left (4 x^3+7 x^2+6 x+1\right )+400\right ) \log ^2(x)+2 \left (400 x^2-40 e^{2 x}+e^{4 x} \left (1-x^2\right )+400\right ) \log (x)-80 e^{2 x} x+2 e^{4 x} x+800 x}{\log ^3(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{25} \int \left (-\frac {25 \left (64 x^3 \log ^3(x)+96 x^2 \log ^3(x)+96 x \log ^3(x)+27 \log ^3(x)+96 x^2 \log ^2(x)+64 x \log ^2(x)+32 \log ^2(x)-32 x^2 \log (x)-32 \log (x)-32 x\right )}{\log ^3(x)}+\frac {80 e^{2 x} \left (\log ^3(x) x^4+4 \log ^3(x) x^3+2 \log ^2(x) x^3+6 \log ^3(x) x^2+5 \log ^2(x) x^2+5 \log ^3(x) x+4 \log ^2(x) x-x+2 \log ^3(x)+\log ^2(x)-\log (x)\right )}{\log ^3(x)}-\frac {2 e^{4 x} \left (2 \log ^3(x) x^4+6 \log ^3(x) x^3+4 \log ^2(x) x^3+9 \log ^3(x) x^2+7 \log ^2(x) x^2+\log (x) x^2+7 \log ^3(x) x+6 \log ^2(x) x-x+3 \log ^3(x)+\log ^2(x)-\log (x)\right )}{\log ^3(x)}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle -\frac {1}{25} \int \left (-\frac {25 \left (64 x^3 \log ^3(x)+96 x^2 \log ^3(x)+96 x \log ^3(x)+27 \log ^3(x)+96 x^2 \log ^2(x)+64 x \log ^2(x)+32 \log ^2(x)-32 x^2 \log (x)-32 \log (x)-32 x\right )}{\log ^3(x)}+\frac {80 e^{2 x} \left (\log ^3(x) x^4+4 \log ^3(x) x^3+2 \log ^2(x) x^3+6 \log ^3(x) x^2+5 \log ^2(x) x^2+5 \log ^3(x) x+4 \log ^2(x) x-x+2 \log ^3(x)+\log ^2(x)-\log (x)\right )}{\log ^3(x)}-\frac {2 e^{4 x} \left (2 \log ^3(x) x^4+6 \log ^3(x) x^3+4 \log ^2(x) x^3+9 \log ^3(x) x^2+7 \log ^2(x) x^2+\log (x) x^2+7 \log ^3(x) x+6 \log ^2(x) x-x+3 \log ^3(x)+\log ^2(x)-\log (x)\right )}{\log ^3(x)}\right )dx\) |
Input:
Int[(-800*x + 80*E^(2*x)*x - 2*E^(4*x)*x + (-800 + 80*E^(2*x) - 800*x^2 + E^(4*x)*(-2 + 2*x^2))*Log[x] + (800 + 1600*x + 2400*x^2 + E^(2*x)*(-80 - 3 20*x - 400*x^2 - 160*x^3) + E^(4*x)*(2 + 12*x + 14*x^2 + 8*x^3))*Log[x]^2 + (675 + 2400*x + 2400*x^2 + 1600*x^3 + E^(2*x)*(-160 - 400*x - 480*x^2 - 320*x^3 - 80*x^4) + E^(4*x)*(6 + 14*x + 18*x^2 + 12*x^3 + 4*x^4))*Log[x]^3 )/(25*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(195\) vs. \(2(29)=58\).
Time = 0.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 6.12
\[\frac {{\mathrm e}^{4 x} x^{4}}{25}+\frac {2 x^{3} {\mathrm e}^{4 x}}{25}-\frac {8 \,{\mathrm e}^{2 x} x^{4}}{5}+\frac {3 \,{\mathrm e}^{4 x} x^{2}}{25}-\frac {16 \,{\mathrm e}^{2 x} x^{3}}{5}+\frac {2 x \,{\mathrm e}^{4 x}}{25}+16 x^{4}-\frac {24 \,{\mathrm e}^{2 x} x^{2}}{5}+\frac {{\mathrm e}^{4 x}}{25}+32 x^{3}-\frac {16 x \,{\mathrm e}^{2 x}}{5}+48 x^{2}-\frac {8 \,{\mathrm e}^{2 x}}{5}+27 x +\frac {x \left (2 \ln \left (x \right ) {\mathrm e}^{4 x} x^{2}+2 \ln \left (x \right ) {\mathrm e}^{4 x} x -80 x^{2} {\mathrm e}^{2 x} \ln \left (x \right )+x \,{\mathrm e}^{4 x}+2 \ln \left (x \right ) {\mathrm e}^{4 x}-80 x \,{\mathrm e}^{2 x} \ln \left (x \right )+800 x^{2} \ln \left (x \right )-40 x \,{\mathrm e}^{2 x}-80 \,{\mathrm e}^{2 x} \ln \left (x \right )+800 x \ln \left (x \right )+400 x +800 \ln \left (x \right )\right )}{25 \ln \left (x \right )^{2}}\]
Input:
int(1/25*(((4*x^4+12*x^3+18*x^2+14*x+6)*exp(x)^4+(-80*x^4-320*x^3-480*x^2- 400*x-160)*exp(x)^2+1600*x^3+2400*x^2+2400*x+675)*ln(x)^3+((8*x^3+14*x^2+1 2*x+2)*exp(x)^4+(-160*x^3-400*x^2-320*x-80)*exp(x)^2+2400*x^2+1600*x+800)* ln(x)^2+((2*x^2-2)*exp(x)^4+80*exp(x)^2-800*x^2-800)*ln(x)-2*x*exp(x)^4+80 *x*exp(x)^2-800*x)/ln(x)^3,x)
Output:
1/25*exp(4*x)*x^4+2/25*x^3*exp(4*x)-8/5*exp(2*x)*x^4+3/25*exp(4*x)*x^2-16/ 5*exp(2*x)*x^3+2/25*x*exp(4*x)+16*x^4-24/5*exp(2*x)*x^2+1/25*exp(4*x)+32*x ^3-16/5*x*exp(2*x)+48*x^2-8/5*exp(2*x)+27*x+1/25*x*(2*ln(x)*exp(4*x)*x^2+2 *ln(x)*exp(4*x)*x-80*x^2*exp(2*x)*ln(x)+x*exp(4*x)+2*ln(x)*exp(4*x)-80*x*e xp(2*x)*ln(x)+800*x^2*ln(x)-40*x*exp(2*x)-80*exp(2*x)*ln(x)+800*x*ln(x)+40 0*x+800*ln(x))/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (28) = 56\).
Time = 0.11 (sec) , antiderivative size = 145, normalized size of antiderivative = 4.53 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=\frac {x^{2} e^{\left (4 \, x\right )} - 40 \, x^{2} e^{\left (2 \, x\right )} + {\left (400 \, x^{4} + 800 \, x^{3} + 1200 \, x^{2} + {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} e^{\left (4 \, x\right )} - 40 \, {\left (x^{4} + 2 \, x^{3} + 3 \, x^{2} + 2 \, x + 1\right )} e^{\left (2 \, x\right )} + 675 \, x\right )} \log \left (x\right )^{2} + 400 \, x^{2} + 2 \, {\left (400 \, x^{3} + 400 \, x^{2} + {\left (x^{3} + x^{2} + x\right )} e^{\left (4 \, x\right )} - 40 \, {\left (x^{3} + x^{2} + x\right )} e^{\left (2 \, x\right )} + 400 \, x\right )} \log \left (x\right )}{25 \, \log \left (x\right )^{2}} \] Input:
integrate(1/25*(((4*x^4+12*x^3+18*x^2+14*x+6)*exp(x)^4+(-80*x^4-320*x^3-48 0*x^2-400*x-160)*exp(x)^2+1600*x^3+2400*x^2+2400*x+675)*log(x)^3+((8*x^3+1 4*x^2+12*x+2)*exp(x)^4+(-160*x^3-400*x^2-320*x-80)*exp(x)^2+2400*x^2+1600* x+800)*log(x)^2+((2*x^2-2)*exp(x)^4+80*exp(x)^2-800*x^2-800)*log(x)-2*x*ex p(x)^4+80*x*exp(x)^2-800*x)/log(x)^3,x, algorithm="fricas")
Output:
1/25*(x^2*e^(4*x) - 40*x^2*e^(2*x) + (400*x^4 + 800*x^3 + 1200*x^2 + (x^4 + 2*x^3 + 3*x^2 + 2*x + 1)*e^(4*x) - 40*(x^4 + 2*x^3 + 3*x^2 + 2*x + 1)*e^ (2*x) + 675*x)*log(x)^2 + 400*x^2 + 2*(400*x^3 + 400*x^2 + (x^3 + x^2 + x) *e^(4*x) - 40*(x^3 + x^2 + x)*e^(2*x) + 400*x)*log(x))/log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (26) = 52\).
Time = 0.25 (sec) , antiderivative size = 233, normalized size of antiderivative = 7.28 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=16 x^{4} + 32 x^{3} + 48 x^{2} + 27 x + \frac {16 x^{2} + \left (32 x^{3} + 32 x^{2} + 32 x\right ) \log {\left (x \right )}}{\log {\left (x \right )}^{2}} + \frac {\left (- 200 x^{4} \log {\left (x \right )}^{4} - 400 x^{3} \log {\left (x \right )}^{4} - 400 x^{3} \log {\left (x \right )}^{3} - 600 x^{2} \log {\left (x \right )}^{4} - 400 x^{2} \log {\left (x \right )}^{3} - 200 x^{2} \log {\left (x \right )}^{2} - 400 x \log {\left (x \right )}^{4} - 400 x \log {\left (x \right )}^{3} - 200 \log {\left (x \right )}^{4}\right ) e^{2 x} + \left (5 x^{4} \log {\left (x \right )}^{4} + 10 x^{3} \log {\left (x \right )}^{4} + 10 x^{3} \log {\left (x \right )}^{3} + 15 x^{2} \log {\left (x \right )}^{4} + 10 x^{2} \log {\left (x \right )}^{3} + 5 x^{2} \log {\left (x \right )}^{2} + 10 x \log {\left (x \right )}^{4} + 10 x \log {\left (x \right )}^{3} + 5 \log {\left (x \right )}^{4}\right ) e^{4 x}}{125 \log {\left (x \right )}^{4}} \] Input:
integrate(1/25*(((4*x**4+12*x**3+18*x**2+14*x+6)*exp(x)**4+(-80*x**4-320*x **3-480*x**2-400*x-160)*exp(x)**2+1600*x**3+2400*x**2+2400*x+675)*ln(x)**3 +((8*x**3+14*x**2+12*x+2)*exp(x)**4+(-160*x**3-400*x**2-320*x-80)*exp(x)** 2+2400*x**2+1600*x+800)*ln(x)**2+((2*x**2-2)*exp(x)**4+80*exp(x)**2-800*x* *2-800)*ln(x)-2*x*exp(x)**4+80*x*exp(x)**2-800*x)/ln(x)**3,x)
Output:
16*x**4 + 32*x**3 + 48*x**2 + 27*x + (16*x**2 + (32*x**3 + 32*x**2 + 32*x) *log(x))/log(x)**2 + ((-200*x**4*log(x)**4 - 400*x**3*log(x)**4 - 400*x**3 *log(x)**3 - 600*x**2*log(x)**4 - 400*x**2*log(x)**3 - 200*x**2*log(x)**2 - 400*x*log(x)**4 - 400*x*log(x)**3 - 200*log(x)**4)*exp(2*x) + (5*x**4*lo g(x)**4 + 10*x**3*log(x)**4 + 10*x**3*log(x)**3 + 15*x**2*log(x)**4 + 10*x **2*log(x)**3 + 5*x**2*log(x)**2 + 10*x*log(x)**4 + 10*x*log(x)**3 + 5*log (x)**4)*exp(4*x))/(125*log(x)**4)
\[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=\int { \frac {{\left (1600 \, x^{3} + 2400 \, x^{2} + 2 \, {\left (2 \, x^{4} + 6 \, x^{3} + 9 \, x^{2} + 7 \, x + 3\right )} e^{\left (4 \, x\right )} - 80 \, {\left (x^{4} + 4 \, x^{3} + 6 \, x^{2} + 5 \, x + 2\right )} e^{\left (2 \, x\right )} + 2400 \, x + 675\right )} \log \left (x\right )^{3} + 2 \, {\left (1200 \, x^{2} + {\left (4 \, x^{3} + 7 \, x^{2} + 6 \, x + 1\right )} e^{\left (4 \, x\right )} - 40 \, {\left (2 \, x^{3} + 5 \, x^{2} + 4 \, x + 1\right )} e^{\left (2 \, x\right )} + 800 \, x + 400\right )} \log \left (x\right )^{2} - 2 \, x e^{\left (4 \, x\right )} + 80 \, x e^{\left (2 \, x\right )} - 2 \, {\left (400 \, x^{2} - {\left (x^{2} - 1\right )} e^{\left (4 \, x\right )} - 40 \, e^{\left (2 \, x\right )} + 400\right )} \log \left (x\right ) - 800 \, x}{25 \, \log \left (x\right )^{3}} \,d x } \] Input:
integrate(1/25*(((4*x^4+12*x^3+18*x^2+14*x+6)*exp(x)^4+(-80*x^4-320*x^3-48 0*x^2-400*x-160)*exp(x)^2+1600*x^3+2400*x^2+2400*x+675)*log(x)^3+((8*x^3+1 4*x^2+12*x+2)*exp(x)^4+(-160*x^3-400*x^2-320*x-80)*exp(x)^2+2400*x^2+1600* x+800)*log(x)^2+((2*x^2-2)*exp(x)^4+80*exp(x)^2-800*x^2-800)*log(x)-2*x*ex p(x)^4+80*x*exp(x)^2-800*x)/log(x)^3,x, algorithm="maxima")
Output:
16*x^4 + 32*x^3 + 48*x^2 + 1/800*(32*x^4 - 32*x^3 + 24*x^2 - 12*x + 3)*e^( 4*x) + 3/800*(32*x^3 - 24*x^2 + 12*x - 3)*e^(4*x) + 9/400*(8*x^2 - 4*x + 1 )*e^(4*x) + 7/200*(4*x - 1)*e^(4*x) - 4/5*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3 )*e^(2*x) - 8/5*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) - 24/5*(2*x^2 - 2*x + 1) *e^(2*x) - 4*(2*x - 1)*e^(2*x) + 27*x + 1/25*((x^2 + 2*(x^3 + x^2 + x)*log (x))*e^(4*x) - 40*(x^2 + 2*(x^3 + x^2 + x)*log(x))*e^(2*x) + 800*(x^3 + x) *log(x))/log(x)^2 + 3/50*e^(4*x) - 16/5*e^(2*x) + 128*gamma(-2, -2*log(x)) + 64*integrate(x/log(x), x)
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (28) = 56\).
Time = 0.19 (sec) , antiderivative size = 294, normalized size of antiderivative = 9.19 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=16 \, x^{4} + 32 \, x^{3} + 48 \, x^{2} + \frac {1}{800} \, {\left (32 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 12 \, x + 3\right )} e^{\left (4 \, x\right )} + \frac {3}{800} \, {\left (32 \, x^{3} - 24 \, x^{2} + 12 \, x - 3\right )} e^{\left (4 \, x\right )} + \frac {9}{400} \, {\left (8 \, x^{2} - 4 \, x + 1\right )} e^{\left (4 \, x\right )} + \frac {7}{200} \, {\left (4 \, x - 1\right )} e^{\left (4 \, x\right )} - \frac {4}{5} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} - \frac {8}{5} \, {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} - \frac {24}{5} \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 27 \, x + \frac {2 \, x^{3} e^{\left (4 \, x\right )} \log \left (x\right ) - 80 \, x^{3} e^{\left (2 \, x\right )} \log \left (x\right ) + 2 \, x^{2} e^{\left (4 \, x\right )} \log \left (x\right ) - 80 \, x^{2} e^{\left (2 \, x\right )} \log \left (x\right ) + x^{2} e^{\left (4 \, x\right )} - 40 \, x^{2} e^{\left (2 \, x\right )} + 2 \, x e^{\left (4 \, x\right )} \log \left (x\right ) - 80 \, x e^{\left (2 \, x\right )} \log \left (x\right )}{25 \, \log \left (x\right )^{2}} + \frac {16 \, {\left (2 \, x^{3} \log \left (x\right ) + 2 \, x^{2} \log \left (x\right ) + x^{2} + 2 \, x \log \left (x\right )\right )}}{\log \left (x\right )^{2}} + \frac {3}{50} \, e^{\left (4 \, x\right )} - \frac {16}{5} \, e^{\left (2 \, x\right )} \] Input:
integrate(1/25*(((4*x^4+12*x^3+18*x^2+14*x+6)*exp(x)^4+(-80*x^4-320*x^3-48 0*x^2-400*x-160)*exp(x)^2+1600*x^3+2400*x^2+2400*x+675)*log(x)^3+((8*x^3+1 4*x^2+12*x+2)*exp(x)^4+(-160*x^3-400*x^2-320*x-80)*exp(x)^2+2400*x^2+1600* x+800)*log(x)^2+((2*x^2-2)*exp(x)^4+80*exp(x)^2-800*x^2-800)*log(x)-2*x*ex p(x)^4+80*x*exp(x)^2-800*x)/log(x)^3,x, algorithm="giac")
Output:
16*x^4 + 32*x^3 + 48*x^2 + 1/800*(32*x^4 - 32*x^3 + 24*x^2 - 12*x + 3)*e^( 4*x) + 3/800*(32*x^3 - 24*x^2 + 12*x - 3)*e^(4*x) + 9/400*(8*x^2 - 4*x + 1 )*e^(4*x) + 7/200*(4*x - 1)*e^(4*x) - 4/5*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3 )*e^(2*x) - 8/5*(4*x^3 - 6*x^2 + 6*x - 3)*e^(2*x) - 24/5*(2*x^2 - 2*x + 1) *e^(2*x) - 4*(2*x - 1)*e^(2*x) + 27*x + 1/25*(2*x^3*e^(4*x)*log(x) - 80*x^ 3*e^(2*x)*log(x) + 2*x^2*e^(4*x)*log(x) - 80*x^2*e^(2*x)*log(x) + x^2*e^(4 *x) - 40*x^2*e^(2*x) + 2*x*e^(4*x)*log(x) - 80*x*e^(2*x)*log(x))/log(x)^2 + 16*(2*x^3*log(x) + 2*x^2*log(x) + x^2 + 2*x*log(x))/log(x)^2 + 3/50*e^(4 *x) - 16/5*e^(2*x)
Time = 4.35 (sec) , antiderivative size = 233, normalized size of antiderivative = 7.28 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=27\,x-\frac {8\,{\mathrm {e}}^{2\,x}}{5}+\frac {{\mathrm {e}}^{4\,x}}{25}-\frac {16\,x\,{\mathrm {e}}^{2\,x}}{5}+\frac {2\,x\,{\mathrm {e}}^{4\,x}}{25}+\frac {32\,x}{\ln \left (x\right )}-\frac {24\,x^2\,{\mathrm {e}}^{2\,x}}{5}-\frac {16\,x^3\,{\mathrm {e}}^{2\,x}}{5}+\frac {3\,x^2\,{\mathrm {e}}^{4\,x}}{25}-\frac {8\,x^4\,{\mathrm {e}}^{2\,x}}{5}+\frac {2\,x^3\,{\mathrm {e}}^{4\,x}}{25}+\frac {x^4\,{\mathrm {e}}^{4\,x}}{25}+\frac {32\,x^2}{\ln \left (x\right )}+\frac {16\,x^2}{{\ln \left (x\right )}^2}+\frac {32\,x^3}{\ln \left (x\right )}+48\,x^2+32\,x^3+16\,x^4-\frac {16\,x\,{\mathrm {e}}^{2\,x}}{5\,\ln \left (x\right )}+\frac {2\,x\,{\mathrm {e}}^{4\,x}}{25\,\ln \left (x\right )}-\frac {16\,x^2\,{\mathrm {e}}^{2\,x}}{5\,\ln \left (x\right )}-\frac {8\,x^2\,{\mathrm {e}}^{2\,x}}{5\,{\ln \left (x\right )}^2}-\frac {16\,x^3\,{\mathrm {e}}^{2\,x}}{5\,\ln \left (x\right )}+\frac {2\,x^2\,{\mathrm {e}}^{4\,x}}{25\,\ln \left (x\right )}+\frac {x^2\,{\mathrm {e}}^{4\,x}}{25\,{\ln \left (x\right )}^2}+\frac {2\,x^3\,{\mathrm {e}}^{4\,x}}{25\,\ln \left (x\right )} \] Input:
int(((16*x*exp(2*x))/5 - 32*x - (2*x*exp(4*x))/25 + (log(x)^3*(2400*x + ex p(4*x)*(14*x + 18*x^2 + 12*x^3 + 4*x^4 + 6) - exp(2*x)*(400*x + 480*x^2 + 320*x^3 + 80*x^4 + 160) + 2400*x^2 + 1600*x^3 + 675))/25 + (log(x)^2*(1600 *x + exp(4*x)*(12*x + 14*x^2 + 8*x^3 + 2) - exp(2*x)*(320*x + 400*x^2 + 16 0*x^3 + 80) + 2400*x^2 + 800))/25 + (log(x)*(80*exp(2*x) + exp(4*x)*(2*x^2 - 2) - 800*x^2 - 800))/25)/log(x)^3,x)
Output:
27*x - (8*exp(2*x))/5 + exp(4*x)/25 - (16*x*exp(2*x))/5 + (2*x*exp(4*x))/2 5 + (32*x)/log(x) - (24*x^2*exp(2*x))/5 - (16*x^3*exp(2*x))/5 + (3*x^2*exp (4*x))/25 - (8*x^4*exp(2*x))/5 + (2*x^3*exp(4*x))/25 + (x^4*exp(4*x))/25 + (32*x^2)/log(x) + (16*x^2)/log(x)^2 + (32*x^3)/log(x) + 48*x^2 + 32*x^3 + 16*x^4 - (16*x*exp(2*x))/(5*log(x)) + (2*x*exp(4*x))/(25*log(x)) - (16*x^ 2*exp(2*x))/(5*log(x)) - (8*x^2*exp(2*x))/(5*log(x)^2) - (16*x^3*exp(2*x)) /(5*log(x)) + (2*x^2*exp(4*x))/(25*log(x)) + (x^2*exp(4*x))/(25*log(x)^2) + (2*x^3*exp(4*x))/(25*log(x))
Time = 0.20 (sec) , antiderivative size = 280, normalized size of antiderivative = 8.75 \[ \int \frac {-800 x+80 e^{2 x} x-2 e^{4 x} x+\left (-800+80 e^{2 x}-800 x^2+e^{4 x} \left (-2+2 x^2\right )\right ) \log (x)+\left (800+1600 x+2400 x^2+e^{2 x} \left (-80-320 x-400 x^2-160 x^3\right )+e^{4 x} \left (2+12 x+14 x^2+8 x^3\right )\right ) \log ^2(x)+\left (675+2400 x+2400 x^2+1600 x^3+e^{2 x} \left (-160-400 x-480 x^2-320 x^3-80 x^4\right )+e^{4 x} \left (6+14 x+18 x^2+12 x^3+4 x^4\right )\right ) \log ^3(x)}{25 \log ^3(x)} \, dx=\frac {400 \mathrm {log}\left (x \right )^{2} x^{4}+800 \mathrm {log}\left (x \right )^{2} x^{3}+e^{4 x} x^{2}-40 e^{2 x} x^{2}+1200 \mathrm {log}\left (x \right )^{2} x^{2}+400 x^{2}+2 e^{4 x} \mathrm {log}\left (x \right )^{2} x^{3}+3 e^{4 x} \mathrm {log}\left (x \right )^{2} x^{2}+2 e^{4 x} \mathrm {log}\left (x \right )^{2} x +2 e^{4 x} \mathrm {log}\left (x \right ) x^{3}+2 e^{4 x} \mathrm {log}\left (x \right ) x^{2}+2 e^{4 x} \mathrm {log}\left (x \right ) x -40 e^{2 x} \mathrm {log}\left (x \right )^{2} x^{4}-80 e^{2 x} \mathrm {log}\left (x \right )^{2} x^{3}-120 e^{2 x} \mathrm {log}\left (x \right )^{2} x^{2}-80 e^{2 x} \mathrm {log}\left (x \right )^{2} x -80 e^{2 x} \mathrm {log}\left (x \right ) x^{3}-80 e^{2 x} \mathrm {log}\left (x \right ) x^{2}-80 e^{2 x} \mathrm {log}\left (x \right ) x +e^{4 x} \mathrm {log}\left (x \right )^{2}-40 e^{2 x} \mathrm {log}\left (x \right )^{2}+800 \,\mathrm {log}\left (x \right ) x^{2}+800 \,\mathrm {log}\left (x \right ) x^{3}+800 \,\mathrm {log}\left (x \right ) x +675 \mathrm {log}\left (x \right )^{2} x +e^{4 x} \mathrm {log}\left (x \right )^{2} x^{4}}{25 \mathrm {log}\left (x \right )^{2}} \] Input:
int(1/25*(((4*x^4+12*x^3+18*x^2+14*x+6)*exp(x)^4+(-80*x^4-320*x^3-480*x^2- 400*x-160)*exp(x)^2+1600*x^3+2400*x^2+2400*x+675)*log(x)^3+((8*x^3+14*x^2+ 12*x+2)*exp(x)^4+(-160*x^3-400*x^2-320*x-80)*exp(x)^2+2400*x^2+1600*x+800) *log(x)^2+((2*x^2-2)*exp(x)^4+80*exp(x)^2-800*x^2-800)*log(x)-2*x*exp(x)^4 +80*x*exp(x)^2-800*x)/log(x)^3,x)
Output:
(e**(4*x)*log(x)**2*x**4 + 2*e**(4*x)*log(x)**2*x**3 + 3*e**(4*x)*log(x)** 2*x**2 + 2*e**(4*x)*log(x)**2*x + e**(4*x)*log(x)**2 + 2*e**(4*x)*log(x)*x **3 + 2*e**(4*x)*log(x)*x**2 + 2*e**(4*x)*log(x)*x + e**(4*x)*x**2 - 40*e* *(2*x)*log(x)**2*x**4 - 80*e**(2*x)*log(x)**2*x**3 - 120*e**(2*x)*log(x)** 2*x**2 - 80*e**(2*x)*log(x)**2*x - 40*e**(2*x)*log(x)**2 - 80*e**(2*x)*log (x)*x**3 - 80*e**(2*x)*log(x)*x**2 - 80*e**(2*x)*log(x)*x - 40*e**(2*x)*x* *2 + 400*log(x)**2*x**4 + 800*log(x)**2*x**3 + 1200*log(x)**2*x**2 + 675*l og(x)**2*x + 800*log(x)*x**3 + 800*log(x)*x**2 + 800*log(x)*x + 400*x**2)/ (25*log(x)**2)