Integrand size = 363, antiderivative size = 28 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=-e^{x-\frac {1}{16} \left (5 \left (e^{3+x}+x\right )+\log (x)\right )^4}+x+\log (x) \] Output:
ln(x)+x-exp(x-1/16*(ln(x)+5*exp(3+x)+5*x)^4)
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(28)=56\).
Time = 1.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.96 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=x-e^{\frac {1}{16} \left (-625 e^{4 (3+x)}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4-150 \left (e^{3+x}+x\right )^2 \log ^2(x)-20 \left (e^{3+x}+x\right ) \log ^3(x)-\log ^4(x)\right )} x^{-\frac {125}{4} \left (e^{3+x}+x\right )^3}+\log (x) \] Input:
Integrate[(4 + 4*x + E^((-625*E^(12 + 4*x) + 16*x - 2500*E^(9 + 3*x)*x - 3 750*E^(6 + 2*x)*x^2 - 2500*E^(3 + x)*x^3 - 625*x^4 + (-500*E^(9 + 3*x) - 1 500*E^(6 + 2*x)*x - 1500*E^(3 + x)*x^2 - 500*x^3)*Log[x] + (-150*E^(6 + 2* x) - 300*E^(3 + x)*x - 150*x^2)*Log[x]^2 + (-20*E^(3 + x) - 20*x)*Log[x]^3 - Log[x]^4)/16)*(-4*x + 625*E^(12 + 4*x)*x + 125*x^3 + 625*x^4 + E^(9 + 3 *x)*(125 + 625*x + 1875*x^2) + E^(6 + 2*x)*(375*x + 1875*x^2 + 1875*x^3) + E^(3 + x)*(375*x^2 + 1875*x^3 + 625*x^4) + (375*E^(9 + 3*x)*x + 75*x^2 + 375*x^3 + E^(6 + 2*x)*(75 + 375*x + 750*x^2) + E^(3 + x)*(150*x + 750*x^2 + 375*x^3))*Log[x] + (15*x + 75*E^(6 + 2*x)*x + 75*x^2 + E^(3 + x)*(15 + 7 5*x + 75*x^2))*Log[x]^2 + (1 + 5*x + 5*E^(3 + x)*x)*Log[x]^3))/(4*x),x]
Output:
x - E^((-625*E^(4*(3 + x)) + 16*x - 2500*E^(9 + 3*x)*x - 3750*E^(6 + 2*x)* x^2 - 2500*E^(3 + x)*x^3 - 625*x^4 - 150*(E^(3 + x) + x)^2*Log[x]^2 - 20*( E^(3 + x) + x)*Log[x]^3 - Log[x]^4)/16)/x^((125*(E^(3 + x) + x)^3)/4) + Lo g[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.
| \(\displaystyle \int \frac {\left (625 x^4+125 x^3+e^{3 x+9} \left (1875 x^2+625 x+125\right )+\left (75 x^2+e^{x+3} \left (75 x^2+75 x+15\right )+75 e^{2 x+6} x+15 x\right ) \log ^2(x)+e^{2 x+6} \left (1875 x^3+1875 x^2+375 x\right )+\left (375 x^3+75 x^2+e^{2 x+6} \left (750 x^2+375 x+75\right )+e^{x+3} \left (375 x^3+750 x^2+150 x\right )+375 e^{3 x+9} x\right ) \log (x)+e^{x+3} \left (625 x^4+1875 x^3+375 x^2\right )+625 e^{4 x+12} x-4 x+\left (5 e^{x+3} x+5 x+1\right ) \log ^3(x)\right ) \exp \left (\frac {1}{16} \left (-625 x^4-2500 e^{x+3} x^3-3750 e^{2 x+6} x^2+\left (-150 x^2-300 e^{x+3} x-150 e^{2 x+6}\right ) \log ^2(x)+\left (-500 x^3-1500 e^{x+3} x^2-1500 e^{2 x+6} x-500 e^{3 x+9}\right ) \log (x)-2500 e^{3 x+9} x+16 x-625 e^{4 x+12}-\log ^4(x)+\left (-20 x-20 e^{x+3}\right ) \log ^3(x)\right )\right )+4 x+4}{4 x} \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {-\exp \left (\frac {1}{16} \left (-625 x^4-2500 e^{x+3} x^3-3750 e^{2 x+6} x^2-2500 e^{3 x+9} x+16 x-625 e^{4 x+12}-\log ^4(x)-20 \left (x+e^{x+3}\right ) \log ^3(x)-150 \left (x^2+2 e^{x+3} x+e^{2 x+6}\right ) \log ^2(x)\right )\right ) \left (-625 x^4-125 x^3-625 e^{4 x+12} x+4 x-\left (5 e^{x+3} x+5 x+1\right ) \log ^3(x)-15 \left (5 x^2+5 e^{2 x+6} x+x+e^{x+3} \left (5 x^2+5 x+1\right )\right ) \log ^2(x)-125 e^{3 x+9} \left (15 x^2+5 x+1\right )-375 e^{2 x+6} \left (5 x^3+5 x^2+x\right )-125 e^{x+3} \left (5 x^4+15 x^3+3 x^2\right )-75 \left (5 x^3+x^2+5 e^{3 x+9} x+e^{2 x+6} \left (10 x^2+5 x+1\right )+e^{x+3} \left (5 x^3+10 x^2+2 x\right )\right ) \log (x)\right ) x^{\frac {1}{16} \left (-500 x^3-1500 e^{x+3} x^2-1500 e^{2 x+6} x-500 e^{3 x+9}\right )}+4 x+4}{x}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \frac {1}{4} \int \left (\exp \left (-\frac {625 x^4}{16}-\frac {625}{4} e^{x+3} x^3-\frac {1875}{8} e^{2 x+6} x^2-\frac {625}{4} e^{3 x+9} x+x-\frac {625}{16} e^{4 x+12}-\frac {\log ^4(x)}{16}-\frac {5}{4} \left (x+e^{x+3}\right ) \log ^3(x)-\frac {75}{8} \left (x+e^{x+3}\right )^2 \log ^2(x)\right ) \left (625 e^{x+3} x^4+625 x^4+1875 e^{x+3} x^3+1875 e^{2 x+6} x^3+375 e^{x+3} \log (x) x^3+375 \log (x) x^3+125 x^3+375 e^{x+3} x^2+1875 e^{2 x+6} x^2+1875 e^{3 x+9} x^2+75 e^{x+3} \log ^2(x) x^2+75 \log ^2(x) x^2+750 e^{x+3} \log (x) x^2+750 e^{2 x+6} \log (x) x^2+75 \log (x) x^2+375 e^{2 x+6} x+625 e^{3 x+9} x+625 e^{4 x+12} x+5 e^{x+3} \log ^3(x) x+5 \log ^3(x) x+75 e^{x+3} \log ^2(x) x+75 e^{2 x+6} \log ^2(x) x+15 \log ^2(x) x+150 e^{x+3} \log (x) x+375 e^{2 x+6} \log (x) x+375 e^{3 x+9} \log (x) x-4 x+125 e^{3 x+9}+\log ^3(x)+15 e^{x+3} \log ^2(x)+75 e^{2 x+6} \log (x)\right ) x^{\frac {1}{4} \left (-125 x^3-375 e^{x+3} x^2-375 e^{2 x+6} x-125 e^{3 x+9}-4\right )}+\frac {4 (x+1)}{x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \frac {1}{4} \int \left (\exp \left (-\frac {625 x^4}{16}-\frac {625}{4} e^{x+3} x^3-\frac {1875}{8} e^{2 x+6} x^2-\frac {625}{4} e^{3 x+9} x+x-\frac {625}{16} e^{4 x+12}-\frac {\log ^4(x)}{16}-\frac {5}{4} \left (x+e^{x+3}\right ) \log ^3(x)-\frac {75}{8} \left (x+e^{x+3}\right )^2 \log ^2(x)\right ) \left (625 e^{x+3} x^4+625 x^4+1875 e^{x+3} x^3+1875 e^{2 x+6} x^3+375 e^{x+3} \log (x) x^3+375 \log (x) x^3+125 x^3+375 e^{x+3} x^2+1875 e^{2 x+6} x^2+1875 e^{3 x+9} x^2+75 e^{x+3} \log ^2(x) x^2+75 \log ^2(x) x^2+750 e^{x+3} \log (x) x^2+750 e^{2 x+6} \log (x) x^2+75 \log (x) x^2+375 e^{2 x+6} x+625 e^{3 x+9} x+625 e^{4 x+12} x+5 e^{x+3} \log ^3(x) x+5 \log ^3(x) x+75 e^{x+3} \log ^2(x) x+75 e^{2 x+6} \log ^2(x) x+15 \log ^2(x) x+150 e^{x+3} \log (x) x+375 e^{2 x+6} \log (x) x+375 e^{3 x+9} \log (x) x-4 x+125 e^{3 x+9}+\log ^3(x)+15 e^{x+3} \log ^2(x)+75 e^{2 x+6} \log (x)\right ) x^{\frac {1}{4} \left (-125 x^3-375 e^{x+3} x^2-375 e^{2 x+6} x-125 e^{3 x+9}-4\right )}+\frac {4 (x+1)}{x}\right )dx\) |
Input:
Int[(4 + 4*x + E^((-625*E^(12 + 4*x) + 16*x - 2500*E^(9 + 3*x)*x - 3750*E^ (6 + 2*x)*x^2 - 2500*E^(3 + x)*x^3 - 625*x^4 + (-500*E^(9 + 3*x) - 1500*E^ (6 + 2*x)*x - 1500*E^(3 + x)*x^2 - 500*x^3)*Log[x] + (-150*E^(6 + 2*x) - 3 00*E^(3 + x)*x - 150*x^2)*Log[x]^2 + (-20*E^(3 + x) - 20*x)*Log[x]^3 - Log [x]^4)/16)*(-4*x + 625*E^(12 + 4*x)*x + 125*x^3 + 625*x^4 + E^(9 + 3*x)*(1 25 + 625*x + 1875*x^2) + E^(6 + 2*x)*(375*x + 1875*x^2 + 1875*x^3) + E^(3 + x)*(375*x^2 + 1875*x^3 + 625*x^4) + (375*E^(9 + 3*x)*x + 75*x^2 + 375*x^ 3 + E^(6 + 2*x)*(75 + 375*x + 750*x^2) + E^(3 + x)*(150*x + 750*x^2 + 375* x^3))*Log[x] + (15*x + 75*E^(6 + 2*x)*x + 75*x^2 + E^(3 + x)*(15 + 75*x + 75*x^2))*Log[x]^2 + (1 + 5*x + 5*E^(3 + x)*x)*Log[x]^3))/(4*x),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs. \(2(25)=50\).
Time = 0.10 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.04
\[\ln \left (x \right )+x -x^{-\frac {375 x^{2} {\mathrm e}^{3+x}}{4}-\frac {125 x^{3}}{4}-\frac {375 x \,{\mathrm e}^{2 x +6}}{4}-\frac {125 \,{\mathrm e}^{3 x +9}}{4}} {\mathrm e}^{-\frac {\ln \left (x \right )^{4}}{16}-\frac {5 \ln \left (x \right )^{3} {\mathrm e}^{3+x}}{4}-\frac {5 x \ln \left (x \right )^{3}}{4}-\frac {75 \ln \left (x \right )^{2} {\mathrm e}^{3+x} x}{4}-\frac {75 x^{2} \ln \left (x \right )^{2}}{8}-\frac {75 \,{\mathrm e}^{2 x +6} \ln \left (x \right )^{2}}{8}-\frac {625 \,{\mathrm e}^{4 x +12}}{16}-\frac {625 x \,{\mathrm e}^{3 x +9}}{4}-\frac {1875 x^{2} {\mathrm e}^{2 x +6}}{8}-\frac {625 x^{3} {\mathrm e}^{3+x}}{4}-\frac {625 x^{4}}{16}+x}\]
Input:
int(1/4*(((5*exp(3+x)*x+1+5*x)*ln(x)^3+(75*x*exp(3+x)^2+(75*x^2+75*x+15)*e xp(3+x)+75*x^2+15*x)*ln(x)^2+(375*x*exp(3+x)^3+(750*x^2+375*x+75)*exp(3+x) ^2+(375*x^3+750*x^2+150*x)*exp(3+x)+375*x^3+75*x^2)*ln(x)+625*x*exp(3+x)^4 +(1875*x^2+625*x+125)*exp(3+x)^3+(1875*x^3+1875*x^2+375*x)*exp(3+x)^2+(625 *x^4+1875*x^3+375*x^2)*exp(3+x)+625*x^4+125*x^3-4*x)*exp(-1/16*ln(x)^4+1/1 6*(-20*exp(3+x)-20*x)*ln(x)^3+1/16*(-150*exp(3+x)^2-300*exp(3+x)*x-150*x^2 )*ln(x)^2+1/16*(-500*exp(3+x)^3-1500*x*exp(3+x)^2-1500*x^2*exp(3+x)-500*x^ 3)*ln(x)-625/16*exp(3+x)^4-625/4*x*exp(3+x)^3-1875/8*x^2*exp(3+x)^2-625/4* x^3*exp(3+x)-625/16*x^4+x)+4*x+4)/x,x)
Output:
ln(x)+x-x^(-375/4*x^2*exp(3+x)-125/4*x^3-375/4*x*exp(2*x+6)-125/4*exp(3*x+ 9))*exp(-1/16*ln(x)^4-5/4*ln(x)^3*exp(3+x)-5/4*x*ln(x)^3-75/4*ln(x)^2*exp( 3+x)*x-75/8*x^2*ln(x)^2-75/8*exp(2*x+6)*ln(x)^2-625/16*exp(4*x+12)-625/4*x *exp(3*x+9)-1875/8*x^2*exp(2*x+6)-625/4*x^3*exp(3+x)-625/16*x^4+x)
Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (25) = 50\).
Time = 0.10 (sec) , antiderivative size = 124, normalized size of antiderivative = 4.43 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=x - e^{\left (-\frac {625}{16} \, x^{4} - \frac {625}{4} \, x^{3} e^{\left (x + 3\right )} - \frac {5}{4} \, {\left (x + e^{\left (x + 3\right )}\right )} \log \left (x\right )^{3} - \frac {1}{16} \, \log \left (x\right )^{4} - \frac {1875}{8} \, x^{2} e^{\left (2 \, x + 6\right )} - \frac {75}{8} \, {\left (x^{2} + 2 \, x e^{\left (x + 3\right )} + e^{\left (2 \, x + 6\right )}\right )} \log \left (x\right )^{2} - \frac {625}{4} \, x e^{\left (3 \, x + 9\right )} - \frac {125}{4} \, {\left (x^{3} + 3 \, x^{2} e^{\left (x + 3\right )} + 3 \, x e^{\left (2 \, x + 6\right )} + e^{\left (3 \, x + 9\right )}\right )} \log \left (x\right ) + x - \frac {625}{16} \, e^{\left (4 \, x + 12\right )}\right )} + \log \left (x\right ) \] Input:
integrate(1/4*(((5*exp(3+x)*x+1+5*x)*log(x)^3+(75*x*exp(3+x)^2+(75*x^2+75* x+15)*exp(3+x)+75*x^2+15*x)*log(x)^2+(375*x*exp(3+x)^3+(750*x^2+375*x+75)* exp(3+x)^2+(375*x^3+750*x^2+150*x)*exp(3+x)+375*x^3+75*x^2)*log(x)+625*x*e xp(3+x)^4+(1875*x^2+625*x+125)*exp(3+x)^3+(1875*x^3+1875*x^2+375*x)*exp(3+ x)^2+(625*x^4+1875*x^3+375*x^2)*exp(3+x)+625*x^4+125*x^3-4*x)*exp(-1/16*lo g(x)^4+1/16*(-20*exp(3+x)-20*x)*log(x)^3+1/16*(-150*exp(3+x)^2-300*exp(3+x )*x-150*x^2)*log(x)^2+1/16*(-500*exp(3+x)^3-1500*x*exp(3+x)^2-1500*x^2*exp (3+x)-500*x^3)*log(x)-625/16*exp(3+x)^4-625/4*x*exp(3+x)^3-1875/8*x^2*exp( 3+x)^2-625/4*x^3*exp(3+x)-625/16*x^4+x)+4*x+4)/x,x, algorithm="fricas")
Output:
x - e^(-625/16*x^4 - 625/4*x^3*e^(x + 3) - 5/4*(x + e^(x + 3))*log(x)^3 - 1/16*log(x)^4 - 1875/8*x^2*e^(2*x + 6) - 75/8*(x^2 + 2*x*e^(x + 3) + e^(2* x + 6))*log(x)^2 - 625/4*x*e^(3*x + 9) - 125/4*(x^3 + 3*x^2*e^(x + 3) + 3* x*e^(2*x + 6) + e^(3*x + 9))*log(x) + x - 625/16*e^(4*x + 12)) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (24) = 48\).
Time = 1.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 6.00 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=x - e^{- \frac {625 x^{4}}{16} - \frac {625 x^{3} e^{x + 3}}{4} - \frac {1875 x^{2} e^{2 x + 6}}{8} - \frac {625 x e^{3 x + 9}}{4} + x + \left (- \frac {5 x}{4} - \frac {5 e^{x + 3}}{4}\right ) \log {\left (x \right )}^{3} + \left (- \frac {75 x^{2}}{8} - \frac {75 x e^{x + 3}}{4} - \frac {75 e^{2 x + 6}}{8}\right ) \log {\left (x \right )}^{2} + \left (- \frac {125 x^{3}}{4} - \frac {375 x^{2} e^{x + 3}}{4} - \frac {375 x e^{2 x + 6}}{4} - \frac {125 e^{3 x + 9}}{4}\right ) \log {\left (x \right )} - \frac {625 e^{4 x + 12}}{16} - \frac {\log {\left (x \right )}^{4}}{16}} + \log {\left (x \right )} \] Input:
integrate(1/4*(((5*exp(3+x)*x+1+5*x)*ln(x)**3+(75*x*exp(3+x)**2+(75*x**2+7 5*x+15)*exp(3+x)+75*x**2+15*x)*ln(x)**2+(375*x*exp(3+x)**3+(750*x**2+375*x +75)*exp(3+x)**2+(375*x**3+750*x**2+150*x)*exp(3+x)+375*x**3+75*x**2)*ln(x )+625*x*exp(3+x)**4+(1875*x**2+625*x+125)*exp(3+x)**3+(1875*x**3+1875*x**2 +375*x)*exp(3+x)**2+(625*x**4+1875*x**3+375*x**2)*exp(3+x)+625*x**4+125*x* *3-4*x)*exp(-1/16*ln(x)**4+1/16*(-20*exp(3+x)-20*x)*ln(x)**3+1/16*(-150*ex p(3+x)**2-300*exp(3+x)*x-150*x**2)*ln(x)**2+1/16*(-500*exp(3+x)**3-1500*x* exp(3+x)**2-1500*x**2*exp(3+x)-500*x**3)*ln(x)-625/16*exp(3+x)**4-625/4*x* exp(3+x)**3-1875/8*x**2*exp(3+x)**2-625/4*x**3*exp(3+x)-625/16*x**4+x)+4*x +4)/x,x)
Output:
x - exp(-625*x**4/16 - 625*x**3*exp(x + 3)/4 - 1875*x**2*exp(2*x + 6)/8 - 625*x*exp(3*x + 9)/4 + x + (-5*x/4 - 5*exp(x + 3)/4)*log(x)**3 + (-75*x**2 /8 - 75*x*exp(x + 3)/4 - 75*exp(2*x + 6)/8)*log(x)**2 + (-125*x**3/4 - 375 *x**2*exp(x + 3)/4 - 375*x*exp(2*x + 6)/4 - 125*exp(3*x + 9)/4)*log(x) - 6 25*exp(4*x + 12)/16 - log(x)**4/16) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (25) = 50\).
Time = 0.76 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.18 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=x - e^{\left (-\frac {625}{16} \, x^{4} - \frac {625}{4} \, x^{3} e^{\left (x + 3\right )} - \frac {125}{4} \, x^{3} \log \left (x\right ) - \frac {375}{4} \, x^{2} e^{\left (x + 3\right )} \log \left (x\right ) - \frac {75}{8} \, x^{2} \log \left (x\right )^{2} - \frac {75}{4} \, x e^{\left (x + 3\right )} \log \left (x\right )^{2} - \frac {5}{4} \, x \log \left (x\right )^{3} - \frac {5}{4} \, e^{\left (x + 3\right )} \log \left (x\right )^{3} - \frac {1}{16} \, \log \left (x\right )^{4} - \frac {1875}{8} \, x^{2} e^{\left (2 \, x + 6\right )} - \frac {375}{4} \, x e^{\left (2 \, x + 6\right )} \log \left (x\right ) - \frac {75}{8} \, e^{\left (2 \, x + 6\right )} \log \left (x\right )^{2} - \frac {625}{4} \, x e^{\left (3 \, x + 9\right )} - \frac {125}{4} \, e^{\left (3 \, x + 9\right )} \log \left (x\right ) + x - \frac {625}{16} \, e^{\left (4 \, x + 12\right )}\right )} + \log \left (x\right ) \] Input:
integrate(1/4*(((5*exp(3+x)*x+1+5*x)*log(x)^3+(75*x*exp(3+x)^2+(75*x^2+75* x+15)*exp(3+x)+75*x^2+15*x)*log(x)^2+(375*x*exp(3+x)^3+(750*x^2+375*x+75)* exp(3+x)^2+(375*x^3+750*x^2+150*x)*exp(3+x)+375*x^3+75*x^2)*log(x)+625*x*e xp(3+x)^4+(1875*x^2+625*x+125)*exp(3+x)^3+(1875*x^3+1875*x^2+375*x)*exp(3+ x)^2+(625*x^4+1875*x^3+375*x^2)*exp(3+x)+625*x^4+125*x^3-4*x)*exp(-1/16*lo g(x)^4+1/16*(-20*exp(3+x)-20*x)*log(x)^3+1/16*(-150*exp(3+x)^2-300*exp(3+x )*x-150*x^2)*log(x)^2+1/16*(-500*exp(3+x)^3-1500*x*exp(3+x)^2-1500*x^2*exp (3+x)-500*x^3)*log(x)-625/16*exp(3+x)^4-625/4*x*exp(3+x)^3-1875/8*x^2*exp( 3+x)^2-625/4*x^3*exp(3+x)-625/16*x^4+x)+4*x+4)/x,x, algorithm="maxima")
Output:
x - e^(-625/16*x^4 - 625/4*x^3*e^(x + 3) - 125/4*x^3*log(x) - 375/4*x^2*e^ (x + 3)*log(x) - 75/8*x^2*log(x)^2 - 75/4*x*e^(x + 3)*log(x)^2 - 5/4*x*log (x)^3 - 5/4*e^(x + 3)*log(x)^3 - 1/16*log(x)^4 - 1875/8*x^2*e^(2*x + 6) - 375/4*x*e^(2*x + 6)*log(x) - 75/8*e^(2*x + 6)*log(x)^2 - 625/4*x*e^(3*x + 9) - 125/4*e^(3*x + 9)*log(x) + x - 625/16*e^(4*x + 12)) + log(x)
\[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=\int { \frac {{\left (625 \, x^{4} + {\left (5 \, x e^{\left (x + 3\right )} + 5 \, x + 1\right )} \log \left (x\right )^{3} + 125 \, x^{3} + 15 \, {\left (5 \, x^{2} + 5 \, x e^{\left (2 \, x + 6\right )} + {\left (5 \, x^{2} + 5 \, x + 1\right )} e^{\left (x + 3\right )} + x\right )} \log \left (x\right )^{2} + 625 \, x e^{\left (4 \, x + 12\right )} + 125 \, {\left (15 \, x^{2} + 5 \, x + 1\right )} e^{\left (3 \, x + 9\right )} + 375 \, {\left (5 \, x^{3} + 5 \, x^{2} + x\right )} e^{\left (2 \, x + 6\right )} + 125 \, {\left (5 \, x^{4} + 15 \, x^{3} + 3 \, x^{2}\right )} e^{\left (x + 3\right )} + 75 \, {\left (5 \, x^{3} + x^{2} + 5 \, x e^{\left (3 \, x + 9\right )} + {\left (10 \, x^{2} + 5 \, x + 1\right )} e^{\left (2 \, x + 6\right )} + {\left (5 \, x^{3} + 10 \, x^{2} + 2 \, x\right )} e^{\left (x + 3\right )}\right )} \log \left (x\right ) - 4 \, x\right )} e^{\left (-\frac {625}{16} \, x^{4} - \frac {625}{4} \, x^{3} e^{\left (x + 3\right )} - \frac {5}{4} \, {\left (x + e^{\left (x + 3\right )}\right )} \log \left (x\right )^{3} - \frac {1}{16} \, \log \left (x\right )^{4} - \frac {1875}{8} \, x^{2} e^{\left (2 \, x + 6\right )} - \frac {75}{8} \, {\left (x^{2} + 2 \, x e^{\left (x + 3\right )} + e^{\left (2 \, x + 6\right )}\right )} \log \left (x\right )^{2} - \frac {625}{4} \, x e^{\left (3 \, x + 9\right )} - \frac {125}{4} \, {\left (x^{3} + 3 \, x^{2} e^{\left (x + 3\right )} + 3 \, x e^{\left (2 \, x + 6\right )} + e^{\left (3 \, x + 9\right )}\right )} \log \left (x\right ) + x - \frac {625}{16} \, e^{\left (4 \, x + 12\right )}\right )} + 4 \, x + 4}{4 \, x} \,d x } \] Input:
integrate(1/4*(((5*exp(3+x)*x+1+5*x)*log(x)^3+(75*x*exp(3+x)^2+(75*x^2+75* x+15)*exp(3+x)+75*x^2+15*x)*log(x)^2+(375*x*exp(3+x)^3+(750*x^2+375*x+75)* exp(3+x)^2+(375*x^3+750*x^2+150*x)*exp(3+x)+375*x^3+75*x^2)*log(x)+625*x*e xp(3+x)^4+(1875*x^2+625*x+125)*exp(3+x)^3+(1875*x^3+1875*x^2+375*x)*exp(3+ x)^2+(625*x^4+1875*x^3+375*x^2)*exp(3+x)+625*x^4+125*x^3-4*x)*exp(-1/16*lo g(x)^4+1/16*(-20*exp(3+x)-20*x)*log(x)^3+1/16*(-150*exp(3+x)^2-300*exp(3+x )*x-150*x^2)*log(x)^2+1/16*(-500*exp(3+x)^3-1500*x*exp(3+x)^2-1500*x^2*exp (3+x)-500*x^3)*log(x)-625/16*exp(3+x)^4-625/4*x*exp(3+x)^3-1875/8*x^2*exp( 3+x)^2-625/4*x^3*exp(3+x)-625/16*x^4+x)+4*x+4)/x,x, algorithm="giac")
Output:
integrate(1/4*((625*x^4 + (5*x*e^(x + 3) + 5*x + 1)*log(x)^3 + 125*x^3 + 1 5*(5*x^2 + 5*x*e^(2*x + 6) + (5*x^2 + 5*x + 1)*e^(x + 3) + x)*log(x)^2 + 6 25*x*e^(4*x + 12) + 125*(15*x^2 + 5*x + 1)*e^(3*x + 9) + 375*(5*x^3 + 5*x^ 2 + x)*e^(2*x + 6) + 125*(5*x^4 + 15*x^3 + 3*x^2)*e^(x + 3) + 75*(5*x^3 + x^2 + 5*x*e^(3*x + 9) + (10*x^2 + 5*x + 1)*e^(2*x + 6) + (5*x^3 + 10*x^2 + 2*x)*e^(x + 3))*log(x) - 4*x)*e^(-625/16*x^4 - 625/4*x^3*e^(x + 3) - 5/4* (x + e^(x + 3))*log(x)^3 - 1/16*log(x)^4 - 1875/8*x^2*e^(2*x + 6) - 75/8*( x^2 + 2*x*e^(x + 3) + e^(2*x + 6))*log(x)^2 - 625/4*x*e^(3*x + 9) - 125/4* (x^3 + 3*x^2*e^(x + 3) + 3*x*e^(2*x + 6) + e^(3*x + 9))*log(x) + x - 625/1 6*e^(4*x + 12)) + 4*x + 4)/x, x)
Time = 3.24 (sec) , antiderivative size = 163, normalized size of antiderivative = 5.82 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx=x+\ln \left (x\right )-\frac {{\mathrm {e}}^{-\frac {{\ln \left (x\right )}^4}{16}}\,{\mathrm {e}}^{-\frac {125\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^9\,\ln \left (x\right )}{4}}\,{\mathrm {e}}^{-\frac {5\,{\mathrm {e}}^3\,{\mathrm {e}}^x\,{\ln \left (x\right )}^3}{4}}\,{\mathrm {e}}^{-\frac {5\,x\,{\ln \left (x\right )}^3}{4}}\,{\mathrm {e}}^{-\frac {125\,x^3\,\ln \left (x\right )}{4}}\,{\mathrm {e}}^{-\frac {75\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6\,{\ln \left (x\right )}^2}{8}}\,{\mathrm {e}}^{-\frac {375\,x\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6\,\ln \left (x\right )}{4}}\,{\mathrm {e}}^{-\frac {75\,x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\,{\ln \left (x\right )}^2}{4}}\,{\mathrm {e}}^{-\frac {375\,x^2\,{\mathrm {e}}^3\,{\mathrm {e}}^x\,\ln \left (x\right )}{4}}\,{\mathrm {e}}^x\,{\mathrm {e}}^{-\frac {75\,x^2\,{\ln \left (x\right )}^2}{8}}}{{\left ({\mathrm {e}}^{x^4}\right )}^{625/16}\,{\left ({\mathrm {e}}^{x^2\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^6}\right )}^{1875/8}\,{\left ({\mathrm {e}}^{{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{12}}\right )}^{625/16}\,{\left ({\mathrm {e}}^{x\,{\mathrm {e}}^{3\,x}\,{\mathrm {e}}^9}\right )}^{625/4}\,{\left ({\mathrm {e}}^{x^3\,{\mathrm {e}}^3\,{\mathrm {e}}^x}\right )}^{625/4}} \] Input:
int((x + (exp(x - (625*exp(4*x + 12))/16 - log(x)^4/16 - (log(x)^2*(150*ex p(2*x + 6) + 300*x*exp(x + 3) + 150*x^2))/16 - (625*x*exp(3*x + 9))/4 - (6 25*x^3*exp(x + 3))/4 - (log(x)^3*(20*x + 20*exp(x + 3)))/16 - (log(x)*(500 *exp(3*x + 9) + 1500*x*exp(2*x + 6) + 1500*x^2*exp(x + 3) + 500*x^3))/16 - (1875*x^2*exp(2*x + 6))/8 - (625*x^4)/16)*(log(x)*(exp(x + 3)*(150*x + 75 0*x^2 + 375*x^3) + exp(2*x + 6)*(375*x + 750*x^2 + 75) + 375*x*exp(3*x + 9 ) + 75*x^2 + 375*x^3) - 4*x + exp(3*x + 9)*(625*x + 1875*x^2 + 125) + 625* x*exp(4*x + 12) + log(x)^2*(15*x + exp(x + 3)*(75*x + 75*x^2 + 15) + 75*x* exp(2*x + 6) + 75*x^2) + exp(2*x + 6)*(375*x + 1875*x^2 + 1875*x^3) + exp( x + 3)*(375*x^2 + 1875*x^3 + 625*x^4) + 125*x^3 + 625*x^4 + log(x)^3*(5*x + 5*x*exp(x + 3) + 1)))/4 + 1)/x,x)
Output:
x + log(x) - (exp(-log(x)^4/16)*exp(-(125*exp(3*x)*exp(9)*log(x))/4)*exp(- (5*exp(3)*exp(x)*log(x)^3)/4)*exp(-(5*x*log(x)^3)/4)*exp(-(125*x^3*log(x)) /4)*exp(-(75*exp(2*x)*exp(6)*log(x)^2)/8)*exp(-(375*x*exp(2*x)*exp(6)*log( x))/4)*exp(-(75*x*exp(3)*exp(x)*log(x)^2)/4)*exp(-(375*x^2*exp(3)*exp(x)*l og(x))/4)*exp(x)*exp(-(75*x^2*log(x)^2)/8))/(exp(x^4)^(625/16)*exp(x^2*exp (2*x)*exp(6))^(1875/8)*exp(exp(4*x)*exp(12))^(625/16)*exp(x*exp(3*x)*exp(9 ))^(625/4)*exp(x^3*exp(3)*exp(x))^(625/4))
Time = 1.48 (sec) , antiderivative size = 493, normalized size of antiderivative = 17.61 \[ \int \frac {4+4 x+e^{\frac {1}{16} \left (-625 e^{12+4 x}+16 x-2500 e^{9+3 x} x-3750 e^{6+2 x} x^2-2500 e^{3+x} x^3-625 x^4+\left (-500 e^{9+3 x}-1500 e^{6+2 x} x-1500 e^{3+x} x^2-500 x^3\right ) \log (x)+\left (-150 e^{6+2 x}-300 e^{3+x} x-150 x^2\right ) \log ^2(x)+\left (-20 e^{3+x}-20 x\right ) \log ^3(x)-\log ^4(x)\right )} \left (-4 x+625 e^{12+4 x} x+125 x^3+625 x^4+e^{9+3 x} \left (125+625 x+1875 x^2\right )+e^{6+2 x} \left (375 x+1875 x^2+1875 x^3\right )+e^{3+x} \left (375 x^2+1875 x^3+625 x^4\right )+\left (375 e^{9+3 x} x+75 x^2+375 x^3+e^{6+2 x} \left (75+375 x+750 x^2\right )+e^{3+x} \left (150 x+750 x^2+375 x^3\right )\right ) \log (x)+\left (15 x+75 e^{6+2 x} x+75 x^2+e^{3+x} \left (15+75 x+75 x^2\right )\right ) \log ^2(x)+\left (1+5 x+5 e^{3+x} x\right ) \log ^3(x)\right )}{4 x} \, dx =\text {Too large to display} \] Input:
int(1/4*(((5*exp(3+x)*x+1+5*x)*log(x)^3+(75*x*exp(3+x)^2+(75*x^2+75*x+15)* exp(3+x)+75*x^2+15*x)*log(x)^2+(375*x*exp(3+x)^3+(750*x^2+375*x+75)*exp(3+ x)^2+(375*x^3+750*x^2+150*x)*exp(3+x)+375*x^3+75*x^2)*log(x)+625*x*exp(3+x )^4+(1875*x^2+625*x+125)*exp(3+x)^3+(1875*x^3+1875*x^2+375*x)*exp(3+x)^2+( 625*x^4+1875*x^3+375*x^2)*exp(3+x)+625*x^4+125*x^3-4*x)*exp(-1/16*log(x)^4 +1/16*(-20*exp(3+x)-20*x)*log(x)^3+1/16*(-150*exp(3+x)^2-300*exp(3+x)*x-15 0*x^2)*log(x)^2+1/16*(-500*exp(3+x)^3-1500*x*exp(3+x)^2-1500*x^2*exp(3+x)- 500*x^3)*log(x)-625/16*exp(3+x)^4-625/4*x*exp(3+x)^3-1875/8*x^2*exp(3+x)^2 -625/4*x^3*exp(3+x)-625/16*x^4+x)+4*x+4)/x,x)
Output:
(x**((125*x**3)/4)*e**((625*e**(4*x)*e**12 + 500*e**(3*x)*log(x)*e**9 + 25 00*e**(3*x)*e**9*x + 150*e**(2*x)*log(x)**2*e**6 + 1500*e**(2*x)*log(x)*e* *6*x + 3750*e**(2*x)*e**6*x**2 + 20*e**x*log(x)**3*e**3 + 300*e**x*log(x)* *2*e**3*x + 1500*e**x*log(x)*e**3*x**2 + 2500*e**x*e**3*x**3 + log(x)**4 + 20*log(x)**3*x + 150*log(x)**2*x**2 + 625*x**4)/16)*log(x) + x**((125*x** 3)/4)*e**((625*e**(4*x)*e**12 + 500*e**(3*x)*log(x)*e**9 + 2500*e**(3*x)*e **9*x + 150*e**(2*x)*log(x)**2*e**6 + 1500*e**(2*x)*log(x)*e**6*x + 3750*e **(2*x)*e**6*x**2 + 20*e**x*log(x)**3*e**3 + 300*e**x*log(x)**2*e**3*x + 1 500*e**x*log(x)*e**3*x**2 + 2500*e**x*e**3*x**3 + log(x)**4 + 20*log(x)**3 *x + 150*log(x)**2*x**2 + 625*x**4)/16)*x - e**x)/(x**((125*x**3)/4)*e**(( 625*e**(4*x)*e**12 + 500*e**(3*x)*log(x)*e**9 + 2500*e**(3*x)*e**9*x + 150 *e**(2*x)*log(x)**2*e**6 + 1500*e**(2*x)*log(x)*e**6*x + 3750*e**(2*x)*e** 6*x**2 + 20*e**x*log(x)**3*e**3 + 300*e**x*log(x)**2*e**3*x + 1500*e**x*lo g(x)*e**3*x**2 + 2500*e**x*e**3*x**3 + log(x)**4 + 20*log(x)**3*x + 150*lo g(x)**2*x**2 + 625*x**4)/16))