Integrand size = 215, antiderivative size = 29 \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx=\left (3+4 e^{\left (\frac {3}{2}+\log \left (e^3 \left (1+e^x\right ) x\right )\right )^2}-\log (x)\right )^2 \] Output:
(3-ln(x)+4*exp((ln((1+exp(x))*x*exp(3))+3/2)^2))^2
Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(29)=58\).
Time = 0.74 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.45 \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx=16 e^{\frac {81}{2}+2 \log ^2\left (\left (1+e^x\right ) x\right )} \left (1+e^x\right )^{18} x^{18}-8 e^{\frac {81}{4}+\log ^2\left (\left (1+e^x\right ) x\right )} \left (1+e^x\right )^9 x^9 (-3+\log (x))-6 \log (x)+\log ^2(x) \] Input:
Integrate[(-6 - 6*E^x + (2 + 2*E^x)*Log[x] + E^((9 + 12*Log[E^3*x + E^(3 + x)*x] + 4*Log[E^3*x + E^(3 + x)*x]^2)/2)*(96 + E^x*(96 + 96*x) + (64 + E^ x*(64 + 64*x))*Log[E^3*x + E^(3 + x)*x]) + E^((9 + 12*Log[E^3*x + E^(3 + x )*x] + 4*Log[E^3*x + E^(3 + x)*x]^2)/4)*(64 + E^x*(64 + 72*x) + (-24 + E^x *(-24 - 24*x))*Log[x] + (48 + E^x*(48 + 48*x) + (-16 + E^x*(-16 - 16*x))*L og[x])*Log[E^3*x + E^(3 + x)*x]))/(x + E^x*x),x]
Output:
16*E^(81/2 + 2*Log[(1 + E^x)*x]^2)*(1 + E^x)^18*x^18 - 8*E^(81/4 + Log[(1 + E^x)*x]^2)*(1 + E^x)^9*x^9*(-3 + Log[x]) - 6*Log[x] + Log[x]^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.
| \(\displaystyle \int \frac {\left (e^x (96 x+96)+\left (e^x (64 x+64)+64\right ) \log \left (e^{x+3} x+e^3 x\right )+96\right ) \exp \left (\frac {1}{2} \left (4 \log ^2\left (e^{x+3} x+e^3 x\right )+12 \log \left (e^{x+3} x+e^3 x\right )+9\right )\right )+\left (e^x (72 x+64)+\left (e^x (-24 x-24)-24\right ) \log (x)+\left (e^x (48 x+48)+\left (e^x (-16 x-16)-16\right ) \log (x)+48\right ) \log \left (e^{x+3} x+e^3 x\right )+64\right ) \exp \left (\frac {1}{4} \left (4 \log ^2\left (e^{x+3} x+e^3 x\right )+12 \log \left (e^{x+3} x+e^3 x\right )+9\right )\right )-6 e^x+\left (2 e^x+2\right ) \log (x)-6}{e^x x+x} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (32 \left (e^x+1\right )^5 \left (e^x x+e^x+1\right ) x^5 e^{2 \left (\log \left (\left (e^x+1\right ) x\right )+3\right )^2+\frac {45}{2}} \left (2 \log \left (\left (e^x+1\right ) x\right )+9\right )-8 \left (e^x+1\right )^2 x^2 e^{\left (\log \left (\left (e^x+1\right ) x\right )+3\right )^2+\frac {45}{4}} \left (-27 e^x x-26 e^x+9 e^x x \log (x)-6 e^x x \log \left (\left (e^x+1\right ) x\right )+2 e^x x \log (x) \log \left (\left (e^x+1\right ) x\right )+9 e^x \log (x)+9 \log (x)-6 e^x \log \left (\left (e^x+1\right ) x\right )+2 e^x \log (x) \log \left (\left (e^x+1\right ) x\right )+2 \log (x) \log \left (\left (e^x+1\right ) x\right )-6 \log \left (\left (e^x+1\right ) x\right )-26\right )+\frac {2 (\log (x)-3)}{x}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (32 \left (e^x+1\right )^5 \left (e^x x+e^x+1\right ) x^5 e^{2 \left (\log \left (\left (e^x+1\right ) x\right )+3\right )^2+\frac {45}{2}} \left (2 \log \left (\left (e^x+1\right ) x\right )+9\right )-8 \left (e^x+1\right )^2 x^2 e^{\left (\log \left (\left (e^x+1\right ) x\right )+3\right )^2+\frac {45}{4}} \left (-27 e^x x-26 e^x+9 e^x x \log (x)-6 e^x x \log \left (\left (e^x+1\right ) x\right )+2 e^x x \log (x) \log \left (\left (e^x+1\right ) x\right )+9 e^x \log (x)+9 \log (x)-6 e^x \log \left (\left (e^x+1\right ) x\right )+2 e^x \log (x) \log \left (\left (e^x+1\right ) x\right )+2 \log (x) \log \left (\left (e^x+1\right ) x\right )-6 \log \left (\left (e^x+1\right ) x\right )-26\right )+\frac {2 (\log (x)-3)}{x}\right )dx\) |
Input:
Int[(-6 - 6*E^x + (2 + 2*E^x)*Log[x] + E^((9 + 12*Log[E^3*x + E^(3 + x)*x] + 4*Log[E^3*x + E^(3 + x)*x]^2)/2)*(96 + E^x*(96 + 96*x) + (64 + E^x*(64 + 64*x))*Log[E^3*x + E^(3 + x)*x]) + E^((9 + 12*Log[E^3*x + E^(3 + x)*x] + 4*Log[E^3*x + E^(3 + x)*x]^2)/4)*(64 + E^x*(64 + 72*x) + (-24 + E^x*(-24 - 24*x))*Log[x] + (48 + E^x*(48 + 48*x) + (-16 + E^x*(-16 - 16*x))*Log[x]) *Log[E^3*x + E^(3 + x)*x]))/(x + E^x*x),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(94\) vs. \(2(24)=48\).
Time = 4.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.28
| method | result | size |
| parallelrisch | \(\ln \left (x \right )^{2}-8 \ln \left (x \right ) {\mathrm e}^{{\ln \left (\left ({\mathrm e}^{x}+1\right ) x \,{\mathrm e}^{3}\right )}^{2}+3 \ln \left (\left ({\mathrm e}^{x}+1\right ) x \,{\mathrm e}^{3}\right )+\frac {9}{4}}+16 \left ({\mathrm e}^{x}+1\right )^{6} x^{6} {\mathrm e}^{\frac {45}{2}+2 {\ln \left (\left ({\mathrm e}^{x}+1\right ) x \,{\mathrm e}^{3}\right )}^{2}}-6 \ln \left (x \right )+24 \,{\mathrm e}^{{\ln \left (\left ({\mathrm e}^{x}+1\right ) x \,{\mathrm e}^{3}\right )}^{2}+3 \ln \left (\left ({\mathrm e}^{x}+1\right ) x \,{\mathrm e}^{3}\right )+\frac {9}{4}}\) | \(95\) |
| risch | \(\text {Expression too large to display}\) | \(971\) |
Input:
int(((((64*x+64)*exp(x)+64)*ln(x*exp(3)*exp(x)+x*exp(3))+(96*x+96)*exp(x)+ 96)*exp(ln(x*exp(3)*exp(x)+x*exp(3))^2+3*ln(x*exp(3)*exp(x)+x*exp(3))+9/4) ^2+((((-16*x-16)*exp(x)-16)*ln(x)+(48*x+48)*exp(x)+48)*ln(x*exp(3)*exp(x)+ x*exp(3))+((-24*x-24)*exp(x)-24)*ln(x)+(72*x+64)*exp(x)+64)*exp(ln(x*exp(3 )*exp(x)+x*exp(3))^2+3*ln(x*exp(3)*exp(x)+x*exp(3))+9/4)+(2*exp(x)+2)*ln(x )-6*exp(x)-6)/(exp(x)*x+x),x,method=_RETURNVERBOSE)
Output:
ln(x)^2-8*ln(x)*exp(ln((exp(x)+1)*x*exp(3))^2+3*ln((exp(x)+1)*x*exp(3))+9/ 4)+16*exp(ln((exp(x)+1)*x*exp(3))^2+3*ln((exp(x)+1)*x*exp(3))+9/4)^2-6*ln( x)+24*exp(ln((exp(x)+1)*x*exp(3))^2+3*ln((exp(x)+1)*x*exp(3))+9/4)
Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (28) = 56\).
Time = 0.09 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.79 \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx=-8 \, {\left (\log \left (x\right ) - 3\right )} e^{\left (\log \left (x e^{3} + x e^{\left (x + 3\right )}\right )^{2} + 3 \, \log \left (x e^{3} + x e^{\left (x + 3\right )}\right ) + \frac {9}{4}\right )} + \log \left (x\right )^{2} + 16 \, e^{\left (2 \, \log \left (x e^{3} + x e^{\left (x + 3\right )}\right )^{2} + 6 \, \log \left (x e^{3} + x e^{\left (x + 3\right )}\right ) + \frac {9}{2}\right )} - 6 \, \log \left (x\right ) \] Input:
integrate(((((64*x+64)*exp(x)+64)*log(x*exp(3)*exp(x)+x*exp(3))+(96*x+96)* exp(x)+96)*exp(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp (3))+9/4)^2+((((-16*x-16)*exp(x)-16)*log(x)+(48*x+48)*exp(x)+48)*log(x*exp (3)*exp(x)+x*exp(3))+((-24*x-24)*exp(x)-24)*log(x)+(72*x+64)*exp(x)+64)*ex p(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp(3))+9/4)+(2* exp(x)+2)*log(x)-6*exp(x)-6)/(exp(x)*x+x),x, algorithm="fricas")
Output:
-8*(log(x) - 3)*e^(log(x*e^3 + x*e^(x + 3))^2 + 3*log(x*e^3 + x*e^(x + 3)) + 9/4) + log(x)^2 + 16*e^(2*log(x*e^3 + x*e^(x + 3))^2 + 6*log(x*e^3 + x* e^(x + 3)) + 9/2) - 6*log(x)
Timed out. \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx=\text {Timed out} \] Input:
integrate(((((64*x+64)*exp(x)+64)*ln(x*exp(3)*exp(x)+x*exp(3))+(96*x+96)*e xp(x)+96)*exp(ln(x*exp(3)*exp(x)+x*exp(3))**2+3*ln(x*exp(3)*exp(x)+x*exp(3 ))+9/4)**2+((((-16*x-16)*exp(x)-16)*ln(x)+(48*x+48)*exp(x)+48)*ln(x*exp(3) *exp(x)+x*exp(3))+((-24*x-24)*exp(x)-24)*ln(x)+(72*x+64)*exp(x)+64)*exp(ln (x*exp(3)*exp(x)+x*exp(3))**2+3*ln(x*exp(3)*exp(x)+x*exp(3))+9/4)+(2*exp(x )+2)*ln(x)-6*exp(x)-6)/(exp(x)*x+x),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (28) = 56\).
Time = 0.22 (sec) , antiderivative size = 474, normalized size of antiderivative = 16.34 \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx =\text {Too large to display} \] Input:
integrate(((((64*x+64)*exp(x)+64)*log(x*exp(3)*exp(x)+x*exp(3))+(96*x+96)* exp(x)+96)*exp(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp (3))+9/4)^2+((((-16*x-16)*exp(x)-16)*log(x)+(48*x+48)*exp(x)+48)*log(x*exp (3)*exp(x)+x*exp(3))+((-24*x-24)*exp(x)-24)*log(x)+(72*x+64)*exp(x)+64)*ex p(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp(3))+9/4)+(2* exp(x)+2)*log(x)-6*exp(x)-6)/(exp(x)*x+x),x, algorithm="maxima")
Output:
16*(x^18*e^(81/2) + x^18*e^(18*x + 81/2) + 18*x^18*e^(17*x + 81/2) + 153*x ^18*e^(16*x + 81/2) + 816*x^18*e^(15*x + 81/2) + 3060*x^18*e^(14*x + 81/2) + 8568*x^18*e^(13*x + 81/2) + 18564*x^18*e^(12*x + 81/2) + 31824*x^18*e^( 11*x + 81/2) + 43758*x^18*e^(10*x + 81/2) + 48620*x^18*e^(9*x + 81/2) + 43 758*x^18*e^(8*x + 81/2) + 31824*x^18*e^(7*x + 81/2) + 18564*x^18*e^(6*x + 81/2) + 8568*x^18*e^(5*x + 81/2) + 3060*x^18*e^(4*x + 81/2) + 816*x^18*e^( 3*x + 81/2) + 153*x^18*e^(2*x + 81/2) + 18*x^18*e^(x + 81/2))*e^(2*log(x)^ 2 + 4*log(x)*log(e^x + 1) + 2*log(e^x + 1)^2) - 8*(x^9*e^(81/4)*log(x) - 3 *x^9*e^(81/4) + (x^9*e^(81/4)*log(x) - 3*x^9*e^(81/4))*e^(9*x) + 9*(x^9*e^ (81/4)*log(x) - 3*x^9*e^(81/4))*e^(8*x) + 36*(x^9*e^(81/4)*log(x) - 3*x^9* e^(81/4))*e^(7*x) + 84*(x^9*e^(81/4)*log(x) - 3*x^9*e^(81/4))*e^(6*x) + 12 6*(x^9*e^(81/4)*log(x) - 3*x^9*e^(81/4))*e^(5*x) + 126*(x^9*e^(81/4)*log(x ) - 3*x^9*e^(81/4))*e^(4*x) + 84*(x^9*e^(81/4)*log(x) - 3*x^9*e^(81/4))*e^ (3*x) + 36*(x^9*e^(81/4)*log(x) - 3*x^9*e^(81/4))*e^(2*x) + 9*(x^9*e^(81/4 )*log(x) - 3*x^9*e^(81/4))*e^x)*e^(log(x)^2 + 2*log(x)*log(e^x + 1) + log( e^x + 1)^2) + log(x)^2 - 6*log(x)
\[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx=\int { \frac {2 \, {\left (16 \, {\left (3 \, {\left (x + 1\right )} e^{x} + 2 \, {\left ({\left (x + 1\right )} e^{x} + 1\right )} \log \left (x e^{3} + x e^{\left (x + 3\right )}\right ) + 3\right )} e^{\left (2 \, \log \left (x e^{3} + x e^{\left (x + 3\right )}\right )^{2} + 6 \, \log \left (x e^{3} + x e^{\left (x + 3\right )}\right ) + \frac {9}{2}\right )} + 4 \, {\left ({\left (9 \, x + 8\right )} e^{x} + 2 \, {\left (3 \, {\left (x + 1\right )} e^{x} - {\left ({\left (x + 1\right )} e^{x} + 1\right )} \log \left (x\right ) + 3\right )} \log \left (x e^{3} + x e^{\left (x + 3\right )}\right ) - 3 \, {\left ({\left (x + 1\right )} e^{x} + 1\right )} \log \left (x\right ) + 8\right )} e^{\left (\log \left (x e^{3} + x e^{\left (x + 3\right )}\right )^{2} + 3 \, \log \left (x e^{3} + x e^{\left (x + 3\right )}\right ) + \frac {9}{4}\right )} + {\left (e^{x} + 1\right )} \log \left (x\right ) - 3 \, e^{x} - 3\right )}}{x e^{x} + x} \,d x } \] Input:
integrate(((((64*x+64)*exp(x)+64)*log(x*exp(3)*exp(x)+x*exp(3))+(96*x+96)* exp(x)+96)*exp(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp (3))+9/4)^2+((((-16*x-16)*exp(x)-16)*log(x)+(48*x+48)*exp(x)+48)*log(x*exp (3)*exp(x)+x*exp(3))+((-24*x-24)*exp(x)-24)*log(x)+(72*x+64)*exp(x)+64)*ex p(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp(3))+9/4)+(2* exp(x)+2)*log(x)-6*exp(x)-6)/(exp(x)*x+x),x, algorithm="giac")
Output:
integrate(2*(16*(3*(x + 1)*e^x + 2*((x + 1)*e^x + 1)*log(x*e^3 + x*e^(x + 3)) + 3)*e^(2*log(x*e^3 + x*e^(x + 3))^2 + 6*log(x*e^3 + x*e^(x + 3)) + 9/ 2) + 4*((9*x + 8)*e^x + 2*(3*(x + 1)*e^x - ((x + 1)*e^x + 1)*log(x) + 3)*l og(x*e^3 + x*e^(x + 3)) - 3*((x + 1)*e^x + 1)*log(x) + 8)*e^(log(x*e^3 + x *e^(x + 3))^2 + 3*log(x*e^3 + x*e^(x + 3)) + 9/4) + (e^x + 1)*log(x) - 3*e ^x - 3)/(x*e^x + x), x)
Time = 2.71 (sec) , antiderivative size = 295, normalized size of antiderivative = 10.17 \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx={\ln \left (x\right )}^2-6\,\ln \left (x\right )+96\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+x+\frac {45}{2}}-\left (8\,\ln \left (x\right )-24\right )\,\left (3\,x^3\,{\mathrm {e}}^{{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+2\,x+\frac {45}{4}}+x^3\,{\mathrm {e}}^{{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+3\,x+\frac {45}{4}}+x^3\,{\mathrm {e}}^{{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+\frac {45}{4}}+3\,x^3\,{\mathrm {e}}^{{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+x+\frac {45}{4}}\right )+240\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+2\,x+\frac {45}{2}}+320\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+3\,x+\frac {45}{2}}+240\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+4\,x+\frac {45}{2}}+96\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+5\,x+\frac {45}{2}}+16\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+6\,x+\frac {45}{2}}+16\,x^6\,{\mathrm {e}}^{2\,{\ln \left (x\,{\mathrm {e}}^3+x\,{\mathrm {e}}^3\,{\mathrm {e}}^x\right )}^2+\frac {45}{2}} \] Input:
int((exp(3*log(x*exp(3) + x*exp(3)*exp(x)) + log(x*exp(3) + x*exp(3)*exp(x ))^2 + 9/4)*(log(x*exp(3) + x*exp(3)*exp(x))*(exp(x)*(48*x + 48) - log(x)* (exp(x)*(16*x + 16) + 16) + 48) - log(x)*(exp(x)*(24*x + 24) + 24) + exp(x )*(72*x + 64) + 64) - 6*exp(x) + exp(6*log(x*exp(3) + x*exp(3)*exp(x)) + 2 *log(x*exp(3) + x*exp(3)*exp(x))^2 + 9/2)*(log(x*exp(3) + x*exp(3)*exp(x)) *(exp(x)*(64*x + 64) + 64) + exp(x)*(96*x + 96) + 96) + log(x)*(2*exp(x) + 2) - 6)/(x + x*exp(x)),x)
Output:
log(x)^2 - 6*log(x) + 96*x^6*exp(x + 2*log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/2) - (8*log(x) - 24)*(3*x^3*exp(2*x + log(x*exp(3) + x*exp(3)*exp(x))^ 2 + 45/4) + x^3*exp(3*x + log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/4) + x^3* exp(log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/4) + 3*x^3*exp(x + log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/4)) + 240*x^6*exp(2*x + 2*log(x*exp(3) + x*exp( 3)*exp(x))^2 + 45/2) + 320*x^6*exp(3*x + 2*log(x*exp(3) + x*exp(3)*exp(x)) ^2 + 45/2) + 240*x^6*exp(4*x + 2*log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/2) + 96*x^6*exp(5*x + 2*log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/2) + 16*x^6*e xp(6*x + 2*log(x*exp(3) + x*exp(3)*exp(x))^2 + 45/2) + 16*x^6*exp(2*log(x* exp(3) + x*exp(3)*exp(x))^2 + 45/2)
Time = 0.23 (sec) , antiderivative size = 496, normalized size of antiderivative = 17.10 \[ \int \frac {-6-6 e^x+\left (2+2 e^x\right ) \log (x)+e^{\frac {1}{2} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (96+e^x (96+96 x)+\left (64+e^x (64+64 x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )+e^{\frac {1}{4} \left (9+12 \log \left (e^3 x+e^{3+x} x\right )+4 \log ^2\left (e^3 x+e^{3+x} x\right )\right )} \left (64+e^x (64+72 x)+\left (-24+e^x (-24-24 x)\right ) \log (x)+\left (48+e^x (48+48 x)+\left (-16+e^x (-16-16 x)\right ) \log (x)\right ) \log \left (e^3 x+e^{3+x} x\right )\right )}{x+e^x x} \, dx=16 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+6 x +\frac {1}{2}} e^{22} x^{6}+96 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+5 x +\frac {1}{2}} e^{22} x^{6}+240 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+4 x +\frac {1}{2}} e^{22} x^{6}+320 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+3 x +\frac {1}{2}} e^{22} x^{6}+240 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+2 x +\frac {1}{2}} e^{22} x^{6}+96 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+x +\frac {1}{2}} e^{22} x^{6}+16 e^{2 \mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+\frac {1}{2}} e^{22} x^{6}-8 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+3 x +\frac {1}{4}} \mathrm {log}\left (x \right ) e^{11} x^{3}+24 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+3 x +\frac {1}{4}} e^{11} x^{3}-24 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+2 x +\frac {1}{4}} \mathrm {log}\left (x \right ) e^{11} x^{3}+72 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+2 x +\frac {1}{4}} e^{11} x^{3}-24 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+x +\frac {1}{4}} \mathrm {log}\left (x \right ) e^{11} x^{3}+72 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+x +\frac {1}{4}} e^{11} x^{3}-8 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+\frac {1}{4}} \mathrm {log}\left (x \right ) e^{11} x^{3}+24 e^{\mathrm {log}\left (e^{x} e^{3} x +e^{3} x \right )^{2}+\frac {1}{4}} e^{11} x^{3}+\mathrm {log}\left (x \right )^{2}-6 \,\mathrm {log}\left (x \right ) \] Input:
int(((((64*x+64)*exp(x)+64)*log(x*exp(3)*exp(x)+x*exp(3))+(96*x+96)*exp(x) +96)*exp(log(x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp(3))+9 /4)^2+((((-16*x-16)*exp(x)-16)*log(x)+(48*x+48)*exp(x)+48)*log(x*exp(3)*ex p(x)+x*exp(3))+((-24*x-24)*exp(x)-24)*log(x)+(72*x+64)*exp(x)+64)*exp(log( x*exp(3)*exp(x)+x*exp(3))^2+3*log(x*exp(3)*exp(x)+x*exp(3))+9/4)+(2*exp(x) +2)*log(x)-6*exp(x)-6)/(exp(x)*x+x),x)
Output:
16*e**((4*log(e**x*e**3*x + e**3*x)**2 + 12*x + 1)/2)*e**22*x**6 + 96*e**( (4*log(e**x*e**3*x + e**3*x)**2 + 10*x + 1)/2)*e**22*x**6 + 240*e**((4*log (e**x*e**3*x + e**3*x)**2 + 8*x + 1)/2)*e**22*x**6 + 320*e**((4*log(e**x*e **3*x + e**3*x)**2 + 6*x + 1)/2)*e**22*x**6 + 240*e**((4*log(e**x*e**3*x + e**3*x)**2 + 4*x + 1)/2)*e**22*x**6 + 96*e**((4*log(e**x*e**3*x + e**3*x) **2 + 2*x + 1)/2)*e**22*x**6 + 16*e**((4*log(e**x*e**3*x + e**3*x)**2 + 1) /2)*e**22*x**6 - 8*e**((4*log(e**x*e**3*x + e**3*x)**2 + 12*x + 1)/4)*log( x)*e**11*x**3 + 24*e**((4*log(e**x*e**3*x + e**3*x)**2 + 12*x + 1)/4)*e**1 1*x**3 - 24*e**((4*log(e**x*e**3*x + e**3*x)**2 + 8*x + 1)/4)*log(x)*e**11 *x**3 + 72*e**((4*log(e**x*e**3*x + e**3*x)**2 + 8*x + 1)/4)*e**11*x**3 - 24*e**((4*log(e**x*e**3*x + e**3*x)**2 + 4*x + 1)/4)*log(x)*e**11*x**3 + 7 2*e**((4*log(e**x*e**3*x + e**3*x)**2 + 4*x + 1)/4)*e**11*x**3 - 8*e**((4* log(e**x*e**3*x + e**3*x)**2 + 1)/4)*log(x)*e**11*x**3 + 24*e**((4*log(e** x*e**3*x + e**3*x)**2 + 1)/4)*e**11*x**3 + log(x)**2 - 6*log(x)