\(\int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} (-5+(-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} (5-5 x-250 x^2-750 x^3-500 x^4)) \log (3 x)+(5-10 e^{2+x+25 x^2+50 x^3+25 x^4}) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x)))}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx\) [478]

Optimal result

Integrand size = 260, antiderivative size = 32 \[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=e^{5 \left (-x+\frac {x}{e^{2+x+25 \left (x+x^2\right )^2}+\log (\log (3 x))}\right )} \] Output:

exp(5*x/(exp(x+2+5*(x^2+x)*(5*x^2+5*x))+ln(ln(3*x)))-5*x)
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(32)=64\).

Time = 1.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.34 \[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=e^{-5 x-\frac {5 \left (-1+e^{2+x+25 x^2+50 x^3+25 x^4}\right ) x}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \log ^{\frac {5 x}{\log (\log (3 x))}-\frac {5 x}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}}(3 x) \] Input:

Integrate[(E^((5*x - 5*E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4)*x - 5*x*Log[Lo 
g[3*x]])/(E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4) + Log[Log[3*x]]))*(-5 + (-5 
*E^(4 + 2*x + 50*x^2 + 100*x^3 + 50*x^4) + E^(2 + x + 25*x^2 + 50*x^3 + 25 
*x^4)*(5 - 5*x - 250*x^2 - 750*x^3 - 500*x^4))*Log[3*x] + (5 - 10*E^(2 + x 
 + 25*x^2 + 50*x^3 + 25*x^4))*Log[3*x]*Log[Log[3*x]] - 5*Log[3*x]*Log[Log[ 
3*x]]^2))/(E^(4 + 2*x + 50*x^2 + 100*x^3 + 50*x^4)*Log[3*x] + 2*E^(2 + x + 
 25*x^2 + 50*x^3 + 25*x^4)*Log[3*x]*Log[Log[3*x]] + Log[3*x]*Log[Log[3*x]] 
^2),x]
 

Output:

E^(-5*x - (5*(-1 + E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4))*x)/(E^(2 + x + 25 
*x^2 + 50*x^3 + 25*x^4) + Log[Log[3*x]]))*Log[3*x]^((5*x)/Log[Log[3*x]] - 
(5*x)/(E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4) + Log[Log[3*x]]))
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (\left (5-10 e^{25 x^4+50 x^3+25 x^2+x+2}\right ) \log (3 x) \log (\log (3 x))+\left (e^{25 x^4+50 x^3+25 x^2+x+2} \left (-500 x^4-750 x^3-250 x^2-5 x+5\right )-5 e^{50 x^4+100 x^3+50 x^2+2 x+4}\right ) \log (3 x)-5 \log (3 x) \log ^2(\log (3 x))-5\right ) \exp \left (\frac {-5 e^{25 x^4+50 x^3+25 x^2+x+2} x+5 x-5 x \log (\log (3 x))}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )}{2 e^{25 x^4+50 x^3+25 x^2+x+2} \log (3 x) \log (\log (3 x))+e^{50 x^4+100 x^3+50 x^2+2 x+4} \log (3 x)+\log (3 x) \log ^2(\log (3 x))} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (\left (5-10 e^{25 x^4+50 x^3+25 x^2+x+2}\right ) \log (3 x) \log (\log (3 x))+\left (e^{25 x^4+50 x^3+25 x^2+x+2} \left (-500 x^4-750 x^3-250 x^2-5 x+5\right )-5 e^{50 x^4+100 x^3+50 x^2+2 x+4}\right ) \log (3 x)-5 \log (3 x) \log ^2(\log (3 x))-5\right ) \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )}{\log (3 x) \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {5 \left (100 x^4+150 x^3+50 x^2+x-1\right ) \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}-5 \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )+\frac {5 \left (100 x^4 \log (3 x) \log (\log (3 x))+150 x^3 \log (3 x) \log (\log (3 x))+50 x^2 \log (3 x) \log (\log (3 x))+x \log (3 x) \log (\log (3 x))-1\right ) \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )}{\log (3 x) \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))\right )^2}\right )dx\)

\(\Big \downarrow \) 7299

\(\displaystyle \int \left (-\frac {5 \left (100 x^4+150 x^3+50 x^2+x-1\right ) \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}-5 \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )+\frac {5 \left (100 x^4 \log (3 x) \log (\log (3 x))+150 x^3 \log (3 x) \log (\log (3 x))+50 x^2 \log (3 x) \log (\log (3 x))+x \log (3 x) \log (\log (3 x))-1\right ) \exp \left (-\frac {5 x \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))-1\right )}{e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))}\right )}{\log (3 x) \left (e^{25 x^4+50 x^3+25 x^2+x+2}+\log (\log (3 x))\right )^2}\right )dx\)

Input:

Int[(E^((5*x - 5*E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4)*x - 5*x*Log[Log[3*x] 
])/(E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4) + Log[Log[3*x]]))*(-5 + (-5*E^(4 
+ 2*x + 50*x^2 + 100*x^3 + 50*x^4) + E^(2 + x + 25*x^2 + 50*x^3 + 25*x^4)* 
(5 - 5*x - 250*x^2 - 750*x^3 - 500*x^4))*Log[3*x] + (5 - 10*E^(2 + x + 25* 
x^2 + 50*x^3 + 25*x^4))*Log[3*x]*Log[Log[3*x]] - 5*Log[3*x]*Log[Log[3*x]]^ 
2))/(E^(4 + 2*x + 50*x^2 + 100*x^3 + 50*x^4)*Log[3*x] + 2*E^(2 + x + 25*x^ 
2 + 50*x^3 + 25*x^4)*Log[3*x]*Log[Log[3*x]] + Log[3*x]*Log[Log[3*x]]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.81

\[{\mathrm e}^{-\frac {5 x \left (\ln \left (\ln \left (3 x \right )\right )+{\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}-1\right )}{\ln \left (\ln \left (3 x \right )\right )+{\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}}}\]

Input:

int((-5*ln(3*x)*ln(ln(3*x))^2+(-10*exp(25*x^4+50*x^3+25*x^2+x+2)+5)*ln(3*x 
)*ln(ln(3*x))+(-5*exp(25*x^4+50*x^3+25*x^2+x+2)^2+(-500*x^4-750*x^3-250*x^ 
2-5*x+5)*exp(25*x^4+50*x^3+25*x^2+x+2))*ln(3*x)-5)*exp((-5*x*ln(ln(3*x))-5 
*x*exp(25*x^4+50*x^3+25*x^2+x+2)+5*x)/(ln(ln(3*x))+exp(25*x^4+50*x^3+25*x^ 
2+x+2)))/(ln(3*x)*ln(ln(3*x))^2+2*exp(25*x^4+50*x^3+25*x^2+x+2)*ln(3*x)*ln 
(ln(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)^2*ln(3*x)),x)
 

Output:

exp(-5*x*(ln(ln(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)-1)/(ln(ln(3*x))+exp(25 
*x^4+50*x^3+25*x^2+x+2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (29) = 58\).

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.94 \[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=e^{\left (-\frac {5 \, {\left (x e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + x \log \left (\log \left (3 \, x\right )\right ) - x\right )}}{e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + \log \left (\log \left (3 \, x\right )\right )}\right )} \] Input:

integrate((-5*log(3*x)*log(log(3*x))^2+(-10*exp(25*x^4+50*x^3+25*x^2+x+2)+ 
5)*log(3*x)*log(log(3*x))+(-5*exp(25*x^4+50*x^3+25*x^2+x+2)^2+(-500*x^4-75 
0*x^3-250*x^2-5*x+5)*exp(25*x^4+50*x^3+25*x^2+x+2))*log(3*x)-5)*exp((-5*x* 
log(log(3*x))-5*x*exp(25*x^4+50*x^3+25*x^2+x+2)+5*x)/(log(log(3*x))+exp(25 
*x^4+50*x^3+25*x^2+x+2)))/(log(3*x)*log(log(3*x))^2+2*exp(25*x^4+50*x^3+25 
*x^2+x+2)*log(3*x)*log(log(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)^2*log(3*x)) 
,x, algorithm="fricas")
 

Output:

e^(-5*(x*e^(25*x^4 + 50*x^3 + 25*x^2 + x + 2) + x*log(log(3*x)) - x)/(e^(2 
5*x^4 + 50*x^3 + 25*x^2 + x + 2) + log(log(3*x))))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).

Time = 18.80 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.03 \[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=e^{\frac {- 5 x e^{25 x^{4} + 50 x^{3} + 25 x^{2} + x + 2} - 5 x \log {\left (\log {\left (3 x \right )} \right )} + 5 x}{e^{25 x^{4} + 50 x^{3} + 25 x^{2} + x + 2} + \log {\left (\log {\left (3 x \right )} \right )}}} \] Input:

integrate((-5*ln(3*x)*ln(ln(3*x))**2+(-10*exp(25*x**4+50*x**3+25*x**2+x+2) 
+5)*ln(3*x)*ln(ln(3*x))+(-5*exp(25*x**4+50*x**3+25*x**2+x+2)**2+(-500*x**4 
-750*x**3-250*x**2-5*x+5)*exp(25*x**4+50*x**3+25*x**2+x+2))*ln(3*x)-5)*exp 
((-5*x*ln(ln(3*x))-5*x*exp(25*x**4+50*x**3+25*x**2+x+2)+5*x)/(ln(ln(3*x))+ 
exp(25*x**4+50*x**3+25*x**2+x+2)))/(ln(3*x)*ln(ln(3*x))**2+2*exp(25*x**4+5 
0*x**3+25*x**2+x+2)*ln(3*x)*ln(ln(3*x))+exp(25*x**4+50*x**3+25*x**2+x+2)** 
2*ln(3*x)),x)
 

Output:

exp((-5*x*exp(25*x**4 + 50*x**3 + 25*x**2 + x + 2) - 5*x*log(log(3*x)) + 5 
*x)/(exp(25*x**4 + 50*x**3 + 25*x**2 + x + 2) + log(log(3*x))))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (29) = 58\).

Time = 0.42 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.75 \[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=e^{\left (-\frac {5 \, x e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )}}{e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + \log \left (\log \left (3\right ) + \log \left (x\right )\right )} - \frac {5 \, x \log \left (\log \left (3\right ) + \log \left (x\right )\right )}{e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + \log \left (\log \left (3\right ) + \log \left (x\right )\right )} + \frac {5 \, x}{e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + \log \left (\log \left (3\right ) + \log \left (x\right )\right )}\right )} \] Input:

integrate((-5*log(3*x)*log(log(3*x))^2+(-10*exp(25*x^4+50*x^3+25*x^2+x+2)+ 
5)*log(3*x)*log(log(3*x))+(-5*exp(25*x^4+50*x^3+25*x^2+x+2)^2+(-500*x^4-75 
0*x^3-250*x^2-5*x+5)*exp(25*x^4+50*x^3+25*x^2+x+2))*log(3*x)-5)*exp((-5*x* 
log(log(3*x))-5*x*exp(25*x^4+50*x^3+25*x^2+x+2)+5*x)/(log(log(3*x))+exp(25 
*x^4+50*x^3+25*x^2+x+2)))/(log(3*x)*log(log(3*x))^2+2*exp(25*x^4+50*x^3+25 
*x^2+x+2)*log(3*x)*log(log(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)^2*log(3*x)) 
,x, algorithm="maxima")
 

Output:

e^(-5*x*e^(25*x^4 + 50*x^3 + 25*x^2 + x + 2)/(e^(25*x^4 + 50*x^3 + 25*x^2 
+ x + 2) + log(log(3) + log(x))) - 5*x*log(log(3) + log(x))/(e^(25*x^4 + 5 
0*x^3 + 25*x^2 + x + 2) + log(log(3) + log(x))) + 5*x/(e^(25*x^4 + 50*x^3 
+ 25*x^2 + x + 2) + log(log(3) + log(x))))
 

Giac [F]

\[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=\int { -\frac {5 \, {\left ({\left (2 \, e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} - 1\right )} \log \left (3 \, x\right ) \log \left (\log \left (3 \, x\right )\right ) + \log \left (3 \, x\right ) \log \left (\log \left (3 \, x\right )\right )^{2} + {\left ({\left (100 \, x^{4} + 150 \, x^{3} + 50 \, x^{2} + x - 1\right )} e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + e^{\left (50 \, x^{4} + 100 \, x^{3} + 50 \, x^{2} + 2 \, x + 4\right )}\right )} \log \left (3 \, x\right ) + 1\right )} e^{\left (-\frac {5 \, {\left (x e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + x \log \left (\log \left (3 \, x\right )\right ) - x\right )}}{e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} + \log \left (\log \left (3 \, x\right )\right )}\right )}}{2 \, e^{\left (25 \, x^{4} + 50 \, x^{3} + 25 \, x^{2} + x + 2\right )} \log \left (3 \, x\right ) \log \left (\log \left (3 \, x\right )\right ) + \log \left (3 \, x\right ) \log \left (\log \left (3 \, x\right )\right )^{2} + e^{\left (50 \, x^{4} + 100 \, x^{3} + 50 \, x^{2} + 2 \, x + 4\right )} \log \left (3 \, x\right )} \,d x } \] Input:

integrate((-5*log(3*x)*log(log(3*x))^2+(-10*exp(25*x^4+50*x^3+25*x^2+x+2)+ 
5)*log(3*x)*log(log(3*x))+(-5*exp(25*x^4+50*x^3+25*x^2+x+2)^2+(-500*x^4-75 
0*x^3-250*x^2-5*x+5)*exp(25*x^4+50*x^3+25*x^2+x+2))*log(3*x)-5)*exp((-5*x* 
log(log(3*x))-5*x*exp(25*x^4+50*x^3+25*x^2+x+2)+5*x)/(log(log(3*x))+exp(25 
*x^4+50*x^3+25*x^2+x+2)))/(log(3*x)*log(log(3*x))^2+2*exp(25*x^4+50*x^3+25 
*x^2+x+2)*log(3*x)*log(log(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)^2*log(3*x)) 
,x, algorithm="giac")
 

Output:

undef
 

Mupad [B] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 138, normalized size of antiderivative = 4.31 \[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=\frac {{\mathrm {e}}^{\frac {5\,x}{\ln \left (\ln \left (3\right )+\ln \left (x\right )\right )+{\mathrm {e}}^2\,{\mathrm {e}}^{25\,x^2}\,{\mathrm {e}}^{25\,x^4}\,{\mathrm {e}}^{50\,x^3}\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {5\,x\,{\mathrm {e}}^2\,{\mathrm {e}}^{25\,x^2}\,{\mathrm {e}}^{25\,x^4}\,{\mathrm {e}}^{50\,x^3}\,{\mathrm {e}}^x}{\ln \left (\ln \left (3\right )+\ln \left (x\right )\right )+{\mathrm {e}}^2\,{\mathrm {e}}^{25\,x^2}\,{\mathrm {e}}^{25\,x^4}\,{\mathrm {e}}^{50\,x^3}\,{\mathrm {e}}^x}}}{{\left (\ln \left (3\right )+\ln \left (x\right )\right )}^{\frac {5\,x}{\ln \left (\ln \left (3\right )+\ln \left (x\right )\right )+{\mathrm {e}}^2\,{\mathrm {e}}^{25\,x^2}\,{\mathrm {e}}^{25\,x^4}\,{\mathrm {e}}^{50\,x^3}\,{\mathrm {e}}^x}}} \] Input:

int(-(exp(-(5*x*log(log(3*x)) - 5*x + 5*x*exp(x + 25*x^2 + 50*x^3 + 25*x^4 
 + 2))/(log(log(3*x)) + exp(x + 25*x^2 + 50*x^3 + 25*x^4 + 2)))*(5*log(3*x 
)*log(log(3*x))^2 + log(3*x)*(5*exp(2*x + 50*x^2 + 100*x^3 + 50*x^4 + 4) + 
 exp(x + 25*x^2 + 50*x^3 + 25*x^4 + 2)*(5*x + 250*x^2 + 750*x^3 + 500*x^4 
- 5)) + log(3*x)*log(log(3*x))*(10*exp(x + 25*x^2 + 50*x^3 + 25*x^4 + 2) - 
 5) + 5))/(log(3*x)*log(log(3*x))^2 + log(3*x)*exp(2*x + 50*x^2 + 100*x^3 
+ 50*x^4 + 4) + 2*log(3*x)*exp(x + 25*x^2 + 50*x^3 + 25*x^4 + 2)*log(log(3 
*x))),x)
 

Output:

(exp((5*x)/(log(log(3) + log(x)) + exp(2)*exp(25*x^2)*exp(25*x^4)*exp(50*x 
^3)*exp(x)))*exp(-(5*x*exp(2)*exp(25*x^2)*exp(25*x^4)*exp(50*x^3)*exp(x))/ 
(log(log(3) + log(x)) + exp(2)*exp(25*x^2)*exp(25*x^4)*exp(50*x^3)*exp(x)) 
))/(log(3) + log(x))^((5*x)/(log(log(3) + log(x)) + exp(2)*exp(25*x^2)*exp 
(25*x^4)*exp(50*x^3)*exp(x)))
 

Reduce [F]

\[ \int \frac {e^{\frac {5 x-5 e^{2+x+25 x^2+50 x^3+25 x^4} x-5 x \log (\log (3 x))}{e^{2+x+25 x^2+50 x^3+25 x^4}+\log (\log (3 x))}} \left (-5+\left (-5 e^{4+2 x+50 x^2+100 x^3+50 x^4}+e^{2+x+25 x^2+50 x^3+25 x^4} \left (5-5 x-250 x^2-750 x^3-500 x^4\right )\right ) \log (3 x)+\left (5-10 e^{2+x+25 x^2+50 x^3+25 x^4}\right ) \log (3 x) \log (\log (3 x))-5 \log (3 x) \log ^2(\log (3 x))\right )}{e^{4+2 x+50 x^2+100 x^3+50 x^4} \log (3 x)+2 e^{2+x+25 x^2+50 x^3+25 x^4} \log (3 x) \log (\log (3 x))+\log (3 x) \log ^2(\log (3 x))} \, dx=\int \frac {\left (-5 \,\mathrm {log}\left (3 x \right ) \mathrm {log}\left (\mathrm {log}\left (3 x \right )\right )^{2}+\left (-10 \,{\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}+5\right ) \mathrm {log}\left (3 x \right ) \mathrm {log}\left (\mathrm {log}\left (3 x \right )\right )+\left (-5 \left ({\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}\right )^{2}+\left (-500 x^{4}-750 x^{3}-250 x^{2}-5 x +5\right ) {\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}\right ) \mathrm {log}\left (3 x \right )-5\right ) {\mathrm e}^{\frac {-5 x \,\mathrm {log}\left (\mathrm {log}\left (3 x \right )\right )-5 x \,{\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}+5 x}{\mathrm {log}\left (\mathrm {log}\left (3 x \right )\right )+{\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}}}}{\mathrm {log}\left (3 x \right ) \mathrm {log}\left (\mathrm {log}\left (3 x \right )\right )^{2}+2 \,{\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2} \mathrm {log}\left (3 x \right ) \mathrm {log}\left (\mathrm {log}\left (3 x \right )\right )+\left ({\mathrm e}^{25 x^{4}+50 x^{3}+25 x^{2}+x +2}\right )^{2} \mathrm {log}\left (3 x \right )}d x \] Input:

int((-5*log(3*x)*log(log(3*x))^2+(-10*exp(25*x^4+50*x^3+25*x^2+x+2)+5)*log 
(3*x)*log(log(3*x))+(-5*exp(25*x^4+50*x^3+25*x^2+x+2)^2+(-500*x^4-750*x^3- 
250*x^2-5*x+5)*exp(25*x^4+50*x^3+25*x^2+x+2))*log(3*x)-5)*exp((-5*x*log(lo 
g(3*x))-5*x*exp(25*x^4+50*x^3+25*x^2+x+2)+5*x)/(log(log(3*x))+exp(25*x^4+5 
0*x^3+25*x^2+x+2)))/(log(3*x)*log(log(3*x))^2+2*exp(25*x^4+50*x^3+25*x^2+x 
+2)*log(3*x)*log(log(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)^2*log(3*x)),x)
 

Output:

int((-5*log(3*x)*log(log(3*x))^2+(-10*exp(25*x^4+50*x^3+25*x^2+x+2)+5)*log 
(3*x)*log(log(3*x))+(-5*exp(25*x^4+50*x^3+25*x^2+x+2)^2+(-500*x^4-750*x^3- 
250*x^2-5*x+5)*exp(25*x^4+50*x^3+25*x^2+x+2))*log(3*x)-5)*exp((-5*x*log(lo 
g(3*x))-5*x*exp(25*x^4+50*x^3+25*x^2+x+2)+5*x)/(log(log(3*x))+exp(25*x^4+5 
0*x^3+25*x^2+x+2)))/(log(3*x)*log(log(3*x))^2+2*exp(25*x^4+50*x^3+25*x^2+x 
+2)*log(3*x)*log(log(3*x))+exp(25*x^4+50*x^3+25*x^2+x+2)^2*log(3*x)),x)