\(\int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+(-240-120 e^4-15 e^8+5 x) \log (5)+(5 x^2+(-240-120 e^4-15 e^8) \log (5)) \log (x) \log (\log (x))}{(2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 (2304 x^2-48 x^3)+e^8 (864 x^2-6 x^3)+(-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 (-1728 x+12 x^2)+e^4 (-4608 x+96 x^2)) \log (5)+(2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2) \log ^2(5)) \log (x)} \, dx\) [1320]

Optimal result

Integrand size = 229, antiderivative size = 29 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=-\frac {5 x \log (\log (x))}{\left (3 \left (4+e^4\right )^2-x\right ) (-x+\log (5))} \] Output:

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

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=\frac {5 x \log (\log (x))}{\left (48+24 e^4+3 e^8-x\right ) (x-\log (5))} \] Input:

Integrate[(240*x + 120*E^4*x + 15*E^8*x - 5*x^2 + (-240 - 120*E^4 - 15*E^8 
 + 5*x)*Log[5] + (5*x^2 + (-240 - 120*E^4 - 15*E^8)*Log[5])*Log[x]*Log[Log 
[x]])/((2304*x^2 + 144*E^12*x^2 + 9*E^16*x^2 - 96*x^3 + x^4 + E^4*(2304*x^ 
2 - 48*x^3) + E^8*(864*x^2 - 6*x^3) + (-4608*x - 288*E^12*x - 18*E^16*x + 
192*x^2 - 2*x^3 + E^8*(-1728*x + 12*x^2) + E^4*(-4608*x + 96*x^2))*Log[5] 
+ (2304 + 144*E^12 + 9*E^16 + E^4*(2304 - 48*x) + E^8*(864 - 6*x) - 96*x + 
 x^2)*Log[5]^2)*Log[x]),x]
 

Output:

(5*x*Log[Log[x]])/((48 + 24*E^4 + 3*E^8 - x)*(x - Log[5]))
 

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.

\(\displaystyle \int \frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+15 e^8 x+120 e^4 x+240 x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (x^4-96 x^3+9 e^{16} x^2+144 e^{12} x^2+2304 x^2+\left (x^2-96 x+e^4 (2304-48 x)+e^8 (864-6 x)+9 e^{16}+144 e^{12}+2304\right ) \log ^2(5)+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-2 x^3+192 x^2+e^8 \left (12 x^2-1728 x\right )+e^4 \left (96 x^2-4608 x\right )-18 e^{16} x-288 e^{12} x-4608 x\right ) \log (5)\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+15 e^8 x+120 e^4 x+240 x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (x^4-96 x^3+\left (2304+144 e^{12}\right ) x^2+9 e^{16} x^2+\left (x^2-96 x+e^4 (2304-48 x)+e^8 (864-6 x)+9 e^{16}+144 e^{12}+2304\right ) \log ^2(5)+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-2 x^3+192 x^2+e^8 \left (12 x^2-1728 x\right )+e^4 \left (96 x^2-4608 x\right )-18 e^{16} x-288 e^{12} x-4608 x\right ) \log (5)\right ) \log (x)}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+15 e^8 x+120 e^4 x+240 x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (x^4-96 x^3+\left (2304+144 e^{12}+9 e^{16}\right ) x^2+\left (x^2-96 x+e^4 (2304-48 x)+e^8 (864-6 x)+9 e^{16}+144 e^{12}+2304\right ) \log ^2(5)+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-2 x^3+192 x^2+e^8 \left (12 x^2-1728 x\right )+e^4 \left (96 x^2-4608 x\right )-18 e^{16} x-288 e^{12} x-4608 x\right ) \log (5)\right ) \log (x)}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (240+120 e^4\right ) x+15 e^8 x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (x^4-96 x^3+\left (2304+144 e^{12}+9 e^{16}\right ) x^2+\left (x^2-96 x+e^4 (2304-48 x)+e^8 (864-6 x)+9 e^{16}+144 e^{12}+2304\right ) \log ^2(5)+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-2 x^3+192 x^2+e^8 \left (12 x^2-1728 x\right )+e^4 \left (96 x^2-4608 x\right )-18 e^{16} x-288 e^{12} x-4608 x\right ) \log (5)\right ) \log (x)}dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (240+120 e^4+15 e^8\right ) x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (x^4-96 x^3+\left (2304+144 e^{12}+9 e^{16}\right ) x^2+\left (x^2-96 x+e^4 (2304-48 x)+e^8 (864-6 x)+9 e^{16}+144 e^{12}+2304\right ) \log ^2(5)+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-2 x^3+192 x^2+e^8 \left (12 x^2-1728 x\right )+e^4 \left (96 x^2-4608 x\right )-18 e^{16} x-288 e^{12} x-4608 x\right ) \log (5)\right ) \log (x)}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (-\frac {2 \left (-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (240+120 e^4+15 e^8\right ) x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)\right )}{\left (48+24 e^4+3 e^8-\log (5)\right )^3 (\log (5)-x) \log (x)}+\frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (240+120 e^4+15 e^8\right ) x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (48+24 e^4+3 e^8-\log (5)\right )^2 (\log (5)-x)^2 \log (x)}+\frac {-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (240+120 e^4+15 e^8\right ) x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)}{\left (-x+3 e^8+24 e^4+48\right )^2 \left (48+24 e^4+3 e^8-\log (5)\right )^2 \log (x)}+\frac {2 \left (-5 x^2+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))+\left (240+120 e^4+15 e^8\right ) x+\left (5 x-15 e^8-120 e^4-240\right ) \log (5)\right )}{\left (-x+3 e^8+24 e^4+48\right ) \left (48+24 e^4+3 e^8-\log (5)\right )^3 \log (x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5 \int \frac {x^2}{\left (-x+3 e^8+24 e^4+48\right )^2 \log (x)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}-\frac {5 \int \frac {x^2}{(x-\log (5))^2 \log (x)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}-\frac {5 \log (5) \int \frac {1}{\left (-x+3 e^8+24 e^4+48\right ) \log (x)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}+\frac {15 \left (4+e^4\right )^2 \int \frac {x}{\left (-x+3 e^8+24 e^4+48\right )^2 \log (x)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}-\frac {5 \log (5) \int \frac {-x+3 e^8+24 e^4+48}{(x-\log (5))^2 \log (x)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}+\frac {15 \left (4+e^4\right )^2 \int \frac {x}{(x-\log (5))^2 \log (x)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}+\frac {15 \left (4+e^4\right )^2 \int \frac {\log (\log (x))}{\left (-x+3 e^8+24 e^4+48\right )^2}dx}{48+24 e^4+3 e^8-\log (5)}-\frac {5 \log (5) \int \frac {\log (\log (x))}{(x-\log (5))^2}dx}{48+24 e^4+3 e^8-\log (5)}+\frac {5 \log (25) \int \frac {\log (\log (x))}{x-\log (5)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}-\frac {10 \log (5) \int \frac {\log (\log (x))}{x-\log (5)}dx}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}-\frac {20 \log (5) \operatorname {LogIntegral}(x)}{\left (48+24 e^4+3 e^8-\log (5)\right )^3}-\frac {10 \operatorname {LogIntegral}(x)}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}+\frac {60 \left (4+e^4\right )^2 \operatorname {LogIntegral}(x)}{\left (48+24 e^4+3 e^8-\log (5)\right )^3}+\frac {10 x \log (5) \log (\log (x))}{\left (48+24 e^4+3 e^8-\log (5)\right )^3}+\frac {10 x \log (\log (x))}{\left (48+24 e^4+3 e^8-\log (5)\right )^2}-\frac {30 \left (4+e^4\right )^2 x \log (\log (x))}{\left (48+24 e^4+3 e^8-\log (5)\right )^3}\)

Input:

Int[(240*x + 120*E^4*x + 15*E^8*x - 5*x^2 + (-240 - 120*E^4 - 15*E^8 + 5*x 
)*Log[5] + (5*x^2 + (-240 - 120*E^4 - 15*E^8)*Log[5])*Log[x]*Log[Log[x]])/ 
((2304*x^2 + 144*E^12*x^2 + 9*E^16*x^2 - 96*x^3 + x^4 + E^4*(2304*x^2 - 48 
*x^3) + E^8*(864*x^2 - 6*x^3) + (-4608*x - 288*E^12*x - 18*E^16*x + 192*x^ 
2 - 2*x^3 + E^8*(-1728*x + 12*x^2) + E^4*(-4608*x + 96*x^2))*Log[5] + (230 
4 + 144*E^12 + 9*E^16 + E^4*(2304 - 48*x) + E^8*(864 - 6*x) - 96*x + x^2)* 
Log[5]^2)*Log[x]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 17.90 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62

method	result	size
risch	\(-\frac {5 x \ln \left (\ln \left (x \right )\right )}{3 \ln \left (5\right ) {\mathrm e}^{8}-3 x \,{\mathrm e}^{8}+24 \,{\mathrm e}^{4} \ln \left (5\right )-24 x \,{\mathrm e}^{4}-x \ln \left (5\right )+x^{2}+48 \ln \left (5\right )-48 x}\)	\(47\)
parallelrisch	\(-\frac {5 x \ln \left (\ln \left (x \right )\right )}{3 \ln \left (5\right ) {\mathrm e}^{8}-3 x \,{\mathrm e}^{8}+24 \,{\mathrm e}^{4} \ln \left (5\right )-24 x \,{\mathrm e}^{4}-x \ln \left (5\right )+x^{2}+48 \ln \left (5\right )-48 x}\)	\(51\)

Input:

int((((-15*exp(4)^2-120*exp(4)-240)*ln(5)+5*x^2)*ln(x)*ln(ln(x))+(-15*exp( 
4)^2-120*exp(4)+5*x-240)*ln(5)+15*x*exp(4)^2+120*x*exp(4)-5*x^2+240*x)/((9 
*exp(4)^4+144*exp(4)^3+(-6*x+864)*exp(4)^2+(-48*x+2304)*exp(4)+x^2-96*x+23 
04)*ln(5)^2+(-18*x*exp(4)^4-288*x*exp(4)^3+(12*x^2-1728*x)*exp(4)^2+(96*x^ 
2-4608*x)*exp(4)-2*x^3+192*x^2-4608*x)*ln(5)+9*x^2*exp(4)^4+144*x^2*exp(4) 
^3+(-6*x^3+864*x^2)*exp(4)^2+(-48*x^3+2304*x^2)*exp(4)+x^4-96*x^3+2304*x^2 
)/ln(x),x,method=_RETURNVERBOSE)
 

Output:

-5*x/(3*ln(5)*exp(8)-3*x*exp(8)+24*exp(4)*ln(5)-24*x*exp(4)-x*ln(5)+x^2+48 
*ln(5)-48*x)*ln(ln(x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=-\frac {5 \, x \log \left (\log \left (x\right )\right )}{x^{2} - 3 \, x e^{8} - 24 \, x e^{4} - {\left (x - 3 \, e^{8} - 24 \, e^{4} - 48\right )} \log \left (5\right ) - 48 \, x} \] Input:

integrate((((-15*exp(4)^2-120*exp(4)-240)*log(5)+5*x^2)*log(x)*log(log(x)) 
+(-15*exp(4)^2-120*exp(4)+5*x-240)*log(5)+15*x*exp(4)^2+120*x*exp(4)-5*x^2 
+240*x)/((9*exp(4)^4+144*exp(4)^3+(-6*x+864)*exp(4)^2+(-48*x+2304)*exp(4)+ 
x^2-96*x+2304)*log(5)^2+(-18*x*exp(4)^4-288*x*exp(4)^3+(12*x^2-1728*x)*exp 
(4)^2+(96*x^2-4608*x)*exp(4)-2*x^3+192*x^2-4608*x)*log(5)+9*x^2*exp(4)^4+1 
44*x^2*exp(4)^3+(-6*x^3+864*x^2)*exp(4)^2+(-48*x^3+2304*x^2)*exp(4)+x^4-96 
*x^3+2304*x^2)/log(x),x, algorithm="fricas")
 

Output:

-5*x*log(log(x))/(x^2 - 3*x*e^8 - 24*x*e^4 - (x - 3*e^8 - 24*e^4 - 48)*log 
(5) - 48*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).

Time = 0.40 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.93 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=- \frac {5 x \log {\left (\log {\left (x \right )} \right )}}{x^{2} - 3 x e^{8} - 24 x e^{4} - 48 x - x \log {\left (5 \right )} + 48 \log {\left (5 \right )} + 24 e^{4} \log {\left (5 \right )} + 3 e^{8} \log {\left (5 \right )}} \] Input:

integrate((((-15*exp(4)**2-120*exp(4)-240)*ln(5)+5*x**2)*ln(x)*ln(ln(x))+( 
-15*exp(4)**2-120*exp(4)+5*x-240)*ln(5)+15*x*exp(4)**2+120*x*exp(4)-5*x**2 
+240*x)/((9*exp(4)**4+144*exp(4)**3+(-6*x+864)*exp(4)**2+(-48*x+2304)*exp( 
4)+x**2-96*x+2304)*ln(5)**2+(-18*x*exp(4)**4-288*x*exp(4)**3+(12*x**2-1728 
*x)*exp(4)**2+(96*x**2-4608*x)*exp(4)-2*x**3+192*x**2-4608*x)*ln(5)+9*x**2 
*exp(4)**4+144*x**2*exp(4)**3+(-6*x**3+864*x**2)*exp(4)**2+(-48*x**3+2304* 
x**2)*exp(4)+x**4-96*x**3+2304*x**2)/ln(x),x)
 

Output:

-5*x*log(log(x))/(x**2 - 3*x*exp(8) - 24*x*exp(4) - 48*x - x*log(5) + 48*l 
og(5) + 24*exp(4)*log(5) + 3*exp(8)*log(5))
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.48 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=-\frac {5 \, x \log \left (\log \left (x\right )\right )}{x^{2} - x {\left (3 \, e^{8} + 24 \, e^{4} + \log \left (5\right ) + 48\right )} + 3 \, e^{8} \log \left (5\right ) + 24 \, e^{4} \log \left (5\right ) + 48 \, \log \left (5\right )} \] Input:

integrate((((-15*exp(4)^2-120*exp(4)-240)*log(5)+5*x^2)*log(x)*log(log(x)) 
+(-15*exp(4)^2-120*exp(4)+5*x-240)*log(5)+15*x*exp(4)^2+120*x*exp(4)-5*x^2 
+240*x)/((9*exp(4)^4+144*exp(4)^3+(-6*x+864)*exp(4)^2+(-48*x+2304)*exp(4)+ 
x^2-96*x+2304)*log(5)^2+(-18*x*exp(4)^4-288*x*exp(4)^3+(12*x^2-1728*x)*exp 
(4)^2+(96*x^2-4608*x)*exp(4)-2*x^3+192*x^2-4608*x)*log(5)+9*x^2*exp(4)^4+1 
44*x^2*exp(4)^3+(-6*x^3+864*x^2)*exp(4)^2+(-48*x^3+2304*x^2)*exp(4)+x^4-96 
*x^3+2304*x^2)/log(x),x, algorithm="maxima")
 

Output:

-5*x*log(log(x))/(x^2 - x*(3*e^8 + 24*e^4 + log(5) + 48) + 3*e^8*log(5) + 
24*e^4*log(5) + 48*log(5))
 

Giac [A] (verification not implemented)

Time = 5.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=-\frac {5 \, x \log \left (\log \left (x\right )\right )}{x^{2} - 3 \, x e^{8} - 24 \, x e^{4} - x \log \left (5\right ) + 3 \, e^{8} \log \left (5\right ) + 24 \, e^{4} \log \left (5\right ) - 48 \, x + 48 \, \log \left (5\right )} \] Input:

integrate((((-15*exp(4)^2-120*exp(4)-240)*log(5)+5*x^2)*log(x)*log(log(x)) 
+(-15*exp(4)^2-120*exp(4)+5*x-240)*log(5)+15*x*exp(4)^2+120*x*exp(4)-5*x^2 
+240*x)/((9*exp(4)^4+144*exp(4)^3+(-6*x+864)*exp(4)^2+(-48*x+2304)*exp(4)+ 
x^2-96*x+2304)*log(5)^2+(-18*x*exp(4)^4-288*x*exp(4)^3+(12*x^2-1728*x)*exp 
(4)^2+(96*x^2-4608*x)*exp(4)-2*x^3+192*x^2-4608*x)*log(5)+9*x^2*exp(4)^4+1 
44*x^2*exp(4)^3+(-6*x^3+864*x^2)*exp(4)^2+(-48*x^3+2304*x^2)*exp(4)+x^4-96 
*x^3+2304*x^2)/log(x),x, algorithm="giac")
 

Output:

-5*x*log(log(x))/(x^2 - 3*x*e^8 - 24*x*e^4 - x*log(5) + 3*e^8*log(5) + 24* 
e^4*log(5) - 48*x + 48*log(5))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=\int \frac {240\,x+120\,x\,{\mathrm {e}}^4+15\,x\,{\mathrm {e}}^8-\ln \left (5\right )\,\left (120\,{\mathrm {e}}^4-5\,x+15\,{\mathrm {e}}^8+240\right )-5\,x^2-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left (\ln \left (5\right )\,\left (120\,{\mathrm {e}}^4+15\,{\mathrm {e}}^8+240\right )-5\,x^2\right )}{\ln \left (x\right )\,\left ({\mathrm {e}}^8\,\left (864\,x^2-6\,x^3\right )-\ln \left (5\right )\,\left (4608\,x+{\mathrm {e}}^8\,\left (1728\,x-12\,x^2\right )+{\mathrm {e}}^4\,\left (4608\,x-96\,x^2\right )+288\,x\,{\mathrm {e}}^{12}+18\,x\,{\mathrm {e}}^{16}-192\,x^2+2\,x^3\right )+{\mathrm {e}}^4\,\left (2304\,x^2-48\,x^3\right )+144\,x^2\,{\mathrm {e}}^{12}+9\,x^2\,{\mathrm {e}}^{16}+{\ln \left (5\right )}^2\,\left (144\,{\mathrm {e}}^{12}-96\,x+9\,{\mathrm {e}}^{16}+x^2-{\mathrm {e}}^8\,\left (6\,x-864\right )-{\mathrm {e}}^4\,\left (48\,x-2304\right )+2304\right )+2304\,x^2-96\,x^3+x^4\right )} \,d x \] Input:

int((240*x + 120*x*exp(4) + 15*x*exp(8) - log(5)*(120*exp(4) - 5*x + 15*ex 
p(8) + 240) - 5*x^2 - log(log(x))*log(x)*(log(5)*(120*exp(4) + 15*exp(8) + 
 240) - 5*x^2))/(log(x)*(exp(8)*(864*x^2 - 6*x^3) - log(5)*(4608*x + exp(8 
)*(1728*x - 12*x^2) + exp(4)*(4608*x - 96*x^2) + 288*x*exp(12) + 18*x*exp( 
16) - 192*x^2 + 2*x^3) + exp(4)*(2304*x^2 - 48*x^3) + 144*x^2*exp(12) + 9* 
x^2*exp(16) + log(5)^2*(144*exp(12) - 96*x + 9*exp(16) + x^2 - exp(8)*(6*x 
 - 864) - exp(4)*(48*x - 2304) + 2304) + 2304*x^2 - 96*x^3 + x^4)),x)
 

Output:

int((240*x + 120*x*exp(4) + 15*x*exp(8) - log(5)*(120*exp(4) - 5*x + 15*ex 
p(8) + 240) - 5*x^2 - log(log(x))*log(x)*(log(5)*(120*exp(4) + 15*exp(8) + 
 240) - 5*x^2))/(log(x)*(exp(8)*(864*x^2 - 6*x^3) - log(5)*(4608*x + exp(8 
)*(1728*x - 12*x^2) + exp(4)*(4608*x - 96*x^2) + 288*x*exp(12) + 18*x*exp( 
16) - 192*x^2 + 2*x^3) + exp(4)*(2304*x^2 - 48*x^3) + 144*x^2*exp(12) + 9* 
x^2*exp(16) + log(5)^2*(144*exp(12) - 96*x + 9*exp(16) + x^2 - exp(8)*(6*x 
 - 864) - exp(4)*(48*x - 2304) + 2304) + 2304*x^2 - 96*x^3 + x^4)), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.72 \[ \int \frac {240 x+120 e^4 x+15 e^8 x-5 x^2+\left (-240-120 e^4-15 e^8+5 x\right ) \log (5)+\left (5 x^2+\left (-240-120 e^4-15 e^8\right ) \log (5)\right ) \log (x) \log (\log (x))}{\left (2304 x^2+144 e^{12} x^2+9 e^{16} x^2-96 x^3+x^4+e^4 \left (2304 x^2-48 x^3\right )+e^8 \left (864 x^2-6 x^3\right )+\left (-4608 x-288 e^{12} x-18 e^{16} x+192 x^2-2 x^3+e^8 \left (-1728 x+12 x^2\right )+e^4 \left (-4608 x+96 x^2\right )\right ) \log (5)+\left (2304+144 e^{12}+9 e^{16}+e^4 (2304-48 x)+e^8 (864-6 x)-96 x+x^2\right ) \log ^2(5)\right ) \log (x)} \, dx=-\frac {5 \,\mathrm {log}\left (\mathrm {log}\left (x \right )\right ) x}{3 \,\mathrm {log}\left (5\right ) e^{8}+24 \,\mathrm {log}\left (5\right ) e^{4}-\mathrm {log}\left (5\right ) x +48 \,\mathrm {log}\left (5\right )-3 e^{8} x -24 e^{4} x +x^{2}-48 x} \] Input:

int((((-15*exp(4)^2-120*exp(4)-240)*log(5)+5*x^2)*log(x)*log(log(x))+(-15* 
exp(4)^2-120*exp(4)+5*x-240)*log(5)+15*x*exp(4)^2+120*x*exp(4)-5*x^2+240*x 
)/((9*exp(4)^4+144*exp(4)^3+(-6*x+864)*exp(4)^2+(-48*x+2304)*exp(4)+x^2-96 
*x+2304)*log(5)^2+(-18*x*exp(4)^4-288*x*exp(4)^3+(12*x^2-1728*x)*exp(4)^2+ 
(96*x^2-4608*x)*exp(4)-2*x^3+192*x^2-4608*x)*log(5)+9*x^2*exp(4)^4+144*x^2 
*exp(4)^3+(-6*x^3+864*x^2)*exp(4)^2+(-48*x^3+2304*x^2)*exp(4)+x^4-96*x^3+2 
304*x^2)/log(x),x)
 

Output:

( - 5*log(log(x))*x)/(3*log(5)*e**8 + 24*log(5)*e**4 - log(5)*x + 48*log(5 
) - 3*e**8*x - 24*e**4*x + x**2 - 48*x)