\(\int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+(10+e^{12-180 x+675 x^2} (720 x-5400 x^2)+e^{4-60 x+225 x^2} (-20+240 x-1800 x^2)+e^{16-240 x+900 x^2} (-240 x+1800 x^2)+e^{8-120 x+450 x^2} (10-720 x+5400 x^2)) \log (x)+(e^{8-120 x+450 x^2} (1200 x-9000 x^2)+e^{4-60 x+225 x^2} (-1200 x+9000 x^2)) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx\) [144]

Optimal result

Integrand size = 217, antiderivative size = 30 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=\log \left (\frac {e^{\left (5-\frac {\left (-1+e^{(2-15 x)^2}\right )^2}{\log (x)}\right )^2}}{x}\right ) \] Output:

ln(exp((5-(exp((2-15*x)^2)-1)^2/ln(x))^2)/x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=\frac {\left (-1+e^{(2-15 x)^2}\right )^4-10 \left (-1+e^{(2-15 x)^2}\right )^2 \log (x)-\log ^3(x)}{\log ^2(x)} \] Input:

Integrate[(-2 + 8*E^(4 - 60*x + 225*x^2) - 12*E^(8 - 120*x + 450*x^2) + 8* 
E^(12 - 180*x + 675*x^2) - 2*E^(16 - 240*x + 900*x^2) + (10 + E^(12 - 180* 
x + 675*x^2)*(720*x - 5400*x^2) + E^(4 - 60*x + 225*x^2)*(-20 + 240*x - 18 
00*x^2) + E^(16 - 240*x + 900*x^2)*(-240*x + 1800*x^2) + E^(8 - 120*x + 45 
0*x^2)*(10 - 720*x + 5400*x^2))*Log[x] + (E^(8 - 120*x + 450*x^2)*(1200*x 
- 9000*x^2) + E^(4 - 60*x + 225*x^2)*(-1200*x + 9000*x^2))*Log[x]^2 - Log[ 
x]^3)/(x*Log[x]^3),x]
 

Output:

((-1 + E^(2 - 15*x)^2)^4 - 10*(-1 + E^(2 - 15*x)^2)^2*Log[x] - Log[x]^3)/L 
og[x]^2
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(207\) vs. \(2(30)=60\).

Time = 2.91 (sec) , antiderivative size = 207, normalized size of antiderivative = 6.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {7293, 2009}

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.

\(\displaystyle \int \frac {8 e^{225 x^2-60 x+4}-12 e^{450 x^2-120 x+8}+8 e^{675 x^2-180 x+12}-2 e^{900 x^2-240 x+16}+\left (e^{450 x^2-120 x+8} \left (1200 x-9000 x^2\right )+e^{225 x^2-60 x+4} \left (9000 x^2-1200 x\right )\right ) \log ^2(x)+\left (e^{675 x^2-180 x+12} \left (720 x-5400 x^2\right )+e^{225 x^2-60 x+4} \left (-1800 x^2+240 x-20\right )+e^{900 x^2-240 x+16} \left (1800 x^2-240 x\right )+e^{450 x^2-120 x+8} \left (5400 x^2-720 x+10\right )+10\right ) \log (x)-\log ^3(x)-2}{x \log ^3(x)} \, dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {8 e^{3 (2-15 x)^2} \left (675 x^2 \log (x)-90 x \log (x)-1\right )}{x \log ^3(x)}+\frac {2 e^{4 (2-15 x)^2} \left (900 x^2 \log (x)-120 x \log (x)-1\right )}{x \log ^3(x)}+\frac {4 e^{225 x^2-60 x+4} \left (2250 x^2 \log ^2(x)-450 x^2 \log (x)-300 x \log ^2(x)+60 x \log (x)-5 \log (x)+2\right )}{x \log ^3(x)}-\frac {2 e^{2 (2-15 x)^2} \left (4500 x^2 \log ^2(x)-2700 x^2 \log (x)-600 x \log ^2(x)+360 x \log (x)-5 \log (x)+6\right )}{x \log ^3(x)}+\frac {-\log ^3(x)+10 \log (x)-2}{x \log ^3(x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {4 e^{3 (2-15 x)^2} \left (2 x \log (x)-15 x^2 \log (x)\right )}{(2-15 x) x \log ^3(x)}+\frac {e^{4 (2-15 x)^2} \left (2 x \log (x)-15 x^2 \log (x)\right )}{(2-15 x) x \log ^3(x)}+\frac {2 e^{2 (2-15 x)^2} \left (75 x^2 \log ^2(x)-45 x^2 \log (x)-10 x \log ^2(x)+6 x \log (x)\right )}{(2-15 x) x \log ^3(x)}-\frac {4 e^{225 x^2-60 x+4} \left (75 x^2 \log ^2(x)-15 x^2 \log (x)-10 x \log ^2(x)+2 x \log (x)\right )}{(2-15 x) x \log ^3(x)}+\frac {1}{\log ^2(x)}-\log (x)-\frac {10}{\log (x)}\)

Input:

Int[(-2 + 8*E^(4 - 60*x + 225*x^2) - 12*E^(8 - 120*x + 450*x^2) + 8*E^(12 
- 180*x + 675*x^2) - 2*E^(16 - 240*x + 900*x^2) + (10 + E^(12 - 180*x + 67 
5*x^2)*(720*x - 5400*x^2) + E^(4 - 60*x + 225*x^2)*(-20 + 240*x - 1800*x^2 
) + E^(16 - 240*x + 900*x^2)*(-240*x + 1800*x^2) + E^(8 - 120*x + 450*x^2) 
*(10 - 720*x + 5400*x^2))*Log[x] + (E^(8 - 120*x + 450*x^2)*(1200*x - 9000 
*x^2) + E^(4 - 60*x + 225*x^2)*(-1200*x + 9000*x^2))*Log[x]^2 - Log[x]^3)/ 
(x*Log[x]^3),x]
 

Output:

Log[x]^(-2) - 10/Log[x] - Log[x] - (4*E^(3*(2 - 15*x)^2)*(2*x*Log[x] - 15* 
x^2*Log[x]))/((2 - 15*x)*x*Log[x]^3) + (E^(4*(2 - 15*x)^2)*(2*x*Log[x] - 1 
5*x^2*Log[x]))/((2 - 15*x)*x*Log[x]^3) + (2*E^(2*(2 - 15*x)^2)*(6*x*Log[x] 
 - 45*x^2*Log[x] - 10*x*Log[x]^2 + 75*x^2*Log[x]^2))/((2 - 15*x)*x*Log[x]^ 
3) - (4*E^(4 - 60*x + 225*x^2)*(2*x*Log[x] - 15*x^2*Log[x] - 10*x*Log[x]^2 
 + 75*x^2*Log[x]^2))/((2 - 15*x)*x*Log[x]^3)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(86\) vs. \(2(28)=56\).

Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.90

\[-\ln \left (x \right )+\frac {{\mathrm e}^{4 \left (15 x -2\right )^{2}}-4 \,{\mathrm e}^{3 \left (15 x -2\right )^{2}}-10 \ln \left (x \right ) {\mathrm e}^{2 \left (15 x -2\right )^{2}}+6 \,{\mathrm e}^{2 \left (15 x -2\right )^{2}}+20 \ln \left (x \right ) {\mathrm e}^{\left (15 x -2\right )^{2}}-4 \,{\mathrm e}^{\left (15 x -2\right )^{2}}-10 \ln \left (x \right )+1}{\ln \left (x \right )^{2}}\]

Input:

int((-ln(x)^3+((-9000*x^2+1200*x)*exp(225*x^2-60*x+4)^2+(9000*x^2-1200*x)* 
exp(225*x^2-60*x+4))*ln(x)^2+((1800*x^2-240*x)*exp(225*x^2-60*x+4)^4+(-540 
0*x^2+720*x)*exp(225*x^2-60*x+4)^3+(5400*x^2-720*x+10)*exp(225*x^2-60*x+4) 
^2+(-1800*x^2+240*x-20)*exp(225*x^2-60*x+4)+10)*ln(x)-2*exp(225*x^2-60*x+4 
)^4+8*exp(225*x^2-60*x+4)^3-12*exp(225*x^2-60*x+4)^2+8*exp(225*x^2-60*x+4) 
-2)/x/ln(x)^3,x)
 

Output:

-ln(x)+(exp(4*(15*x-2)^2)-4*exp(3*(15*x-2)^2)-10*ln(x)*exp(2*(15*x-2)^2)+6 
*exp(2*(15*x-2)^2)+20*ln(x)*exp((15*x-2)^2)-4*exp((15*x-2)^2)-10*ln(x)+1)/ 
ln(x)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 3.13 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=-\frac {\log \left (x\right )^{3} + 10 \, {\left (e^{\left (450 \, x^{2} - 120 \, x + 8\right )} - 2 \, e^{\left (225 \, x^{2} - 60 \, x + 4\right )} + 1\right )} \log \left (x\right ) - e^{\left (900 \, x^{2} - 240 \, x + 16\right )} + 4 \, e^{\left (675 \, x^{2} - 180 \, x + 12\right )} - 6 \, e^{\left (450 \, x^{2} - 120 \, x + 8\right )} + 4 \, e^{\left (225 \, x^{2} - 60 \, x + 4\right )} - 1}{\log \left (x\right )^{2}} \] Input:

integrate((-log(x)^3+((-9000*x^2+1200*x)*exp(225*x^2-60*x+4)^2+(9000*x^2-1 
200*x)*exp(225*x^2-60*x+4))*log(x)^2+((1800*x^2-240*x)*exp(225*x^2-60*x+4) 
^4+(-5400*x^2+720*x)*exp(225*x^2-60*x+4)^3+(5400*x^2-720*x+10)*exp(225*x^2 
-60*x+4)^2+(-1800*x^2+240*x-20)*exp(225*x^2-60*x+4)+10)*log(x)-2*exp(225*x 
^2-60*x+4)^4+8*exp(225*x^2-60*x+4)^3-12*exp(225*x^2-60*x+4)^2+8*exp(225*x^ 
2-60*x+4)-2)/x/log(x)^3,x, algorithm="fricas")
 

Output:

-(log(x)^3 + 10*(e^(450*x^2 - 120*x + 8) - 2*e^(225*x^2 - 60*x + 4) + 1)*l 
og(x) - e^(900*x^2 - 240*x + 16) + 4*e^(675*x^2 - 180*x + 12) - 6*e^(450*x 
^2 - 120*x + 8) + 4*e^(225*x^2 - 60*x + 4) - 1)/log(x)^2
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (22) = 44\).

Time = 0.22 (sec) , antiderivative size = 105, normalized size of antiderivative = 3.50 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=\frac {1 - 10 \log {\left (x \right )}}{\log {\left (x \right )}^{2}} + \frac {\left (- 10 \log {\left (x \right )}^{7} + 6 \log {\left (x \right )}^{6}\right ) e^{450 x^{2} - 120 x + 8} + \left (20 \log {\left (x \right )}^{7} - 4 \log {\left (x \right )}^{6}\right ) e^{225 x^{2} - 60 x + 4} - 4 e^{675 x^{2} - 180 x + 12} \log {\left (x \right )}^{6} + e^{900 x^{2} - 240 x + 16} \log {\left (x \right )}^{6}}{\log {\left (x \right )}^{8}} - \log {\left (x \right )} \] Input:

integrate((-ln(x)**3+((-9000*x**2+1200*x)*exp(225*x**2-60*x+4)**2+(9000*x* 
*2-1200*x)*exp(225*x**2-60*x+4))*ln(x)**2+((1800*x**2-240*x)*exp(225*x**2- 
60*x+4)**4+(-5400*x**2+720*x)*exp(225*x**2-60*x+4)**3+(5400*x**2-720*x+10) 
*exp(225*x**2-60*x+4)**2+(-1800*x**2+240*x-20)*exp(225*x**2-60*x+4)+10)*ln 
(x)-2*exp(225*x**2-60*x+4)**4+8*exp(225*x**2-60*x+4)**3-12*exp(225*x**2-60 
*x+4)**2+8*exp(225*x**2-60*x+4)-2)/x/ln(x)**3,x)
 

Output:

(1 - 10*log(x))/log(x)**2 + ((-10*log(x)**7 + 6*log(x)**6)*exp(450*x**2 - 
120*x + 8) + (20*log(x)**7 - 4*log(x)**6)*exp(225*x**2 - 60*x + 4) - 4*exp 
(675*x**2 - 180*x + 12)*log(x)**6 + exp(900*x**2 - 240*x + 16)*log(x)**6)/ 
log(x)**8 - log(x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (27) = 54\).

Time = 0.10 (sec) , antiderivative size = 97, normalized size of antiderivative = 3.23 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=-\frac {{\left (2 \, {\left (5 \, e^{8} \log \left (x\right ) - 3 \, e^{8}\right )} e^{\left (450 \, x^{2} + 120 \, x\right )} - 4 \, {\left (5 \, e^{4} \log \left (x\right ) - e^{4}\right )} e^{\left (225 \, x^{2} + 180 \, x\right )} + 10 \, e^{\left (240 \, x\right )} \log \left (x\right ) - e^{\left (900 \, x^{2} + 16\right )} + 4 \, e^{\left (675 \, x^{2} + 60 \, x + 12\right )}\right )} e^{\left (-240 \, x\right )}}{\log \left (x\right )^{2}} + \frac {1}{\log \left (x\right )^{2}} - \log \left (x\right ) \] Input:

integrate((-log(x)^3+((-9000*x^2+1200*x)*exp(225*x^2-60*x+4)^2+(9000*x^2-1 
200*x)*exp(225*x^2-60*x+4))*log(x)^2+((1800*x^2-240*x)*exp(225*x^2-60*x+4) 
^4+(-5400*x^2+720*x)*exp(225*x^2-60*x+4)^3+(5400*x^2-720*x+10)*exp(225*x^2 
-60*x+4)^2+(-1800*x^2+240*x-20)*exp(225*x^2-60*x+4)+10)*log(x)-2*exp(225*x 
^2-60*x+4)^4+8*exp(225*x^2-60*x+4)^3-12*exp(225*x^2-60*x+4)^2+8*exp(225*x^ 
2-60*x+4)-2)/x/log(x)^3,x, algorithm="maxima")
 

Output:

-(2*(5*e^8*log(x) - 3*e^8)*e^(450*x^2 + 120*x) - 4*(5*e^4*log(x) - e^4)*e^ 
(225*x^2 + 180*x) + 10*e^(240*x)*log(x) - e^(900*x^2 + 16) + 4*e^(675*x^2 
+ 60*x + 12))*e^(-240*x)/log(x)^2 + 1/log(x)^2 - log(x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (27) = 54\).

Time = 0.20 (sec) , antiderivative size = 98, normalized size of antiderivative = 3.27 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=-\frac {\log \left (x\right )^{3} + 10 \, e^{\left (450 \, x^{2} - 120 \, x + 8\right )} \log \left (x\right ) - 20 \, e^{\left (225 \, x^{2} - 60 \, x + 4\right )} \log \left (x\right ) - e^{\left (900 \, x^{2} - 240 \, x + 16\right )} + 4 \, e^{\left (675 \, x^{2} - 180 \, x + 12\right )} - 6 \, e^{\left (450 \, x^{2} - 120 \, x + 8\right )} + 4 \, e^{\left (225 \, x^{2} - 60 \, x + 4\right )} + 10 \, \log \left (x\right ) - 1}{\log \left (x\right )^{2}} \] Input:

integrate((-log(x)^3+((-9000*x^2+1200*x)*exp(225*x^2-60*x+4)^2+(9000*x^2-1 
200*x)*exp(225*x^2-60*x+4))*log(x)^2+((1800*x^2-240*x)*exp(225*x^2-60*x+4) 
^4+(-5400*x^2+720*x)*exp(225*x^2-60*x+4)^3+(5400*x^2-720*x+10)*exp(225*x^2 
-60*x+4)^2+(-1800*x^2+240*x-20)*exp(225*x^2-60*x+4)+10)*log(x)-2*exp(225*x 
^2-60*x+4)^4+8*exp(225*x^2-60*x+4)^3-12*exp(225*x^2-60*x+4)^2+8*exp(225*x^ 
2-60*x+4)-2)/x/log(x)^3,x, algorithm="giac")
 

Output:

-(log(x)^3 + 10*e^(450*x^2 - 120*x + 8)*log(x) - 20*e^(225*x^2 - 60*x + 4) 
*log(x) - e^(900*x^2 - 240*x + 16) + 4*e^(675*x^2 - 180*x + 12) - 6*e^(450 
*x^2 - 120*x + 8) + 4*e^(225*x^2 - 60*x + 4) + 10*log(x) - 1)/log(x)^2
 

Mupad [B] (verification not implemented)

Time = 3.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 3.37 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=\frac {{\mathrm {e}}^{900\,x^2-240\,x+16}}{{\ln \left (x\right )}^2}-\frac {5}{\ln \left (x\right )}-\frac {5\,\ln \left (x\right )-1}{{\ln \left (x\right )}^2}-\frac {4\,{\mathrm {e}}^{675\,x^2-180\,x+12}}{{\ln \left (x\right )}^2}-\ln \left (x\right )+\frac {{\mathrm {e}}^{225\,x^2-60\,x+4}\,\left (20\,\ln \left (x\right )-4\right )}{{\ln \left (x\right )}^2}-\frac {{\mathrm {e}}^{450\,x^2-120\,x+8}\,\left (10\,\ln \left (x\right )-6\right )}{{\ln \left (x\right )}^2} \] Input:

int(-(12*exp(450*x^2 - 120*x + 8) - 8*exp(225*x^2 - 60*x + 4) - 8*exp(675* 
x^2 - 180*x + 12) + 2*exp(900*x^2 - 240*x + 16) - log(x)*(exp(450*x^2 - 12 
0*x + 8)*(5400*x^2 - 720*x + 10) - exp(900*x^2 - 240*x + 16)*(240*x - 1800 
*x^2) - exp(225*x^2 - 60*x + 4)*(1800*x^2 - 240*x + 20) + exp(675*x^2 - 18 
0*x + 12)*(720*x - 5400*x^2) + 10) + log(x)^2*(exp(225*x^2 - 60*x + 4)*(12 
00*x - 9000*x^2) - exp(450*x^2 - 120*x + 8)*(1200*x - 9000*x^2)) + log(x)^ 
3 + 2)/(x*log(x)^3),x)
 

Output:

exp(900*x^2 - 240*x + 16)/log(x)^2 - 5/log(x) - (5*log(x) - 1)/log(x)^2 - 
(4*exp(675*x^2 - 180*x + 12))/log(x)^2 - log(x) + (exp(225*x^2 - 60*x + 4) 
*(20*log(x) - 4))/log(x)^2 - (exp(450*x^2 - 120*x + 8)*(10*log(x) - 6))/lo 
g(x)^2
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.43 \[ \int \frac {-2+8 e^{4-60 x+225 x^2}-12 e^{8-120 x+450 x^2}+8 e^{12-180 x+675 x^2}-2 e^{16-240 x+900 x^2}+\left (10+e^{12-180 x+675 x^2} \left (720 x-5400 x^2\right )+e^{4-60 x+225 x^2} \left (-20+240 x-1800 x^2\right )+e^{16-240 x+900 x^2} \left (-240 x+1800 x^2\right )+e^{8-120 x+450 x^2} \left (10-720 x+5400 x^2\right )\right ) \log (x)+\left (e^{8-120 x+450 x^2} \left (1200 x-9000 x^2\right )+e^{4-60 x+225 x^2} \left (-1200 x+9000 x^2\right )\right ) \log ^2(x)-\log ^3(x)}{x \log ^3(x)} \, dx=\frac {e^{900 x^{2}} e^{16}-4 e^{675 x^{2}+60 x} e^{12}-10 e^{450 x^{2}+120 x} \mathrm {log}\left (x \right ) e^{8}+6 e^{450 x^{2}+120 x} e^{8}+20 e^{225 x^{2}+180 x} \mathrm {log}\left (x \right ) e^{4}-4 e^{225 x^{2}+180 x} e^{4}-e^{240 x} \mathrm {log}\left (x \right )^{3}-10 e^{240 x} \mathrm {log}\left (x \right )+e^{240 x}}{e^{240 x} \mathrm {log}\left (x \right )^{2}} \] Input:

int((-log(x)^3+((-9000*x^2+1200*x)*exp(225*x^2-60*x+4)^2+(9000*x^2-1200*x) 
*exp(225*x^2-60*x+4))*log(x)^2+((1800*x^2-240*x)*exp(225*x^2-60*x+4)^4+(-5 
400*x^2+720*x)*exp(225*x^2-60*x+4)^3+(5400*x^2-720*x+10)*exp(225*x^2-60*x+ 
4)^2+(-1800*x^2+240*x-20)*exp(225*x^2-60*x+4)+10)*log(x)-2*exp(225*x^2-60* 
x+4)^4+8*exp(225*x^2-60*x+4)^3-12*exp(225*x^2-60*x+4)^2+8*exp(225*x^2-60*x 
+4)-2)/x/log(x)^3,x)
 

Output:

(e**(900*x**2)*e**16 - 4*e**(675*x**2 + 60*x)*e**12 - 10*e**(450*x**2 + 12 
0*x)*log(x)*e**8 + 6*e**(450*x**2 + 120*x)*e**8 + 20*e**(225*x**2 + 180*x) 
*log(x)*e**4 - 4*e**(225*x**2 + 180*x)*e**4 - e**(240*x)*log(x)**3 - 10*e* 
*(240*x)*log(x) + e**(240*x))/(e**(240*x)*log(x)**2)