\(\int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} (3600-2280 x+361 x^2+(1140 x-361 x^2) \log (x)+200 \log ^2(x)+200 \log ^3(x))}{100 \log ^2(x)} \, dx\) [2418]

Optimal result

Integrand size = 62, antiderivative size = 24 \[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=-5+2 e^{-\frac {\left (3-\frac {19 x}{20}\right )^2}{\log ^2(x)}} x \log (x) \] Output:

2*ln(x)*x/exp((3-19/20*x)^2/ln(x)^2)-5
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=2 e^{-\frac {(60-19 x)^2}{400 \log ^2(x)}} x \log (x) \] Input:

Integrate[(3600 - 2280*x + 361*x^2 + (1140*x - 361*x^2)*Log[x] + 200*Log[x 
]^2 + 200*Log[x]^3)/(100*E^((3600 - 2280*x + 361*x^2)/(400*Log[x]^2))*Log[ 
x]^2),x]
 

Output:

(2*x*Log[x])/E^((60 - 19*x)^2/(400*Log[x]^2))
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(81\) vs. \(2(24)=48\).

Time = 0.39 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.38, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {27, 2726}

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.

\(\displaystyle \int \frac {e^{-\frac {361 x^2-2280 x+3600}{400 \log ^2(x)}} \left (361 x^2+\left (1140 x-361 x^2\right ) \log (x)-2280 x+200 \log ^3(x)+200 \log ^2(x)+3600\right )}{100 \log ^2(x)} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{100} \int \frac {e^{-\frac {361 x^2-2280 x+3600}{400 \log ^2(x)}} \left (200 \log ^3(x)+200 \log ^2(x)+19 \left (60 x-19 x^2\right ) \log (x)+361 x^2-2280 x+3600\right )}{\log ^2(x)}dx\)

\(\Big \downarrow \) 2726

\(\displaystyle \frac {2 e^{-\frac {361 x^2-2280 x+3600}{400 \log ^2(x)}} \left (361 x^2+19 \left (60 x-19 x^2\right ) \log (x)-2280 x+3600\right )}{\left (\frac {361 x^2-2280 x+3600}{x \log ^3(x)}+\frac {19 (60-19 x)}{\log ^2(x)}\right ) \log ^2(x)}\)

Input:

Int[(3600 - 2280*x + 361*x^2 + (1140*x - 361*x^2)*Log[x] + 200*Log[x]^2 + 
200*Log[x]^3)/(100*E^((3600 - 2280*x + 361*x^2)/(400*Log[x]^2))*Log[x]^2), 
x]
 

Output:

(2*(3600 - 2280*x + 361*x^2 + 19*(60*x - 19*x^2)*Log[x]))/(E^((3600 - 2280 
*x + 361*x^2)/(400*Log[x]^2))*((3600 - 2280*x + 361*x^2)/(x*Log[x]^3) + (1 
9*(60 - 19*x))/Log[x]^2)*Log[x]^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2726

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, 
 x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

method	result	size
risch	\(2 \ln \left (x \right ) x \,{\mathrm e}^{-\frac {\left (19 x -60\right )^{2}}{400 \ln \left (x \right )^{2}}}\)	\(20\)
parallelrisch	\(2 \ln \left (x \right ) x \,{\mathrm e}^{-\frac {361 x^{2}-2280 x +3600}{400 \ln \left (x \right )^{2}}}\)	\(25\)

Input:

int(1/100*(200*ln(x)^3+200*ln(x)^2+(-361*x^2+1140*x)*ln(x)+361*x^2-2280*x+ 
3600)/ln(x)^2/exp(1/400*(361*x^2-2280*x+3600)/ln(x)^2),x,method=_RETURNVER 
BOSE)
 

Output:

2*ln(x)*x*exp(-1/400*(19*x-60)^2/ln(x)^2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=2 \, x e^{\left (-\frac {361 \, x^{2} - 2280 \, x + 3600}{400 \, \log \left (x\right )^{2}}\right )} \log \left (x\right ) \] Input:

integrate(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^ 
2-2280*x+3600)/log(x)^2/exp(1/400*(361*x^2-2280*x+3600)/log(x)^2),x, algor 
ithm="fricas")
 

Output:

2*x*e^(-1/400*(361*x^2 - 2280*x + 3600)/log(x)^2)*log(x)
 

Sympy [A] (verification not implemented)

Time = 1.62 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=2 x e^{- \frac {\frac {361 x^{2}}{400} - \frac {57 x}{10} + 9}{\log {\left (x \right )}^{2}}} \log {\left (x \right )} \] Input:

integrate(1/100*(200*ln(x)**3+200*ln(x)**2+(-361*x**2+1140*x)*ln(x)+361*x* 
*2-2280*x+3600)/ln(x)**2/exp(1/400*(361*x**2-2280*x+3600)/ln(x)**2),x)
 

Output:

2*x*exp(-(361*x**2/400 - 57*x/10 + 9)/log(x)**2)*log(x)
 

Maxima [F]

\[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=\int { \frac {{\left (200 \, \log \left (x\right )^{3} + 361 \, x^{2} - 19 \, {\left (19 \, x^{2} - 60 \, x\right )} \log \left (x\right ) + 200 \, \log \left (x\right )^{2} - 2280 \, x + 3600\right )} e^{\left (-\frac {361 \, x^{2} - 2280 \, x + 3600}{400 \, \log \left (x\right )^{2}}\right )}}{100 \, \log \left (x\right )^{2}} \,d x } \] Input:

integrate(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^ 
2-2280*x+3600)/log(x)^2/exp(1/400*(361*x^2-2280*x+3600)/log(x)^2),x, algor 
ithm="maxima")
 

Output:

1/100*integrate((200*log(x)^3 + 361*x^2 - 19*(19*x^2 - 60*x)*log(x) + 200* 
log(x)^2 - 2280*x + 3600)*e^(-1/400*(361*x^2 - 2280*x + 3600)/log(x)^2)/lo 
g(x)^2, x)
 

Giac [F]

\[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=\int { \frac {{\left (200 \, \log \left (x\right )^{3} + 361 \, x^{2} - 19 \, {\left (19 \, x^{2} - 60 \, x\right )} \log \left (x\right ) + 200 \, \log \left (x\right )^{2} - 2280 \, x + 3600\right )} e^{\left (-\frac {361 \, x^{2} - 2280 \, x + 3600}{400 \, \log \left (x\right )^{2}}\right )}}{100 \, \log \left (x\right )^{2}} \,d x } \] Input:

integrate(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^ 
2-2280*x+3600)/log(x)^2/exp(1/400*(361*x^2-2280*x+3600)/log(x)^2),x, algor 
ithm="giac")
 

Output:

undef
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 4.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=2\,x\,{\mathrm {e}}^{-\frac {361\,x^2-2280\,x+3600}{400\,{\ln \left (x\right )}^2}}\,\ln \left (x\right ) \] Input:

int((exp(-((361*x^2)/400 - (57*x)/10 + 9)/log(x)^2)*(2*log(x)^2 - (114*x)/ 
5 + 2*log(x)^3 + (log(x)*(1140*x - 361*x^2))/100 + (361*x^2)/100 + 36))/lo 
g(x)^2,x)
 

Output:

2*x*exp(-(361*x^2 - 2280*x + 3600)/(400*log(x)^2))*log(x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {e^{-\frac {3600-2280 x+361 x^2}{400 \log ^2(x)}} \left (3600-2280 x+361 x^2+\left (1140 x-361 x^2\right ) \log (x)+200 \log ^2(x)+200 \log ^3(x)\right )}{100 \log ^2(x)} \, dx=\frac {2 e^{\frac {57 x}{10 \mathrm {log}\left (x \right )^{2}}} \mathrm {log}\left (x \right ) x}{e^{\frac {361 x^{2}+3600}{400 \mathrm {log}\left (x \right )^{2}}}} \] Input:

int(1/100*(200*log(x)^3+200*log(x)^2+(-361*x^2+1140*x)*log(x)+361*x^2-2280 
*x+3600)/log(x)^2/exp(1/400*(361*x^2-2280*x+3600)/log(x)^2),x)
 

Output:

(2*e**((57*x)/(10*log(x)**2))*log(x)*x)/e**((361*x**2 + 3600)/(400*log(x)* 
*2))