\(\int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} (-3 e^5 x-6 x^2+(-3 e^5 x-6 x^2) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+(e^5 (2-3 x)+4 x-6 x^2-2 x^3) \log ^2(x))}{(e^{10} x^3+4 e^5 x^4+4 x^5) \log ^2(x)} \, dx\) [433]

Optimal result

Integrand size = 137, antiderivative size = 35 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {x+e^{-\frac {-3+\frac {1}{x}-\frac {3}{\log (x)}}{x}} x}{x \left (e^5+2 x\right )} \] Output:

(x+x/exp((1/x-3-3/ln(x))/x))/x/(2*x+exp(5))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {1+e^{\frac {3 x+(-1+3 x) \log (x)}{x^2 \log (x)}}}{e^5+2 x} \] Input:

Integrate[(-3*E^5*x - 6*x^2 + (-3*E^5*x - 6*x^2)*Log[x] - 2*E^((-3*x + (1 
- 3*x)*Log[x])/(x^2*Log[x]))*x^3*Log[x]^2 + (E^5*(2 - 3*x) + 4*x - 6*x^2 - 
 2*x^3)*Log[x]^2)/(E^((-3*x + (1 - 3*x)*Log[x])/(x^2*Log[x]))*(E^10*x^3 + 
4*E^5*x^4 + 4*x^5)*Log[x]^2),x]
 

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(35)=70\).

Time = 17.82 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2026, 2007, 7293, 7292, 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.

\(\displaystyle \int \frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-6 x^2+\left (-6 x^2-3 e^5 x\right ) \log (x)-2 x^3 e^{\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \log ^2(x)+\left (-2 x^3-6 x^2+4 x+e^5 (2-3 x)\right ) \log ^2(x)-3 e^5 x\right )}{\left (4 x^5+4 e^5 x^4+e^{10} x^3\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-6 x^2+\left (-6 x^2-3 e^5 x\right ) \log (x)-2 x^3 e^{\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \log ^2(x)+\left (-2 x^3-6 x^2+4 x+e^5 (2-3 x)\right ) \log ^2(x)-3 e^5 x\right )}{x^3 \left (4 x^2+4 e^5 x+e^{10}\right ) \log ^2(x)}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-6 x^2+\left (-6 x^2-3 e^5 x\right ) \log (x)-2 x^3 e^{\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \log ^2(x)+\left (-2 x^3-6 x^2+4 x+e^5 (2-3 x)\right ) \log ^2(x)-3 e^5 x\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-2 x^3 \log ^2(x)-6 x^2-6 x^2 \log ^2(x)-6 x^2 \log (x)-3 e^5 x+4 \left (1-\frac {3 e^5}{4}\right ) x \log ^2(x)+2 e^5 \log ^2(x)-3 e^5 x \log (x)\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x)}-\frac {2 \exp \left (\frac {-3 x-3 x \log (x)+\log (x)}{x^2 \log (x)}-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}\right )}{\left (2 x+e^5\right )^2}\right )dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \left (\frac {e^{-\frac {(1-3 x) \log (x)-3 x}{x^2 \log (x)}} \left (-2 x^3 \log ^2(x)-6 x^2-6 x^2 \log ^2(x)-6 x^2 \log (x)-3 e^5 x+4 \left (1-\frac {3 e^5}{4}\right ) x \log ^2(x)+2 e^5 \log ^2(x)-3 e^5 x \log (x)\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x)}-\frac {2}{\left (2 x+e^5\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2 x+e^5}-\frac {e^{\frac {3}{x \log (x)}-\frac {1-3 x}{x^2}} \left (6 x^2+6 x^2 \log ^2(x)+6 x^2 \log (x)+3 e^5 x-\left (4-3 e^5\right ) x \log ^2(x)-2 e^5 \log ^2(x)+3 e^5 x \log (x)\right )}{x^3 \left (2 x+e^5\right )^2 \log ^2(x) \left (-\frac {3 x-(1-3 x) \log (x)}{x^3 \log ^2(x)}-\frac {2 (3 x-(1-3 x) \log (x))}{x^3 \log (x)}+\frac {-\frac {1-3 x}{x}+3 \log (x)+3}{x^2 \log (x)}\right )}\)

Input:

Int[(-3*E^5*x - 6*x^2 + (-3*E^5*x - 6*x^2)*Log[x] - 2*E^((-3*x + (1 - 3*x) 
*Log[x])/(x^2*Log[x]))*x^3*Log[x]^2 + (E^5*(2 - 3*x) + 4*x - 6*x^2 - 2*x^3 
)*Log[x]^2)/(E^((-3*x + (1 - 3*x)*Log[x])/(x^2*Log[x]))*(E^10*x^3 + 4*E^5* 
x^4 + 4*x^5)*Log[x]^2),x]
 

Output:

(E^5 + 2*x)^(-1) - (E^(-((1 - 3*x)/x^2) + 3/(x*Log[x]))*(3*E^5*x + 6*x^2 + 
 3*E^5*x*Log[x] + 6*x^2*Log[x] - 2*E^5*Log[x]^2 - (4 - 3*E^5)*x*Log[x]^2 + 
 6*x^2*Log[x]^2))/(x^3*(E^5 + 2*x)^2*Log[x]^2*((3 - (1 - 3*x)/x + 3*Log[x] 
)/(x^2*Log[x]) - (3*x - (1 - 3*x)*Log[x])/(x^3*Log[x]^2) - (2*(3*x - (1 - 
3*x)*Log[x]))/(x^3*Log[x])))
 

Defintions of rubi rules used

rule 2007

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex 
pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol 
yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 2009

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

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

rule 7292

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

rule 7293

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

Maple [A] (verified)

Time = 10.59 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.17

method	result	size
risch	\(\frac {1}{2 x +{\mathrm e}^{5}}+\frac {{\mathrm e}^{\frac {3 x \ln \left (x \right )-\ln \left (x \right )+3 x}{\ln \left (x \right ) x^{2}}}}{2 x +{\mathrm e}^{5}}\)	\(41\)
parallelrisch	\(\frac {\left (2 x^{2} \ln \left (x \right ) {\mathrm e}^{\frac {\left (1-3 x \right ) \ln \left (x \right )-3 x}{x^{2} \ln \left (x \right )}}+2 x^{2} \ln \left (x \right )\right ) {\mathrm e}^{-\frac {\left (1-3 x \right ) \ln \left (x \right )-3 x}{x^{2} \ln \left (x \right )}}}{2 \ln \left (x \right ) x^{2} \left (2 x +{\mathrm e}^{5}\right )}\)	\(77\)

Input:

int((-2*x^3*ln(x)^2*exp(((1-3*x)*ln(x)-3*x)/x^2/ln(x))+((2-3*x)*exp(5)-2*x 
^3-6*x^2+4*x)*ln(x)^2+(-3*x*exp(5)-6*x^2)*ln(x)-3*x*exp(5)-6*x^2)/(x^3*exp 
(5)^2+4*x^4*exp(5)+4*x^5)/ln(x)^2/exp(((1-3*x)*ln(x)-3*x)/x^2/ln(x)),x,met 
hod=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {e^{\left (\frac {{\left (3 \, x - 1\right )} \log \left (x\right ) + 3 \, x}{x^{2} \log \left (x\right )}\right )} + 1}{2 \, x + e^{5}} \] Input:

integrate((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*e 
xp(5)-2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^ 
2)/(x^3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2 
/log(x)),x, algorithm="fricas")
 

Output:

(e^(((3*x - 1)*log(x) + 3*x)/(x^2*log(x))) + 1)/(2*x + e^5)
 

Sympy [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {2}{4 x + 2 e^{5}} + \frac {e^{- \frac {- 3 x + \left (1 - 3 x\right ) \log {\left (x \right )}}{x^{2} \log {\left (x \right )}}}}{2 x + e^{5}} \] Input:

integrate((-2*x**3*ln(x)**2*exp(((1-3*x)*ln(x)-3*x)/x**2/ln(x))+((2-3*x)*e 
xp(5)-2*x**3-6*x**2+4*x)*ln(x)**2+(-3*x*exp(5)-6*x**2)*ln(x)-3*x*exp(5)-6* 
x**2)/(x**3*exp(5)**2+4*x**4*exp(5)+4*x**5)/ln(x)**2/exp(((1-3*x)*ln(x)-3* 
x)/x**2/ln(x)),x)
 

Output:

2/(4*x + 2*exp(5)) + exp(-(-3*x + (1 - 3*x)*log(x))/(x**2*log(x)))/(2*x + 
exp(5))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.03 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {{\left (e^{\left (\frac {3}{x} + \frac {3}{x \log \left (x\right )}\right )} + e^{\left (\frac {1}{x^{2}}\right )}\right )} e^{\left (-\frac {1}{x^{2}}\right )}}{2 \, x + e^{5}} \] Input:

integrate((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*e 
xp(5)-2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^ 
2)/(x^3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2 
/log(x)),x, algorithm="maxima")
 

Output:

(e^(3/x + 3/(x*log(x))) + e^(x^(-2)))*e^(-1/x^2)/(2*x + e^5)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*e 
xp(5)-2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^ 
2)/(x^3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2 
/log(x)),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-13824,[0,25,0]%%%}+%%%{-48384,[0,24,1]%%%}+%%%{-72576,[0, 
23,2]%%%}
 

Mupad [B] (verification not implemented)

Time = 0.73 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\frac {1}{2\,x+{\mathrm {e}}^5}+\frac {{\mathrm {e}}^{-\frac {1}{x^2}}\,{\mathrm {e}}^{3/x}\,{\mathrm {e}}^{\frac {3}{x\,\ln \left (x\right )}}}{2\,x+{\mathrm {e}}^5} \] Input:

int(-(exp((3*x + log(x)*(3*x - 1))/(x^2*log(x)))*(log(x)*(3*x*exp(5) + 6*x 
^2) + 3*x*exp(5) + log(x)^2*(6*x^2 - 4*x + 2*x^3 + exp(5)*(3*x - 2)) + 6*x 
^2 + 2*x^3*exp(-(3*x + log(x)*(3*x - 1))/(x^2*log(x)))*log(x)^2))/(log(x)^ 
2*(4*x^4*exp(5) + x^3*exp(10) + 4*x^5)),x)
 

Output:

1/(2*x + exp(5)) + (exp(-1/x^2)*exp(3/x)*exp(3/(x*log(x))))/(2*x + exp(5))
 

Reduce [F]

\[ \int \frac {e^{-\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} \left (-3 e^5 x-6 x^2+\left (-3 e^5 x-6 x^2\right ) \log (x)-2 e^{\frac {-3 x+(1-3 x) \log (x)}{x^2 \log (x)}} x^3 \log ^2(x)+\left (e^5 (2-3 x)+4 x-6 x^2-2 x^3\right ) \log ^2(x)\right )}{\left (e^{10} x^3+4 e^5 x^4+4 x^5\right ) \log ^2(x)} \, dx=\int \frac {-2 x^{3} \mathrm {log}\left (x \right )^{2} {\mathrm e}^{\frac {\left (1-3 x \right ) \mathrm {log}\left (x \right )-3 x}{x^{2} \mathrm {log}\left (x \right )}}+\left (\left (2-3 x \right ) {\mathrm e}^{5}-2 x^{3}-6 x^{2}+4 x \right ) \mathrm {log}\left (x \right )^{2}+\left (-3 x \,{\mathrm e}^{5}-6 x^{2}\right ) \mathrm {log}\left (x \right )-3 x \,{\mathrm e}^{5}-6 x^{2}}{\left (x^{3} \left ({\mathrm e}^{5}\right )^{2}+4 x^{4} {\mathrm e}^{5}+4 x^{5}\right ) \mathrm {log}\left (x \right )^{2} {\mathrm e}^{\frac {\left (1-3 x \right ) \mathrm {log}\left (x \right )-3 x}{x^{2} \mathrm {log}\left (x \right )}}}d x \] Input:

int((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*exp(5)- 
2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^2)/(x^ 
3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2/log(x 
)),x)
 

Output:

int((-2*x^3*log(x)^2*exp(((1-3*x)*log(x)-3*x)/x^2/log(x))+((2-3*x)*exp(5)- 
2*x^3-6*x^2+4*x)*log(x)^2+(-3*x*exp(5)-6*x^2)*log(x)-3*x*exp(5)-6*x^2)/(x^ 
3*exp(5)^2+4*x^4*exp(5)+4*x^5)/log(x)^2/exp(((1-3*x)*log(x)-3*x)/x^2/log(x 
)),x)