3.8.28 \(\int \frac {e^{\frac {-(\frac {e^{-x}}{x})^{e^{-e^{15/x}}}+x^3}{x}} (2 x^4 \log (\frac {e^{-x}}{x})+(\frac {e^{-x}}{x})^{e^{-e^{15/x}}} (x \log (\frac {e^{-x}}{x})+e^{-e^{15/x}} \log (\frac {e^{-x}}{x}) (x+x^2-15 e^{15/x} \log (\frac {e^{-x}}{x}))))}{x^3 \log (\frac {e^{-x}}{x})} \, dx\) [728]

3.8.28.1 Optimal result

Integrand size = 146, antiderivative size = 32 \begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \end {dmath*}

output

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

3.8.28.2 Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}+x^2} \end {dmath*}

input

Integrate[(E^((-(1/(E^x*x))^E^(-E^(15/x)) + x^3)/x)*(2*x^4*Log[1/(E^x*x)] 
+ (1/(E^x*x))^E^(-E^(15/x))*(x*Log[1/(E^x*x)] + (Log[1/(E^x*x)]*(x + x^2 - 
 15*E^(15/x)*Log[1/(E^x*x)]))/E^E^(15/x))))/(x^3*Log[1/(E^x*x)]),x]
 

output

E^(-((1/(E^x*x))^E^(-E^(15/x))/x) + x^2)
 

3.8.28.3 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.

\(\displaystyle \int \frac {e^{\frac {x^3-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (e^{-e^{15/x}} \left (x^2+x-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right ) \log \left (\frac {e^{-x}}{x}\right )+x \log \left (\frac {e^{-x}}{x}\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^{x^2-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (e^{-e^{15/x}} \left (x^2+x-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right ) \log \left (\frac {e^{-x}}{x}\right )+x \log \left (\frac {e^{-x}}{x}\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (2 e^{x^2-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}} x+\frac {e^{x^2-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}-e^{15/x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x^2+e^{e^{15/x}} x+x-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {e^{x^2-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2}dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}-e^{15/x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^2}dx+\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}-e^{15/x}+x^2} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}dx+2 \int e^{x^2-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}} xdx-15 \int \int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}-e^{15/x}+x^2+\frac {15}{x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3}dxdx-15 \int \frac {\int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}-e^{15/x}+x^2+\frac {15}{x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3}dx}{x}dx-15 \log \left (\frac {e^{-x}}{x}\right ) \int \frac {e^{-\frac {\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x}-e^{15/x}+x^2+\frac {15}{x}} \left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}}{x^3}dx\)

input

Int[(E^((-(1/(E^x*x))^E^(-E^(15/x)) + x^3)/x)*(2*x^4*Log[1/(E^x*x)] + (1/( 
E^x*x))^E^(-E^(15/x))*(x*Log[1/(E^x*x)] + (Log[1/(E^x*x)]*(x + x^2 - 15*E^ 
(15/x)*Log[1/(E^x*x)]))/E^E^(15/x))))/(x^3*Log[1/(E^x*x)]),x]
 

output

$Aborted
 

3.8.28.3.1 Defintions of rubi rules used

rule 2009

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

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] 
]
 

3.8.28.4 Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.54 (sec) , antiderivative size = 130, normalized size of antiderivative = 4.06

\[{\mathrm e}^{\frac {-{\mathrm e}^{-\frac {\left (i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{3}-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{-x}}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{-x}\right )+2 \ln \left (x \right )+2 \ln \left ({\mathrm e}^{x}\right )\right ) {\mathrm e}^{-{\mathrm e}^{\frac {15}{x}}}}{2}}+x^{3}}{x}}\]

input

int((((-15*exp(15/x)*ln(1/exp(x)/x)+x^2+x)*exp(ln(ln(1/exp(x)/x))-exp(15/x 
))+x*ln(1/exp(x)/x))*exp(exp(ln(ln(1/exp(x)/x))-exp(15/x)))+2*x^4*ln(1/exp 
(x)/x))*exp((-exp(exp(ln(ln(1/exp(x)/x))-exp(15/x)))+x^3)/x)/x^3/ln(1/exp( 
x)/x),x)
 

output

exp((-exp(-1/2*(I*Pi*csgn(I*exp(-x)/x)^3-I*Pi*csgn(I*exp(-x)/x)^2*csgn(I/x 
)-I*Pi*csgn(I*exp(-x)/x)^2*csgn(I*exp(-x))+I*Pi*csgn(I*exp(-x)/x)*csgn(I/x 
)*csgn(I*exp(-x))+2*ln(x)+2*ln(exp(x)))*exp(-exp(15/x)))+x^3)/x)
 

3.8.28.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{\left (\frac {x^{3} - e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}}{x}\right )} \end {dmath*}

input

integrate((((-15*exp(15/x)*log(1/exp(x)/x)+x^2+x)*exp(log(log(1/exp(x)/x)) 
-exp(15/x))+x*log(1/exp(x)/x))*exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+2* 
x^4*log(1/exp(x)/x))*exp((-exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+x^3)/x 
)/x^3/log(1/exp(x)/x),x, algorithm=\
 

output

e^((x^3 - e^(e^(-e^(15/x) + log(log(e^(-x)/x)))))/x)
 

3.8.28.6 Sympy [F(-1)]

Timed out. \begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=\text {Timed out} \end {dmath*}

input

integrate((((-15*exp(15/x)*ln(1/exp(x)/x)+x**2+x)*exp(ln(ln(1/exp(x)/x))-e 
xp(15/x))+x*ln(1/exp(x)/x))*exp(exp(ln(ln(1/exp(x)/x))-exp(15/x)))+2*x**4* 
ln(1/exp(x)/x))*exp((-exp(exp(ln(ln(1/exp(x)/x))-exp(15/x)))+x**3)/x)/x**3 
/ln(1/exp(x)/x),x)
 

output

Timed out
 

3.8.28.7 Maxima [A] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=e^{\left (x^{2} - \frac {e^{\left (-x e^{\left (-e^{\frac {15}{x}}\right )} - e^{\left (-e^{\frac {15}{x}}\right )} \log \left (x\right )\right )}}{x}\right )} \end {dmath*}

input

integrate((((-15*exp(15/x)*log(1/exp(x)/x)+x^2+x)*exp(log(log(1/exp(x)/x)) 
-exp(15/x))+x*log(1/exp(x)/x))*exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+2* 
x^4*log(1/exp(x)/x))*exp((-exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+x^3)/x 
)/x^3/log(1/exp(x)/x),x, algorithm=\
 

output

e^(x^2 - e^(-x*e^(-e^(15/x)) - e^(-e^(15/x))*log(x))/x)
 

3.8.28.8 Giac [F]

\begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx=\int { \frac {{\left (2 \, x^{4} \log \left (\frac {e^{\left (-x\right )}}{x}\right ) + {\left ({\left (x^{2} - 15 \, e^{\frac {15}{x}} \log \left (\frac {e^{\left (-x\right )}}{x}\right ) + x\right )} e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )} + x \log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )} e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}\right )} e^{\left (\frac {x^{3} - e^{\left (e^{\left (-e^{\frac {15}{x}} + \log \left (\log \left (\frac {e^{\left (-x\right )}}{x}\right )\right )\right )}\right )}}{x}\right )}}{x^{3} \log \left (\frac {e^{\left (-x\right )}}{x}\right )} \,d x } \end {dmath*}

input

integrate((((-15*exp(15/x)*log(1/exp(x)/x)+x^2+x)*exp(log(log(1/exp(x)/x)) 
-exp(15/x))+x*log(1/exp(x)/x))*exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+2* 
x^4*log(1/exp(x)/x))*exp((-exp(exp(log(log(1/exp(x)/x))-exp(15/x)))+x^3)/x 
)/x^3/log(1/exp(x)/x),x, algorithm=\
 

output

integrate((2*x^4*log(e^(-x)/x) + ((x^2 - 15*e^(15/x)*log(e^(-x)/x) + x)*e^ 
(-e^(15/x) + log(log(e^(-x)/x))) + x*log(e^(-x)/x))*e^(e^(-e^(15/x) + log( 
log(e^(-x)/x)))))*e^((x^3 - e^(e^(-e^(15/x) + log(log(e^(-x)/x)))))/x)/(x^ 
3*log(e^(-x)/x)), x)
 

3.8.28.9 Mupad [B] (verification not implemented)

Time = 15.53 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \begin {dmath*} \int \frac {e^{\frac {-\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}}+x^3}{x}} \left (2 x^4 \log \left (\frac {e^{-x}}{x}\right )+\left (\frac {e^{-x}}{x}\right )^{e^{-e^{15/x}}} \left (x \log \left (\frac {e^{-x}}{x}\right )+e^{-e^{15/x}} \log \left (\frac {e^{-x}}{x}\right ) \left (x+x^2-15 e^{15/x} \log \left (\frac {e^{-x}}{x}\right )\right )\right )\right )}{x^3 \log \left (\frac {e^{-x}}{x}\right )} \, dx={\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{-x\,{\mathrm {e}}^{-{\mathrm {e}}^{15/x}}}\,{\left (\frac {1}{x}\right )}^{{\mathrm {e}}^{-{\mathrm {e}}^{15/x}}}}{x}} \end {dmath*}

input

int((exp(-(exp(exp(log(log(exp(-x)/x)) - exp(15/x))) - x^3)/x)*(2*x^4*log( 
exp(-x)/x) + exp(exp(log(log(exp(-x)/x)) - exp(15/x)))*(x*log(exp(-x)/x) + 
 exp(log(log(exp(-x)/x)) - exp(15/x))*(x + x^2 - 15*exp(15/x)*log(exp(-x)/ 
x)))))/(x^3*log(exp(-x)/x)),x)
 

output

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