\(\int \frac {(5+e^{(7-x) \log ^2(x)} (-1+(14-2 x) \log (x)-x \log ^2(x))) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx\) [7788]

Optimal result

Integrand size = 68, antiderivative size = 26 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=-1+\frac {x \log (\log (\log (5)))}{5-e^{(7-x) \log ^2(x)}} \]

[Out]

x*ln(ln(ln(5)))/(5-exp((-x+7)*ln(x)^2))-1

Rubi [F]

\[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=\int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx \]

[In]

Int[((5 + E^((7 - x)*Log[x]^2)*(-1 + (14 - 2*x)*Log[x] - x*Log[x]^2))*Log[Log[Log[5]]])/(25 - 10*E^((7 - x)*Lo
g[x]^2) + E^(2*(7 - x)*Log[x]^2)),x]

[Out]

(x*Log[Log[Log[5]]])/5 - ((14*Log[x] - 2*x*Log[x] - x*Log[x]^2)*Log[Log[Log[5]]])/(E^((7 - x)*Log[x]^2)*((2*(7
- x)*Log[x])/x - Log[x]^2)) + Log[Log[Log[5]]]*Defer[Int][E^((-7 + x)*Log[x]^2), x] + (Log[Log[Log[5]]]*Defer
[Int][(-1 + 5*E^((-7 + x)*Log[x]^2))^(-1), x])/5 + 70*Log[Log[Log[5]]]*Defer[Int][(E^(2*(-7 + x)*Log[x]^2)*Log
[x])/(-1 + 5*E^((-7 + x)*Log[x]^2))^2, x] - 70*Log[Log[Log[5]]]*Defer[Int][(E^(2*(-7 + x)*Log[x]^2)*Log[x])/(-
1 + 5*E^((-7 + x)*Log[x]^2)), x] - 10*Log[Log[Log[5]]]*Defer[Int][(E^(2*(-7 + x)*Log[x]^2)*x*Log[x])/(-1 + 5*E
^((-7 + x)*Log[x]^2))^2, x] + 10*Log[Log[Log[5]]]*Defer[Int][(E^(2*(-7 + x)*Log[x]^2)*x*Log[x])/(-1 + 5*E^((-7
+ x)*Log[x]^2)), x] - 5*Log[Log[Log[5]]]*Defer[Int][(E^(2*(-7 + x)*Log[x]^2)*x*Log[x]^2)/(-1 + 5*E^((-7 + x)*
Log[x]^2))^2, x] + 5*Log[Log[Log[5]]]*Defer[Int][(E^(2*(-7 + x)*Log[x]^2)*x*Log[x]^2)/(-1 + 5*E^((-7 + x)*Log[
x]^2)), x]

Rubi steps \begin{align*} \text {integral}& = \log (\log (\log (5))) \int \frac {5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx \\ & = \log (\log (\log (5))) \int \frac {e^{2 (-7+x) \log ^2(x)} \left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right )}{\left (1-5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx \\ & = \log (\log (\log (5))) \int \left (-\frac {5 e^{2 (-7+x) \log ^2(x)} \log (x) (-14+2 x+x \log (x))}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2}-e^{(-7+x) \log ^2(x)} \left (1-14 \log (x)+2 x \log (x)+x \log ^2(x)\right )+\frac {5 e^{2 (-7+x) \log ^2(x)} \left (1-14 \log (x)+2 x \log (x)+x \log ^2(x)\right )}{-1+5 e^{(-7+x) \log ^2(x)}}\right ) \, dx \\ & = -\left (\log (\log (\log (5))) \int e^{(-7+x) \log ^2(x)} \left (1-14 \log (x)+2 x \log (x)+x \log ^2(x)\right ) \, dx\right )-(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x) (-14+2 x+x \log (x))}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \left (1-14 \log (x)+2 x \log (x)+x \log ^2(x)\right )}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx \\ & = -\frac {e^{-\left ((7-x) \log ^2(x)\right )} \left (14 \log (x)-2 x \log (x)-x \log ^2(x)\right ) \log (\log (\log (5)))}{\frac {2 (7-x) \log (x)}{x}-\log ^2(x)}-(5 \log (\log (\log (5)))) \int \left (-\frac {14 e^{2 (-7+x) \log ^2(x)} \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2}+\frac {2 e^{2 (-7+x) \log ^2(x)} x \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2}+\frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2}\right ) \, dx+(5 \log (\log (\log (5)))) \int \left (\frac {e^{2 (-7+x) \log ^2(x)}}{-1+5 e^{(-7+x) \log ^2(x)}}-\frac {14 e^{2 (-7+x) \log ^2(x)} \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}}+\frac {2 e^{2 (-7+x) \log ^2(x)} x \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}}+\frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{-1+5 e^{(-7+x) \log ^2(x)}}\right ) \, dx \\ & = -\frac {e^{-\left ((7-x) \log ^2(x)\right )} \left (14 \log (x)-2 x \log (x)-x \log ^2(x)\right ) \log (\log (\log (5)))}{\frac {2 (7-x) \log (x)}{x}-\log ^2(x)}+(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)}}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx-(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx-(10 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(10 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx+(70 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx-(70 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx \\ & = -\frac {e^{-\left ((7-x) \log ^2(x)\right )} \left (14 \log (x)-2 x \log (x)-x \log ^2(x)\right ) \log (\log (\log (5)))}{\frac {2 (7-x) \log (x)}{x}-\log ^2(x)}+(5 \log (\log (\log (5)))) \int \left (\frac {1}{25}+\frac {1}{5} e^{(-7+x) \log ^2(x)}+\frac {1}{25 \left (-1+5 e^{(-7+x) \log ^2(x)}\right )}\right ) \, dx-(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx-(10 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(10 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx+(70 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx-(70 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx \\ & = \frac {1}{5} x \log (\log (\log (5)))-\frac {e^{-\left ((7-x) \log ^2(x)\right )} \left (14 \log (x)-2 x \log (x)-x \log ^2(x)\right ) \log (\log (\log (5)))}{\frac {2 (7-x) \log (x)}{x}-\log ^2(x)}+\frac {1}{5} \log (\log (\log (5))) \int \frac {1}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx+\log (\log (\log (5))) \int e^{(-7+x) \log ^2(x)} \, dx-(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(5 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log ^2(x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx-(10 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx+(10 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} x \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx+(70 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x)}{\left (-1+5 e^{(-7+x) \log ^2(x)}\right )^2} \, dx-(70 \log (\log (\log (5)))) \int \frac {e^{2 (-7+x) \log ^2(x)} \log (x)}{-1+5 e^{(-7+x) \log ^2(x)}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=\frac {1}{5} \left (1+\frac {1}{-1+5 e^{(-7+x) \log ^2(x)}}\right ) x \log (\log (\log (5))) \]

[In]

Integrate[((5 + E^((7 - x)*Log[x]^2)*(-1 + (14 - 2*x)*Log[x] - x*Log[x]^2))*Log[Log[Log[5]]])/(25 - 10*E^((7 -
x)*Log[x]^2) + E^(2*(7 - x)*Log[x]^2)),x]

[Out]

((1 + (-1 + 5*E^((-7 + x)*Log[x]^2))^(-1))*x*Log[Log[Log[5]]])/5

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85

method	result	size
risch	\(-\frac {\ln \left (\ln \left (\ln \left (5\right )\right )\right ) x}{{\mathrm e}^{-\left (-7+x \right ) \ln \left (x \right )^{2}}-5}\)	\(22\)
norman	\(-\frac {\ln \left (\ln \left (\ln \left (5\right )\right )\right ) x}{{\mathrm e}^{\left (-x +7\right ) \ln \left (x \right )^{2}}-5}\)	\(23\)
parallelrisch	\(-\frac {\ln \left (\ln \left (\ln \left (5\right )\right )\right ) x}{{\mathrm e}^{\left (-x +7\right ) \ln \left (x \right )^{2}}-5}\)	\(23\)

[In]

int(((-x*ln(x)^2+(-2*x+14)*ln(x)-1)*exp((-x+7)*ln(x)^2)+5)*ln(ln(ln(5)))/(exp((-x+7)*ln(x)^2)^2-10*exp((-x+7)*
ln(x)^2)+25),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(ln(5)))*x/(exp(-(-7+x)*ln(x)^2)-5)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=-\frac {x \log \left (\log \left (\log \left (5\right )\right )\right )}{e^{\left (-{\left (x - 7\right )} \log \left (x\right )^{2}\right )} - 5} \]

[In]

integrate(((-x*log(x)^2+(-2*x+14)*log(x)-1)*exp((-x+7)*log(x)^2)+5)*log(log(log(5)))/(exp((-x+7)*log(x)^2)^2-1
0*exp((-x+7)*log(x)^2)+25),x, algorithm="fricas")

[Out]

-x*log(log(log(5)))/(e^(-(x - 7)*log(x)^2) - 5)

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=- \frac {x \log {\left (\log {\left (\log {\left (5 \right )} \right )} \right )}}{e^{\left (7 - x\right ) \log {\left (x \right )}^{2}} - 5} \]

[In]

integrate(((-x*ln(x)**2+(-2*x+14)*ln(x)-1)*exp((-x+7)*ln(x)**2)+5)*ln(ln(ln(5)))/(exp((-x+7)*ln(x)**2)**2-10*e
xp((-x+7)*ln(x)**2)+25),x)

[Out]

-x*log(log(log(5)))/(exp((7 - x)*log(x)**2) - 5)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=\frac {x e^{\left (x \log \left (x\right )^{2}\right )} \log \left (\log \left (\log \left (5\right )\right )\right )}{5 \, e^{\left (x \log \left (x\right )^{2}\right )} - e^{\left (7 \, \log \left (x\right )^{2}\right )}} \]

[In]

integrate(((-x*log(x)^2+(-2*x+14)*log(x)-1)*exp((-x+7)*log(x)^2)+5)*log(log(log(5)))/(exp((-x+7)*log(x)^2)^2-1
0*exp((-x+7)*log(x)^2)+25),x, algorithm="maxima")

[Out]

x*e^(x*log(x)^2)*log(log(log(5)))/(5*e^(x*log(x)^2) - e^(7*log(x)^2))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=-\frac {x \log \left (\log \left (\log \left (5\right )\right )\right )}{e^{\left (-x \log \left (x\right )^{2} + 7 \, \log \left (x\right )^{2}\right )} - 5} \]

[In]

integrate(((-x*log(x)^2+(-2*x+14)*log(x)-1)*exp((-x+7)*log(x)^2)+5)*log(log(log(5)))/(exp((-x+7)*log(x)^2)^2-1
0*exp((-x+7)*log(x)^2)+25),x, algorithm="giac")

[Out]

-x*log(log(log(5)))/(e^(-x*log(x)^2 + 7*log(x)^2) - 5)

Mupad [B] (verification not implemented)

Time = 13.75 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {\left (5+e^{(7-x) \log ^2(x)} \left (-1+(14-2 x) \log (x)-x \log ^2(x)\right )\right ) \log (\log (\log (5)))}{25-10 e^{(7-x) \log ^2(x)}+e^{2 (7-x) \log ^2(x)}} \, dx=-\frac {x\,\ln \left (\ln \left (\ln \left (5\right )\right )\right )}{{\mathrm {e}}^{7\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{-x\,{\ln \left (x\right )}^2}-5} \]

[In]

int(-(log(log(log(5)))*(exp(-log(x)^2*(x - 7))*(x*log(x)^2 + log(x)*(2*x - 14) + 1) - 5))/(exp(-2*log(x)^2*(x
- 7)) - 10*exp(-log(x)^2*(x - 7)) + 25),x)

[Out]

-(x*log(log(log(5))))/(exp(7*log(x)^2)*exp(-x*log(x)^2) - 5)