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

Optimal. Leaf size=26 \[ -1+\frac {x \log (\log (\log (5)))}{5-e^{(7-x) \log ^2(x)}} \]

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Rubi [F]  time = 5.00, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 1.17, size = 27, normalized size = 1.04 \begin {gather*} \frac {1}{5} \left (1+\frac {1}{-1+5 e^{(-7+x) \log ^2(x)}}\right ) x \log (\log (\log (5))) \end {gather*}

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.50, size = 21, normalized size = 0.81 \begin {gather*} -\frac {x \log \left (\log \left (\log \relax (5)\right )\right )}{e^{\left (-{\left (x - 7\right )} \log \relax (x)^{2}\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.30, size = 26, normalized size = 1.00 \begin {gather*} -\frac {x \log \left (\log \left (\log \relax (5)\right )\right )}{e^{\left (-x \log \relax (x)^{2} + 7 \, \log \relax (x)^{2}\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.12, size = 22, normalized size = 0.85

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

Verification of antiderivative is not currently implemented for this CAS.

[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(-(x-7)*ln(x)^2)-5)

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maxima [A]  time = 0.41, size = 34, normalized size = 1.31 \begin {gather*} \frac {x e^{\left (x \log \relax (x)^{2}\right )} \log \left (\log \left (\log \relax (5)\right )\right )}{5 \, e^{\left (x \log \relax (x)^{2}\right )} - e^{\left (7 \, \log \relax (x)^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 5.67, size = 27, normalized size = 1.04 \begin {gather*} -\frac {x\,\ln \left (\ln \left (\ln \relax (5)\right )\right )}{{\mathrm {e}}^{7\,{\ln \relax (x)}^2}\,{\mathrm {e}}^{-x\,{\ln \relax (x)}^2}-5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.33, size = 20, normalized size = 0.77 \begin {gather*} - \frac {x \log {\left (\log {\left (\log {\relax (5 )} \right )} \right )}}{e^{\left (7 - x\right ) \log {\relax (x )}^{2}} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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