\(\int \frac {e^{\frac {-x \log (9)+(-5+e^{5 e^4 x} x) \log (x)}{\log (x)}} (\log (9)-\log (9) \log (x)+e^{5 e^4 x} (1+5 e^4 x) \log ^2(x))}{\log ^2(x)} \, dx\) [7439]

Optimal result

Integrand size = 63, antiderivative size = 23 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{-5+e^{5 e^4 x} x-\frac {x \log (9)}{\log (x)}} \]

[Out]

exp(x*exp(5*x*exp(4))-5-2*x*ln(3)/ln(x))

Rubi [F]

\[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=\int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx \]

[In]

Int[(E^((-(x*Log[9]) + (-5 + E^(5*E^4*x)*x)*Log[x])/Log[x])*(Log[9] - Log[9]*Log[x] + E^(5*E^4*x)*(1 + 5*E^4*x
)*Log[x]^2))/Log[x]^2,x]

[Out]

Defer[Int][E^((-(x*Log[9]) - 5*Log[x] + 5*E^4*x*Log[x] + E^(5*E^4*x)*x*Log[x])/Log[x]), x] + 5*Defer[Int][E^((
-(x*Log[9]) - Log[x] + 5*E^4*x*Log[x] + E^(5*E^4*x)*x*Log[x])/Log[x])*x, x] + Log[9]*Defer[Int][E^((-(x*Log[9]
) + (-5 + E^(5*E^4*x)*x)*Log[x])/Log[x])/Log[x]^2, x] - Log[9]*Defer[Int][E^((-(x*Log[9]) + (-5 + E^(5*E^4*x)*
x)*Log[x])/Log[x])/Log[x], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\exp \left (5 e^4 x+\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \left (1+5 e^4 x\right )-\frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \log (9) (-1+\log (x))}{\log ^2(x)}\right ) \, dx \\ & = -\left (\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) (-1+\log (x))}{\log ^2(x)} \, dx\right )+\int \exp \left (5 e^4 x+\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right ) \left (1+5 e^4 x\right ) \, dx \\ & = -\left (\log (9) \int \left (-\frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)}+\frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)}\right ) \, dx\right )+\int \exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) \left (1+5 e^4 x\right ) \, dx \\ & = \log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)} \, dx-\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)} \, dx+\int \left (\exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right )+5 \exp \left (4+\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) x\right ) \, dx \\ & = 5 \int \exp \left (4+\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) x \, dx+\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)} \, dx-\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)} \, dx+\int \exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) \, dx \\ & = 5 \int \exp \left (\frac {-x \log (9)-\log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) x \, dx+\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log ^2(x)} \, dx-\log (9) \int \frac {\exp \left (\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}\right )}{\log (x)} \, dx+\int \exp \left (\frac {-x \log (9)-5 \log (x)+5 e^4 x \log (x)+e^{5 e^4 x} x \log (x)}{\log (x)}\right ) \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=9^{-\frac {x}{\log (x)}} e^{-5+e^{5 e^4 x} x} \]

[In]

Integrate[(E^((-(x*Log[9]) + (-5 + E^(5*E^4*x)*x)*Log[x])/Log[x])*(Log[9] - Log[9]*Log[x] + E^(5*E^4*x)*(1 + 5
*E^4*x)*Log[x]^2))/Log[x]^2,x]

[Out]

E^(-5 + E^(5*E^4*x)*x)/9^(x/Log[x])

Maple [A] (verified)

Time = 24.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91

method	result	size
risch	\(\left (\frac {1}{9}\right )^{\frac {x}{\ln \left (x \right )}} {\mathrm e}^{x \,{\mathrm e}^{5 x \,{\mathrm e}^{4}}-5}\)	\(21\)
parallelrisch	\({\mathrm e}^{\frac {\ln \left (x \right ) {\mathrm e}^{5 x \,{\mathrm e}^{4}} x -2 x \ln \left (3\right )-5 \ln \left (x \right )}{\ln \left (x \right )}}\)	\(27\)

[In]

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

[Out]

(1/9)^(x/ln(x))*exp(x*exp(5*x*exp(4))-5)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\left (-\frac {2 \, x \log \left (3\right ) - {\left (x e^{\left (5 \, x e^{4}\right )} - 5\right )} \log \left (x\right )}{\log \left (x\right )}\right )} \]

[In]

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

[Out]

e^(-(2*x*log(3) - (x*e^(5*x*e^4) - 5)*log(x))/log(x))

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\frac {- 2 x \log {\left (3 \right )} + \left (x e^{5 x e^{4}} - 5\right ) \log {\left (x \right )}}{\log {\left (x \right )}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.42 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\left (x e^{\left (5 \, x e^{4}\right )} - \frac {2 \, x \log \left (3\right )}{\log \left (x\right )} - 5\right )} \]

[In]

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

[Out]

e^(x*e^(5*x*e^4) - 2*x*log(3)/log(x) - 5)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=e^{\left (x e^{\left (5 \, x e^{4}\right )} - \frac {2 \, x \log \left (3\right )}{\log \left (x\right )} - 5\right )} \]

[In]

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

[Out]

e^(x*e^(5*x*e^4) - 2*x*log(3)/log(x) - 5)

Mupad [B] (verification not implemented)

Time = 11.61 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\frac {-x \log (9)+\left (-5+e^{5 e^4 x} x\right ) \log (x)}{\log (x)}} \left (\log (9)-\log (9) \log (x)+e^{5 e^4 x} \left (1+5 e^4 x\right ) \log ^2(x)\right )}{\log ^2(x)} \, dx=\frac {{\mathrm {e}}^{-5}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{5\,x\,{\mathrm {e}}^4}}}{3^{\frac {2\,x}{\ln \left (x\right )}}} \]

[In]

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

[Out]

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