\(\int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log (x^2)+(e^{2 x} x^2+4 x^7) \log ^2(x^2)}{4 x^6 \log ^2(x^2)}} (2 \log ^2(x)+((-1+2 e^x x) \log (x)+3 \log ^2(x)) \log (x^2)+(-e^x x+e^x (5 x-x^2) \log (x)) \log ^2(x^2)+(2 x^6-2 x^7+e^{2 x} (2 x^2-x^3)) \log ^3(x^2))}{2 x^6 \log ^3(x^2)} \, dx\) [1941]

Optimal result

Integrand size = 161, antiderivative size = 33 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=e^{-x-\frac {\left (e^x+\frac {\log (x)}{x \log \left (x^2\right )}\right )^2}{4 x^4}} x \]

[Out]

x/exp(x+1/4*(ln(x)/x/ln(x^2)+exp(x))^2/x^4)

Rubi [F]

\[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx \]

[In]

Int[(2*Log[x]^2 + ((-1 + 2*E^x*x)*Log[x] + 3*Log[x]^2)*Log[x^2] + (-(E^x*x) + E^x*(5*x - x^2)*Log[x])*Log[x^2]
^2 + (2*x^6 - 2*x^7 + E^(2*x)*(2*x^2 - x^3))*Log[x^2]^3)/(2*E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*
x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3),x]

[Out]

Defer[Int][E^(-1/4*(Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(x^6*Log[x^2]^2)),
x] + Defer[Int][E^(2*x - (Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^
2]^2))/x^4, x] - Defer[Int][E^(2*x - (Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(
4*x^6*Log[x^2]^2))/x^3, x]/2 - Defer[Int][x/E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log
[x^2]^2)/(4*x^6*Log[x^2]^2)), x] + Defer[Int][Log[x]^2/(E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2
+ 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3), x] - Defer[Int][Log[x]/(E^((Log[x]^2 + 2*E^x*x*Log[x
]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^2), x]/2 + Defer[Int][Log[x]/(
E^((Log[x] + E^x*x*Log[x^2])^2/(4*x^6*Log[x^2]^2))*x^5*Log[x^2]^2), x] + (3*Defer[Int][Log[x]^2/(E^((Log[x]^2
+ 2*E^x*x*Log[x]*Log[x^2] + (E^(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^2), x])/2 - Def
er[Int][1/(E^((Log[x] + E^x*x*Log[x^2])^2/(4*x^6*Log[x^2]^2))*x^5*Log[x^2]), x]/2 + (5*Defer[Int][Log[x]/(E^((
Log[x] + E^x*x*Log[x^2])^2/(4*x^6*Log[x^2]^2))*x^5*Log[x^2]), x])/2 - Defer[Int][Log[x]/(E^((Log[x] + E^x*x*Lo
g[x^2])^2/(4*x^6*Log[x^2]^2))*x^4*Log[x^2]), x]/2

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )} \, dx \\ & = \frac {1}{2} \int \left (-\frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-2+x)}{x^4}-\frac {\exp \left (x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )}+\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)-\log (x) \log \left (x^2\right )+3 \log ^2(x) \log \left (x^2\right )+2 x^6 \log ^3\left (x^2\right )-2 x^7 \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-2+x)}{x^4} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx+\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (2 \log ^2(x)-\log (x) \log \left (x^2\right )+3 \log ^2(x) \log \left (x^2\right )+2 x^6 \log ^3\left (x^2\right )-2 x^7 \log ^3\left (x^2\right )\right )}{x^6 \log ^3\left (x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \left (-\frac {2 \exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4}+\frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3}\right ) \, dx\right )+\frac {1}{2} \int \left (-2 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-1+x)+\frac {2 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )}+\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x) (-1+3 \log (x))}{x^6 \log ^2\left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (-2 \log (x)+\log \left (x^2\right )-5 \log (x) \log \left (x^2\right )+x \log (x) \log \left (x^2\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x) (-1+3 \log (x))}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \left (\log \left (x^2\right )+\log (x) \left (-2+(-5+x) \log \left (x^2\right )\right )\right )}{x^5 \log ^2\left (x^2\right )} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) (-1+x) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )+\frac {1}{2} \int \left (-\frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )}+\frac {3 \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \left (-\frac {2 \exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )}+\frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) (1-5 \log (x)+x \log (x))}{x^5 \log \left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \left (-\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )+\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x\right ) \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) (1-5 \log (x)+x \log (x))}{x^5 \log \left (x^2\right )} \, dx+\frac {3}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )} \, dx+\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx+\int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )-\frac {1}{2} \int \left (\frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^5 \log \left (x^2\right )}-\frac {5 \exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log \left (x^2\right )}+\frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^4 \log \left (x^2\right )}\right ) \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )} \, dx+\frac {3}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )} \, dx+\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx+\int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )} \, dx \\ & = -\left (\frac {1}{2} \int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^3} \, dx\right )-\frac {1}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^6 \log ^2\left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^5 \log \left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^4 \log \left (x^2\right )} \, dx+\frac {3}{2} \int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^2\left (x^2\right )} \, dx+\frac {5}{2} \int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log \left (x^2\right )} \, dx+\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \, dx+\int \frac {\exp \left (2 x-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right )}{x^4} \, dx-\int \exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) x \, dx+\int \frac {\exp \left (-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}\right ) \log ^2(x)}{x^6 \log ^3\left (x^2\right )} \, dx+\int \frac {\exp \left (-\frac {\left (\log (x)+e^x x \log \left (x^2\right )\right )^2}{4 x^6 \log ^2\left (x^2\right )}\right ) \log (x)}{x^5 \log ^2\left (x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.70 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=e^{-\frac {e^{2 x}}{4 x^4}-x-\frac {\log ^2(x)}{4 x^6 \log ^2\left (x^2\right )}} x^{1-\frac {e^x}{2 x^5 \log \left (x^2\right )}} \]

[In]

Integrate[(2*Log[x]^2 + ((-1 + 2*E^x*x)*Log[x] + 3*Log[x]^2)*Log[x^2] + (-(E^x*x) + E^x*(5*x - x^2)*Log[x])*Lo
g[x^2]^2 + (2*x^6 - 2*x^7 + E^(2*x)*(2*x^2 - x^3))*Log[x^2]^3)/(2*E^((Log[x]^2 + 2*E^x*x*Log[x]*Log[x^2] + (E^
(2*x)*x^2 + 4*x^7)*Log[x^2]^2)/(4*x^6*Log[x^2]^2))*x^6*Log[x^2]^3),x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 2.27 (sec) , antiderivative size = 549, normalized size of antiderivative = 16.64

\[x \,{\mathrm e}^{-\frac {-4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{6} x^{7}+16 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) x^{7}-24 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} x^{7}+16 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} x^{7}-4 \pi ^{2} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} x^{7}-8 i \ln \left (x \right ) \pi \,{\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{3} x^{2}-32 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{3} x^{7}+64 i \ln \left (x \right ) \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) x^{7}-\pi ^{2} {\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{6} x^{2}+4 \pi ^{2} {\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{5} \operatorname {csgn}\left (i x \right ) x^{2}-6 \pi ^{2} {\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{4} \operatorname {csgn}\left (i x \right )^{2} x^{2}+4 \pi ^{2} {\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{3} \operatorname {csgn}\left (i x \right )^{3} x^{2}-\pi ^{2} {\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )^{4} x^{2}+64 x^{7} \ln \left (x \right )^{2}+16 i \ln \left (x \right ) \pi \,{\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right ) x^{2}-32 i \ln \left (x \right ) \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} x^{7}-4 i \ln \left (x \right ) {\mathrm e}^{x} x \pi \operatorname {csgn}\left (i x^{2}\right )^{3}-4 i \ln \left (x \right ) {\mathrm e}^{x} x \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}-8 i \ln \left (x \right ) \pi \,{\mathrm e}^{2 x} \operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2} x^{2}+8 i \ln \left (x \right ) {\mathrm e}^{x} x \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+16 \,{\mathrm e}^{2 x} \ln \left (x \right )^{2} x^{2}+16 x \,{\mathrm e}^{x} \ln \left (x \right )^{2}+4 \ln \left (x \right )^{2}}{4 x^{6} {\left (-i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}+2 i \pi \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )-i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+4 \ln \left (x \right )\right )}^{2}}}\]

[In]

int(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*ln(x^2)^3+((-x^2+5*x)*exp(x)*ln(x)-exp(x)*x)*ln(x^2)^2+(3*ln(x)^2
+(2*exp(x)*x-1)*ln(x))*ln(x^2)+2*ln(x)^2)/x^6/ln(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*ln(x^2)^2+2*x*exp(x)*ln(
x)*ln(x^2)+ln(x)^2)/x^6/ln(x^2)^2),x)

[Out]

x*exp(-1/4*(-4*Pi^2*csgn(I*x^2)^6*x^7+16*Pi^2*csgn(I*x^2)^5*csgn(I*x)*x^7-24*Pi^2*csgn(I*x^2)^4*csgn(I*x)^2*x^
7+16*Pi^2*csgn(I*x^2)^3*csgn(I*x)^3*x^7-4*Pi^2*csgn(I*x^2)^2*csgn(I*x)^4*x^7-8*I*ln(x)*Pi*exp(2*x)*csgn(I*x^2)
^3*x^2-32*I*ln(x)*Pi*csgn(I*x^2)^3*x^7+64*I*ln(x)*Pi*csgn(I*x^2)^2*csgn(I*x)*x^7-Pi^2*exp(2*x)*csgn(I*x^2)^6*x
^2+4*Pi^2*exp(2*x)*csgn(I*x^2)^5*csgn(I*x)*x^2-6*Pi^2*exp(2*x)*csgn(I*x^2)^4*csgn(I*x)^2*x^2+4*Pi^2*exp(2*x)*c
sgn(I*x^2)^3*csgn(I*x)^3*x^2-Pi^2*exp(2*x)*csgn(I*x^2)^2*csgn(I*x)^4*x^2+64*x^7*ln(x)^2+16*I*ln(x)*Pi*exp(2*x)
*csgn(I*x^2)^2*csgn(I*x)*x^2-32*I*ln(x)*Pi*csgn(I*x^2)*csgn(I*x)^2*x^7-4*I*ln(x)*exp(x)*x*Pi*csgn(I*x^2)^3-4*I
*ln(x)*exp(x)*x*Pi*csgn(I*x^2)*csgn(I*x)^2-8*I*ln(x)*Pi*exp(2*x)*csgn(I*x^2)*csgn(I*x)^2*x^2+8*I*ln(x)*exp(x)*
x*Pi*csgn(I*x^2)^2*csgn(I*x)+16*exp(2*x)*ln(x)^2*x^2+16*x*exp(x)*ln(x)^2+4*ln(x)^2)/x^6/(-I*Pi*csgn(I*x^2)^3+2
*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*x^2)*csgn(I*x)^2+4*ln(x))^2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x e^{\left (-\frac {16 \, x^{7} + 4 \, x^{2} e^{\left (2 \, x\right )} + 4 \, x e^{x} + 1}{16 \, x^{6}}\right )} \]

[In]

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(
3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2
+2*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="fricas")

[Out]

x*e^(-1/16*(16*x^7 + 4*x^2*e^(2*x) + 4*x*e^x + 1)/x^6)

Sympy [A] (verification not implemented)

Time = 1.65 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x e^{- \frac {x e^{x} \log {\left (x \right )}^{2} + \left (4 x^{7} + x^{2} e^{2 x}\right ) \log {\left (x \right )}^{2} + \frac {\log {\left (x \right )}^{2}}{4}}{4 x^{6} \log {\left (x \right )}^{2}}} \]

[In]

integrate(1/2*(((-x**3+2*x**2)*exp(x)**2-2*x**7+2*x**6)*ln(x**2)**3+((-x**2+5*x)*exp(x)*ln(x)-exp(x)*x)*ln(x**
2)**2+(3*ln(x)**2+(2*exp(x)*x-1)*ln(x))*ln(x**2)+2*ln(x)**2)/x**6/ln(x**2)**3/exp(1/4*((exp(x)**2*x**2+4*x**7)
*ln(x**2)**2+2*x*exp(x)*ln(x)*ln(x**2)+ln(x)**2)/x**6/ln(x**2)**2),x)

[Out]

x*exp(-(x*exp(x)*log(x)**2 + (4*x**7 + x**2*exp(2*x))*log(x)**2 + log(x)**2/4)/(4*x**6*log(x)**2))

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x e^{\left (-x - \frac {e^{\left (2 \, x\right )}}{4 \, x^{4}} - \frac {e^{x}}{4 \, x^{5}} - \frac {1}{16 \, x^{6}}\right )} \]

[In]

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(
3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2
+2*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="maxima")

[Out]

x*e^(-x - 1/4*e^(2*x)/x^4 - 1/4*e^x/x^5 - 1/16/x^6)

Giac [F]

\[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=\int { -\frac {{\left ({\left (2 \, x^{7} - 2 \, x^{6} + {\left (x^{3} - 2 \, x^{2}\right )} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{3} + {\left ({\left (x^{2} - 5 \, x\right )} e^{x} \log \left (x\right ) + x e^{x}\right )} \log \left (x^{2}\right )^{2} - {\left ({\left (2 \, x e^{x} - 1\right )} \log \left (x\right ) + 3 \, \log \left (x\right )^{2}\right )} \log \left (x^{2}\right ) - 2 \, \log \left (x\right )^{2}\right )} e^{\left (-\frac {2 \, x e^{x} \log \left (x^{2}\right ) \log \left (x\right ) + {\left (4 \, x^{7} + x^{2} e^{\left (2 \, x\right )}\right )} \log \left (x^{2}\right )^{2} + \log \left (x\right )^{2}}{4 \, x^{6} \log \left (x^{2}\right )^{2}}\right )}}{2 \, x^{6} \log \left (x^{2}\right )^{3}} \,d x } \]

[In]

integrate(1/2*(((-x^3+2*x^2)*exp(x)^2-2*x^7+2*x^6)*log(x^2)^3+((-x^2+5*x)*exp(x)*log(x)-exp(x)*x)*log(x^2)^2+(
3*log(x)^2+(2*exp(x)*x-1)*log(x))*log(x^2)+2*log(x)^2)/x^6/log(x^2)^3/exp(1/4*((exp(x)^2*x^2+4*x^7)*log(x^2)^2
+2*x*exp(x)*log(x)*log(x^2)+log(x)^2)/x^6/log(x^2)^2),x, algorithm="giac")

[Out]

integrate(-1/2*((2*x^7 - 2*x^6 + (x^3 - 2*x^2)*e^(2*x))*log(x^2)^3 + ((x^2 - 5*x)*e^x*log(x) + x*e^x)*log(x^2)
^2 - ((2*x*e^x - 1)*log(x) + 3*log(x)^2)*log(x^2) - 2*log(x)^2)*e^(-1/4*(2*x*e^x*log(x^2)*log(x) + (4*x^7 + x^
2*e^(2*x))*log(x^2)^2 + log(x)^2)/(x^6*log(x^2)^2))/(x^6*log(x^2)^3), x)

Mupad [B] (verification not implemented)

Time = 10.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^{-\frac {\log ^2(x)+2 e^x x \log (x) \log \left (x^2\right )+\left (e^{2 x} x^2+4 x^7\right ) \log ^2\left (x^2\right )}{4 x^6 \log ^2\left (x^2\right )}} \left (2 \log ^2(x)+\left (\left (-1+2 e^x x\right ) \log (x)+3 \log ^2(x)\right ) \log \left (x^2\right )+\left (-e^x x+e^x \left (5 x-x^2\right ) \log (x)\right ) \log ^2\left (x^2\right )+\left (2 x^6-2 x^7+e^{2 x} \left (2 x^2-x^3\right )\right ) \log ^3\left (x^2\right )\right )}{2 x^6 \log ^3\left (x^2\right )} \, dx=x\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^x\,\ln \left (x\right )}{2\,x^5\,\ln \left (x^2\right )}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-\frac {{\ln \left (x\right )}^2}{4\,x^6\,{\ln \left (x^2\right )}^2}}\,{\mathrm {e}}^{-\frac {{\mathrm {e}}^{2\,x}}{4\,x^4}} \]

[In]

int((exp(-((log(x^2)^2*(x^2*exp(2*x) + 4*x^7))/4 + log(x)^2/4 + (x*log(x^2)*exp(x)*log(x))/2)/(x^6*log(x^2)^2)
)*((log(x^2)^3*(exp(2*x)*(2*x^2 - x^3) + 2*x^6 - 2*x^7))/2 + log(x)^2 - (log(x^2)^2*(x*exp(x) - exp(x)*log(x)*
(5*x - x^2)))/2 + (log(x^2)*(3*log(x)^2 + log(x)*(2*x*exp(x) - 1)))/2))/(x^6*log(x^2)^3),x)

[Out]

x*exp(-(exp(x)*log(x))/(2*x^5*log(x^2)))*exp(-x)*exp(-log(x)^2/(4*x^6*log(x^2)^2))*exp(-exp(2*x)/(4*x^4))