\(\int \frac {e^{\frac {e^{2 x^4} (1-4 x+8 x^3+4 x^4) \log ^2(2)+e^{2 x^4} (-2+8 x-16 x^3-8 x^4) \log (2) \log (x)+e^{2 x^4} (1-4 x+8 x^3+4 x^4) \log ^2(x)}{x^2}} (e^{2 x^4} ((-2+8 x-16 x^3-8 x^4) \log (2)+(-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8) \log ^2(2))+e^{2 x^4} (2-8 x+16 x^3+8 x^4+(4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8) \log (2)) \log (x)+e^{2 x^4} (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8) \log ^2(x))}{x^3} \, dx\) [2637]

Optimal result

Integrand size = 255, antiderivative size = 31 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{e^{2 x^4} \left (2-\frac {1}{x}+2 x\right )^2 (-\log (2)+\log (x))^2} \]

[Out]

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

Rubi [F]

\[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=\int \frac {\exp \left (\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}\right ) \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx \]

[In]

Int[(E^((E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[2]^2 + E^(2*x^4)*(-2 + 8*x - 16*x^3 - 8*x^4)*Log[2]*Log[x] +
E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[x]^2)/x^2)*(E^(2*x^4)*((-2 + 8*x - 16*x^3 - 8*x^4)*Log[2] + (-2 + 4*x
+ 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[2]^2) + E^(2*x^4)*(2 - 8*x + 16*x^3 + 8*x^4 + (4 - 8*x - 16*x
^3 - 32*x^4 + 64*x^5 - 128*x^7 - 64*x^8)*Log[2])*Log[x] + E^(2*x^4)*(-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x
^7 + 32*x^8)*Log[x]^2))/x^3,x]

[Out]

-8*(2 - Log[2])*Log[2]*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2), x] - 2*Log[2]*(
1 + Log[2])*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)/x^3, x] + 2*Log[2]*(4 + Log
[4])*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)/x^2, x] - 8*Log[2]*(1 - Log[4])*De
fer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x, x] - 32*Log[2]^2*Defer[Int][E^(2*x^4 +
(E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^2, x] + 64*Log[2]^2*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 +
2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^4, x] + 32*Log[2]^2*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*L
og[x/2]^2)/x^2)*x^5, x] + 8*(2 - Log[4])*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2
)*Log[x], x] + 2*(1 + Log[4])*Defer[Int][(E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*Log[x])/
x^3, x] - 4*(2 + Log[4])*Defer[Int][(E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*Log[x])/x^2,
x] + 8*(1 - Log[16])*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x*Log[x], x] + 64*
Log[2]*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^2*Log[x], x] - 128*Log[2]*Defe
r[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^4*Log[x], x] - 64*Log[2]*Defer[Int][E^(2*
x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^5*Log[x], x] + 8*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1
+ 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*Log[x]^2, x] - 2*Defer[Int][(E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log
[x/2]^2)/x^2)*Log[x]^2)/x^3, x] + 4*Defer[Int][(E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*Lo
g[x]^2)/x^2, x] + 16*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x*Log[x]^2, x] - 3
2*Defer[Int][E^(2*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^2*Log[x]^2, x] + 64*Defer[Int][E^(2
*x^4 + (E^(2*x^4)*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^4*Log[x]^2, x] + 32*Defer[Int][E^(2*x^4 + (E^(2*x^4)
*(-1 + 2*x + 2*x^2)^2*Log[x/2]^2)/x^2)*x^5*Log[x]^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {2 \exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (1-2 x-2 x^2\right ) (\log (2)-\log (x)) \left (-1+2 x-\log (2)+4 x^4 \log (2)-8 x^5 \log (2)-8 x^6 \log (2)-x^2 (-2+\log (4))+\left (1+2 x^2-4 x^4+8 x^5+8 x^6\right ) \log (x)\right )}{x^3} \, dx \\ & = 2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (1-2 x-2 x^2\right ) (\log (2)-\log (x)) \left (-1+2 x-\log (2)+4 x^4 \log (2)-8 x^5 \log (2)-8 x^6 \log (2)-x^2 (-2+\log (4))+\left (1+2 x^2-4 x^4+8 x^5+8 x^6\right ) \log (x)\right )}{x^3} \, dx \\ & = 2 \int \left (\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (1-2 x-2 x^2\right ) \log (2) \left (-1+2 x+2 x^2 (1-\log (2))-\log (2)+4 x^4 \log (2)-8 x^5 \log (2)-8 x^6 \log (2)\right )}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (1-2 x-2 x^2\right ) \left (1-2 x-8 x^4 \log (2)+16 x^5 \log (2)+16 x^6 \log (2)-2 x^2 (1-\log (4))+\log (4)\right ) \log (x)}{x^3}+\frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-1+2 x+2 x^2\right ) \left (1+2 x^2-4 x^4+8 x^5+8 x^6\right ) \log ^2(x)}{x^3}\right ) \, dx \\ & = 2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (1-2 x-2 x^2\right ) \left (1-2 x-8 x^4 \log (2)+16 x^5 \log (2)+16 x^6 \log (2)-2 x^2 (1-\log (4))+\log (4)\right ) \log (x)}{x^3} \, dx+2 \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (-1+2 x+2 x^2\right ) \left (1+2 x^2-4 x^4+8 x^5+8 x^6\right ) \log ^2(x)}{x^3} \, dx+(2 \log (2)) \int \frac {\exp \left (2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}\right ) \left (1-2 x-2 x^2\right ) \left (-1+2 x+2 x^2 (1-\log (2))-\log (2)+4 x^4 \log (2)-8 x^5 \log (2)-8 x^6 \log (2)\right )}{x^3} \, dx \\ & = 2 \int \left (32 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^2 \log (2) \log (x)-64 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^4 \log (2) \log (x)-32 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^5 \log (2) \log (x)-4 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} (-2+\log (4)) \log (x)+\frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} (1+\log (4)) \log (x)}{x^3}-\frac {2 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} (2+\log (4)) \log (x)}{x^2}-4 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x (-1+\log (16)) \log (x)\right ) \, dx+2 \int \left (4 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log ^2(x)-\frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log ^2(x)}{x^3}+\frac {2 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log ^2(x)}{x^2}+8 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x \log ^2(x)-16 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^2 \log ^2(x)+32 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^4 \log ^2(x)+16 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^5 \log ^2(x)\right ) \, dx+(2 \log (2)) \int \left (\frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} (-1-\log (2))}{x^3}-8 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \left (1-\frac {\log (2)}{2}\right )-16 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^2 \log (2)+32 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^4 \log (2)+16 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^5 \log (2)+4 e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x (-1+\log (4))+\frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} (4+\log (4))}{x^2}\right ) \, dx \\ & = -\left (2 \int \frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log ^2(x)}{x^3} \, dx\right )+4 \int \frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log ^2(x)}{x^2} \, dx+8 \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log ^2(x) \, dx+16 \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x \log ^2(x) \, dx-32 \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^2 \log ^2(x) \, dx+32 \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^5 \log ^2(x) \, dx+64 \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^4 \log ^2(x) \, dx+(64 \log (2)) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^2 \log (x) \, dx-(64 \log (2)) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^5 \log (x) \, dx-(128 \log (2)) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^4 \log (x) \, dx-(8 (2-\log (2)) \log (2)) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \, dx-\left (32 \log ^2(2)\right ) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^2 \, dx+\left (32 \log ^2(2)\right ) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^5 \, dx+\left (64 \log ^2(2)\right ) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x^4 \, dx-(2 \log (2) (1+\log (2))) \int \frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}}}{x^3} \, dx-(8 \log (2) (1-\log (4))) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x \, dx+(8 (2-\log (4))) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log (x) \, dx+(2 (1+\log (4))) \int \frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log (x)}{x^3} \, dx-(4 (2+\log (4))) \int \frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} \log (x)}{x^2} \, dx+(2 \log (2) (4+\log (4))) \int \frac {e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}}}{x^2} \, dx+(8 (1-\log (16))) \int e^{2 x^4+\frac {e^{2 x^4} \left (-1+2 x+2 x^2\right )^2 \log ^2\left (\frac {x}{2}\right )}{x^2}} x \log (x) \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=\int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx \]

[In]

Integrate[(E^((E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[2]^2 + E^(2*x^4)*(-2 + 8*x - 16*x^3 - 8*x^4)*Log[2]*Log
[x] + E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[x]^2)/x^2)*(E^(2*x^4)*((-2 + 8*x - 16*x^3 - 8*x^4)*Log[2] + (-2
+ 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[2]^2) + E^(2*x^4)*(2 - 8*x + 16*x^3 + 8*x^4 + (4 - 8*x
- 16*x^3 - 32*x^4 + 64*x^5 - 128*x^7 - 64*x^8)*Log[2])*Log[x] + E^(2*x^4)*(-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5
+ 64*x^7 + 32*x^8)*Log[x]^2))/x^3,x]

[Out]

Integrate[(E^((E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[2]^2 + E^(2*x^4)*(-2 + 8*x - 16*x^3 - 8*x^4)*Log[2]*Log
[x] + E^(2*x^4)*(1 - 4*x + 8*x^3 + 4*x^4)*Log[x]^2)/x^2)*(E^(2*x^4)*((-2 + 8*x - 16*x^3 - 8*x^4)*Log[2] + (-2
+ 4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64*x^7 + 32*x^8)*Log[2]^2) + E^(2*x^4)*(2 - 8*x + 16*x^3 + 8*x^4 + (4 - 8*x
- 16*x^3 - 32*x^4 + 64*x^5 - 128*x^7 - 64*x^8)*Log[2])*Log[x] + E^(2*x^4)*(-2 + 4*x + 8*x^3 + 16*x^4 - 32*x^5
+ 64*x^7 + 32*x^8)*Log[x]^2))/x^3, x]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06

\[{\mathrm e}^{\frac {{\mathrm e}^{2 x^{4}} \left (2 x^{2}+2 x -1\right )^{2} \left (-\ln \left (x \right )+\ln \left (2\right )\right )^{2}}{x^{2}}}\]

[In]

int(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*ln(x)^2+((-64*x^8-128*x^7+64*x^5-32*x^4-16*x^3-8*x+4
)*ln(2)+8*x^4+16*x^3-8*x+2)*exp(x^4)^2*ln(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*ln(2)^2+(-8*x^4-16*x^3
+8*x-2)*ln(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*ln(x)^2+(-8*x^4-16*x^3+8*x-2)*ln(2)*exp(x^4)^2*
ln(x)+(4*x^4+8*x^3-4*x+1)*ln(2)^2*exp(x^4)^2)/x^2)/x^3,x)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (29) = 58\).

Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.74 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\left (\frac {{\left (4 \, x^{4} + 8 \, x^{3} - 4 \, x + 1\right )} e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 2 \, {\left (4 \, x^{4} + 8 \, x^{3} - 4 \, x + 1\right )} e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + {\left (4 \, x^{4} + 8 \, x^{3} - 4 \, x + 1\right )} e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x^{2}}\right )} \]

[In]

integrate(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+((-64*x^8-128*x^7+64*x^5-32*x^4-16*x^
3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*exp(x^4)^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*
x^4-16*x^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*log(x)^2+(-8*x^4-16*x^3+8*x-2)*log(2
)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)*log(2)^2*exp(x^4)^2)/x^2)/x^3,x, algorithm="fricas")

[Out]

e^(((4*x^4 + 8*x^3 - 4*x + 1)*e^(2*x^4)*log(2)^2 - 2*(4*x^4 + 8*x^3 - 4*x + 1)*e^(2*x^4)*log(2)*log(x) + (4*x^
4 + 8*x^3 - 4*x + 1)*e^(2*x^4)*log(x)^2)/x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (26) = 52\).

Time = 1.05 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\frac {\left (- 8 x^{4} - 16 x^{3} + 8 x - 2\right ) e^{2 x^{4}} \log {\left (2 \right )} \log {\left (x \right )} + \left (4 x^{4} + 8 x^{3} - 4 x + 1\right ) e^{2 x^{4}} \log {\left (x \right )}^{2} + \left (4 x^{4} + 8 x^{3} - 4 x + 1\right ) e^{2 x^{4}} \log {\left (2 \right )}^{2}}{x^{2}}} \]

[In]

integrate(((32*x**8+64*x**7-32*x**5+16*x**4+8*x**3+4*x-2)*exp(x**4)**2*ln(x)**2+((-64*x**8-128*x**7+64*x**5-32
*x**4-16*x**3-8*x+4)*ln(2)+8*x**4+16*x**3-8*x+2)*exp(x**4)**2*ln(x)+((32*x**8+64*x**7-32*x**5+16*x**4+8*x**3+4
*x-2)*ln(2)**2+(-8*x**4-16*x**3+8*x-2)*ln(2))*exp(x**4)**2)*exp(((4*x**4+8*x**3-4*x+1)*exp(x**4)**2*ln(x)**2+(
-8*x**4-16*x**3+8*x-2)*ln(2)*exp(x**4)**2*ln(x)+(4*x**4+8*x**3-4*x+1)*ln(2)**2*exp(x**4)**2)/x**2)/x**3,x)

[Out]

exp(((-8*x**4 - 16*x**3 + 8*x - 2)*exp(2*x**4)*log(2)*log(x) + (4*x**4 + 8*x**3 - 4*x + 1)*exp(2*x**4)*log(x)*
*2 + (4*x**4 + 8*x**3 - 4*x + 1)*exp(2*x**4)*log(2)**2)/x**2)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (29) = 58\).

Time = 0.83 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.61 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\left (4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 8 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 16 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x} + \frac {8 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x^{2}} - \frac {2 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x^{2}}\right )} \]

[In]

integrate(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+((-64*x^8-128*x^7+64*x^5-32*x^4-16*x^
3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*exp(x^4)^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*
x^4-16*x^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*log(x)^2+(-8*x^4-16*x^3+8*x-2)*log(2
)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)*log(2)^2*exp(x^4)^2)/x^2)/x^3,x, algorithm="maxima")

[Out]

e^(4*x^2*e^(2*x^4)*log(2)^2 - 8*x^2*e^(2*x^4)*log(2)*log(x) + 4*x^2*e^(2*x^4)*log(x)^2 + 8*x*e^(2*x^4)*log(2)^
2 - 16*x*e^(2*x^4)*log(2)*log(x) + 8*x*e^(2*x^4)*log(x)^2 - 4*e^(2*x^4)*log(2)^2/x + 8*e^(2*x^4)*log(2)*log(x)
/x - 4*e^(2*x^4)*log(x)^2/x + e^(2*x^4)*log(2)^2/x^2 - 2*e^(2*x^4)*log(2)*log(x)/x^2 + e^(2*x^4)*log(x)^2/x^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (29) = 58\).

Time = 0.63 (sec) , antiderivative size = 174, normalized size of antiderivative = 5.61 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=e^{\left (4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 8 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 4 \, x^{2} e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2} - 16 \, x e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right ) + 8 \, x e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x} + \frac {8 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x} - \frac {4 \, e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (2\right )^{2}}{x^{2}} - \frac {2 \, e^{\left (2 \, x^{4}\right )} \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {e^{\left (2 \, x^{4}\right )} \log \left (x\right )^{2}}{x^{2}}\right )} \]

[In]

integrate(((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*exp(x^4)^2*log(x)^2+((-64*x^8-128*x^7+64*x^5-32*x^4-16*x^
3-8*x+4)*log(2)+8*x^4+16*x^3-8*x+2)*exp(x^4)^2*log(x)+((32*x^8+64*x^7-32*x^5+16*x^4+8*x^3+4*x-2)*log(2)^2+(-8*
x^4-16*x^3+8*x-2)*log(2))*exp(x^4)^2)*exp(((4*x^4+8*x^3-4*x+1)*exp(x^4)^2*log(x)^2+(-8*x^4-16*x^3+8*x-2)*log(2
)*exp(x^4)^2*log(x)+(4*x^4+8*x^3-4*x+1)*log(2)^2*exp(x^4)^2)/x^2)/x^3,x, algorithm="giac")

[Out]

e^(4*x^2*e^(2*x^4)*log(2)^2 - 8*x^2*e^(2*x^4)*log(2)*log(x) + 4*x^2*e^(2*x^4)*log(x)^2 + 8*x*e^(2*x^4)*log(2)^
2 - 16*x*e^(2*x^4)*log(2)*log(x) + 8*x*e^(2*x^4)*log(x)^2 - 4*e^(2*x^4)*log(2)^2/x + 8*e^(2*x^4)*log(2)*log(x)
/x - 4*e^(2*x^4)*log(x)^2/x + e^(2*x^4)*log(2)^2/x^2 - 2*e^(2*x^4)*log(2)*log(x)/x^2 + e^(2*x^4)*log(x)^2/x^2)

Mupad [B] (verification not implemented)

Time = 9.21 (sec) , antiderivative size = 155, normalized size of antiderivative = 5.00 \[ \int \frac {e^{\frac {e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(2)+e^{2 x^4} \left (-2+8 x-16 x^3-8 x^4\right ) \log (2) \log (x)+e^{2 x^4} \left (1-4 x+8 x^3+4 x^4\right ) \log ^2(x)}{x^2}} \left (e^{2 x^4} \left (\left (-2+8 x-16 x^3-8 x^4\right ) \log (2)+\left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(2)\right )+e^{2 x^4} \left (2-8 x+16 x^3+8 x^4+\left (4-8 x-16 x^3-32 x^4+64 x^5-128 x^7-64 x^8\right ) \log (2)\right ) \log (x)+e^{2 x^4} \left (-2+4 x+8 x^3+16 x^4-32 x^5+64 x^7+32 x^8\right ) \log ^2(x)\right )}{x^3} \, dx=\frac {{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}{x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{\frac {{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}{x^2}}\,{\mathrm {e}}^{-\frac {4\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}{x}}\,{\mathrm {e}}^{4\,x^2\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (x\right )}^2}\,{\mathrm {e}}^{8\,x\,{\mathrm {e}}^{2\,x^4}\,{\ln \left (2\right )}^2}}{x^{\frac {2\,{\mathrm {e}}^{2\,x^4}\,\ln \left (2\right )\,\left (4\,x^4+8\,x^3-4\,x+1\right )}{x^2}}} \]

[In]

int((exp((exp(2*x^4)*log(2)^2*(8*x^3 - 4*x + 4*x^4 + 1) + exp(2*x^4)*log(x)^2*(8*x^3 - 4*x + 4*x^4 + 1) - exp(
2*x^4)*log(2)*log(x)*(16*x^3 - 8*x + 8*x^4 + 2))/x^2)*(exp(2*x^4)*log(x)^2*(4*x + 8*x^3 + 16*x^4 - 32*x^5 + 64
*x^7 + 32*x^8 - 2) - exp(2*x^4)*(log(2)*(16*x^3 - 8*x + 8*x^4 + 2) - log(2)^2*(4*x + 8*x^3 + 16*x^4 - 32*x^5 +
64*x^7 + 32*x^8 - 2)) + exp(2*x^4)*log(x)*(16*x^3 - log(2)*(8*x + 16*x^3 + 32*x^4 - 64*x^5 + 128*x^7 + 64*x^8
- 4) - 8*x + 8*x^4 + 2)))/x^3,x)

[Out]

(exp((exp(2*x^4)*log(2)^2)/x^2)*exp(-(4*exp(2*x^4)*log(2)^2)/x)*exp(4*x^2*exp(2*x^4)*log(2)^2)*exp(8*x*exp(2*x
^4)*log(x)^2)*exp((exp(2*x^4)*log(x)^2)/x^2)*exp(-(4*exp(2*x^4)*log(x)^2)/x)*exp(4*x^2*exp(2*x^4)*log(x)^2)*ex
p(8*x*exp(2*x^4)*log(2)^2))/x^((2*exp(2*x^4)*log(2)*(8*x^3 - 4*x + 4*x^4 + 1))/x^2)