\(\int \frac {(-4-2 x+e^{4 e^{2 x}} x)^{\frac {1}{x}} (-2 x+e^{4 e^{2 x}} (x+8 e^{2 x} x^2)+(4+2 x-e^{4 e^{2 x}} x) \log (-4-2 x+e^{4 e^{2 x}} x))}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx\) [9231]

Optimal result

Integrand size = 108, antiderivative size = 22 \[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx=\left (\left (-2+e^{4 e^{2 x}}-\frac {4}{x}\right ) x\right )^{\frac {1}{x}} \]

[Out]

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

Rubi [F]

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

[In]

Int[((-4 - 2*x + E^(4*E^(2*x))*x)^x^(-1)*(-2*x + E^(4*E^(2*x))*(x + 8*E^(2*x)*x^2) + (4 + 2*x - E^(4*E^(2*x))*
x)*Log[-4 - 2*x + E^(4*E^(2*x))*x]))/(-4*x^2 - 2*x^3 + E^(4*E^(2*x))*x^3),x]

[Out]

8*Defer[Int][E^(2*(2*E^(2*x) + x))*(-4 - 2*x + E^(4*E^(2*x))*x)^(-1 + x^(-1)), x] + 4*Log[-4 - (2 - E^(4*E^(2*
x)))*x]*Defer[Int][(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1))/x^2, x] - 2*Defer[Int][(-4 + (-2 + E^(4*E^(2*x)
))*x)^(-1 + x^(-1))/x, x] + 2*Log[-4 - (2 - E^(4*E^(2*x)))*x]*Defer[Int][(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x
^(-1))/x, x] + Defer[Int][(E^(4*E^(2*x))*(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1)))/x, x] - Log[-4 - (2 - E^
(4*E^(2*x)))*x]*Defer[Int][(E^(4*E^(2*x))*(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1)))/x, x] - 4*Defer[Int][De
fer[Int][(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1))/x^2, x]/x, x] - 16*Defer[Int][Defer[Int][(-4 + (-2 + E^(4
*E^(2*x)))*x)^(-1 + x^(-1))/x^2, x]/(x*(-4 - 2*x + E^(4*E^(2*x))*x)), x] - 32*Defer[Int][(E^(2*(2*E^(2*x) + x)
)*x*Defer[Int][(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1))/x^2, x])/(-4 - 2*x + E^(4*E^(2*x))*x), x] - 2*Defer
[Int][Defer[Int][(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1))/x, x]/x, x] - 8*Defer[Int][Defer[Int][(-4 + (-2 +
E^(4*E^(2*x)))*x)^(-1 + x^(-1))/x, x]/(x*(-4 - 2*x + E^(4*E^(2*x))*x)), x] - 16*Defer[Int][(E^(2*(2*E^(2*x) +
x))*x*Defer[Int][(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1))/x, x])/(-4 - 2*x + E^(4*E^(2*x))*x), x] + Defer[
Int][Defer[Int][(E^(4*E^(2*x))*(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1)))/x, x]/x, x] + 4*Defer[Int][Defer[I
nt][(E^(4*E^(2*x))*(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1)))/x, x]/(x*(-4 - 2*x + E^(4*E^(2*x))*x)), x] + 8
*Defer[Int][(E^(2*(2*E^(2*x) + x))*x*Defer[Int][(E^(4*E^(2*x))*(-4 + (-2 + E^(4*E^(2*x)))*x)^(-1 + x^(-1)))/x,
x])/(-4 - 2*x + E^(4*E^(2*x))*x), x]

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx=\left (-4+\left (-2+e^{4 e^{2 x}}\right ) x\right )^{\frac {1}{x}} \]

[In]

Integrate[((-4 - 2*x + E^(4*E^(2*x))*x)^x^(-1)*(-2*x + E^(4*E^(2*x))*(x + 8*E^(2*x)*x^2) + (4 + 2*x - E^(4*E^(
2*x))*x)*Log[-4 - 2*x + E^(4*E^(2*x))*x]))/(-4*x^2 - 2*x^3 + E^(4*E^(2*x))*x^3),x]

[Out]

(-4 + (-2 + E^(4*E^(2*x)))*x)^x^(-1)

Maple [A] (verified)

Time = 14.79 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86

method	result	size
risch	\(\left (x \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}-2 x -4\right )^{\frac {1}{x}}\)	\(19\)
parallelrisch	\({\mathrm e}^{\frac {\ln \left (x \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}-2 x -4\right )}{x}}\)	\(21\)

[In]

int(((-x*exp(4*exp(2*x))+2*x+4)*ln(x*exp(4*exp(2*x))-2*x-4)+(8*exp(2*x)*x^2+x)*exp(4*exp(2*x))-2*x)*exp(ln(x*e
xp(4*exp(2*x))-2*x-4)/x)/(x^3*exp(4*exp(2*x))-2*x^3-4*x^2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx={\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )}^{\left (\frac {1}{x}\right )} \]

[In]

integrate(((-x*exp(4*exp(2*x))+2*x+4)*log(x*exp(4*exp(2*x))-2*x-4)+(8*exp(2*x)*x^2+x)*exp(4*exp(2*x))-2*x)*exp
(log(x*exp(4*exp(2*x))-2*x-4)/x)/(x^3*exp(4*exp(2*x))-2*x^3-4*x^2),x, algorithm="fricas")

[Out]

(x*e^(4*e^(2*x)) - 2*x - 4)^(1/x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx=\text {Timed out} \]

[In]

integrate(((-x*exp(4*exp(2*x))+2*x+4)*ln(x*exp(4*exp(2*x))-2*x-4)+(8*exp(2*x)*x**2+x)*exp(4*exp(2*x))-2*x)*exp
(ln(x*exp(4*exp(2*x))-2*x-4)/x)/(x**3*exp(4*exp(2*x))-2*x**3-4*x**2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx={\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )}^{\left (\frac {1}{x}\right )} \]

[In]

integrate(((-x*exp(4*exp(2*x))+2*x+4)*log(x*exp(4*exp(2*x))-2*x-4)+(8*exp(2*x)*x^2+x)*exp(4*exp(2*x))-2*x)*exp
(log(x*exp(4*exp(2*x))-2*x-4)/x)/(x^3*exp(4*exp(2*x))-2*x^3-4*x^2),x, algorithm="maxima")

[Out]

(x*e^(4*e^(2*x)) - 2*x - 4)^(1/x)

Giac [F]

\[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx=\int { \frac {{\left ({\left (8 \, x^{2} e^{\left (2 \, x\right )} + x\right )} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - {\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )} \log \left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right ) - 2 \, x\right )} {\left (x e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x - 4\right )}^{\left (\frac {1}{x}\right )}}{x^{3} e^{\left (4 \, e^{\left (2 \, x\right )}\right )} - 2 \, x^{3} - 4 \, x^{2}} \,d x } \]

[In]

integrate(((-x*exp(4*exp(2*x))+2*x+4)*log(x*exp(4*exp(2*x))-2*x-4)+(8*exp(2*x)*x^2+x)*exp(4*exp(2*x))-2*x)*exp
(log(x*exp(4*exp(2*x))-2*x-4)/x)/(x^3*exp(4*exp(2*x))-2*x^3-4*x^2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.93 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {\left (-4-2 x+e^{4 e^{2 x}} x\right )^{\frac {1}{x}} \left (-2 x+e^{4 e^{2 x}} \left (x+8 e^{2 x} x^2\right )+\left (4+2 x-e^{4 e^{2 x}} x\right ) \log \left (-4-2 x+e^{4 e^{2 x}} x\right )\right )}{-4 x^2-2 x^3+e^{4 e^{2 x}} x^3} \, dx={\left (x\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}-2\,x-4\right )}^{1/x} \]

[In]

int(-(exp(log(x*exp(4*exp(2*x)) - 2*x - 4)/x)*(exp(4*exp(2*x))*(x + 8*x^2*exp(2*x)) - 2*x + log(x*exp(4*exp(2*
x)) - 2*x - 4)*(2*x - x*exp(4*exp(2*x)) + 4)))/(4*x^2 - x^3*exp(4*exp(2*x)) + 2*x^3),x)

[Out]

(x*exp(4*exp(2*x)) - 2*x - 4)^(1/x)