\(\int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} (2 x-8 e^{x^4} x^4)}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx\) [1876]

Optimal result

Integrand size = 69, antiderivative size = 26 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=\log \left (\frac {1}{50} \left (-1-e^{e^{-4-e^{x^4}+x}}+x\right )^2\right ) \]

[Out]

ln(1/10*(x-exp(exp(-exp(x^4)+x-4))-1)*(1/5*x-1/5*exp(exp(-exp(x^4)+x-4))-1/5))

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 e^{-4-e^{x^4}} \left (e^{4+e^{x^4}}-e^{e^{-4-e^{x^4}+x}+x}\right )}{1+e^{e^{-4-e^{x^4}+x}}-x}-\frac {8 e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x}\right ) \, dx \\ & = -\left (2 \int \frac {e^{-4-e^{x^4}} \left (e^{4+e^{x^4}}-e^{e^{-4-e^{x^4}+x}+x}\right )}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx\right )-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ & = -\left (2 \int \frac {1-e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x}}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx\right )-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ & = -\left (2 \int \left (\frac {1}{1+e^{e^{-4-e^{x^4}+x}}-x}-\frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x}}{1+e^{e^{-4-e^{x^4}+x}}-x}\right ) \, dx\right )-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ & = -\left (2 \int \frac {1}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx\right )+2 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x}}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx-8 \int \frac {e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x+x^4} x^3}{1+e^{e^{-4-e^{x^4}+x}}-x} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \log \left (1+e^{e^{-4-e^{x^4}+x}}-x\right ) \]

[In]

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

[Out]

2*Log[1 + E^E^(-4 - E^x^4 + x) - x]

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.77

method	result	size
risch	\(2 \ln \left ({\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x^{4}}+x -4}}-x +1\right )\)	\(20\)
parallelrisch	\(2 \ln \left (x -{\mathrm e}^{{\mathrm e}^{-{\mathrm e}^{x^{4}}+x -4}}-1\right )\)	\(20\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \, \log \left (-x + e^{\left (e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.65 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \log {\left (- x + e^{e^{x - e^{x^{4}} - 4}} + 1 \right )} \]

[In]

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

[Out]

2*log(-x + exp(exp(x - exp(x**4) - 4)) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.73 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2 \, \log \left (-x + e^{\left (e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + 1\right ) \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (21) = 42\).

Time = 0.34 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.08 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=-2 \, x + 2 \, e^{\left (x^{4}\right )} + 2 \, \log \left (-x e^{\left (x - e^{\left (x^{4}\right )}\right )} + e^{\left (x - e^{\left (x^{4}\right )} + e^{\left (x - e^{\left (x^{4}\right )} - 4\right )}\right )} + e^{\left (x - e^{\left (x^{4}\right )}\right )}\right ) \]

[In]

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

[Out]

-2*x + 2*e^(x^4) + 2*log(-x*e^(x - e^(x^4)) + e^(x - e^(x^4) + e^(x - e^(x^4) - 4)) + e^(x - e^(x^4)))

Mupad [B] (verification not implemented)

Time = 8.63 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.81 \[ \int \frac {-2 x+e^{-4-e^{x^4}+e^{-4-e^{x^4}+x}+x} \left (2 x-8 e^{x^4} x^4\right )}{x+e^{e^{-4-e^{x^4}+x}} x-x^2} \, dx=2\,\ln \left ({\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{x^4}}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}-x+1\right ) \]

[In]

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

[Out]

2*log(exp(exp(-exp(x^4))*exp(-4)*exp(x)) - x + 1)