3.4.80 \(\int \frac {-2-5 x+2 x^2+e^{e^x} (1+2 x-2 e^x x)+(6+e^{e^x} (-2+2 e^x)-2 x) \log (2-e^{e^x}-x)}{-2+e^{e^x}+x} \, dx\)

Optimal. Leaf size=24 \[ -\frac {1}{3}+x+\left (-x+\log \left (2-e^{e^x}-x\right )\right )^2 \]

________________________________________________________________________________________

Rubi [F]  time = 2.83, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-2-5 x+2 x^2+e^{e^x} \left (1+2 x-2 e^x x\right )+\left (6+e^{e^x} \left (-2+2 e^x\right )-2 x\right ) \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-2*Defer[Int][(-2 + E^E^x + x)^(-1), x] + 6*Log[2 - E^E^x - x]*Defer[Int][(-2 + E^E^x + x)^(-1), x] + Defer[In
t][E^E^x/(-2 + E^E^x + x), x] - 2*Log[2 - E^E^x - x]*Defer[Int][E^E^x/(-2 + E^E^x + x), x] + 2*Log[2 - E^E^x -
x]*Defer[Int][E^(E^x + x)/(-2 + E^E^x + x), x] - 5*Defer[Int][x/(-2 + E^E^x + x), x] - 2*Log[2 - E^E^x - x]*D
efer[Int][x/(-2 + E^E^x + x), x] + 2*Defer[Int][(E^E^x*x)/(-2 + E^E^x + x), x] - 2*Defer[Int][(E^(E^x + x)*x)/
(-2 + E^E^x + x), x] + 2*Defer[Int][x^2/(-2 + E^E^x + x), x] - 6*Defer[Int][Defer[Int][(-2 + E^E^x + x)^(-1),
x]/(-2 + E^E^x + x), x] - 6*Defer[Int][(E^(E^x + x)*Defer[Int][(-2 + E^E^x + x)^(-1), x])/(-2 + E^E^x + x), x]
+ 2*Defer[Int][Defer[Int][E^E^x/(-2 + E^E^x + x), x]/(-2 + E^E^x + x), x] + 2*Defer[Int][(E^(E^x + x)*Defer[I
nt][E^E^x/(-2 + E^E^x + x), x])/(-2 + E^E^x + x), x] - 2*Defer[Int][Defer[Int][E^(E^x + x)/(-2 + E^E^x + x), x
]/(-2 + E^E^x + x), x] - 2*Defer[Int][(E^(E^x + x)*Defer[Int][E^(E^x + x)/(-2 + E^E^x + x), x])/(-2 + E^E^x +
x), x] + 2*Defer[Int][Defer[Int][x/(-2 + E^E^x + x), x]/(-2 + E^E^x + x), x] + 2*Defer[Int][(E^(E^x + x)*Defer
[Int][x/(-2 + E^E^x + x), x])/(-2 + E^E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{-2+e^{e^x}+x}+\frac {e^{e^x}}{-2+e^{e^x}+x}-\frac {5 x}{-2+e^{e^x}+x}+\frac {2 e^{e^x} x}{-2+e^{e^x}+x}+\frac {2 x^2}{-2+e^{e^x}+x}-\frac {2 e^{e^x+x} \left (x-\log \left (2-e^{e^x}-x\right )\right )}{-2+e^{e^x}+x}+\frac {6 \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x}-\frac {2 e^{e^x} \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x}-\frac {2 x \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x}\right ) \, dx\\ &=-\left (2 \int \frac {1}{-2+e^{e^x}+x} \, dx\right )+2 \int \frac {e^{e^x} x}{-2+e^{e^x}+x} \, dx+2 \int \frac {x^2}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x+x} \left (x-\log \left (2-e^{e^x}-x\right )\right )}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x} \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x} \, dx-2 \int \frac {x \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x} \, dx-5 \int \frac {x}{-2+e^{e^x}+x} \, dx+6 \int \frac {\log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x} \, dx+\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx\\ &=-\left (2 \int \frac {1}{-2+e^{e^x}+x} \, dx\right )+2 \int \frac {e^{e^x} x}{-2+e^{e^x}+x} \, dx+2 \int \frac {x^2}{-2+e^{e^x}+x} \, dx-2 \int \left (\frac {e^{e^x+x} x}{-2+e^{e^x}+x}-\frac {e^{e^x+x} \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x}\right ) \, dx+2 \int \frac {\left (-1-e^{e^x+x}\right ) \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{2-e^{e^x}-x} \, dx+2 \int \frac {\left (-1-e^{e^x+x}\right ) \int \frac {x}{-2+e^{e^x}+x} \, dx}{2-e^{e^x}-x} \, dx-5 \int \frac {x}{-2+e^{e^x}+x} \, dx-6 \int \frac {\left (-1-e^{e^x+x}\right ) \int \frac {1}{-2+e^{e^x}+x} \, dx}{2-e^{e^x}-x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {x}{-2+e^{e^x}+x} \, dx+\left (6 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {1}{-2+e^{e^x}+x} \, dx+\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx\\ &=-\left (2 \int \frac {1}{-2+e^{e^x}+x} \, dx\right )+2 \int \frac {e^{e^x} x}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x+x} x}{-2+e^{e^x}+x} \, dx+2 \int \frac {x^2}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \log \left (2-e^{e^x}-x\right )}{-2+e^{e^x}+x} \, dx+2 \int \left (\frac {\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}+\frac {e^{e^x+x} \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}\right ) \, dx+2 \int \left (\frac {\int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}+\frac {e^{e^x+x} \int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}\right ) \, dx-5 \int \frac {x}{-2+e^{e^x}+x} \, dx-6 \int \left (\frac {\int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}+\frac {e^{e^x+x} \int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}\right ) \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {x}{-2+e^{e^x}+x} \, dx+\left (6 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {1}{-2+e^{e^x}+x} \, dx+\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx\\ &=-\left (2 \int \frac {1}{-2+e^{e^x}+x} \, dx\right )+2 \int \frac {e^{e^x} x}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x+x} x}{-2+e^{e^x}+x} \, dx+2 \int \frac {x^2}{-2+e^{e^x}+x} \, dx+2 \int \frac {\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-2 \int \frac {\left (-1-e^{e^x+x}\right ) \int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx}{2-e^{e^x}-x} \, dx+2 \int \frac {\int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-5 \int \frac {x}{-2+e^{e^x}+x} \, dx-6 \int \frac {\int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-6 \int \frac {e^{e^x+x} \int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx+\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {x}{-2+e^{e^x}+x} \, dx+\left (6 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {1}{-2+e^{e^x}+x} \, dx+\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx\\ &=-\left (2 \int \frac {1}{-2+e^{e^x}+x} \, dx\right )+2 \int \frac {e^{e^x} x}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x+x} x}{-2+e^{e^x}+x} \, dx+2 \int \frac {x^2}{-2+e^{e^x}+x} \, dx+2 \int \frac {\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-2 \int \left (\frac {\int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}+\frac {e^{e^x+x} \int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x}\right ) \, dx+2 \int \frac {\int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-5 \int \frac {x}{-2+e^{e^x}+x} \, dx-6 \int \frac {\int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-6 \int \frac {e^{e^x+x} \int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx+\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {x}{-2+e^{e^x}+x} \, dx+\left (6 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {1}{-2+e^{e^x}+x} \, dx+\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx\\ &=-\left (2 \int \frac {1}{-2+e^{e^x}+x} \, dx\right )+2 \int \frac {e^{e^x} x}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x+x} x}{-2+e^{e^x}+x} \, dx+2 \int \frac {x^2}{-2+e^{e^x}+x} \, dx+2 \int \frac {\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-2 \int \frac {\int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-2 \int \frac {e^{e^x+x} \int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {\int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx+2 \int \frac {e^{e^x+x} \int \frac {x}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-5 \int \frac {x}{-2+e^{e^x}+x} \, dx-6 \int \frac {\int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-6 \int \frac {e^{e^x+x} \int \frac {1}{-2+e^{e^x}+x} \, dx}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx+\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {e^{e^x+x}}{-2+e^{e^x}+x} \, dx-\left (2 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {x}{-2+e^{e^x}+x} \, dx+\left (6 \log \left (2-e^{e^x}-x\right )\right ) \int \frac {1}{-2+e^{e^x}+x} \, dx+\int \frac {e^{e^x}}{-2+e^{e^x}+x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.37, size = 36, normalized size = 1.50 \begin {gather*} x+x^2-2 x \log \left (2-e^{e^x}-x\right )+\log ^2\left (2-e^{e^x}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - 5*x + 2*x^2 + E^E^x*(1 + 2*x - 2*E^x*x) + (6 + E^E^x*(-2 + 2*E^x) - 2*x)*Log[2 - E^E^x - x])/(
-2 + E^E^x + x),x]

[Out]

x + x^2 - 2*x*Log[2 - E^E^x - x] + Log[2 - E^E^x - x]^2

________________________________________________________________________________________

fricas [A]  time = 0.91, size = 32, normalized size = 1.33 \begin {gather*} x^{2} - 2 \, x \log \left (-x - e^{\left (e^{x}\right )} + 2\right ) + \log \left (-x - e^{\left (e^{x}\right )} + 2\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{2} - {\left (2 \, x e^{x} - 2 \, x - 1\right )} e^{\left (e^{x}\right )} + 2 \, {\left ({\left (e^{x} - 1\right )} e^{\left (e^{x}\right )} - x + 3\right )} \log \left (-x - e^{\left (e^{x}\right )} + 2\right ) - 5 \, x - 2}{x + e^{\left (e^{x}\right )} - 2}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.05, size = 33, normalized size = 1.38

method	result	size
risch	\(x^{2}-2 \ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+2-x \right ) x +\ln \left (-{\mathrm e}^{{\mathrm e}^{x}}+2-x \right )^{2}+x\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*exp(x)-2)*exp(exp(x))+6-2*x)*ln(-exp(exp(x))+2-x)+(-2*exp(x)*x+2*x+1)*exp(exp(x))+2*x^2-5*x-2)/(exp(e
xp(x))+x-2),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 32, normalized size = 1.33 \begin {gather*} x^{2} - 2 \, x \log \left (-x - e^{\left (e^{x}\right )} + 2\right ) + \log \left (-x - e^{\left (e^{x}\right )} + 2\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.15, size = 32, normalized size = 1.33 \begin {gather*} x^2-2\,x\,\ln \left (2-{\mathrm {e}}^{{\mathrm {e}}^x}-x\right )+x+{\ln \left (2-{\mathrm {e}}^{{\mathrm {e}}^x}-x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 0.51, size = 29, normalized size = 1.21 \begin {gather*} x^{2} - 2 x \log {\left (- x - e^{e^{x}} + 2 \right )} + x + \log {\left (- x - e^{e^{x}} + 2 \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________