3.45.79 \(\int \frac {e^{-x+e^{e^x} x} (-4 x+e^{e^x} (4 x+4 e^x x^2))+(-4+4 e^{-x+e^{e^x} x}) \log (-1+e^{-x+e^{e^x} x})}{-1+e^{-x+e^{e^x} x}} \, dx\)

Optimal. Leaf size=17 \[ 4 x \log \left (-1+e^{\left (-1+e^{e^x}\right ) x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 3.51, antiderivative size = 20, normalized size of antiderivative = 1.18, number of steps used = 15, number of rules used = 4, integrand size = 89, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6742, 2548, 6688, 14} \begin {gather*} 4 x \log \left (e^{-\left (\left (1-e^{e^x}\right ) x\right )}-1\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*x*Log[-1 + E^(-((1 - E^E^x)*x))]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {4 e^x x \left (-1+e^{e^x}+e^{e^x+x} x\right )}{e^x-e^{e^{e^x} x}}+4 \left (-x+e^{e^x} x+e^{e^x+x} x^2+\log \left (-1+e^{\left (-1+e^{e^x}\right ) x}\right )\right )\right ) \, dx\\ &=-\left (4 \int \frac {e^x x \left (-1+e^{e^x}+e^{e^x+x} x\right )}{e^x-e^{e^{e^x} x}} \, dx\right )+4 \int \left (-x+e^{e^x} x+e^{e^x+x} x^2+\log \left (-1+e^{\left (-1+e^{e^x}\right ) x}\right )\right ) \, dx\\ &=-2 x^2+4 \int e^{e^x} x \, dx+4 \int e^{e^x+x} x^2 \, dx-4 \int \left (-\frac {e^x x}{e^x-e^{e^{e^x} x}}+\frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}}+\frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}}\right ) \, dx+4 \int \log \left (-1+e^{\left (-1+e^{e^x}\right ) x}\right ) \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx+4 \int \frac {e^x x}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}} \, dx+4 \int e^{e^x+x} x^2 \, dx-4 \int \frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {e^{-x+e^{e^x} x} x \left (1-e^{e^x}-e^{e^x+x} x\right )}{1-e^{\left (-1+e^{e^x}\right ) x}} \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx+4 \int \frac {e^x x}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}} \, dx+4 \int e^{e^x+x} x^2 \, dx-4 \int \frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {x \left (-1+e^{e^x}+e^{e^x+x} x\right )}{1-e^{x-e^{e^x} x}} \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx+4 \int \frac {e^x x}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}} \, dx+4 \int e^{e^x+x} x^2 \, dx-4 \int \frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}} \, dx-4 \int \left (x \left (-1+e^{e^x}+e^{e^x+x} x\right )-\frac {e^x x \left (-1+e^{e^x}+e^{e^x+x} x\right )}{e^x-e^{e^{e^x} x}}\right ) \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx+4 \int \frac {e^x x}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}} \, dx+4 \int e^{e^x+x} x^2 \, dx-4 \int \frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}} \, dx-4 \int x \left (-1+e^{e^x}+e^{e^x+x} x\right ) \, dx+4 \int \frac {e^x x \left (-1+e^{e^x}+e^{e^x+x} x\right )}{e^x-e^{e^{e^x} x}} \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx+4 \int \frac {e^x x}{e^x-e^{e^{e^x} x}} \, dx-4 \int \frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}} \, dx+4 \int e^{e^x+x} x^2 \, dx-4 \int \frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}} \, dx-4 \int \left (\left (-1+e^{e^x}\right ) x+e^{e^x+x} x^2\right ) \, dx+4 \int \left (-\frac {e^x x}{e^x-e^{e^{e^x} x}}+\frac {e^{e^x+x} x}{e^x-e^{e^{e^x} x}}+\frac {e^{e^x+2 x} x^2}{e^x-e^{e^{e^x} x}}\right ) \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx-4 \int \left (-1+e^{e^x}\right ) x \, dx\\ &=-2 x^2+4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )+4 \int e^{e^x} x \, dx-4 \int \left (-x+e^{e^x} x\right ) \, dx\\ &=4 x \log \left (-1+e^{-\left (\left (1-e^{e^x}\right ) x\right )}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.56, size = 17, normalized size = 1.00 \begin {gather*} 4 x \log \left (-1+e^{\left (-1+e^{e^x}\right ) x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*x*Log[-1 + E^((-1 + E^E^x)*x)]

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 16, normalized size = 0.94 \begin {gather*} 4 \, x \log \left (e^{\left (x e^{\left (e^{x}\right )} - x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x*exp(exp(x))-x)-4)*log(exp(x*exp(exp(x))-x)-1)+((4*exp(x)*x^2+4*x)*exp(exp(x))-4*x)*exp(x*e
xp(exp(x))-x))/(exp(x*exp(exp(x))-x)-1),x, algorithm="fricas")

[Out]

4*x*log(e^(x*e^(e^x) - x) - 1)

________________________________________________________________________________________

giac [A]  time = 0.36, size = 16, normalized size = 0.94 \begin {gather*} 4 \, x \log \left (e^{\left (x e^{\left (e^{x}\right )} - x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x*exp(exp(x))-x)-4)*log(exp(x*exp(exp(x))-x)-1)+((4*exp(x)*x^2+4*x)*exp(exp(x))-4*x)*exp(x*e
xp(exp(x))-x))/(exp(x*exp(exp(x))-x)-1),x, algorithm="giac")

[Out]

4*x*log(e^(x*e^(e^x) - x) - 1)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 15, normalized size = 0.88

method	result	size
risch	\(4 \ln \left (-1+{\mathrm e}^{x \left ({\mathrm e}^{{\mathrm e}^{x}}-1\right )}\right ) x\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*exp(x*exp(exp(x))-x)-4)*ln(exp(x*exp(exp(x))-x)-1)+((4*exp(x)*x^2+4*x)*exp(exp(x))-4*x)*exp(x*exp(exp(
x))-x))/(exp(x*exp(exp(x))-x)-1),x,method=_RETURNVERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.41, size = 21, normalized size = 1.24 \begin {gather*} -4 \, x^{2} + 4 \, x \log \left (e^{\left (x e^{\left (e^{x}\right )}\right )} - e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x*exp(exp(x))-x)-4)*log(exp(x*exp(exp(x))-x)-1)+((4*exp(x)*x^2+4*x)*exp(exp(x))-4*x)*exp(x*e
xp(exp(x))-x))/(exp(x*exp(exp(x))-x)-1),x, algorithm="maxima")

[Out]

-4*x^2 + 4*x*log(e^(x*e^(e^x)) - e^x)

________________________________________________________________________________________

mupad [B]  time = 3.61, size = 16, normalized size = 0.94 \begin {gather*} 4\,x\,\ln \left ({\mathrm {e}}^{x\,{\mathrm {e}}^{{\mathrm {e}}^x}-x}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x*log(exp(x*exp(exp(x)) - x) - 1)

________________________________________________________________________________________

sympy [A]  time = 6.71, size = 15, normalized size = 0.88 \begin {gather*} 4 x \log {\left (e^{x e^{e^{x}} - x} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*exp(x*exp(exp(x))-x)-4)*ln(exp(x*exp(exp(x))-x)-1)+((4*exp(x)*x**2+4*x)*exp(exp(x))-4*x)*exp(x*e
xp(exp(x))-x))/(exp(x*exp(exp(x))-x)-1),x)

[Out]

4*x*log(exp(x*exp(exp(x)) - x) - 1)

________________________________________________________________________________________