3.22.63 \(\int \frac {e^{-20+5 x \log (-1-e+e^{e^x+x})} (e^{e^x+x} (5 x+5 e^x x)+(-5-5 e+5 e^{e^x+x}) \log (-1-e+e^{e^x+x}))}{-1-e+e^{e^x+x}} \, dx\)

Optimal. Leaf size=20 \[ e^{-20+5 x \log \left (-1-e+e^{e^x+x}\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 0.59, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {6706} \begin {gather*} \frac {\left (e^{x+e^x}-1-e\right )^{5 x}}{e^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(-20 + 5*x*Log[-1 - E + E^(E^x + x)])*(E^(E^x + x)*(5*x + 5*E^x*x) + (-5 - 5*E + 5*E^(E^x + x))*Log[-1
- E + E^(E^x + x)]))/(-1 - E + E^(E^x + x)),x]

[Out]

(-1 - E + E^(E^x + x))^(5*x)/E^20

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\left (-1-e+e^{e^x+x}\right )^{5 x}}{e^{20}}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.56, size = 20, normalized size = 1.00 \begin {gather*} \frac {\left (-1-e+e^{e^x+x}\right )^{5 x}}{e^{20}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-20 + 5*x*Log[-1 - E + E^(E^x + x)])*(E^(E^x + x)*(5*x + 5*E^x*x) + (-5 - 5*E + 5*E^(E^x + x))*L
og[-1 - E + E^(E^x + x)]))/(-1 - E + E^(E^x + x)),x]

[Out]

(-1 - E + E^(E^x + x))^(5*x)/E^20

________________________________________________________________________________________

fricas [A]  time = 0.59, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (5 \, x \log \left (-e + e^{\left (x + e^{x}\right )} - 1\right ) - 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(exp(x)+x)-5*exp(1)-5)*log(exp(exp(x)+x)-exp(1)-1)+(5*exp(x)*x+5*x)*exp(exp(x)+x))*exp(5*x*lo
g(exp(exp(x)+x)-exp(1)-1)-20)/(exp(exp(x)+x)-exp(1)-1),x, algorithm="fricas")

[Out]

e^(5*x*log(-e + e^(x + e^x) - 1) - 20)

________________________________________________________________________________________

giac [A]  time = 0.64, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (5 \, x \log \left (-e + e^{\left (x + e^{x}\right )} - 1\right ) - 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(exp(x)+x)-5*exp(1)-5)*log(exp(exp(x)+x)-exp(1)-1)+(5*exp(x)*x+5*x)*exp(exp(x)+x))*exp(5*x*lo
g(exp(exp(x)+x)-exp(1)-1)-20)/(exp(exp(x)+x)-exp(1)-1),x, algorithm="giac")

[Out]

e^(5*x*log(-e + e^(x + e^x) - 1) - 20)

________________________________________________________________________________________

maple [A]  time = 0.08, size = 19, normalized size = 0.95

method	result	size
risch	\(\left ({\mathrm e}^{{\mathrm e}^{x}+x}-{\mathrm e}-1\right )^{5 x} {\mathrm e}^{-20}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*exp(exp(x)+x)-5*exp(1)-5)*ln(exp(exp(x)+x)-exp(1)-1)+(5*exp(x)*x+5*x)*exp(exp(x)+x))*exp(5*x*ln(exp(ex
p(x)+x)-exp(1)-1)-20)/(exp(exp(x)+x)-exp(1)-1),x,method=_RETURNVERBOSE)

[Out]

(exp(exp(x)+x)-exp(1)-1)^(5*x)*exp(-20)

________________________________________________________________________________________

maxima [A]  time = 0.54, size = 18, normalized size = 0.90 \begin {gather*} e^{\left (5 \, x \log \left (-e + e^{\left (x + e^{x}\right )} - 1\right ) - 20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(exp(x)+x)-5*exp(1)-5)*log(exp(exp(x)+x)-exp(1)-1)+(5*exp(x)*x+5*x)*exp(exp(x)+x))*exp(5*x*lo
g(exp(exp(x)+x)-exp(1)-1)-20)/(exp(exp(x)+x)-exp(1)-1),x, algorithm="maxima")

[Out]

e^(5*x*log(-e + e^(x + e^x) - 1) - 20)

________________________________________________________________________________________

mupad [B]  time = 1.52, size = 18, normalized size = 0.90 \begin {gather*} {\mathrm {e}}^{-20}\,{\left ({\mathrm {e}}^{x+{\mathrm {e}}^x}-\mathrm {e}-1\right )}^{5\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(5*x*log(exp(x + exp(x)) - exp(1) - 1) - 20)*(exp(x + exp(x))*(5*x + 5*x*exp(x)) - log(exp(x + exp(x)
) - exp(1) - 1)*(5*exp(1) - 5*exp(x + exp(x)) + 5)))/(exp(1) - exp(x + exp(x)) + 1),x)

[Out]

exp(-20)*(exp(x + exp(x)) - exp(1) - 1)^(5*x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(exp(x)+x)-5*exp(1)-5)*ln(exp(exp(x)+x)-exp(1)-1)+(5*exp(x)*x+5*x)*exp(exp(x)+x))*exp(5*x*ln(
exp(exp(x)+x)-exp(1)-1)-20)/(exp(exp(x)+x)-exp(1)-1),x)

[Out]

Timed out

________________________________________________________________________________________