3.63.46 \(\int e^{-e^{625} x} (-24+e^{e^{625} x}+24 e^{625} x) \, dx\)

Optimal. Leaf size=20 \[ \log \left (\frac {6}{5} e^{x-24 e^{-e^{625} x} x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 13, normalized size of antiderivative = 0.65, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6742, 2194, 2176} \begin {gather*} x-24 e^{-e^{625} x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-24 + E^(E^625*x) + 24*E^625*x)/E^(E^625*x),x]

[Out]

x - (24*x)/E^(E^625*x)

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, 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} &=\frac {\operatorname {Subst}\left (\int e^{-x} \left (-24+e^x+24 x\right ) \, dx,x,e^{625} x\right )}{e^{625}}\\ &=\frac {\operatorname {Subst}\left (\int \left (1-24 e^{-x}+24 e^{-x} x\right ) \, dx,x,e^{625} x\right )}{e^{625}}\\ &=x-\frac {24 \operatorname {Subst}\left (\int e^{-x} \, dx,x,e^{625} x\right )}{e^{625}}+\frac {24 \operatorname {Subst}\left (\int e^{-x} x \, dx,x,e^{625} x\right )}{e^{625}}\\ &=24 e^{-625-e^{625} x}+x-24 e^{-e^{625} x} x+\frac {24 \operatorname {Subst}\left (\int e^{-x} \, dx,x,e^{625} x\right )}{e^{625}}\\ &=x-24 e^{-e^{625} x} x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 13, normalized size = 0.65 \begin {gather*} x-24 e^{-e^{625} x} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-24 + E^(E^625*x) + 24*E^625*x)/E^(E^625*x),x]

[Out]

x - (24*x)/E^(E^625*x)

________________________________________________________________________________________

fricas [A]  time = 0.77, size = 18, normalized size = 0.90 \begin {gather*} {\left (x e^{\left (x e^{625}\right )} - 24 \, x\right )} e^{\left (-x e^{625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x*exp(625))+24*x*exp(625)-24)/exp(x*exp(625)),x, algorithm="fricas")

[Out]

(x*e^(x*e^625) - 24*x)*e^(-x*e^625)

________________________________________________________________________________________

giac [A]  time = 0.21, size = 28, normalized size = 1.40 \begin {gather*} -24 \, {\left (x e^{625} + 1\right )} e^{\left (-x e^{625} - 625\right )} + x + 24 \, e^{\left (-x e^{625} - 625\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x*exp(625))+24*x*exp(625)-24)/exp(x*exp(625)),x, algorithm="giac")

[Out]

-24*(x*e^625 + 1)*e^(-x*e^625 - 625) + x + 24*e^(-x*e^625 - 625)

________________________________________________________________________________________

maple [A]  time = 0.10, size = 12, normalized size = 0.60

method	result	size
risch	\(x -24 x \,{\mathrm e}^{-x \,{\mathrm e}^{625}}\)	\(12\)
norman	\(\left (x \,{\mathrm e}^{x \,{\mathrm e}^{625}}-24 x \right ) {\mathrm e}^{-x \,{\mathrm e}^{625}}\)	\(20\)
derivativedivides	\({\mathrm e}^{-625} \left (x \,{\mathrm e}^{625}-24 \,{\mathrm e}^{-x \,{\mathrm e}^{625}} x \,{\mathrm e}^{625}\right )\)	\(23\)
default	\({\mathrm e}^{-625} \left (x \,{\mathrm e}^{625}-24 \,{\mathrm e}^{-x \,{\mathrm e}^{625}} x \,{\mathrm e}^{625}\right )\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*exp(625))+24*x*exp(625)-24)/exp(x*exp(625)),x,method=_RETURNVERBOSE)

[Out]

x-24*x*exp(-x*exp(625))

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 28, normalized size = 1.40 \begin {gather*} -24 \, {\left (x e^{625} + 1\right )} e^{\left (-x e^{625} - 625\right )} + x + 24 \, e^{\left (-x e^{625} - 625\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x*exp(625))+24*x*exp(625)-24)/exp(x*exp(625)),x, algorithm="maxima")

[Out]

-24*(x*e^625 + 1)*e^(-x*e^625 - 625) + x + 24*e^(-x*e^625 - 625)

________________________________________________________________________________________

mupad [B]  time = 0.07, size = 13, normalized size = 0.65 \begin {gather*} -x\,\left (24\,{\mathrm {e}}^{-x\,{\mathrm {e}}^{625}}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x*exp(625))*(exp(x*exp(625)) + 24*x*exp(625) - 24),x)

[Out]

-x*(24*exp(-x*exp(625)) - 1)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 10, normalized size = 0.50 \begin {gather*} x - 24 x e^{- x e^{625}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((exp(x*exp(625))+24*x*exp(625)-24)/exp(x*exp(625)),x)

[Out]

x - 24*x*exp(-x*exp(625))

________________________________________________________________________________________