3.63.46 \(\int e^{-e^{625} x} (-24+e^{e^{625} x}+24 e^{625} x) \, dx\) [6246]

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

[Out]

ln(6/5/exp(24*x/exp(x*exp(625))-x))

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Rubi [A]
time = 0.06, 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 = {6874, 2225, 2207} \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 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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 6874

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\text {Subst}\left (\int e^{-x} \left (-24+e^x+24 x\right ) \, dx,x,e^{625} x\right )}{e^{625}}\\ &=\frac {\text {Subst}\left (\int \left (1-24 e^{-x}+24 e^{-x} x\right ) \, dx,x,e^{625} x\right )}{e^{625}}\\ &=x-\frac {24 \text {Subst}\left (\int e^{-x} \, dx,x,e^{625} x\right )}{e^{625}}+\frac {24 \text {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 \text {Subst}\left (\int e^{-x} \, dx,x,e^{625} x\right )}{e^{625}}\\ &=x-24 e^{-e^{625} x} x\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.11, 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)

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Maple [A]
time = 1.36, size = 23, normalized size = 1.15

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]

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

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Maxima [A]
time = 0.28, 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)

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Fricas [A]
time = 0.33, 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)

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Sympy [A]
time = 0.04, 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))

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Giac [A]
time = 0.40, size = 19, normalized size = 0.95 \begin {gather*} {\left (x e^{625} - 24 \, x e^{\left (-x e^{625} + 625\right )}\right )} e^{\left (-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]

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

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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)

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