3.47.56 \(\int \frac {1}{99} (99+800 e^{e^{-\frac {400 x^2}{99}}-\frac {400 x^2}{99}} x) \, dx\)

Optimal. Leaf size=16 \[ -4-e^{e^{-\frac {400 x^2}{99}}}+x \]

________________________________________________________________________________________

Rubi [A]  time = 0.07, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 6715, 2282, 2194} \begin {gather*} x-e^{e^{-\frac {400 x^2}{99}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(99 + 800*E^(E^((-400*x^2)/99) - (400*x^2)/99)*x)/99,x]

[Out]

-E^E^((-400*x^2)/99) + x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{99} \int \left (99+800 e^{e^{-\frac {400 x^2}{99}}-\frac {400 x^2}{99}} x\right ) \, dx\\ &=x+\frac {800}{99} \int e^{e^{-\frac {400 x^2}{99}}-\frac {400 x^2}{99}} x \, dx\\ &=x+\frac {400}{99} \operatorname {Subst}\left (\int e^{e^{-400 x/99}-\frac {400 x}{99}} \, dx,x,x^2\right )\\ &=x-\operatorname {Subst}\left (\int e^x \, dx,x,e^{-\frac {400 x^2}{99}}\right )\\ &=-e^{e^{-\frac {400 x^2}{99}}}+x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 15, normalized size = 0.94 \begin {gather*} -e^{e^{-\frac {400 x^2}{99}}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(99 + 800*E^(E^((-400*x^2)/99) - (400*x^2)/99)*x)/99,x]

[Out]

-E^E^((-400*x^2)/99) + x

________________________________________________________________________________________

fricas [B]  time = 0.97, size = 31, normalized size = 1.94 \begin {gather*} {\left (x e^{\left (-\frac {400}{99} \, x^{2}\right )} - e^{\left (-\frac {400}{99} \, x^{2} + e^{\left (-\frac {400}{99} \, x^{2}\right )}\right )}\right )} e^{\left (\frac {400}{99} \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(800/99*x*exp(-400/99*x^2)*exp(exp(-400/99*x^2))+1,x, algorithm="fricas")

[Out]

(x*e^(-400/99*x^2) - e^(-400/99*x^2 + e^(-400/99*x^2)))*e^(400/99*x^2)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 11, normalized size = 0.69 \begin {gather*} x - e^{\left (e^{\left (-\frac {400}{99} \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(800/99*x*exp(-400/99*x^2)*exp(exp(-400/99*x^2))+1,x, algorithm="giac")

[Out]

x - e^(e^(-400/99*x^2))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 12, normalized size = 0.75

method	result	size
default	\(x -{\mathrm e}^{{\mathrm e}^{-\frac {400 x^{2}}{99}}}\)	\(12\)
norman	\(x -{\mathrm e}^{{\mathrm e}^{-\frac {400 x^{2}}{99}}}\)	\(12\)
risch	\(x -{\mathrm e}^{{\mathrm e}^{-\frac {400 x^{2}}{99}}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(800/99*x*exp(-400/99*x^2)*exp(exp(-400/99*x^2))+1,x,method=_RETURNVERBOSE)

[Out]

x-exp(exp(-400/99*x^2))

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 11, normalized size = 0.69 \begin {gather*} x - e^{\left (e^{\left (-\frac {400}{99} \, x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(800/99*x*exp(-400/99*x^2)*exp(exp(-400/99*x^2))+1,x, algorithm="maxima")

[Out]

x - e^(e^(-400/99*x^2))

________________________________________________________________________________________

mupad [B]  time = 0.08, size = 11, normalized size = 0.69 \begin {gather*} x-{\mathrm {e}}^{{\mathrm {e}}^{-\frac {400\,x^2}{99}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((800*x*exp(exp(-(400*x^2)/99))*exp(-(400*x^2)/99))/99 + 1,x)

[Out]

x - exp(exp(-(400*x^2)/99))

________________________________________________________________________________________

sympy [A]  time = 0.25, size = 12, normalized size = 0.75 \begin {gather*} x - e^{e^{- \frac {400 x^{2}}{99}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(800/99*x*exp(-400/99*x**2)*exp(exp(-400/99*x**2))+1,x)

[Out]

x - exp(exp(-400*x**2/99))

________________________________________________________________________________________