3.35.54 \(\int \frac {-120-120 e^2+e^{e^{3 x^2}} (-24-24 e^2-288 e^{3 x^2} x)}{125 e^{x+e^2 x}+75 e^{e^{3 x^2}+x+e^2 x}+15 e^{2 e^{3 x^2}+x+e^2 x}+e^{3 e^{3 x^2}+x+e^2 x}} \, dx\)

Optimal. Leaf size=27 \[ \frac {24 e^{-x-e^2 x}}{\left (5+e^{e^{3 x^2}}\right )^2} \]

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Rubi [A]  time = 0.48, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, integrand size = 105, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 12, 2288} \begin {gather*} \frac {24 e^{-\left (\left (1+e^2\right ) x\right )}}{\left (e^{e^{3 x^2}}+5\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-120 - 120*E^2 + E^E^(3*x^2)*(-24 - 24*E^2 - 288*E^(3*x^2)*x))/(125*E^(x + E^2*x) + 75*E^(E^(3*x^2) + x +
E^2*x) + 15*E^(2*E^(3*x^2) + x + E^2*x) + E^(3*E^(3*x^2) + x + E^2*x)),x]

[Out]

24/(E^((1 + E^2)*x)*(5 + E^E^(3*x^2))^2)

Rule 12

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

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {24 e^{-\left (\left (1+e^2\right ) x\right )} \left (-5 \left (1+e^2\right )-e^{e^{3 x^2}} \left (1+e^2\right )-12 e^{e^{3 x^2}+3 x^2} x\right )}{\left (5+e^{e^{3 x^2}}\right )^3} \, dx\\ &=24 \int \frac {e^{-\left (\left (1+e^2\right ) x\right )} \left (-5 \left (1+e^2\right )-e^{e^{3 x^2}} \left (1+e^2\right )-12 e^{e^{3 x^2}+3 x^2} x\right )}{\left (5+e^{e^{3 x^2}}\right )^3} \, dx\\ &=\frac {24 e^{-\left (\left (1+e^2\right ) x\right )}}{\left (5+e^{e^{3 x^2}}\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.53, size = 25, normalized size = 0.93 \begin {gather*} \frac {24 e^{-\left (\left (1+e^2\right ) x\right )}}{\left (5+e^{e^{3 x^2}}\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-120 - 120*E^2 + E^E^(3*x^2)*(-24 - 24*E^2 - 288*E^(3*x^2)*x))/(125*E^(x + E^2*x) + 75*E^(E^(3*x^2)
+ x + E^2*x) + 15*E^(2*E^(3*x^2) + x + E^2*x) + E^(3*E^(3*x^2) + x + E^2*x)),x]

[Out]

24/(E^((1 + E^2)*x)*(5 + E^E^(3*x^2))^2)

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fricas [B]  time = 0.59, size = 60, normalized size = 2.22 \begin {gather*} \frac {24 \, e^{\left (x e^{2} + x\right )}}{e^{\left (2 \, x e^{2} + 2 \, x + 2 \, e^{\left (3 \, x^{2}\right )}\right )} + 10 \, e^{\left (2 \, x e^{2} + 2 \, x + e^{\left (3 \, x^{2}\right )}\right )} + 25 \, e^{\left (2 \, x e^{2} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-288*x*exp(3*x^2)-24*exp(2)-24)*exp(exp(3*x^2))-120*exp(2)-120)/(exp(x+exp(2)*x)*exp(exp(3*x^2))^3
+15*exp(x+exp(2)*x)*exp(exp(3*x^2))^2+75*exp(x+exp(2)*x)*exp(exp(3*x^2))+125*exp(x+exp(2)*x)),x, algorithm="fr
icas")

[Out]

24*e^(x*e^2 + x)/(e^(2*x*e^2 + 2*x + 2*e^(3*x^2)) + 10*e^(2*x*e^2 + 2*x + e^(3*x^2)) + 25*e^(2*x*e^2 + 2*x))

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giac [B]  time = 0.24, size = 134, normalized size = 4.96 \begin {gather*} \frac {24 \, {\left (x e^{\left (9 \, x^{2} + x e^{2} + x + e^{\left (3 \, x^{2}\right )}\right )} + 5 \, x e^{\left (9 \, x^{2} + x e^{2} + x\right )}\right )}}{x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x + 3 \, e^{\left (3 \, x^{2}\right )}\right )} + 15 \, x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x + 2 \, e^{\left (3 \, x^{2}\right )}\right )} + 75 \, x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x + e^{\left (3 \, x^{2}\right )}\right )} + 125 \, x e^{\left (9 \, x^{2} + 2 \, x e^{2} + 2 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-288*x*exp(3*x^2)-24*exp(2)-24)*exp(exp(3*x^2))-120*exp(2)-120)/(exp(x+exp(2)*x)*exp(exp(3*x^2))^3
+15*exp(x+exp(2)*x)*exp(exp(3*x^2))^2+75*exp(x+exp(2)*x)*exp(exp(3*x^2))+125*exp(x+exp(2)*x)),x, algorithm="gi
ac")

[Out]

24*(x*e^(9*x^2 + x*e^2 + x + e^(3*x^2)) + 5*x*e^(9*x^2 + x*e^2 + x))/(x*e^(9*x^2 + 2*x*e^2 + 2*x + 3*e^(3*x^2)
) + 15*x*e^(9*x^2 + 2*x*e^2 + 2*x + 2*e^(3*x^2)) + 75*x*e^(9*x^2 + 2*x*e^2 + 2*x + e^(3*x^2)) + 125*x*e^(9*x^2
+ 2*x*e^2 + 2*x))

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maple [A]  time = 0.07, size = 22, normalized size = 0.81

method	result	size
risch	\(\frac {24 \,{\mathrm e}^{-\left ({\mathrm e}^{2}+1\right ) x}}{\left (5+{\mathrm e}^{{\mathrm e}^{3 x^{2}}}\right )^{2}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-288*x*exp(3*x^2)-24*exp(2)-24)*exp(exp(3*x^2))-120*exp(2)-120)/(exp(x+exp(2)*x)*exp(exp(3*x^2))^3+15*ex
p(x+exp(2)*x)*exp(exp(3*x^2))^2+75*exp(x+exp(2)*x)*exp(exp(3*x^2))+125*exp(x+exp(2)*x)),x,method=_RETURNVERBOS
E)

[Out]

24*exp(-(exp(2)+1)*x)/(5+exp(exp(3*x^2)))^2

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maxima [A]  time = 0.64, size = 44, normalized size = 1.63 \begin {gather*} \frac {24}{e^{\left (x e^{2} + x + 2 \, e^{\left (3 \, x^{2}\right )}\right )} + 10 \, e^{\left (x e^{2} + x + e^{\left (3 \, x^{2}\right )}\right )} + 25 \, e^{\left (x e^{2} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-288*x*exp(3*x^2)-24*exp(2)-24)*exp(exp(3*x^2))-120*exp(2)-120)/(exp(x+exp(2)*x)*exp(exp(3*x^2))^3
+15*exp(x+exp(2)*x)*exp(exp(3*x^2))^2+75*exp(x+exp(2)*x)*exp(exp(3*x^2))+125*exp(x+exp(2)*x)),x, algorithm="ma
xima")

[Out]

24/(e^(x*e^2 + x + 2*e^(3*x^2)) + 10*e^(x*e^2 + x + e^(3*x^2)) + 25*e^(x*e^2 + x))

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mupad [B]  time = 2.16, size = 34, normalized size = 1.26 \begin {gather*} \frac {24\,{\mathrm {e}}^{-x-x\,{\mathrm {e}}^2}}{10\,{\mathrm {e}}^{{\mathrm {e}}^{3\,x^2}}+{\mathrm {e}}^{2\,{\mathrm {e}}^{3\,x^2}}+25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(120*exp(2) + exp(exp(3*x^2))*(24*exp(2) + 288*x*exp(3*x^2) + 24) + 120)/(125*exp(x + x*exp(2)) + 15*exp(
2*exp(3*x^2))*exp(x + x*exp(2)) + exp(3*exp(3*x^2))*exp(x + x*exp(2)) + 75*exp(exp(3*x^2))*exp(x + x*exp(2))),
x)

[Out]

(24*exp(- x - x*exp(2)))/(10*exp(exp(3*x^2)) + exp(2*exp(3*x^2)) + 25)

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sympy [A]  time = 0.34, size = 48, normalized size = 1.78 \begin {gather*} \frac {24}{e^{x + x e^{2}} e^{2 e^{3 x^{2}}} + 10 e^{x + x e^{2}} e^{e^{3 x^{2}}} + 25 e^{x + x e^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-288*x*exp(3*x**2)-24*exp(2)-24)*exp(exp(3*x**2))-120*exp(2)-120)/(exp(x+exp(2)*x)*exp(exp(3*x**2)
)**3+15*exp(x+exp(2)*x)*exp(exp(3*x**2))**2+75*exp(x+exp(2)*x)*exp(exp(3*x**2))+125*exp(x+exp(2)*x)),x)

[Out]

24/(exp(x + x*exp(2))*exp(2*exp(3*x**2)) + 10*exp(x + x*exp(2))*exp(exp(3*x**2)) + 25*exp(x + x*exp(2)))

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