3.38.80 \(\int \frac {e^{2-e^{10}+e^{\frac {2 x}{-6+2 e^{x^2}}}+x} (9-6 e^{x^2}+e^{2 x^2}+e^{\frac {2 x}{-6+2 e^{x^2}}} (-3+e^{x^2} (1-2 x^2)))}{9-6 e^{x^2}+e^{2 x^2}} \, dx\)

Optimal. Leaf size=23 \[ e^{2-e^{10}+e^{\frac {x}{-3+e^{x^2}}}+x} \]

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Rubi [A]  time = 2.77, antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 1, number of rules used = 1, integrand size = 93, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {6706} \begin {gather*} e^{e^{-\frac {x}{3-e^{x^2}}}+x-e^{10}+2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(2 - E^10 + E^((2*x)/(-6 + 2*E^x^2)) + x)*(9 - 6*E^x^2 + E^(2*x^2) + E^((2*x)/(-6 + 2*E^x^2))*(-3 + E^x
^2*(1 - 2*x^2))))/(9 - 6*E^x^2 + E^(2*x^2)),x]

[Out]

E^(2 - E^10 + E^(-(x/(3 - E^x^2))) + x)

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} &=e^{2-e^{10}+e^{-\frac {x}{3-e^{x^2}}}+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 23, normalized size = 1.00 \begin {gather*} e^{2-e^{10}+e^{\frac {x}{-3+e^{x^2}}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2 - E^10 + E^((2*x)/(-6 + 2*E^x^2)) + x)*(9 - 6*E^x^2 + E^(2*x^2) + E^((2*x)/(-6 + 2*E^x^2))*(-3
+ E^x^2*(1 - 2*x^2))))/(9 - 6*E^x^2 + E^(2*x^2)),x]

[Out]

E^(2 - E^10 + E^(x/(-3 + E^x^2)) + x)

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fricas [A]  time = 0.63, size = 19, normalized size = 0.83 \begin {gather*} e^{\left (x - e^{10} + e^{\left (\frac {x}{e^{\left (x^{2}\right )} - 3}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2+1)*exp(x^2)-3)*exp(x/(2*exp(x^2)-6))^2+exp(x^2)^2-6*exp(x^2)+9)*exp(exp(x/(2*exp(x^2)-6))^
2-exp(5)^2+2+x)/(exp(x^2)^2-6*exp(x^2)+9),x, algorithm="fricas")

[Out]

e^(x - e^10 + e^(x/(e^(x^2) - 3)) + 2)

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giac [A]  time = 0.24, size = 19, normalized size = 0.83 \begin {gather*} e^{\left (x - e^{10} + e^{\left (\frac {x}{e^{\left (x^{2}\right )} - 3}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2+1)*exp(x^2)-3)*exp(x/(2*exp(x^2)-6))^2+exp(x^2)^2-6*exp(x^2)+9)*exp(exp(x/(2*exp(x^2)-6))^
2-exp(5)^2+2+x)/(exp(x^2)^2-6*exp(x^2)+9),x, algorithm="giac")

[Out]

e^(x - e^10 + e^(x/(e^(x^2) - 3)) + 2)

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maple [A]  time = 0.09, size = 20, normalized size = 0.87

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {x}{{\mathrm e}^{x^{2}}-3}}-{\mathrm e}^{10}+2+x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x^2+1)*exp(x^2)-3)*exp(x/(2*exp(x^2)-6))^2+exp(x^2)^2-6*exp(x^2)+9)*exp(exp(x/(2*exp(x^2)-6))^2-exp(
5)^2+2+x)/(exp(x^2)^2-6*exp(x^2)+9),x,method=_RETURNVERBOSE)

[Out]

exp(exp(x/(exp(x^2)-3))-exp(10)+2+x)

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maxima [A]  time = 0.76, size = 19, normalized size = 0.83 \begin {gather*} e^{\left (x - e^{10} + e^{\left (\frac {x}{e^{\left (x^{2}\right )} - 3}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x^2+1)*exp(x^2)-3)*exp(x/(2*exp(x^2)-6))^2+exp(x^2)^2-6*exp(x^2)+9)*exp(exp(x/(2*exp(x^2)-6))^
2-exp(5)^2+2+x)/(exp(x^2)^2-6*exp(x^2)+9),x, algorithm="maxima")

[Out]

e^(x - e^10 + e^(x/(e^(x^2) - 3)) + 2)

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mupad [B]  time = 2.51, size = 22, normalized size = 0.96 \begin {gather*} {\mathrm {e}}^{-{\mathrm {e}}^{10}}\,{\mathrm {e}}^2\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{{\mathrm {e}}^{x^2}-3}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x - exp(10) + exp((2*x)/(2*exp(x^2) - 6)) + 2)*(6*exp(x^2) - exp(2*x^2) + exp((2*x)/(2*exp(x^2) - 6)
)*(exp(x^2)*(2*x^2 - 1) + 3) - 9))/(exp(2*x^2) - 6*exp(x^2) + 9),x)

[Out]

exp(-exp(10))*exp(2)*exp(exp(x/(exp(x^2) - 3)))*exp(x)

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sympy [A]  time = 1.48, size = 20, normalized size = 0.87 \begin {gather*} e^{x + e^{\frac {2 x}{2 e^{x^{2}} - 6}} - e^{10} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x**2+1)*exp(x**2)-3)*exp(x/(2*exp(x**2)-6))**2+exp(x**2)**2-6*exp(x**2)+9)*exp(exp(x/(2*exp(x*
*2)-6))**2-exp(5)**2+2+x)/(exp(x**2)**2-6*exp(x**2)+9),x)

[Out]

exp(x + exp(2*x/(2*exp(x**2) - 6)) - exp(10) + 2)

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