3.101.90 \(\int \frac {e^{-\frac {x^2}{1+3 x+e^{4 x} x-x^2-x^3}} (-2 x-3 x^2-x^4+e^{4 x} (-x^2+4 x^3))}{1+6 x+7 x^2+e^{8 x} x^2-8 x^3-5 x^4+2 x^5+x^6+e^{4 x} (2 x+6 x^2-2 x^3-2 x^4)} \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {x}{-3-e^{4 x}-\frac {1}{x}+x+x^2}} \]

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Rubi [A]  time = 2.02, antiderivative size = 30, normalized size of antiderivative = 1.25, number of steps used = 2, number of rules used = 2, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.016, Rules used = {6688, 6706} \begin {gather*} e^{-\frac {x^2}{-x^3-x^2+\left (e^{4 x}+3\right ) x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x - 3*x^2 - x^4 + E^(4*x)*(-x^2 + 4*x^3))/(E^(x^2/(1 + 3*x + E^(4*x)*x - x^2 - x^3))*(1 + 6*x + 7*x^2
+ E^(8*x)*x^2 - 8*x^3 - 5*x^4 + 2*x^5 + x^6 + E^(4*x)*(2*x + 6*x^2 - 2*x^3 - 2*x^4))),x]

[Out]

E^(-(x^2/(1 + (3 + E^(4*x))*x - x^2 - x^3)))

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \frac {e^{\frac {x^2}{-1-\left (3+e^{4 x}\right ) x+x^2+x^3}} x \left (-2-\left (3+e^{4 x}\right ) x+4 e^{4 x} x^2-x^3\right )}{\left (1+\left (3+e^{4 x}\right ) x-x^2-x^3\right )^2} \, dx\\ &=e^{-\frac {x^2}{1+\left (3+e^{4 x}\right ) x-x^2-x^3}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.92, size = 26, normalized size = 1.08 \begin {gather*} e^{\frac {x^2}{-1-\left (3+e^{4 x}\right ) x+x^2+x^3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - 3*x^2 - x^4 + E^(4*x)*(-x^2 + 4*x^3))/(E^(x^2/(1 + 3*x + E^(4*x)*x - x^2 - x^3))*(1 + 6*x +
7*x^2 + E^(8*x)*x^2 - 8*x^3 - 5*x^4 + 2*x^5 + x^6 + E^(4*x)*(2*x + 6*x^2 - 2*x^3 - 2*x^4))),x]

[Out]

E^(x^2/(-1 - (3 + E^(4*x))*x + x^2 + x^3))

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fricas [A]  time = 0.96, size = 25, normalized size = 1.04 \begin {gather*} e^{\left (\frac {x^{2}}{x^{3} + x^{2} - x e^{\left (4 \, x\right )} - 3 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-x^2)*exp(4*x)-x^4-3*x^2-2*x)*exp(-x^2/(x*exp(4*x)-x^3-x^2+3*x+1))/(x^2*exp(4*x)^2+(-2*x^4-2*
x^3+6*x^2+2*x)*exp(4*x)+x^6+2*x^5-5*x^4-8*x^3+7*x^2+6*x+1),x, algorithm="fricas")

[Out]

e^(x^2/(x^3 + x^2 - x*e^(4*x) - 3*x - 1))

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giac [A]  time = 0.14, size = 25, normalized size = 1.04 \begin {gather*} e^{\left (\frac {x^{2}}{x^{3} + x^{2} - x e^{\left (4 \, x\right )} - 3 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-x^2)*exp(4*x)-x^4-3*x^2-2*x)*exp(-x^2/(x*exp(4*x)-x^3-x^2+3*x+1))/(x^2*exp(4*x)^2+(-2*x^4-2*
x^3+6*x^2+2*x)*exp(4*x)+x^6+2*x^5-5*x^4-8*x^3+7*x^2+6*x+1),x, algorithm="giac")

[Out]

e^(x^2/(x^3 + x^2 - x*e^(4*x) - 3*x - 1))

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maple [A]  time = 0.16, size = 26, normalized size = 1.08

method	result	size
risch	\({\mathrm e}^{\frac {x^{2}}{x^{3}+x^{2}-x \,{\mathrm e}^{4 x}-3 x -1}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-x^2)*exp(4*x)-x^4-3*x^2-2*x)*exp(-x^2/(x*exp(4*x)-x^3-x^2+3*x+1))/(x^2*exp(4*x)^2+(-2*x^4-2*x^3+6*
x^2+2*x)*exp(4*x)+x^6+2*x^5-5*x^4-8*x^3+7*x^2+6*x+1),x,method=_RETURNVERBOSE)

[Out]

exp(x^2/(x^3+x^2-x*exp(4*x)-3*x-1))

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maxima [A]  time = 0.52, size = 25, normalized size = 1.04 \begin {gather*} e^{\left (\frac {x^{2}}{x^{3} + x^{2} - x e^{\left (4 \, x\right )} - 3 \, x - 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-x^2)*exp(4*x)-x^4-3*x^2-2*x)*exp(-x^2/(x*exp(4*x)-x^3-x^2+3*x+1))/(x^2*exp(4*x)^2+(-2*x^4-2*
x^3+6*x^2+2*x)*exp(4*x)+x^6+2*x^5-5*x^4-8*x^3+7*x^2+6*x+1),x, algorithm="maxima")

[Out]

e^(x^2/(x^3 + x^2 - x*e^(4*x) - 3*x - 1))

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mupad [B]  time = 7.06, size = 28, normalized size = 1.17 \begin {gather*} {\mathrm {e}}^{-\frac {x^2}{x\,\left ({\mathrm {e}}^{4\,x}+3\right )-x^2-x^3+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(-x^2/(3*x + x*exp(4*x) - x^2 - x^3 + 1))*(2*x + exp(4*x)*(x^2 - 4*x^3) + 3*x^2 + x^4))/(6*x + x^2*ex
p(8*x) + exp(4*x)*(2*x + 6*x^2 - 2*x^3 - 2*x^4) + 7*x^2 - 8*x^3 - 5*x^4 + 2*x^5 + x^6 + 1),x)

[Out]

exp(-x^2/(x*(exp(4*x) + 3) - x^2 - x^3 + 1))

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sympy [A]  time = 0.58, size = 24, normalized size = 1.00 \begin {gather*} e^{- \frac {x^{2}}{- x^{3} - x^{2} + x e^{4 x} + 3 x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-x**2)*exp(4*x)-x**4-3*x**2-2*x)*exp(-x**2/(x*exp(4*x)-x**3-x**2+3*x+1))/(x**2*exp(4*x)**2+(
-2*x**4-2*x**3+6*x**2+2*x)*exp(4*x)+x**6+2*x**5-5*x**4-8*x**3+7*x**2+6*x+1),x)

[Out]

exp(-x**2/(-x**3 - x**2 + x*exp(4*x) + 3*x + 1))

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