3.82.93 \(\int \frac {-6+6 x+24 x^2+e^{-4+e^x} (2-2 x-2 e^x x-8 x^2)}{9 e^{x+2 x^2}-6 e^{-4+e^x+x+2 x^2}+e^{-8+2 e^x+x+2 x^2}} \, dx\)

Optimal. Leaf size=25 \[ \frac {2 e^{-x-2 x^2} x}{-3+e^{-4+e^x}} \]

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Rubi [F]  time = 2.43, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-6+6 x+24 x^2+e^{-4+e^x} \left (2-2 x-2 e^x x-8 x^2\right )}{9 e^{x+2 x^2}-6 e^{-4+e^x+x+2 x^2}+e^{-8+2 e^x+x+2 x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6 + 6*x + 24*x^2 + E^(-4 + E^x)*(2 - 2*x - 2*E^x*x - 8*x^2))/(9*E^(x + 2*x^2) - 6*E^(-4 + E^x + x + 2*x^
2) + E^(-8 + 2*E^x + x + 2*x^2)),x]

[Out]

-6*Defer[Int][E^(8 - x - 2*x^2)/(-3*E^4 + E^E^x)^2, x] + 2*Defer[Int][E^(4 + E^x - x - 2*x^2)/(-3*E^4 + E^E^x)
^2, x] - 2*Defer[Int][(E^(4 + E^x - 2*x^2)*x)/(-3*E^4 + E^E^x)^2, x] + 6*Defer[Int][(E^(8 - x - 2*x^2)*x)/(-3*
E^4 + E^E^x)^2, x] - 2*Defer[Int][(E^(4 + E^x - x - 2*x^2)*x)/(-3*E^4 + E^E^x)^2, x] + 24*Defer[Int][(E^(8 - x
- 2*x^2)*x^2)/(-3*E^4 + E^E^x)^2, x] - 8*Defer[Int][(E^(4 + E^x - x - 2*x^2)*x^2)/(-3*E^4 + E^E^x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{8-x-2 x^2} \left (-6+6 x+24 x^2+e^{-4+e^x} \left (2-2 x-2 e^x x-8 x^2\right )\right )}{\left (3 e^4-e^{e^x}\right )^2} \, dx\\ &=\int \left (-\frac {6 e^{8-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2}+\frac {2 e^{4+e^x-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2}-\frac {2 e^{4+e^x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2}+\frac {6 e^{8-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2}-\frac {2 e^{4+e^x-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2}+\frac {24 e^{8-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2}-\frac {8 e^{4+e^x-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2}\right ) \, dx\\ &=2 \int \frac {e^{4+e^x-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-2 \int \frac {e^{4+e^x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-2 \int \frac {e^{4+e^x-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-6 \int \frac {e^{8-x-2 x^2}}{\left (-3 e^4+e^{e^x}\right )^2} \, dx+6 \int \frac {e^{8-x-2 x^2} x}{\left (-3 e^4+e^{e^x}\right )^2} \, dx-8 \int \frac {e^{4+e^x-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2} \, dx+24 \int \frac {e^{8-x-2 x^2} x^2}{\left (-3 e^4+e^{e^x}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.69, size = 28, normalized size = 1.12 \begin {gather*} \frac {2 e^{4-x-2 x^2} x}{-3 e^4+e^{e^x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6 + 6*x + 24*x^2 + E^(-4 + E^x)*(2 - 2*x - 2*E^x*x - 8*x^2))/(9*E^(x + 2*x^2) - 6*E^(-4 + E^x + x
+ 2*x^2) + E^(-8 + 2*E^x + x + 2*x^2)),x]

[Out]

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

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fricas [A]  time = 0.87, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 \, x}{e^{\left (2 \, x^{2} + x + e^{x} - 4\right )} - 3 \, e^{\left (2 \, x^{2} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)*x-8*x^2-2*x+2)*exp(exp(x)-4)+24*x^2+6*x-6)/(exp(2*x^2+x)*exp(exp(x)-4)^2-6*exp(2*x^2+x)*
exp(exp(x)-4)+9*exp(2*x^2+x)),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 29, normalized size = 1.16 \begin {gather*} \frac {2 \, x e^{4}}{e^{\left (2 \, x^{2} + x + e^{x}\right )} - 3 \, e^{\left (2 \, x^{2} + x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)*x-8*x^2-2*x+2)*exp(exp(x)-4)+24*x^2+6*x-6)/(exp(2*x^2+x)*exp(exp(x)-4)^2-6*exp(2*x^2+x)*
exp(exp(x)-4)+9*exp(2*x^2+x)),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.20, size = 22, normalized size = 0.88

method	result	size
risch	\(\frac {2 x \,{\mathrm e}^{-\left (2 x +1\right ) x}}{{\mathrm e}^{{\mathrm e}^{x}-4}-3}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(x)*x-8*x^2-2*x+2)*exp(exp(x)-4)+24*x^2+6*x-6)/(exp(2*x^2+x)*exp(exp(x)-4)^2-6*exp(2*x^2+x)*exp(ex
p(x)-4)+9*exp(2*x^2+x)),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.44, size = 29, normalized size = 1.16 \begin {gather*} \frac {2 \, x e^{4}}{e^{\left (2 \, x^{2} + x + e^{x}\right )} - 3 \, e^{\left (2 \, x^{2} + x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)*x-8*x^2-2*x+2)*exp(exp(x)-4)+24*x^2+6*x-6)/(exp(2*x^2+x)*exp(exp(x)-4)^2-6*exp(2*x^2+x)*
exp(exp(x)-4)+9*exp(2*x^2+x)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 5.48, size = 22, normalized size = 0.88 \begin {gather*} \frac {2\,x\,{\mathrm {e}}^{-2\,x^2-x}}{{\mathrm {e}}^{{\mathrm {e}}^x-4}-3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x - exp(exp(x) - 4)*(2*x + 2*x*exp(x) + 8*x^2 - 2) + 24*x^2 - 6)/(9*exp(x + 2*x^2) - 6*exp(x + 2*x^2)*e
xp(exp(x) - 4) + exp(2*exp(x) - 8)*exp(x + 2*x^2)),x)

[Out]

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

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sympy [A]  time = 0.20, size = 27, normalized size = 1.08 \begin {gather*} \frac {2 x}{e^{2 x^{2} + x} e^{e^{x} - 4} - 3 e^{2 x^{2} + x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(x)*x-8*x**2-2*x+2)*exp(exp(x)-4)+24*x**2+6*x-6)/(exp(2*x**2+x)*exp(exp(x)-4)**2-6*exp(2*x**
2+x)*exp(exp(x)-4)+9*exp(2*x**2+x)),x)

[Out]

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

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