3.11.62 \(\int \frac {1}{2} e^{-e^{x^2}-2 x} (20 x-14 x^2-8 x^3+2 x^4+e (10-16 x-7 x^2+2 x^3)+e^{x^2} (-20 x^3-4 x^4+2 x^5+e (-20 x^2-4 x^3+2 x^4))) \, dx\)

Optimal. Leaf size=30 \[ e^{-e^{x^2}-2 x} x (e+x) \left (5+\frac {1}{2} (2-x) x\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 2.64, 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 {1}{2} e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-E^x^2 - 2*x)*(20*x - 14*x^2 - 8*x^3 + 2*x^4 + E*(10 - 16*x - 7*x^2 + 2*x^3) + E^x^2*(-20*x^3 - 4*x^4
+ 2*x^5 + E*(-20*x^2 - 4*x^3 + 2*x^4))))/2,x]

[Out]

5*Defer[Int][E^(1 - E^x^2 - 2*x), x] + 10*Defer[Int][E^(-E^x^2 - 2*x)*x, x] - 8*Defer[Int][E^(1 - E^x^2 - 2*x)
*x, x] - 7*Defer[Int][E^(-E^x^2 - 2*x)*x^2, x] - (7*Defer[Int][E^(1 - E^x^2 - 2*x)*x^2, x])/2 - 10*Defer[Int][
E^(1 - E^x^2 - 2*x + x^2)*x^2, x] - 4*Defer[Int][E^(-E^x^2 - 2*x)*x^3, x] + Defer[Int][E^(1 - E^x^2 - 2*x)*x^3
, x] - 2*(5 + E)*Defer[Int][E^(-E^x^2 - 2*x + x^2)*x^3, x] + Defer[Int][E^(-E^x^2 - 2*x)*x^4, x] - (2 - E)*Def
er[Int][E^(-E^x^2 - 2*x + x^2)*x^4, x] + Defer[Int][E^(-E^x^2 - 2*x + x^2)*x^5, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{-e^{x^2}-2 x} \left (20 x-14 x^2-8 x^3+2 x^4+e \left (10-16 x-7 x^2+2 x^3\right )+e^{x^2} \left (-20 x^3-4 x^4+2 x^5+e \left (-20 x^2-4 x^3+2 x^4\right )\right )\right ) \, dx\\ &=\frac {1}{2} \int \left (20 e^{-e^{x^2}-2 x} x-14 e^{-e^{x^2}-2 x} x^2-8 e^{-e^{x^2}-2 x} x^3+2 e^{-e^{x^2}-2 x} x^4+2 e^{-e^{x^2}-2 x+x^2} x^2 (e+x) \left (-10-2 x+x^2\right )+e^{1-e^{x^2}-2 x} \left (10-16 x-7 x^2+2 x^3\right )\right ) \, dx\\ &=\frac {1}{2} \int e^{1-e^{x^2}-2 x} \left (10-16 x-7 x^2+2 x^3\right ) \, dx-4 \int e^{-e^{x^2}-2 x} x^3 \, dx-7 \int e^{-e^{x^2}-2 x} x^2 \, dx+10 \int e^{-e^{x^2}-2 x} x \, dx+\int e^{-e^{x^2}-2 x} x^4 \, dx+\int e^{-e^{x^2}-2 x+x^2} x^2 (e+x) \left (-10-2 x+x^2\right ) \, dx\\ &=\frac {1}{2} \int \left (10 e^{1-e^{x^2}-2 x}-16 e^{1-e^{x^2}-2 x} x-7 e^{1-e^{x^2}-2 x} x^2+2 e^{1-e^{x^2}-2 x} x^3\right ) \, dx-4 \int e^{-e^{x^2}-2 x} x^3 \, dx-7 \int e^{-e^{x^2}-2 x} x^2 \, dx+10 \int e^{-e^{x^2}-2 x} x \, dx+\int e^{-e^{x^2}-2 x} x^4 \, dx+\int \left (-10 e^{1-e^{x^2}-2 x+x^2} x^2-2 e^{-e^{x^2}-2 x+x^2} (5+e) x^3+(-2+e) e^{-e^{x^2}-2 x+x^2} x^4+e^{-e^{x^2}-2 x+x^2} x^5\right ) \, dx\\ &=-\left (\frac {7}{2} \int e^{1-e^{x^2}-2 x} x^2 \, dx\right )-4 \int e^{-e^{x^2}-2 x} x^3 \, dx+5 \int e^{1-e^{x^2}-2 x} \, dx-7 \int e^{-e^{x^2}-2 x} x^2 \, dx-8 \int e^{1-e^{x^2}-2 x} x \, dx+10 \int e^{-e^{x^2}-2 x} x \, dx-10 \int e^{1-e^{x^2}-2 x+x^2} x^2 \, dx+(-2+e) \int e^{-e^{x^2}-2 x+x^2} x^4 \, dx-(2 (5+e)) \int e^{-e^{x^2}-2 x+x^2} x^3 \, dx+\int e^{1-e^{x^2}-2 x} x^3 \, dx+\int e^{-e^{x^2}-2 x} x^4 \, dx+\int e^{-e^{x^2}-2 x+x^2} x^5 \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 29, normalized size = 0.97 \begin {gather*} -\frac {1}{2} e^{-e^{x^2}-2 x} x (e+x) \left (-10-2 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-E^x^2 - 2*x)*(20*x - 14*x^2 - 8*x^3 + 2*x^4 + E*(10 - 16*x - 7*x^2 + 2*x^3) + E^x^2*(-20*x^3 -
4*x^4 + 2*x^5 + E*(-20*x^2 - 4*x^3 + 2*x^4))))/2,x]

[Out]

-1/2*(E^(-E^x^2 - 2*x)*x*(E + x)*(-10 - 2*x + x^2))

________________________________________________________________________________________

fricas [A]  time = 0.85, size = 42, normalized size = 1.40 \begin {gather*} -\frac {1}{2} \, {\left (x^{4} - 2 \, x^{3} - 10 \, x^{2} + {\left (x^{3} - 2 \, x^{2} - 10 \, x\right )} e\right )} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+(2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*
x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x),x, algorithm="fricas")

[Out]

-1/2*(x^4 - 2*x^3 - 10*x^2 + (x^3 - 2*x^2 - 10*x)*e)*e^(-2*x - e^(x^2))

________________________________________________________________________________________

giac [B]  time = 0.29, size = 96, normalized size = 3.20 \begin {gather*} -\frac {1}{2} \, x^{4} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} - \frac {1}{2} \, x^{3} e^{\left (-2 \, x - e^{\left (x^{2}\right )} + 1\right )} + x^{3} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} + x^{2} e^{\left (-2 \, x - e^{\left (x^{2}\right )} + 1\right )} + 5 \, x^{2} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} + 5 \, x e^{\left (-2 \, x - e^{\left (x^{2}\right )} + 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+(2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*
x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x),x, algorithm="giac")

[Out]

-1/2*x^4*e^(-2*x - e^(x^2)) - 1/2*x^3*e^(-2*x - e^(x^2) + 1) + x^3*e^(-2*x - e^(x^2)) + x^2*e^(-2*x - e^(x^2)
+ 1) + 5*x^2*e^(-2*x - e^(x^2)) + 5*x*e^(-2*x - e^(x^2) + 1)

________________________________________________________________________________________

maple [A]  time = 0.05, size = 42, normalized size = 1.40

method	result	size
risch	\(-\frac {\left (x^{2} {\mathrm e}+x^{3}-2 x \,{\mathrm e}-2 x^{2}-10 \,{\mathrm e}-10 x \right ) x \,{\mathrm e}^{-2 x -{\mathrm e}^{x^{2}}}}{2}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+(2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*x^3-14
*x^2+20*x)/exp(x)/exp(exp(x^2)+x),x,method=_RETURNVERBOSE)

[Out]

-1/2*(x^2*exp(1)+x^3-2*x*exp(1)-2*x^2-10*exp(1)-10*x)*x*exp(-2*x-exp(x^2))

________________________________________________________________________________________

maxima [A]  time = 0.62, size = 39, normalized size = 1.30 \begin {gather*} -\frac {1}{2} \, {\left (x^{4} + x^{3} {\left (e - 2\right )} - 2 \, x^{2} {\left (e + 5\right )} - 10 \, x e\right )} e^{\left (-2 \, x - e^{\left (x^{2}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((2*x^4-4*x^3-20*x^2)*exp(1)+2*x^5-4*x^4-20*x^3)*exp(x^2)+(2*x^3-7*x^2-16*x+10)*exp(1)+2*x^4-8*
x^3-14*x^2+20*x)/exp(x)/exp(exp(x^2)+x),x, algorithm="maxima")

[Out]

-1/2*(x^4 + x^3*(e - 2) - 2*x^2*(e + 5) - 10*x*e)*e^(-2*x - e^(x^2))

________________________________________________________________________________________

mupad [B]  time = 0.88, size = 28, normalized size = 0.93 \begin {gather*} \frac {x\,{\mathrm {e}}^{-2\,x-{\mathrm {e}}^{x^2}}\,\left (x+\mathrm {e}\right )\,\left (-x^2+2\,x+10\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- x - exp(x^2))*exp(-x)*((exp(x^2)*(exp(1)*(20*x^2 + 4*x^3 - 2*x^4) + 20*x^3 + 4*x^4 - 2*x^5))/2 - 10
*x + (exp(1)*(16*x + 7*x^2 - 2*x^3 - 10))/2 + 7*x^2 + 4*x^3 - x^4),x)

[Out]

(x*exp(- 2*x - exp(x^2))*(x + exp(1))*(2*x - x^2 + 10))/2

________________________________________________________________________________________

sympy [B]  time = 82.20, size = 66, normalized size = 2.20 \begin {gather*} \frac {\left (- x^{4} e^{- x} - e x^{3} e^{- x} + 2 x^{3} e^{- x} + 2 e x^{2} e^{- x} + 10 x^{2} e^{- x} + 10 e x e^{- x}\right ) e^{- x - e^{x^{2}}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((2*x**4-4*x**3-20*x**2)*exp(1)+2*x**5-4*x**4-20*x**3)*exp(x**2)+(2*x**3-7*x**2-16*x+10)*exp(1)
+2*x**4-8*x**3-14*x**2+20*x)/exp(x)/exp(exp(x**2)+x),x)

[Out]

(-x**4*exp(-x) - E*x**3*exp(-x) + 2*x**3*exp(-x) + 2*E*x**2*exp(-x) + 10*x**2*exp(-x) + 10*E*x*exp(-x))*exp(-x
- exp(x**2))/2

________________________________________________________________________________________