3.79 \(\int e^{-x^2} x^3 \, dx\)

Optimal. Leaf size=26 \[ -\frac{1}{2} e^{-x^2} x^2-\frac{e^{-x^2}}{2} \]

[Out]

-1/(2*E^x^2) - x^2/(2*E^x^2)

_______________________________________________________________________________________

Rubi [A]  time = 0.0340855, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ -\frac{1}{2} e^{-x^2} x^2-\frac{e^{-x^2}}{2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/E^x^2,x]

[Out]

-1/(2*E^x^2) - x^2/(2*E^x^2)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.91424, size = 19, normalized size = 0.73 \[ - \frac{x^{2} e^{- x^{2}}}{2} - \frac{e^{- x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/exp(x**2),x)

[Out]

-x**2*exp(-x**2)/2 - exp(-x**2)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.00428297, size = 18, normalized size = 0.69 \[ \frac{1}{2} e^{-x^2} \left (-x^2-1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/E^x^2,x]

[Out]

(-1 - x^2)/(2*E^x^2)

_______________________________________________________________________________________

Maple [A]  time = 0.004, size = 14, normalized size = 0.5 \[ -{\frac{{x}^{2}+1}{2\,{{\rm e}^{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/exp(x^2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.40734, size = 18, normalized size = 0.69 \[ -\frac{1}{2} \,{\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(-x^2),x, algorithm="maxima")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.212363, size = 18, normalized size = 0.69 \[ -\frac{1}{2} \,{\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(-x^2),x, algorithm="fricas")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

_______________________________________________________________________________________

Sympy [A]  time = 0.079511, size = 12, normalized size = 0.46 \[ \frac{\left (- x^{2} - 1\right ) e^{- x^{2}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/exp(x**2),x)

[Out]

(-x**2 - 1)*exp(-x**2)/2

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.236536, size = 18, normalized size = 0.69 \[ -\frac{1}{2} \,{\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3*e^(-x^2),x, algorithm="giac")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)