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

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Rubi [A]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2212, 2209} \[ -\frac {1}{2} e^{-x^2} x^2-\frac {e^{-x^2}}{2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/E^x^2,x]

[Out]

-1/(2*E^x^2) - x^2/(2*E^x^2)

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rubi steps

\begin {align*} \int e^{-x^2} x^3 \, dx &=-\frac {1}{2} e^{-x^2} x^2+\int e^{-x^2} x \, dx\\ &=-\frac {e^{-x^2}}{2}-\frac {1}{2} e^{-x^2} x^2\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.62 \[ -\frac {1}{2} e^{-x^2} \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/E^x^2,x]

[Out]

-1/2*(1 + x^2)/E^x^2

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fricas [A]  time = 0.38, size = 13, normalized size = 0.50 \[ -\frac {1}{2} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(x^2),x, algorithm="fricas")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

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giac [A]  time = 0.01, size = 13, normalized size = 0.50 \[ -\frac {1}{2} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(x^2),x, algorithm="giac")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

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maple [A]  time = 0.00, size = 14, normalized size = 0.54 \[ -\frac {\left (x^{2}+1\right ) {\mathrm e}^{-x^{2}}}{2} \]

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)

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maxima [A]  time = 0.51, size = 13, normalized size = 0.50 \[ -\frac {1}{2} \, {\left (x^{2} + 1\right )} e^{\left (-x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/exp(x^2),x, algorithm="maxima")

[Out]

-1/2*(x^2 + 1)*e^(-x^2)

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mupad [B]  time = 0.11, size = 13, normalized size = 0.50 \[ -\frac {{\mathrm {e}}^{-x^2}\,\left (x^2+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*exp(-x^2),x)

[Out]

-(exp(-x^2)*(x^2 + 1))/2

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sympy [A]  time = 0.09, 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

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