3.8.27 \(\int e^{-x^2} (x^4+x^6+x^8) \, dx\) [727]

Optimal. Leaf size=66 \[ -\frac {147}{16} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {147}{32} \sqrt {\pi } \text {erf}(x) \]

[Out]

-147/16*x/exp(x^2)-49/8*x^3/exp(x^2)-9/4*x^5/exp(x^2)-1/2*x^7/exp(x^2)+147/32*erf(x)*Pi^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1608, 2258, 2243, 2236} \begin {gather*} \frac {147}{32} \sqrt {\pi } \text {Erf}(x)-\frac {147}{16} e^{-x^2} x-\frac {1}{2} e^{-x^2} x^7-\frac {9}{4} e^{-x^2} x^5-\frac {49}{8} e^{-x^2} x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4 + x^6 + x^8)/E^x^2,x]

[Out]

(-147*x)/(16*E^x^2) - (49*x^3)/(8*E^x^2) - (9*x^5)/(4*E^x^2) - x^7/(2*E^x^2) + (147*Sqrt[Pi]*Erf[x])/32

Rule 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2243

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])

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {align*} \int e^{-x^2} \left (x^4+x^6+x^8\right ) \, dx &=\int e^{-x^2} x^4 \left (1+x^2+x^4\right ) \, dx\\ &=\int \left (e^{-x^2} x^4+e^{-x^2} x^6+e^{-x^2} x^8\right ) \, dx\\ &=\int e^{-x^2} x^4 \, dx+\int e^{-x^2} x^6 \, dx+\int e^{-x^2} x^8 \, dx\\ &=-\frac {1}{2} e^{-x^2} x^3-\frac {1}{2} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {3}{2} \int e^{-x^2} x^2 \, dx+\frac {5}{2} \int e^{-x^2} x^4 \, dx+\frac {7}{2} \int e^{-x^2} x^6 \, dx\\ &=-\frac {3}{4} e^{-x^2} x-\frac {7}{4} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {3}{4} \int e^{-x^2} \, dx+\frac {15}{4} \int e^{-x^2} x^2 \, dx+\frac {35}{4} \int e^{-x^2} x^4 \, dx\\ &=-\frac {21}{8} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {3}{8} \sqrt {\pi } \text {erf}(x)+\frac {15}{8} \int e^{-x^2} \, dx+\frac {105}{8} \int e^{-x^2} x^2 \, dx\\ &=-\frac {147}{16} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {21}{16} \sqrt {\pi } \text {erf}(x)+\frac {105}{16} \int e^{-x^2} \, dx\\ &=-\frac {147}{16} e^{-x^2} x-\frac {49}{8} e^{-x^2} x^3-\frac {9}{4} e^{-x^2} x^5-\frac {1}{2} e^{-x^2} x^7+\frac {147}{32} \sqrt {\pi } \text {erf}(x)\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 41, normalized size = 0.62 \begin {gather*} \frac {1}{32} \left (-2 e^{-x^2} x \left (147+98 x^2+36 x^4+8 x^6\right )+147 \sqrt {\pi } \text {erf}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^4 + x^6 + x^8)/E^x^2,x]

[Out]

((-2*x*(147 + 98*x^2 + 36*x^4 + 8*x^6))/E^x^2 + 147*Sqrt[Pi]*Erf[x])/32

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Maple [A]
time = 0.03, size = 51, normalized size = 0.77

method	result	size
default	\(-\frac {147 x \,{\mathrm e}^{-x^{2}}}{16}-\frac {49 x^{3} {\mathrm e}^{-x^{2}}}{8}-\frac {9 x^{5} {\mathrm e}^{-x^{2}}}{4}-\frac {x^{7} {\mathrm e}^{-x^{2}}}{2}+\frac {147 \erf \left (x \right ) \sqrt {\pi }}{32}\)	\(51\)
risch	\(-\frac {147 x \,{\mathrm e}^{-x^{2}}}{16}-\frac {49 x^{3} {\mathrm e}^{-x^{2}}}{8}-\frac {9 x^{5} {\mathrm e}^{-x^{2}}}{4}-\frac {x^{7} {\mathrm e}^{-x^{2}}}{2}+\frac {147 \erf \left (x \right ) \sqrt {\pi }}{32}\)	\(51\)
meijerg	\(-\frac {x \left (72 x^{6}+252 x^{4}+630 x^{2}+945\right ) {\mathrm e}^{-x^{2}}}{144}+\frac {147 \erf \left (x \right ) \sqrt {\pi }}{32}-\frac {x \left (28 x^{4}+70 x^{2}+105\right ) {\mathrm e}^{-x^{2}}}{56}-\frac {x \left (10 x^{2}+15\right ) {\mathrm e}^{-x^{2}}}{20}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8+x^6+x^4)/exp(x^2),x,method=_RETURNVERBOSE)

[Out]

-147/16*x/exp(x^2)-49/8*x^3/exp(x^2)-9/4*x^5/exp(x^2)-1/2*x^7/exp(x^2)+147/32*erf(x)*Pi^(1/2)

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Maxima [A]
time = 0.29, size = 74, normalized size = 1.12 \begin {gather*} -\frac {1}{16} \, {\left (8 \, x^{7} + 28 \, x^{5} + 70 \, x^{3} + 105 \, x\right )} e^{\left (-x^{2}\right )} - \frac {1}{8} \, {\left (4 \, x^{5} + 10 \, x^{3} + 15 \, x\right )} e^{\left (-x^{2}\right )} - \frac {1}{4} \, {\left (2 \, x^{3} + 3 \, x\right )} e^{\left (-x^{2}\right )} + \frac {147}{32} \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(8*x^7 + 28*x^5 + 70*x^3 + 105*x)*e^(-x^2) - 1/8*(4*x^5 + 10*x^3 + 15*x)*e^(-x^2) - 1/4*(2*x^3 + 3*x)*e^
(-x^2) + 147/32*sqrt(pi)*erf(x)

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Fricas [A]
time = 0.36, size = 35, normalized size = 0.53 \begin {gather*} -\frac {1}{16} \, {\left (8 \, x^{7} + 36 \, x^{5} + 98 \, x^{3} + 147 \, x\right )} e^{\left (-x^{2}\right )} + \frac {147}{32} \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(8*x^7 + 36*x^5 + 98*x^3 + 147*x)*e^(-x^2) + 147/32*sqrt(pi)*erf(x)

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Sympy [A]
time = 6.50, size = 54, normalized size = 0.82 \begin {gather*} - \frac {x^{7} e^{- x^{2}}}{2} - \frac {9 x^{5} e^{- x^{2}}}{4} - \frac {49 x^{3} e^{- x^{2}}}{8} - \frac {147 x e^{- x^{2}}}{16} + \frac {147 \sqrt {\pi } \operatorname {erf}{\left (x \right )}}{32} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x**7*exp(-x**2)/2 - 9*x**5*exp(-x**2)/4 - 49*x**3*exp(-x**2)/8 - 147*x*exp(-x**2)/16 + 147*sqrt(pi)*erf(x)/32

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Giac [A]
time = 3.35, size = 35, normalized size = 0.53 \begin {gather*} -\frac {1}{16} \, {\left (8 \, x^{7} + 36 \, x^{5} + 98 \, x^{3} + 147 \, x\right )} e^{\left (-x^{2}\right )} + \frac {147}{32} \, \sqrt {\pi } \operatorname {erf}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(8*x^7 + 36*x^5 + 98*x^3 + 147*x)*e^(-x^2) + 147/32*sqrt(pi)*erf(x)

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Mupad [B]
time = 3.62, size = 50, normalized size = 0.76 \begin {gather*} \frac {147\,\sqrt {\pi }\,\mathrm {erf}\left (x\right )}{32}-\frac {49\,x^3\,{\mathrm {e}}^{-x^2}}{8}-\frac {9\,x^5\,{\mathrm {e}}^{-x^2}}{4}-\frac {x^7\,{\mathrm {e}}^{-x^2}}{2}-\frac {147\,x\,{\mathrm {e}}^{-x^2}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x^2)*(x^4 + x^6 + x^8),x)

[Out]

(147*pi^(1/2)*erf(x))/32 - (49*x^3*exp(-x^2))/8 - (9*x^5*exp(-x^2))/4 - (x^7*exp(-x^2))/2 - (147*x*exp(-x^2))/
16

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