3.57.84 \(\int \frac {1}{2} (-12-3 e^{18}-6 x+e^{2 x^2} (-2-8 x^2)) \, dx\)

Optimal. Leaf size=26 \[ x \left (-e^{2 x^2}+\frac {1}{2} \left (-3-3 \left (3+e^{18}+x\right )\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {12, 2226, 2204, 2212} \begin {gather*} -\frac {3 x^2}{2}-e^{2 x^2} x-\frac {3}{2} \left (4+e^{18}\right ) x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12 - 3*E^18 - 6*x + E^(2*x^2)*(-2 - 8*x^2))/2,x]

[Out]

-(E^(2*x^2)*x) - (3*(4 + E^18)*x)/2 - (3*x^2)/2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2204

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

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])

Rule 2226

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (-12-3 e^{18}-6 x+e^{2 x^2} \left (-2-8 x^2\right )\right ) \, dx\\ &=-\frac {3}{2} \left (4+e^{18}\right ) x-\frac {3 x^2}{2}+\frac {1}{2} \int e^{2 x^2} \left (-2-8 x^2\right ) \, dx\\ &=-\frac {3}{2} \left (4+e^{18}\right ) x-\frac {3 x^2}{2}+\frac {1}{2} \int \left (-2 e^{2 x^2}-8 e^{2 x^2} x^2\right ) \, dx\\ &=-\frac {3}{2} \left (4+e^{18}\right ) x-\frac {3 x^2}{2}-4 \int e^{2 x^2} x^2 \, dx-\int e^{2 x^2} \, dx\\ &=-e^{2 x^2} x-\frac {3}{2} \left (4+e^{18}\right ) x-\frac {3 x^2}{2}-\frac {1}{2} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\sqrt {2} x\right )+\int e^{2 x^2} \, dx\\ &=-e^{2 x^2} x-\frac {3}{2} \left (4+e^{18}\right ) x-\frac {3 x^2}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 29, normalized size = 1.12 \begin {gather*} -6 x-\frac {3 e^{18} x}{2}-e^{2 x^2} x-\frac {3 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12 - 3*E^18 - 6*x + E^(2*x^2)*(-2 - 8*x^2))/2,x]

[Out]

-6*x - (3*E^18*x)/2 - E^(2*x^2)*x - (3*x^2)/2

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fricas [A]  time = 0.46, size = 23, normalized size = 0.88 \begin {gather*} -\frac {3}{2} \, x^{2} - \frac {3}{2} \, x e^{18} - x e^{\left (2 \, x^{2}\right )} - 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-8*x^2-2)*exp(x^2)^2-3/2*exp(18)-3*x-6,x, algorithm="fricas")

[Out]

-3/2*x^2 - 3/2*x*e^18 - x*e^(2*x^2) - 6*x

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giac [A]  time = 0.14, size = 23, normalized size = 0.88 \begin {gather*} -\frac {3}{2} \, x^{2} - \frac {3}{2} \, x e^{18} - x e^{\left (2 \, x^{2}\right )} - 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-8*x^2-2)*exp(x^2)^2-3/2*exp(18)-3*x-6,x, algorithm="giac")

[Out]

-3/2*x^2 - 3/2*x*e^18 - x*e^(2*x^2) - 6*x

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maple [A]  time = 0.03, size = 24, normalized size = 0.92

method	result	size
default	\(-6 x -\frac {3 x^{2}}{2}-x \,{\mathrm e}^{2 x^{2}}-\frac {3 \,{\mathrm e}^{18} x}{2}\)	\(24\)
norman	\(\left (-\frac {3 \,{\mathrm e}^{18}}{2}-6\right ) x -\frac {3 x^{2}}{2}-x \,{\mathrm e}^{2 x^{2}}\)	\(24\)
risch	\(-6 x -\frac {3 x^{2}}{2}-x \,{\mathrm e}^{2 x^{2}}-\frac {3 \,{\mathrm e}^{18} x}{2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-8*x^2-2)*exp(x^2)^2-3/2*exp(18)-3*x-6,x,method=_RETURNVERBOSE)

[Out]

-6*x-3/2*x^2-x*exp(x^2)^2-3/2*exp(18)*x

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maxima [A]  time = 0.36, size = 23, normalized size = 0.88 \begin {gather*} -\frac {3}{2} \, x^{2} - \frac {3}{2} \, x e^{18} - x e^{\left (2 \, x^{2}\right )} - 6 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-8*x^2-2)*exp(x^2)^2-3/2*exp(18)-3*x-6,x, algorithm="maxima")

[Out]

-3/2*x^2 - 3/2*x*e^18 - x*e^(2*x^2) - 6*x

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mupad [B]  time = 3.48, size = 20, normalized size = 0.77 \begin {gather*} -\frac {x\,\left (3\,x+3\,{\mathrm {e}}^{18}+2\,{\mathrm {e}}^{2\,x^2}+12\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(- 3*x - (3*exp(18))/2 - (exp(2*x^2)*(8*x^2 + 2))/2 - 6,x)

[Out]

-(x*(3*x + 3*exp(18) + 2*exp(2*x^2) + 12))/2

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sympy [A]  time = 0.10, size = 26, normalized size = 1.00 \begin {gather*} - \frac {3 x^{2}}{2} - x e^{2 x^{2}} + x \left (- \frac {3 e^{18}}{2} - 6\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-8*x**2-2)*exp(x**2)**2-3/2*exp(18)-3*x-6,x)

[Out]

-3*x**2/2 - x*exp(2*x**2) + x*(-3*exp(18)/2 - 6)

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