3.1 \(\int x^2 \sinh (a+b x+c x^2) \, dx\)

Optimal. Leaf size=225 \[ -\frac{\sqrt{\pi } b^2 e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sqrt{\pi } b^2 e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{b \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac{x \cosh \left (a+b x+c x^2\right )}{2 c} \]

[Out]

-(b*Cosh[a + b*x + c*x^2])/(4*c^2) + (x*Cosh[a + b*x + c*x^2])/(2*c) - (b^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b
+ 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))
+ (b^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(a - b^2/(4*c))*Sqrt[Pi]*Er
fi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))

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Rubi [A]  time = 0.140079, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {5386, 5375, 2234, 2204, 2205, 5382, 5374} \[ -\frac{\sqrt{\pi } b^2 e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sqrt{\pi } b^2 e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}-\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{b \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac{x \cosh \left (a+b x+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[x^2*Sinh[a + b*x + c*x^2],x]

[Out]

-(b*Cosh[a + b*x + c*x^2])/(4*c^2) + (x*Cosh[a + b*x + c*x^2])/(2*c) - (b^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b
+ 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))
+ (b^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (E^(a - b^2/(4*c))*Sqrt[Pi]*Er
fi[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2))

Rule 5386

Int[((d_.) + (e_.)*(x_))^(m_)*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)*
Cosh[a + b*x + c*x^2])/(2*c), x] + (-Dist[(e^2*(m - 1))/(2*c), Int[(d + e*x)^(m - 2)*Cosh[a + b*x + c*x^2], x]
, x] - Dist[(b*e - 2*c*d)/(2*c), Int[(d + e*x)^(m - 1)*Sinh[a + b*x + c*x^2], x], x]) /; FreeQ[{a, b, c, d, e}
, x] && GtQ[m, 1] && NeQ[b*e - 2*c*d, 0]

Rule 5375

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] + Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, 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 2205

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 5382

Int[((d_.) + (e_.)*(x_))*Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[(e*Cosh[a + b*x + c*x^2])/
(2*c), x] - Dist[(b*e - 2*c*d)/(2*c), Int[Sinh[a + b*x + c*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
e - 2*c*d, 0]

Rule 5374

Int[Sinh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[1/2, Int[E^(a + b*x + c*x^2), x], x] - Dist[1/2
, Int[E^(-a - b*x - c*x^2), x], x] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int x^2 \sinh \left (a+b x+c x^2\right ) \, dx &=\frac{x \cosh \left (a+b x+c x^2\right )}{2 c}-\frac{\int \cosh \left (a+b x+c x^2\right ) \, dx}{2 c}-\frac{b \int x \sinh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=-\frac{b \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac{x \cosh \left (a+b x+c x^2\right )}{2 c}+\frac{b^2 \int \sinh \left (a+b x+c x^2\right ) \, dx}{4 c^2}-\frac{\int e^{-a-b x-c x^2} \, dx}{4 c}-\frac{\int e^{a+b x+c x^2} \, dx}{4 c}\\ &=-\frac{b \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac{x \cosh \left (a+b x+c x^2\right )}{2 c}-\frac{b^2 \int e^{-a-b x-c x^2} \, dx}{8 c^2}+\frac{b^2 \int e^{a+b x+c x^2} \, dx}{8 c^2}-\frac{e^{a-\frac{b^2}{4 c}} \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c}-\frac{e^{-a+\frac{b^2}{4 c}} \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx}{4 c}\\ &=-\frac{b \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac{x \cosh \left (a+b x+c x^2\right )}{2 c}-\frac{e^{-a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{e^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\left (b^2 e^{a-\frac{b^2}{4 c}}\right ) \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx}{8 c^2}-\frac{\left (b^2 e^{-a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx}{8 c^2}\\ &=-\frac{b \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac{x \cosh \left (a+b x+c x^2\right )}{2 c}-\frac{b^2 e^{-a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}-\frac{e^{-a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{b^2 e^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}-\frac{e^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.308602, size = 149, normalized size = 0.66 \[ \frac{\sqrt{\pi } \left (b^2+2 c\right ) \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )-\cosh \left (a-\frac{b^2}{4 c}\right )\right )+\sqrt{\pi } \left (b^2-2 c\right ) \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )+4 \sqrt{c} (2 c x-b) \cosh (a+x (b+c x))}{16 c^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sinh[a + b*x + c*x^2],x]

[Out]

(4*Sqrt[c]*(-b + 2*c*x)*Cosh[a + x*(b + c*x)] + (b^2 + 2*c)*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])]*(-Cosh[a - b
^2/(4*c)] + Sinh[a - b^2/(4*c)]) + (b^2 - 2*c)*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])]*(Cosh[a - b^2/(4*c)] + S
inh[a - b^2/(4*c)]))/(16*c^(5/2))

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Maple [A]  time = 0.099, size = 252, normalized size = 1.1 \begin{align*}{\frac{x{{\rm e}^{-c{x}^{2}-bx-a}}}{4\,c}}-{\frac{b{{\rm e}^{-c{x}^{2}-bx-a}}}{8\,{c}^{2}}}-{\frac{{b}^{2}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}-{\frac{\sqrt{\pi }}{8}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}+{\frac{x{{\rm e}^{c{x}^{2}+bx+a}}}{4\,c}}-{\frac{b{{\rm e}^{c{x}^{2}+bx+a}}}{8\,{c}^{2}}}-{\frac{{b}^{2}\sqrt{\pi }}{16\,{c}^{2}}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}}+{\frac{\sqrt{\pi }}{8\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sinh(c*x^2+b*x+a),x)

[Out]

1/4/c*x*exp(-c*x^2-b*x-a)-1/8*b/c^2*exp(-c*x^2-b*x-a)-1/16*b^2/c^(5/2)*Pi^(1/2)*exp(-1/4*(4*a*c-b^2)/c)*erf(c^
(1/2)*x+1/2*b/c^(1/2))-1/8/c^(3/2)*Pi^(1/2)*exp(-1/4*(4*a*c-b^2)/c)*erf(c^(1/2)*x+1/2*b/c^(1/2))+1/4/c*x*exp(c
*x^2+b*x+a)-1/8*b/c^2*exp(c*x^2+b*x+a)-1/16*b^2/c^2*Pi^(1/2)*exp(1/4*(4*a*c-b^2)/c)/(-c)^(1/2)*erf(-(-c)^(1/2)
*x+1/2*b/(-c)^(1/2))+1/8/c*Pi^(1/2)*exp(1/4*(4*a*c-b^2)/c)/(-c)^(1/2)*erf(-(-c)^(1/2)*x+1/2*b/(-c)^(1/2))

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Maxima [B]  time = 1.64044, size = 1019, normalized size = 4.53 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(c*x^2+b*x+a),x, algorithm="maxima")

[Out]

1/3*x^3*sinh(c*x^2 + b*x + a) + 1/96*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2
*c*x + b)^2/c)*c^(7/2)) - 6*b^2*e^(1/4*(2*c*x + b)^2/c)/c^(5/2) - 12*(2*c*x + b)^3*b*gamma(3/2, -1/4*(2*c*x +
b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(7/2)) + 8*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(3/2))*b*e^(a - 1/4*b^2/c)/sqr
t(c) - 1/96*(sqrt(pi)*(2*c*x + b)*b^4*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(9/2)) -
8*b^3*e^(1/4*(2*c*x + b)^2/c)/c^(7/2) - 24*(2*c*x + b)^3*b^2*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^
2/c)^(3/2)*c^(9/2)) + 32*b*gamma(2, -1/4*(2*c*x + b)^2/c)/c^(5/2) - 16*(2*c*x + b)^5*gamma(5/2, -1/4*(2*c*x +
b)^2/c)/((-(2*c*x + b)^2/c)^(5/2)*c^(9/2)))*sqrt(c)*e^(a - 1/4*b^2/c) + 1/96*(sqrt(pi)*(2*c*x + b)*b^3*(erf(1/
2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(7/2)) + 6*b^2*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(7/2)
- 12*(2*c*x + b)^3*b*gamma(3/2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)*(-c)^(7/2)) + 8*c^2*gamma(2, 1/
4*(2*c*x + b)^2/c)/(-c)^(7/2))*b*e^(-a + 1/4*b^2/c)/sqrt(-c) + 1/96*(sqrt(pi)*(2*c*x + b)*b^4*(erf(1/2*sqrt((2
*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(9/2)) + 8*b^3*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(9/2) - 24*(2*
c*x + b)^3*b^2*gamma(3/2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)*(-c)^(9/2)) + 32*b*c^2*gamma(2, 1/4*(2
*c*x + b)^2/c)/(-c)^(9/2) - 16*(2*c*x + b)^5*gamma(5/2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(5/2)*(-c)^(9/
2)))*c*e^(-a + 1/4*b^2/c)/sqrt(-c)

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Fricas [B]  time = 2.05521, size = 1075, normalized size = 4.78 \begin{align*} \frac{4 \, c^{2} x + 2 \,{\left (2 \, c^{2} x - b c\right )} \cosh \left (c x^{2} + b x + a\right )^{2} - \sqrt{\pi }{\left ({\left (b^{2} - 2 \, c\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (b^{2} - 2 \, c\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left ({\left (b^{2} - 2 \, c\right )} \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (b^{2} - 2 \, c\right )} \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt{-c} \operatorname{erf}\left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, c}\right ) - \sqrt{\pi }{\left ({\left (b^{2} + 2 \, c\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) -{\left (b^{2} + 2 \, c\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left ({\left (b^{2} + 2 \, c\right )} \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) -{\left (b^{2} + 2 \, c\right )} \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt{c} \operatorname{erf}\left (\frac{2 \, c x + b}{2 \, \sqrt{c}}\right ) + 4 \,{\left (2 \, c^{2} x - b c\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \,{\left (2 \, c^{2} x - b c\right )} \sinh \left (c x^{2} + b x + a\right )^{2} - 2 \, b c}{16 \,{\left (c^{3} \cosh \left (c x^{2} + b x + a\right ) + c^{3} \sinh \left (c x^{2} + b x + a\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

1/16*(4*c^2*x + 2*(2*c^2*x - b*c)*cosh(c*x^2 + b*x + a)^2 - sqrt(pi)*((b^2 - 2*c)*cosh(c*x^2 + b*x + a)*cosh(-
1/4*(b^2 - 4*a*c)/c) + (b^2 - 2*c)*cosh(c*x^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + ((b^2 - 2*c)*cosh(-1/4*(
b^2 - 4*a*c)/c) + (b^2 - 2*c)*sinh(-1/4*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a))*sqrt(-c)*erf(1/2*(2*c*x + b)*
sqrt(-c)/c) - sqrt(pi)*((b^2 + 2*c)*cosh(c*x^2 + b*x + a)*cosh(-1/4*(b^2 - 4*a*c)/c) - (b^2 + 2*c)*cosh(c*x^2
+ b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + ((b^2 + 2*c)*cosh(-1/4*(b^2 - 4*a*c)/c) - (b^2 + 2*c)*sinh(-1/4*(b^2 -
4*a*c)/c))*sinh(c*x^2 + b*x + a))*sqrt(c)*erf(1/2*(2*c*x + b)/sqrt(c)) + 4*(2*c^2*x - b*c)*cosh(c*x^2 + b*x +
a)*sinh(c*x^2 + b*x + a) + 2*(2*c^2*x - b*c)*sinh(c*x^2 + b*x + a)^2 - 2*b*c)/(c^3*cosh(c*x^2 + b*x + a) + c^
3*sinh(c*x^2 + b*x + a))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sinh{\left (a + b x + c x^{2} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sinh(c*x**2+b*x+a),x)

[Out]

Integral(x**2*sinh(a + b*x + c*x**2), x)

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Giac [A]  time = 1.34842, size = 220, normalized size = 0.98 \begin{align*} \frac{\frac{\sqrt{\pi }{\left (b^{2} + 2 \, c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt{c}} + 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (-c x^{2} - b x - a\right )}}{16 \, c^{2}} - \frac{\frac{\sqrt{\pi }{\left (b^{2} - 2 \, c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{-c}{\left (2 \, x + \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt{-c}} - 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (c x^{2} + b x + a\right )}}{16 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*sinh(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/16*(sqrt(pi)*(b^2 + 2*c)*erf(-1/2*sqrt(c)*(2*x + b/c))*e^(1/4*(b^2 - 4*a*c)/c)/sqrt(c) + 2*(c*(2*x + b/c) -
2*b)*e^(-c*x^2 - b*x - a))/c^2 - 1/16*(sqrt(pi)*(b^2 - 2*c)*erf(-1/2*sqrt(-c)*(2*x + b/c))*e^(-1/4*(b^2 - 4*a*
c)/c)/sqrt(-c) - 2*(c*(2*x + b/c) - 2*b)*e^(c*x^2 + b*x + a))/c^2