### 3.28 $$\int (d+e x)^2 \cosh (a+b x+c x^2) \, dx$$

Optimal. Leaf size=261 $\frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^2 e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } e^2 e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}$

[Out]

(e^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(-a + b^2/(4*c
))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (e^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*S
qrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)
) + (e*(2*c*d - b*e)*Sinh[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sinh[a + b*x + c*x^2])/(2*c)

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Rubi [A]  time = 0.172052, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.368, Rules used = {5387, 5374, 2234, 2204, 2205, 5383, 5375} $\frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^2 e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } e^2 e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^2*Cosh[a + b*x + c*x^2],x]

[Out]

(e^2*E^(-a + b^2/(4*c))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(-a + b^2/(4*c
))*Sqrt[Pi]*Erf[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)) - (e^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*S
qrt[c])])/(8*c^(3/2)) + ((2*c*d - b*e)^2*E^(a - b^2/(4*c))*Sqrt[Pi]*Erfi[(b + 2*c*x)/(2*Sqrt[c])])/(16*c^(5/2)
) + (e*(2*c*d - b*e)*Sinh[a + b*x + c*x^2])/(4*c^2) + (e*(d + e*x)*Sinh[a + b*x + c*x^2])/(2*c)

Rule 5387

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

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 5383

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

Rubi steps

\begin{align*} \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx &=\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}-\frac{e^2 \int \sinh \left (a+b x+c x^2\right ) \, dx}{2 c}-\frac{(-2 c d+b e) \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}+\frac{e^2 \int e^{-a-b x-c x^2} \, dx}{4 c}-\frac{e^2 \int e^{a+b x+c x^2} \, dx}{4 c}+\frac{(2 c d-b e)^2 \int \cosh \left (a+b x+c x^2\right ) \, dx}{4 c^2}\\ &=\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}+\frac{(2 c d-b e)^2 \int e^{-a-b x-c x^2} \, dx}{8 c^2}+\frac{(2 c d-b e)^2 \int e^{a+b x+c x^2} \, dx}{8 c^2}-\frac{\left (e^2 e^{a-\frac{b^2}{4 c}}\right ) \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx}{4 c}+\frac{\left (e^2 e^{-a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx}{4 c}\\ &=\frac{e^2 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^2 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{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}+\frac{\left ((2 c d-b e)^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 ((2 c d-b e)^2 e^{-a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx}{8 c^2}\\ &=\frac{e^2 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{(2 c d-b e)^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^2 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{(2 c d-b e)^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 (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\\ \end{align*}

Mathematica [A]  time = 0.527869, size = 194, normalized size = 0.74 $\frac{\sqrt{\pi } \left (b^2 e^2+2 c e (e-2 b d)+4 c^2 d^2\right ) \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\cosh \left (a-\frac{b^2}{4 c}\right )-\sinh \left (a-\frac{b^2}{4 c}\right )\right )+\sqrt{\pi } \left (b^2 e^2-2 c e (2 b d+e)+4 c^2 d^2\right ) \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )+4 \sqrt{c} e \sinh (a+x (b+c x)) (-b e+4 c d+2 c e x)}{16 c^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^2*Cosh[a + b*x + c*x^2],x]

[Out]

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

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Maple [B]  time = 0.056, size = 493, normalized size = 1.9 \begin{align*}{\frac{{d}^{2}\sqrt{\pi }}{4}{{\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{{e}^{2}x{{\rm e}^{-c{x}^{2}-bx-a}}}{4\,c}}+{\frac{{e}^{2}b{{\rm e}^{-c{x}^{2}-bx-a}}}{8\,{c}^{2}}}+{\frac{{b}^{2}{e}^{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{{e}^{2}\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{de{{\rm e}^{-c{x}^{2}-bx-a}}}{2\,c}}-{\frac{bde\sqrt{\pi }}{4}{{\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{{d}^{2}\sqrt{\pi }}{4}{{\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{{e}^{2}x{{\rm e}^{c{x}^{2}+bx+a}}}{4\,c}}-{\frac{{e}^{2}b{{\rm e}^{c{x}^{2}+bx+a}}}{8\,{c}^{2}}}-{\frac{{b}^{2}{e}^{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{{e}^{2}\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}}}}+{\frac{de{{\rm e}^{c{x}^{2}+bx+a}}}{2\,c}}+{\frac{bde\sqrt{\pi }}{4\,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((e*x+d)^2*cosh(c*x^2+b*x+a),x)

[Out]

1/4*d^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))-1/4*e^2/c*x*exp(-c*x^2-b*x-a)+1/
8*e^2*b/c^2*exp(-c*x^2-b*x-a)+1/16*e^2*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*e^2/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))-1/2*d*e/c*exp(-c*x^2-b*x-a)-1
/4*d*e*b/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*d^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))+1/4*e^2/c*x*exp(c*x^2+b*x+a)-1/8*e^2*b/c^2*exp(c*x^2+b
*x+a)-1/16*e^2*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*e^2/
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))+1/2*d*e/c*exp(c*x^2+b*x+a)+1/
4*d*e*b/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 = 1.50754, size = 724, normalized size = 2.77 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(pi)*d^2*erf(sqrt(-c)*x - 1/2*b/sqrt(-c))*e^(a - 1/4*b^2/c)/sqrt(-c) + 1/4*sqrt(pi)*d^2*erf(sqrt(c)*x
+ 1/2*b/sqrt(c))*e^(-a + 1/4*b^2/c)/sqrt(c) - 1/4*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1
)/(sqrt(-(2*c*x + b)^2/c)*c^(3/2)) - 2*e^(1/4*(2*c*x + b)^2/c)/sqrt(c))*d*e*e^(a - 1/4*b^2/c)/sqrt(c) + 1/16*(
sqrt(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(5/2)) - 4*b*e^(1/4*(
2*c*x + b)^2/c)/c^(3/2) - 4*(2*c*x + b)^3*gamma(3/2, -1/4*(2*c*x + b)^2/c)/((-(2*c*x + b)^2/c)^(3/2)*c^(5/2)))
*e^2*e^(a - 1/4*b^2/c)/sqrt(c) - 1/4*(sqrt(pi)*(2*c*x + b)*b*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x
+ b)^2/c)*(-c)^(3/2)) + 2*c*e^(-1/4*(2*c*x + b)^2/c)/(-c)^(3/2))*d*e*e^(-a + 1/4*b^2/c)/sqrt(-c) - 1/16*(sqrt
(pi)*(2*c*x + b)*b^2*(erf(1/2*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(5/2)) + 4*b*c*e^(-1/4*(
2*c*x + b)^2/c)/(-c)^(5/2) - 4*(2*c*x + b)^3*gamma(3/2, 1/4*(2*c*x + b)^2/c)/(((2*c*x + b)^2/c)^(3/2)*(-c)^(5/
2)))*e^2*e^(-a + 1/4*b^2/c)/sqrt(-c)

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Fricas [B]  time = 2.23463, size = 1508, normalized size = 5.78 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*c^2*e^2*x + 8*c^2*d*e - 2*b*c*e^2 - 2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)^2 + s
qrt(pi)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*cosh(c*x^2 + b*x + a)*cosh(-1/4*(b^2 - 4*a*c)/c) + (4*c^2*d
^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*cosh(c*x^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + ((4*c^2*d^2 - 4*b*c*d*e +
(b^2 - 2*c)*e^2)*cosh(-1/4*(b^2 - 4*a*c)/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - 2*c)*e^2)*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)*((4*c^2*d^2 - 4*b*c*d*e + (b
^2 + 2*c)*e^2)*cosh(c*x^2 + b*x + a)*cosh(-1/4*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2*c)*e^2)*co
sh(c*x^2 + b*x + a)*sinh(-1/4*(b^2 - 4*a*c)/c) + ((4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2*c)*e^2)*cosh(-1/4*(b^2 - 4
*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + 2*c)*e^2)*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*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)*sinh(c*x^2 + b*x
+ a) - 2*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*sinh(c*x^2 + b*x + a)^2)/(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 \left (d + e x\right )^{2} \cosh{\left (a + b x + c x^{2} \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*cosh(c*x**2+b*x+a),x)

[Out]

Integral((d + e*x)**2*cosh(a + b*x + c*x**2), x)

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Giac [A]  time = 1.27384, size = 522, normalized size = 2. \begin{align*} -\frac{\sqrt{\pi } d^{2} \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 )}}{4 \, \sqrt{c}} - \frac{\sqrt{\pi } d^{2} \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 )}}{4 \, \sqrt{-c}} + \frac{\frac{\sqrt{\pi } b d \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}{4 \, c}\right )}}{\sqrt{c}} - 2 \, d e^{\left (-c x^{2} - b x - a + 1\right )}}{4 \, c} + \frac{\frac{\sqrt{\pi } b d \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}{4 \, c}\right )}}{\sqrt{-c}} + 2 \, d e^{\left (c x^{2} + b x + a + 1\right )}}{4 \, c} - \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 + 8 \, 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 + 2\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 - 8 \, 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 + 2\right )}}{16 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(pi)*d^2*erf(-1/2*sqrt(c)*(2*x + b/c))*e^(1/4*(b^2 - 4*a*c)/c)/sqrt(c) - 1/4*sqrt(pi)*d^2*erf(-1/2*sq
rt(-c)*(2*x + b/c))*e^(-1/4*(b^2 - 4*a*c)/c)/sqrt(-c) + 1/4*(sqrt(pi)*b*d*erf(-1/2*sqrt(c)*(2*x + b/c))*e^(1/4
*(b^2 - 4*a*c + 4*c)/c)/sqrt(c) - 2*d*e^(-c*x^2 - b*x - a + 1))/c + 1/4*(sqrt(pi)*b*d*erf(-1/2*sqrt(-c)*(2*x +
b/c))*e^(-1/4*(b^2 - 4*a*c - 4*c)/c)/sqrt(-c) + 2*d*e^(c*x^2 + b*x + a + 1))/c - 1/16*(sqrt(pi)*(b^2 + 2*c)*e
rf(-1/2*sqrt(c)*(2*x + b/c))*e^(1/4*(b^2 - 4*a*c + 8*c)/c)/sqrt(c) + 2*(c*(2*x + b/c) - 2*b)*e^(-c*x^2 - b*x -
a + 2))/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 - 8*c)/c)/sqrt(-
c) - 2*(c*(2*x + b/c) - 2*b)*e^(c*x^2 + b*x + a + 2))/c^2