### 3.20 $$\int x^2 \cosh ^2(a+b x-c x^2) \, dx$$

Optimal. Leaf size=268 $-\frac{\sqrt{\frac{\pi }{2}} e^{2 a+\frac{b^2}{2 c}} \text{Erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} b^2 e^{2 a+\frac{b^2}{2 c}} \text{Erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{-2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} b^2 e^{-2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac{x^3}{6}$

[Out]

x^3/6 - (b^2*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a + b^2/(
2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b^2*E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[
(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt
[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b*x - 2*c*x^2])/(16*c^2) - (x*Sinh[2*a + 2*b*x - 2*c*x^2])/(8*c)

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Rubi [A]  time = 0.23522, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.444, Rules used = {5395, 5387, 5374, 2234, 2205, 2204, 5383, 5375} $-\frac{\sqrt{\frac{\pi }{2}} e^{2 a+\frac{b^2}{2 c}} \text{Erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} b^2 e^{2 a+\frac{b^2}{2 c}} \text{Erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{-2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} b^2 e^{-2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac{x^3}{6}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^2*Cosh[a + b*x - c*x^2]^2,x]

[Out]

x^3/6 - (b^2*E^(2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) - (E^(2*a + b^2/(
2*c))*Sqrt[Pi/2]*Erf[(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) - (b^2*E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[
(b - 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(5/2)) + (E^(-2*a - b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b - 2*c*x)/(Sqrt[2]*Sqrt
[c])])/(32*c^(3/2)) - (b*Sinh[2*a + 2*b*x - 2*c*x^2])/(16*c^2) - (x*Sinh[2*a + 2*b*x - 2*c*x^2])/(8*c)

Rule 5395

Int[Cosh[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce
[(d + e*x)^m, Cosh[a + b*x + c*x^2]^n, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 1]

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 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 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 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 x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx &=\int \left (\frac{x^2}{2}+\frac{1}{2} x^2 \cosh \left (2 a+2 b x-2 c x^2\right )\right ) \, dx\\ &=\frac{x^3}{6}+\frac{1}{2} \int x^2 \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx\\ &=\frac{x^3}{6}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac{\int \sinh \left (2 a+2 b x-2 c x^2\right ) \, dx}{8 c}+\frac{b \int x \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx}{4 c}\\ &=\frac{x^3}{6}-\frac{b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac{b^2 \int \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx}{8 c^2}+\frac{\int e^{2 a+2 b x-2 c x^2} \, dx}{16 c}-\frac{\int e^{-2 a-2 b x+2 c x^2} \, dx}{16 c}\\ &=\frac{x^3}{6}-\frac{b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac{b^2 \int e^{2 a+2 b x-2 c x^2} \, dx}{16 c^2}+\frac{b^2 \int e^{-2 a-2 b x+2 c x^2} \, dx}{16 c^2}-\frac{e^{-2 a-\frac{b^2}{2 c}} \int e^{\frac{(-2 b+4 c x)^2}{8 c}} \, dx}{16 c}+\frac{e^{2 a+\frac{b^2}{2 c}} \int e^{-\frac{(2 b-4 c x)^2}{8 c}} \, dx}{16 c}\\ &=\frac{x^3}{6}-\frac{e^{2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{e^{-2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac{\left (b^2 e^{-2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(-2 b+4 c x)^2}{8 c}} \, dx}{16 c^2}+\frac{\left (b^2 e^{2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(2 b-4 c x)^2}{8 c}} \, dx}{16 c^2}\\ &=\frac{x^3}{6}-\frac{b^2 e^{2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}-\frac{e^{2 a+\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erf}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{b^2 e^{-2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{e^{-2 a-\frac{b^2}{2 c}} \sqrt{\frac{\pi }{2}} \text{erfi}\left (\frac{b-2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac{x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}\\ \end{align*}

Mathematica [A]  time = 0.723499, size = 181, normalized size = 0.68 $\frac{3 \sqrt{2 \pi } \left (b^2+c\right ) \text{Erf}\left (\frac{2 c x-b}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a+\frac{b^2}{2 c}\right )+\cosh \left (2 a+\frac{b^2}{2 c}\right )\right )+3 \sqrt{2 \pi } \left (b^2-c\right ) \text{Erfi}\left (\frac{2 c x-b}{\sqrt{2} \sqrt{c}}\right ) \left (\cosh \left (2 a+\frac{b^2}{2 c}\right )-\sinh \left (2 a+\frac{b^2}{2 c}\right )\right )+4 \sqrt{c} \left (8 c^2 x^3-3 (b+2 c x) \sinh (2 (a+x (b-c x)))\right )}{192 c^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(3*(b^2 - c)*Sqrt[2*Pi]*Erfi[(-b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a + b^2/(2*c)] - Sinh[2*a + b^2/(2*c)]) +
3*(b^2 + c)*Sqrt[2*Pi]*Erf[(-b + 2*c*x)/(Sqrt[2]*Sqrt[c])]*(Cosh[2*a + b^2/(2*c)] + Sinh[2*a + b^2/(2*c)]) +
4*Sqrt[c]*(8*c^2*x^3 - 3*(b + 2*c*x)*Sinh[2*(a + x*(b - c*x))]))/(192*c^(5/2))

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Maple [A]  time = 0.063, size = 273, normalized size = 1. \begin{align*}{\frac{{x}^{3}}{6}}+{\frac{x{{\rm e}^{2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}+{\frac{b{{\rm e}^{2\,c{x}^{2}-2\,bx-2\,a}}}{32\,{c}^{2}}}+{\frac{{b}^{2}\sqrt{\pi }}{32\,{c}^{2}}{{\rm e}^{-{\frac{4\,ac+{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}-{\frac{\sqrt{\pi }}{32\,c}{{\rm e}^{-{\frac{4\,ac+{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}-{\frac{x{{\rm e}^{-2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}-{\frac{b{{\rm e}^{-2\,c{x}^{2}+2\,bx+2\,a}}}{32\,{c}^{2}}}-{\frac{{b}^{2}\sqrt{\pi }\sqrt{2}}{64}{{\rm e}^{{\frac{4\,ac+{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}-{\frac{\sqrt{\pi }\sqrt{2}}{64}{{\rm e}^{{\frac{4\,ac+{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3+1/16/c*x*exp(2*c*x^2-2*b*x-2*a)+1/32*b/c^2*exp(2*c*x^2-2*b*x-2*a)+1/32*b^2/c^2*Pi^(1/2)*exp(-1/2*(4*a*
c+b^2)/c)/(-2*c)^(1/2)*erf((-2*c)^(1/2)*x+b/(-2*c)^(1/2))-1/32/c*Pi^(1/2)*exp(-1/2*(4*a*c+b^2)/c)/(-2*c)^(1/2)
*erf((-2*c)^(1/2)*x+b/(-2*c)^(1/2))-1/16/c*x*exp(-2*c*x^2+2*b*x+2*a)-1/32*b/c^2*exp(-2*c*x^2+2*b*x+2*a)-1/64*b
^2/c^(5/2)*Pi^(1/2)*exp(1/2*(4*a*c+b^2)/c)*2^(1/2)*erf(-2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))-1/64/c^(3/2)*
Pi^(1/2)*exp(1/2*(4*a*c+b^2)/c)*2^(1/2)*erf(-2^(1/2)*c^(1/2)*x+1/2*b*2^(1/2)/c^(1/2))

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Maxima [A]  time = 1.3943, size = 435, normalized size = 1.62 \begin{align*} \frac{1}{6} \, x^{3} - \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x - b\right )} b^{2}{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{\frac{{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt{\frac{{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac{5}{2}}} - \frac{2 \, \sqrt{2} b c e^{\left (-\frac{{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac{5}{2}}} - \frac{2 \,{\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac{3}{2}, \frac{{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}{\left (\frac{{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac{3}{2}} \left (-c\right )^{\frac{5}{2}}}\right )} e^{\left (2 \, a + \frac{b^{2}}{2 \, c}\right )}}{64 \, \sqrt{-c}} + \frac{\sqrt{2}{\left (\frac{\sqrt{\pi }{\left (2 \, c x - b\right )} b^{2}{\left (\operatorname{erf}\left (\sqrt{\frac{1}{2}} \sqrt{-\frac{{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt{-\frac{{\left (2 \, c x - b\right )}^{2}}{c}} c^{\frac{5}{2}}} + \frac{2 \, \sqrt{2} b e^{\left (\frac{{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}}{c^{\frac{3}{2}}} - \frac{2 \,{\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac{3}{2}, -\frac{{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}{\left (-\frac{{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac{3}{2}} c^{\frac{5}{2}}}\right )} e^{\left (-2 \, a - \frac{b^{2}}{2 \, c}\right )}}{64 \, \sqrt{c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3 - 1/64*sqrt(2)*(sqrt(pi)*(2*c*x - b)*b^2*(erf(sqrt(1/2)*sqrt((2*c*x - b)^2/c)) - 1)/(sqrt((2*c*x - b)^
2/c)*(-c)^(5/2)) - 2*sqrt(2)*b*c*e^(-1/2*(2*c*x - b)^2/c)/(-c)^(5/2) - 2*(2*c*x - b)^3*gamma(3/2, 1/2*(2*c*x -
b)^2/c)/(((2*c*x - b)^2/c)^(3/2)*(-c)^(5/2)))*e^(2*a + 1/2*b^2/c)/sqrt(-c) + 1/64*sqrt(2)*(sqrt(pi)*(2*c*x -
b)*b^2*(erf(sqrt(1/2)*sqrt(-(2*c*x - b)^2/c)) - 1)/(sqrt(-(2*c*x - b)^2/c)*c^(5/2)) + 2*sqrt(2)*b*e^(1/2*(2*c*
x - b)^2/c)/c^(3/2) - 2*(2*c*x - b)^3*gamma(3/2, -1/2*(2*c*x - b)^2/c)/((-(2*c*x - b)^2/c)^(3/2)*c^(5/2)))*e^(
-2*a - 1/2*b^2/c)/sqrt(c)

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Fricas [B]  time = 2.18093, size = 1887, normalized size = 7.04 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(32*c^3*x^3*cosh(c*x^2 - b*x - a)^2 + 6*(2*c^2*x + b*c)*cosh(c*x^2 - b*x - a)^4 + 24*(2*c^2*x + b*c)*cos
h(c*x^2 - b*x - a)*sinh(c*x^2 - b*x - a)^3 + 6*(2*c^2*x + b*c)*sinh(c*x^2 - b*x - a)^4 - 3*sqrt(2)*sqrt(pi)*((
b^2 - c)*cosh(c*x^2 - b*x - a)^2*cosh(1/2*(b^2 + 4*a*c)/c) - (b^2 - c)*cosh(c*x^2 - b*x - a)^2*sinh(1/2*(b^2 +
4*a*c)/c) + ((b^2 - c)*cosh(1/2*(b^2 + 4*a*c)/c) - (b^2 - c)*sinh(1/2*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a)
^2 + 2*((b^2 - c)*cosh(c*x^2 - b*x - a)*cosh(1/2*(b^2 + 4*a*c)/c) - (b^2 - c)*cosh(c*x^2 - b*x - a)*sinh(1/2*(
b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a))*sqrt(-c)*erf(1/2*sqrt(2)*(2*c*x - b)*sqrt(-c)/c) + 3*sqrt(2)*sqrt(pi)*
((b^2 + c)*cosh(c*x^2 - b*x - a)^2*cosh(1/2*(b^2 + 4*a*c)/c) + (b^2 + c)*cosh(c*x^2 - b*x - a)^2*sinh(1/2*(b^2
+ 4*a*c)/c) + ((b^2 + c)*cosh(1/2*(b^2 + 4*a*c)/c) + (b^2 + c)*sinh(1/2*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x -
a)^2 + 2*((b^2 + c)*cosh(c*x^2 - b*x - a)*cosh(1/2*(b^2 + 4*a*c)/c) + (b^2 + c)*cosh(c*x^2 - b*x - a)*sinh(1/2
*(b^2 + 4*a*c)/c))*sinh(c*x^2 - b*x - a))*sqrt(c)*erf(1/2*sqrt(2)*(2*c*x - b)/sqrt(c)) - 12*c^2*x + 4*(8*c^3*x
^3 + 9*(2*c^2*x + b*c)*cosh(c*x^2 - b*x - a)^2)*sinh(c*x^2 - b*x - a)^2 - 6*b*c + 8*(8*c^3*x^3*cosh(c*x^2 - b*
x - a) + 3*(2*c^2*x + b*c)*cosh(c*x^2 - b*x - a)^3)*sinh(c*x^2 - b*x - a))/(c^3*cosh(c*x^2 - b*x - a)^2 + 2*c^
3*cosh(c*x^2 - b*x - a)*sinh(c*x^2 - b*x - a) + c^3*sinh(c*x^2 - b*x - a)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \cosh ^{2}{\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*cosh(-c*x**2+b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.31472, size = 251, normalized size = 0.94 \begin{align*} \frac{1}{6} \, x^{3} - \frac{\frac{\sqrt{2} \sqrt{\pi }{\left (b^{2} + c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{c}{\left (2 \, x - \frac{b}{c}\right )}\right ) e^{\left (\frac{b^{2} + 4 \, a c}{2 \, c}\right )}}{\sqrt{c}} + 2 \,{\left (c{\left (2 \, x - \frac{b}{c}\right )} + 2 \, b\right )} e^{\left (-2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{64 \, c^{2}} - \frac{\frac{\sqrt{2} \sqrt{\pi }{\left (b^{2} - c\right )} \operatorname{erf}\left (-\frac{1}{2} \, \sqrt{2} \sqrt{-c}{\left (2 \, x - \frac{b}{c}\right )}\right ) e^{\left (-\frac{b^{2} + 4 \, a c}{2 \, c}\right )}}{\sqrt{-c}} - 2 \,{\left (c{\left (2 \, x - \frac{b}{c}\right )} + 2 \, b\right )} e^{\left (2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{64 \, c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*x^3 - 1/64*(sqrt(2)*sqrt(pi)*(b^2 + c)*erf(-1/2*sqrt(2)*sqrt(c)*(2*x - b/c))*e^(1/2*(b^2 + 4*a*c)/c)/sqrt(
c) + 2*(c*(2*x - b/c) + 2*b)*e^(-2*c*x^2 + 2*b*x + 2*a))/c^2 - 1/64*(sqrt(2)*sqrt(pi)*(b^2 - c)*erf(-1/2*sqrt(
2)*sqrt(-c)*(2*x - b/c))*e^(-1/2*(b^2 + 4*a*c)/c)/sqrt(-c) - 2*(c*(2*x - b/c) + 2*b)*e^(2*c*x^2 - 2*b*x - 2*a)
)/c^2