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

Optimal. Leaf size=311 \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^2 e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^2 e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^3}{6 e} \]

[Out]

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

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Rubi [A]  time = 0.385622, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {5395, 5387, 5374, 2234, 2204, 2205, 5383, 5375} \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^2 e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^2 e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^3}{6 e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d + e*x)^3/(6*e) + (e^2*E^(-2*a + b^2/(2*c))*Sqrt[Pi/2]*Erf[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((
2*c*d - b*e)^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*E^(2*a
- b^2/(2*c))*Sqrt[Pi/2]*Erfi[(b + 2*c*x)/(Sqrt[2]*Sqrt[c])])/(32*c^(3/2)) + ((2*c*d - b*e)^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*c*d - b*e)*Sinh[2*a + 2*b*x + 2*c*x^2]
)/(16*c^2) + (e*(d + e*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 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 ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{1}{2} (d+e x)^2+\frac{1}{2} (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{(d+e x)^3}{6 e}+\frac{1}{2} \int (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{e^2 \int \sinh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}+\frac{(2 c d-b e) \int (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{e^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c}-\frac{e^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c}+\frac{(2 c d-b e)^2 \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c^2}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(2 c d-b e)^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c^2}+\frac{(2 c d-b e)^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c^2}-\frac{\left (e^2 e^{2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{16 c}+\frac{\left (e^2 e^{-2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{16 c}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e^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^{3/2}}-\frac{e^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^{3/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{\left ((2 c d-b e)^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 ((2 c d-b e)^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{(d+e x)^3}{6 e}+\frac{e^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^{3/2}}+\frac{(2 c d-b e)^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 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{(2 c d-b e)^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 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}

Mathematica [A]  time = 1.33651, size = 240, normalized size = 0.77 \[ \frac{3 \sqrt{2 \pi } \left (b^2 e^2+c e (e-4 b d)+4 c^2 d^2\right ) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\cosh \left (2 a-\frac{b^2}{2 c}\right )-\sinh \left (2 a-\frac{b^2}{2 c}\right )\right )+3 \sqrt{2 \pi } \left (b^2 e^2-c e (4 b d+e)+4 c^2 d^2\right ) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )+\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )+4 \sqrt{c} \left (3 e \sinh (2 (a+x (b+c x))) (-b e+4 c d+2 c e x)+8 c^2 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{192 c^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.08, size = 558, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}{x}^{3}}{6}}+{\frac{de{x}^{2}}{2}}+{\frac{{d}^{2}x}{2}}+{\frac{{d}^{2}\sqrt{\pi }\sqrt{2}}{16}{{\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 ){\frac{1}{\sqrt{c}}}}-{\frac{{e}^{2}x{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}+{\frac{{e}^{2}b{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{32\,{c}^{2}}}+{\frac{{b}^{2}{e}^{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{{e}^{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{3}{2}}}}-{\frac{de{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{8\,c}}-{\frac{bde\sqrt{\pi }\sqrt{2}}{16}{{\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}}}}-{\frac{{d}^{2}\sqrt{\pi }}{8}{{\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{{e}^{2}x{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}-{\frac{{e}^{2}b{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{32\,{c}^{2}}}-{\frac{{b}^{2}{e}^{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{{e}^{2}\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{de{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{8\,c}}+{\frac{bde\sqrt{\pi }}{8\,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}}}} \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)^2,x)

[Out]

1/6*e^2*x^3+1/2*d*e*x^2+1/2*d^2*x+1/16*d^2*Pi^(1/2)*exp(-1/2*(4*a*c-b^2)/c)*2^(1/2)/c^(1/2)*erf(2^(1/2)*c^(1/2
)*x+1/2*b*2^(1/2)/c^(1/2))-1/16*e^2/c*x*exp(-2*c*x^2-2*b*x-2*a)+1/32*e^2*b/c^2*exp(-2*c*x^2-2*b*x-2*a)+1/64*e^
2*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*e^2/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))-1/8*d*e/c*exp(-2*
c*x^2-2*b*x-2*a)-1/16*d*e*b/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))-1/8*d^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))+1/16*e^2/
c*x*exp(2*c*x^2+2*b*x+2*a)-1/32*e^2*b/c^2*exp(2*c*x^2+2*b*x+2*a)-1/32*e^2*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*e^2/c*Pi^(1/2)*exp(1/2*(4*a*c-b^2)/c)/(-2*c)^(1/2)*e
rf(-(-2*c)^(1/2)*x+b/(-2*c)^(1/2))+1/8*d*e/c*exp(2*c*x^2+2*b*x+2*a)+1/8*d*e*b/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))

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Maxima [B]  time = 2.10776, size = 814, normalized size = 2.62 \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)^2,x, algorithm="maxima")

[Out]

1/16*(sqrt(2)*sqrt(pi)*erf(sqrt(2)*sqrt(-c)*x - 1/2*sqrt(2)*b/sqrt(-c))*e^(2*a - 1/2*b^2/c)/sqrt(-c) + sqrt(2)
*sqrt(pi)*erf(sqrt(2)*sqrt(c)*x + 1/2*sqrt(2)*b/sqrt(c))*e^(-2*a + 1/2*b^2/c)/sqrt(c) + 8*x)*d^2 + 1/16*(8*x^2
- sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(erf(sqrt(1/2)*sqrt(-(2*c*x + b)^2/c)) - 1)/(sqrt(-(2*c*x + b)^2/c)*c^(3/2)
) - sqrt(2)*e^(1/2*(2*c*x + b)^2/c)/sqrt(c))*e^(2*a - 1/2*b^2/c)/sqrt(c) - sqrt(2)*(sqrt(pi)*(2*c*x + b)*b*(er
f(sqrt(1/2)*sqrt((2*c*x + b)^2/c)) - 1)/(sqrt((2*c*x + b)^2/c)*(-c)^(3/2)) + sqrt(2)*c*e^(-1/2*(2*c*x + b)^2/c
)/(-c)^(3/2))*e^(-2*a + 1/2*b^2/c)/sqrt(-c))*d*e + 1/192*(32*x^3 + 3*sqrt(2)*(sqrt(pi)*(2*c*x + b)*b^2*(erf(sq
rt(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) - 3*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))*e^2

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Fricas [B]  time = 2.10577, size = 2700, normalized size = 8.68 \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)^2,x, algorithm="fricas")

[Out]

-1/192*(12*c^2*e^2*x - 6*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)^4 - 24*(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)^3 - 6*(2*c^2*e^2*x + 4*c^2*d*e - b*c*e^2)*sinh(c*x
^2 + b*x + a)^4 + 24*c^2*d*e - 6*b*c*e^2 + 3*sqrt(2)*sqrt(pi)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(c*
x^2 + b*x + a)^2*cosh(-1/2*(b^2 - 4*a*c)/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(c*x^2 + b*x + a)^2*
sinh(-1/2*(b^2 - 4*a*c)/c) + ((4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cosh(-1/2*(b^2 - 4*a*c)/c) + (4*c^2*d^2
- 4*b*c*d*e + (b^2 - c)*e^2)*sinh(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((4*c^2*d^2 - 4*b*c*d*e +
(b^2 - c)*e^2)*cosh(c*x^2 + b*x + a)*cosh(-1/2*(b^2 - 4*a*c)/c) + (4*c^2*d^2 - 4*b*c*d*e + (b^2 - c)*e^2)*cos
h(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)*sqr
t(-c)/c) - 3*sqrt(2)*sqrt(pi)*((4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(c*x^2 + b*x + a)^2*cosh(-1/2*(b^2
- 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(c*x^2 + b*x + a)^2*sinh(-1/2*(b^2 - 4*a*c)/c) + ((4
*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(-1/2*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*sin
h(-1/2*(b^2 - 4*a*c)/c))*sinh(c*x^2 + b*x + a)^2 + 2*((4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*cosh(c*x^2 + b*x
+ a)*cosh(-1/2*(b^2 - 4*a*c)/c) - (4*c^2*d^2 - 4*b*c*d*e + (b^2 + c)*e^2)*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)) - 32*(c^3*e^2*x^3 + 3*c^3*d
*e*x^2 + 3*c^3*d^2*x)*cosh(c*x^2 + b*x + a)^2 - 4*(8*c^3*e^2*x^3 + 24*c^3*d*e*x^2 + 24*c^3*d^2*x + 9*(2*c^2*e^
2*x + 4*c^2*d*e - b*c*e^2)*cosh(c*x^2 + b*x + a)^2)*sinh(c*x^2 + b*x + a)^2 - 8*(3*(2*c^2*e^2*x + 4*c^2*d*e -
b*c*e^2)*cosh(c*x^2 + b*x + a)^3 + 8*(c^3*e^2*x^3 + 3*c^3*d*e*x^2 + 3*c^3*d^2*x)*cosh(c*x^2 + b*x + a))*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 \left (d + e x\right )^{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((e*x+d)**2*cosh(c*x**2+b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.30703, size = 608, normalized size = 1.95 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi } d^{2} \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 )}}{16 \, \sqrt{c}} - \frac{\sqrt{2} \sqrt{\pi } d^{2} \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 )}}{16 \, \sqrt{-c}} + \frac{1}{6} \, x^{3} e^{2} + \frac{1}{2} \, d x^{2} e + \frac{1}{2} \, d^{2} x + \frac{\frac{\sqrt{2} \sqrt{\pi } b d \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}{2 \, c}\right )}}{\sqrt{c}} - 2 \, d e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a + 1\right )}}{16 \, c} + \frac{\frac{\sqrt{2} \sqrt{\pi } b d \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}{2 \, c}\right )}}{\sqrt{-c}} + 2 \, d e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a + 1\right )}}{16 \, c} - \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 + 4 \, 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 + 2\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 - 4 \, 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 + 2\right )}}{64 \, 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)^2,x, algorithm="giac")

[Out]

-1/16*sqrt(2)*sqrt(pi)*d^2*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c)/c)/sqrt(c) - 1/16*sqrt(2
)*sqrt(pi)*d^2*erf(-1/2*sqrt(2)*sqrt(-c)*(2*x + b/c))*e^(-1/2*(b^2 - 4*a*c)/c)/sqrt(-c) + 1/6*x^3*e^2 + 1/2*d*
x^2*e + 1/2*d^2*x + 1/16*(sqrt(2)*sqrt(pi)*b*d*erf(-1/2*sqrt(2)*sqrt(c)*(2*x + b/c))*e^(1/2*(b^2 - 4*a*c + 2*c
)/c)/sqrt(c) - 2*d*e^(-2*c*x^2 - 2*b*x - 2*a + 1))/c + 1/16*(sqrt(2)*sqrt(pi)*b*d*erf(-1/2*sqrt(2)*sqrt(-c)*(2
*x + b/c))*e^(-1/2*(b^2 - 4*a*c - 2*c)/c)/sqrt(-c) + 2*d*e^(2*c*x^2 + 2*b*x + 2*a + 1))/c - 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 + 4*c)/c)/sqrt(c) + 2*(c*(2*x + b/c)
- 2*b)*e^(-2*c*x^2 - 2*b*x - 2*a + 2))/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 - 4*c)/c)/sqrt(-c) - 2*(c*(2*x + b/c) - 2*b)*e^(2*c*x^2 + 2*b*x + 2*a + 2))/c^2