3.561 \(\int \frac{(d+e x^2)^2}{\sqrt{a+b \cosh ^{-1}(c x)}} \, dx\)

Optimal. Leaf size=608 \[ -\frac{\sqrt{\pi } d e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{3}} d e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}+\frac{\sqrt{\pi } d e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} d e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}-\frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 \sqrt{b} c^5}-\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}-\frac{\sqrt{\frac{\pi }{5}} e^2 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 \sqrt{b} c^5}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}+\frac{\sqrt{\frac{\pi }{5}} e^2 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}-\frac{\sqrt{\pi } d^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c}+\frac{\sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c} \]

[Out]

-(d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a
+ b*ArcCosh[c*x]]/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(1
6*Sqrt[b]*c^5) - (d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3)
- (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) - (e^2*E^((5*a
)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) + (d^2*Sqrt[Pi]*Erfi[Sqrt[a
+ b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) + (d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*S
qrt[b]*c^3*E^(a/b)) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*Sqrt[b]*c^5*E^(a/b)) + (d*e*Sq
rt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3*E^((3*a)/b)) + (e^2*Sqrt[3*Pi]*Erfi[
(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((3*a)/b)) + (e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt
[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((5*a)/b))

________________________________________________________________________________________

Rubi [A]  time = 1.14779, antiderivative size = 608, normalized size of antiderivative = 1., number of steps used = 39, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5707, 5658, 3308, 2180, 2205, 2204, 5670, 5448} \[ -\frac{\sqrt{\pi } d e e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}-\frac{\sqrt{\frac{\pi }{3}} d e e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}+\frac{\sqrt{\pi } d e e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}+\frac{\sqrt{\frac{\pi }{3}} d e e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{4 \sqrt{b} c^3}-\frac{\sqrt{\pi } e^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 \sqrt{b} c^5}-\frac{\sqrt{3 \pi } e^2 e^{\frac{3 a}{b}} \text{Erf}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}-\frac{\sqrt{\frac{\pi }{5}} e^2 e^{\frac{5 a}{b}} \text{Erf}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}+\frac{\sqrt{\pi } e^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{16 \sqrt{b} c^5}+\frac{\sqrt{3 \pi } e^2 e^{-\frac{3 a}{b}} \text{Erfi}\left (\frac{\sqrt{3} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}+\frac{\sqrt{\frac{\pi }{5}} e^2 e^{-\frac{5 a}{b}} \text{Erfi}\left (\frac{\sqrt{5} \sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{32 \sqrt{b} c^5}-\frac{\sqrt{\pi } d^2 e^{a/b} \text{Erf}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c}+\frac{\sqrt{\pi } d^2 e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \cosh ^{-1}(c x)}}{\sqrt{b}}\right )}{2 \sqrt{b} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c) - (d*e*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a
+ b*ArcCosh[c*x]]/Sqrt[b]])/(4*Sqrt[b]*c^3) - (e^2*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(1
6*Sqrt[b]*c^5) - (d*e*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3)
- (e^2*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) - (e^2*E^((5*a
)/b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5) + (d^2*Sqrt[Pi]*Erfi[Sqrt[a
+ b*ArcCosh[c*x]]/Sqrt[b]])/(2*Sqrt[b]*c*E^(a/b)) + (d*e*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(4*S
qrt[b]*c^3*E^(a/b)) + (e^2*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcCosh[c*x]]/Sqrt[b]])/(16*Sqrt[b]*c^5*E^(a/b)) + (d*e*Sq
rt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(4*Sqrt[b]*c^3*E^((3*a)/b)) + (e^2*Sqrt[3*Pi]*Erfi[
(Sqrt[3]*Sqrt[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((3*a)/b)) + (e^2*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt
[a + b*ArcCosh[c*x]])/Sqrt[b]])/(32*Sqrt[b]*c^5*E^((5*a)/b))

Rule 5707

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcCosh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p
] && (p > 0 || IGtQ[n, 0])

Rule 5658

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

Rule 3308

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/E^(I*(e + f*x))
, x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

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 5670

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*
Cosh[x]^m*Sinh[x], x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 1.09197, size = 530, normalized size = 0.87 \[ \frac{e^{-\frac{5 a}{b}} \left (30 e^{\frac{6 a}{b}} \left (8 c^4 d^2+4 c^2 d e+e^2\right ) \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}+\cosh ^{-1}(c x)\right )+240 c^4 d^2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+40 \sqrt{3} c^2 d e e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+120 c^2 d e e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+40 \sqrt{3} c^2 d e e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3 \sqrt{5} e^2 \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+15 \sqrt{3} e^2 e^{\frac{2 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+30 e^2 e^{\frac{4 a}{b}} \sqrt{-\frac{a+b \cosh ^{-1}(c x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \cosh ^{-1}(c x)}{b}\right )+15 \sqrt{3} e^2 e^{\frac{8 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )+3 \sqrt{5} e^2 e^{\frac{10 a}{b}} \sqrt{\frac{a}{b}+\cosh ^{-1}(c x)} \text{Gamma}\left (\frac{1}{2},\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )\right )}{480 c^5 \sqrt{a+b \cosh ^{-1}(c x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(30*(8*c^4*d^2 + 4*c^2*d*e + e^2)*E^((6*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, a/b + ArcCosh[c*x]] + 3*Sqrt
[5]*e^2*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-5*(a + b*ArcCosh[c*x]))/b] + 40*Sqrt[3]*c^2*d*e*E^((2*a)/
b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 15*Sqrt[3]*e^2*E^((2*a)/b)*Sqrt[-
((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, (-3*(a + b*ArcCosh[c*x]))/b] + 240*c^4*d^2*E^((4*a)/b)*Sqrt[-((a + b*ArcC
osh[c*x])/b)]*Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 120*c^2*d*e*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*
Gamma[1/2, -((a + b*ArcCosh[c*x])/b)] + 30*e^2*E^((4*a)/b)*Sqrt[-((a + b*ArcCosh[c*x])/b)]*Gamma[1/2, -((a + b
*ArcCosh[c*x])/b)] + 40*Sqrt[3]*c^2*d*e*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]
))/b] + 15*Sqrt[3]*e^2*E^((8*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (3*(a + b*ArcCosh[c*x]))/b] + 3*Sqrt[5]
*e^2*E^((10*a)/b)*Sqrt[a/b + ArcCosh[c*x]]*Gamma[1/2, (5*(a + b*ArcCosh[c*x]))/b])/(480*c^5*E^((5*a)/b)*Sqrt[a
+ b*ArcCosh[c*x]])

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Maple [F]  time = 0.246, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e{x}^{2}+d \right ) ^{2}{\frac{1}{\sqrt{a+b{\rm arccosh} \left (cx\right )}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^2/(a+b*arccosh(c*x))^(1/2),x)

[Out]

int((e*x^2+d)^2/(a+b*arccosh(c*x))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{2} + d\right )}^{2}}{\sqrt{b \operatorname{arcosh}\left (c x\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2/(a+b*arccosh(c*x))^(1/2),x, algorithm="maxima")

[Out]

integrate((e*x^2 + d)^2/sqrt(b*arccosh(c*x) + a), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2/(a+b*arccosh(c*x))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**2/(a+b*acosh(c*x))**(1/2),x)

[Out]

Integral((d + e*x**2)**2/sqrt(a + b*acosh(c*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^2/(a+b*arccosh(c*x))^(1/2),x, algorithm="giac")

[Out]

sage0*x