3.384 \(\int (c-a^2 c x^2)^{3/2} \cosh ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=511 \[ -\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{3/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{a x-1} \sqrt{a x+1}} \]

[Out]

(27*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (9*a*c*x^2*Sqrt[c - a^2*c
*x^2]*Sqrt[ArcCosh[a*x]])/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*Sqrt[Ar
cCosh[a*x]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/8 + (x*(c -
a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2))/4 - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(20*a*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(2048*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x
]) + (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])
- (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(2048*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*S
qrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

________________________________________________________________________________________

Rubi [A]  time = 1.09759, antiderivative size = 523, normalized size of antiderivative = 1.02, number of steps used = 27, number of rules used = 13, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.542, Rules used = {5713, 5685, 5683, 5676, 5664, 5781, 3312, 3307, 2180, 2204, 2205, 5716, 5701} \[ -\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2),x]

[Out]

(27*c*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (9*a*c*x^2*Sqrt[c - a^2*c
*x^2]*Sqrt[ArcCosh[a*x]])/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*Sqrt[Ar
cCosh[a*x]])/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/8 + (c*x*(1
- a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/4 - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/(20*
a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(2048*a*Sqrt[-1
+ a*x]*Sqrt[1 + a*x]) + (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 +
a*x]*Sqrt[1 + a*x]) - (3*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(2048*a*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]) + (3*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(64*a*Sqrt[-1 + a*x]*Sqrt[
1 + a*x])

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5685

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Simp[(x*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n)/(2*p + 1), x] + (Dist[(2*d1*d2*p)/(2*p + 1),
Int[(d1 + e1*x)^(p - 1)*(d2 + e2*x)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^(p - 1/2)
*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((2*p + 1)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(-1 + c^2*x^2)^(p - 1/2)*(a
+ b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)]
&& GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p - 1/2]

Rule 5683

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5664

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

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3307

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

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 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 5716

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)
^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*(-d)^p)/(2*c*(p + 1)), Int[(1 + c*x)^(p + 1/2)*
(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0]
&& GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p]

Rule 5701

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Dist[(-(d1*d2))^p/c, Subst[Int[(a + b*x)^n*Sinh[x]^(2*p + 1), x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c
, d1, e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && IGtQ[p + 1/2, 0] && (GtQ[d1, 0] && LtQ[d2, 0]
)

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{3/2} \, dx &=-\frac{\left (c \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^{3/2} \, dx}{\sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^{3/2} \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 a c \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \sqrt{\cosh ^{-1}(a x)} \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{(-1+a x)^{3/2} (1+a x)^{3/2}}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(a x)^{3/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (9 a c \sqrt{c-a^2 c x^2}\right ) \int x \sqrt{\cosh ^{-1}(a x)} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh ^4(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 a^2 c \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2}{\sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)}} \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}-\frac{\cosh (2 x)}{2 \sqrt{x}}+\frac{\cosh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{9 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (4 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cosh (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{4 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x}}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt{\cosh ^{-1}(a x)}\right )}{128 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{27 c \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}}{32 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}}{20 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{3 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{3 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{64 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.461414, size = 198, normalized size = 0.39 \[ \frac{c \sqrt{c-a^2 c x^2} \left (5 \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},4 \cosh ^{-1}(a x)\right )-5 \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-4 \cosh ^{-1}(a x)\right )+60 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )+60 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-384 \cosh ^{-1}(a x)^3-480 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)+640 \cosh ^{-1}(a x)^2 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{2560 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2),x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-384*ArcCosh[a*x]^3 - 480*ArcCosh[a*x]*Cosh[2*ArcCosh[a*x]] + 60*Sqrt[2*Pi]*Sqrt[ArcCo
sh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] + 60*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]] -
5*Sqrt[-ArcCosh[a*x]]*Gamma[5/2, -4*ArcCosh[a*x]] + 5*Sqrt[ArcCosh[a*x]]*Gamma[5/2, 4*ArcCosh[a*x]] + 640*Arc
Cosh[a*x]^2*Sinh[2*ArcCosh[a*x]]))/(2560*a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*Sqrt[ArcCosh[a*x]])

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Maple [F]  time = 0.309, size = 0, normalized size = 0. \begin{align*} \int \left ( -{a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}} \left ({\rm arccosh} \left (ax\right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(3/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arcosh}\left (a x\right )^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(3/2), 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((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(3/2)*acosh(a*x)**(3/2),x)

[Out]

Timed out

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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((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

sage0*x