3.388 \(\int (c-a^2 c x^2)^{3/2} \cosh ^{-1}(a x)^{5/2} \, dx\)

Optimal. Leaf size=580 \[ -\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]

[Out]

(225*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/512 + (15*c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*Sqrt[Ar
cCosh[a*x]])/256 + (45*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*a*
c*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c*(1 - a^2*x^2)^2*Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]
^(5/2))/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2))/4 - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(7/2))/(28*
a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*x]]])/(16384*a*Sqrt[
-1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-
1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(16384*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])/(256*a*Sqrt[-1 + a*
x]*Sqrt[1 + a*x])

________________________________________________________________________________________

Rubi [A]  time = 1.52602, antiderivative size = 592, normalized size of antiderivative = 1.02, number of steps used = 40, number of rules used = 15, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {5713, 5685, 5683, 5676, 5664, 5759, 5670, 5448, 12, 3308, 2180, 2204, 2205, 5716, 5780} \[ -\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{15 \sqrt{\pi } c \sqrt{c-a^2 c x^2} \text{Erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 \sqrt{\frac{\pi }{2}} c \sqrt{c-a^2 c x^2} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{256 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac{1}{4} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{a x-1} \sqrt{a x+1}}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2),x]

[Out]

(225*c*x*Sqrt[c - a^2*c*x^2]*Sqrt[ArcCosh[a*x]])/512 + (15*c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*Sqrt[Ar
cCosh[a*x]])/256 + (45*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*a*
c*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c*(1 - a^2*x^2)^2*Sqrt[c
- a^2*c*x^2]*ArcCosh[a*x]^(3/2))/(32*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]
^(5/2))/8 + (c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2))/4 - (3*c*Sqrt[c - a^2*c*x^2]*ArcC
osh[a*x]^(7/2))/(28*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erf[2*Sqrt[ArcCosh[a*
x]]])/(16384*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erf[Sqrt[2]*Sqrt[ArcCosh[a
*x]]])/(256*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (15*c*Sqrt[Pi]*Sqrt[c - a^2*c*x^2]*Erfi[2*Sqrt[ArcCosh[a*x]]])/(
16384*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*c*Sqrt[Pi/2]*Sqrt[c - a^2*c*x^2]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*x]]])
/(256*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 5759

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

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]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 5780

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

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{5/2} \, 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)^{5/2} \, 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)^{5/2}+\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)^{5/2} \, dx}{4 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (5 a c \sqrt{c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)^{3/2} \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{\left (15 c \sqrt{c-a^2 c x^2}\right ) \int (-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)^{5/2}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 a c \sqrt{c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^{3/2} \, dx}{16 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \int \sqrt{-1+a x} \sqrt{1+a x} \sqrt{\cosh ^{-1}(a x)} \, dx}{256 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x \left (-1+a^2 x^2\right )}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{512 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 a^2 c \sqrt{c-a^2 c x^2}\right ) \int \frac{x^2 \sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{512 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\sqrt{\cosh ^{-1}(a x)}}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{128 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh ^3(x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{1024 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 a c \sqrt{c-a^2 c x^2}\right ) \int \frac{x}{\sqrt{\cosh ^{-1}(a x)}} \, dx}{256 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (2 x)}{4 \sqrt{x}}+\frac{\sinh (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh (x) \sinh (x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{1024 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{2 \sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{256 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 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 )}{8192 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 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 )}{8192 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 c \sqrt{c-a^2 c x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (2 x)}{\sqrt{x}} \, dx,x,\cosh ^{-1}(a x)\right )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (15 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 )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (15 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 )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 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 )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 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 )}{1024 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 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 )}{1024 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{15 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{15 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 c \sqrt{\frac{\pi }{2}} \sqrt{c-a^2 c x^2} \text{erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )}{4096 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 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 )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 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 )}{2048 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (45 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 )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (45 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 )}{512 a \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{225}{512} c x \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{15}{256} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \sqrt{\cosh ^{-1}(a x)}+\frac{45 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{256 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{32 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{5 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}{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)^{5/2}+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^{7/2}}{28 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erf}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{15 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{15 c \sqrt{\pi } \sqrt{c-a^2 c x^2} \text{erfi}\left (2 \sqrt{\cosh ^{-1}(a x)}\right )}{16384 a \sqrt{-1+a x} \sqrt{1+a x}}-\frac{15 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}}\\ \end{align*}

Mathematica [A]  time = 0.484845, size = 213, normalized size = 0.37 \[ \frac{c \sqrt{c-a^2 c x^2} \left (7 \sqrt{\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},4 \cosh ^{-1}(a x)\right )+7 \sqrt{-\cosh ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-4 \cosh ^{-1}(a x)\right )+420 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{Erf}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-420 \sqrt{2 \pi } \sqrt{\cosh ^{-1}(a x)} \text{Erfi}\left (\sqrt{2} \sqrt{\cosh ^{-1}(a x)}\right )-1536 \cosh ^{-1}(a x)^4-4480 \cosh \left (2 \cosh ^{-1}(a x)\right ) \cosh ^{-1}(a x)^2+3584 \cosh ^{-1}(a x)^3 \sinh \left (2 \cosh ^{-1}(a x)\right )+3360 \cosh ^{-1}(a x) \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{14336 a \sqrt{\frac{a x-1}{a x+1}} (a x+1) \sqrt{\cosh ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(5/2),x]

[Out]

(c*Sqrt[c - a^2*c*x^2]*(-1536*ArcCosh[a*x]^4 - 4480*ArcCosh[a*x]^2*Cosh[2*ArcCosh[a*x]] + 420*Sqrt[2*Pi]*Sqrt[
ArcCosh[a*x]]*Erf[Sqrt[2]*Sqrt[ArcCosh[a*x]]] - 420*Sqrt[2*Pi]*Sqrt[ArcCosh[a*x]]*Erfi[Sqrt[2]*Sqrt[ArcCosh[a*
x]]] + 7*Sqrt[-ArcCosh[a*x]]*Gamma[7/2, -4*ArcCosh[a*x]] + 7*Sqrt[ArcCosh[a*x]]*Gamma[7/2, 4*ArcCosh[a*x]] + 3
360*ArcCosh[a*x]*Sinh[2*ArcCosh[a*x]] + 3584*ArcCosh[a*x]^3*Sinh[2*ArcCosh[a*x]]))/(14336*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{5}{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)^(5/2),x)

[Out]

int((-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(5/2),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{5}{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)^(5/2),x, algorithm="maxima")

[Out]

integrate((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(5/2), 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)^(5/2),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)**(5/2),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)^(5/2),x, algorithm="giac")

[Out]

sage0*x