3.4.84 \(\int (c-a^2 c x^2)^{3/2} \cosh ^{-1}(a x)^{3/2} \, dx\) [384]

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

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^(3/2)+3/8*c*x*arccosh(a*x)^(3/2)*(-a^2*c*x^2+c)^(1/2)-3/20*c*arccosh(a
*x)^(5/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/128*c*erf(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*P
i^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/128*c*erfi(2^(1/2)*arccosh(a*x)^(1/2))*2^(1/2)*Pi
^(1/2)*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-3/2048*c*erf(2*arccosh(a*x)^(1/2))*Pi^(1/2)*(-a^2*c*
x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-3/2048*c*erfi(2*arccosh(a*x)^(1/2))*Pi^(1/2)*(-a^2*c*x^2+c)^(1/2)/a
/(a*x-1)^(1/2)/(a*x+1)^(1/2)+27/256*c*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-9/
32*a*c*x^2*(-a^2*c*x^2+c)^(1/2)*arccosh(a*x)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/32*c*(-a^2*x^2+1)^2*(-a^2*c*x
^2+c)^(1/2)*arccosh(a*x)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 511, normalized size of antiderivative = 1.00, 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 = {5897, 5895, 5893, 5884, 5953, 3393, 3388, 2211, 2235, 2236, 5912, 5914, 5907} \begin {gather*} -\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}} \end {gather*}

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 + (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])

Rule 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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 2236

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 3388

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 3393

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 5884

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 5893

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

Rule 5895

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

Rule 5897

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

Rule 5907

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

Rule 5912

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

Rule 5914

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], 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]

Rule 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.35, size = 198, normalized size = 0.39 \begin {gather*} \frac {c \sqrt {c-a^2 c x^2} \left (-384 \cosh ^{-1}(a x)^3-480 \cosh ^{-1}(a x) \cosh \left (2 \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 )-5 \sqrt {-\cosh ^{-1}(a x)} \Gamma \left (\frac {5}{2},-4 \cosh ^{-1}(a x)\right )+5 \sqrt {\cosh ^{-1}(a x)} \Gamma \left (\frac {5}{2},4 \cosh ^{-1}(a x)\right )+640 \cosh ^{-1}(a x)^2 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )}{2560 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \sqrt {\cosh ^{-1}(a x)}} \end {gather*}

Antiderivative was successfully verified.

[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]])

________________________________________________________________________________________

Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}\, dx\]

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)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

________________________________________________________________________________________

Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^{3/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________