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

Optimal. Leaf size=605 \[ -\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {a x-1} \sqrt {a x+1}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (a x+1)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {65}{576} c^2 x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {5 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {115 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3+\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {a x-1} \sqrt {a x+1}} \]

[Out]

5/24*c*x*(-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^3+1/6*x*(-a^2*c*x^2+c)^(5/2)*arccosh(a*x)^3+245/384*c^2*x*arccosh(a
*x)*(-a^2*c*x^2+c)^(1/2)+65/576*c^2*x*(-a*x+1)*(a*x+1)*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+1/36*c^2*x*(-a*x+1)^2
*(a*x+1)^2*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+5/16*c^2*x*arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2)-865/2304*a*c^2*x^2
*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+65/2304*a^3*c^2*x^4*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+
1)^(1/2)+1/216*c^2*(-a^2*x^2+1)^3*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+115/768*c^2*arccosh(a*x)^
2*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-15/32*a*c^2*x^2*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/(a*x-
1)^(1/2)/(a*x+1)^(1/2)+5/32*c^2*(-a^2*x^2+1)^2*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/
2)+1/12*c^2*(-a^2*x^2+1)^3*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-5/64*c^2*arccosh(
a*x)^4*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.50, antiderivative size = 636, normalized size of antiderivative = 1.05, number of steps used = 25, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {5713, 5685, 5683, 5676, 5662, 5759, 30, 5716, 14, 261} \[ \frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {a x-1} \sqrt {a x+1}}-\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {a x-1} \sqrt {a x+1}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{6} c^2 x (1-a x)^2 (a x+1)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {5}{24} c^2 x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (a x+1)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {65}{576} c^2 x (1-a x) (a x+1) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {5 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {115 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt {a x-1} \sqrt {a x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-865*a*c^2*x^2*Sqrt[c - a^2*c*x^2])/(2304*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (65*a^3*c^2*x^4*Sqrt[c - a^2*c*x^2]
)/(2304*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*(1 - a^2*x^2)^3*Sqrt[c - a^2*c*x^2])/(216*a*Sqrt[-1 + a*x]*Sqrt[1
+ a*x]) + (245*c^2*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/384 + (65*c^2*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^2
]*ArcCosh[a*x])/576 + (c^2*x*(1 - a*x)^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/36 + (115*c^2*Sqrt[c -
a^2*c*x^2]*ArcCosh[a*x]^2)/(768*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) - (15*a*c^2*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*
x]^2)/(32*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (5*c^2*(1 - a^2*x^2)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(32*a*Sqr
t[-1 + a*x]*Sqrt[1 + a*x]) + (c^2*(1 - a^2*x^2)^3*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(12*a*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]) + (5*c^2*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/16 + (5*c^2*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^
2]*ArcCosh[a*x]^3)/24 + (c^2*x*(1 - a*x)^2*(1 + a*x)^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^3)/6 - (5*c^2*Sqrt[c -
a^2*c*x^2]*ArcCosh[a*x]^4)/(64*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 5662

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

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 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 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 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 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 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]

Rubi steps

\begin {align*} \int \left (c-a^2 c x^2\right )^{5/2} \cosh ^{-1}(a x)^3 \, dx &=\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^3 \, dx}{\sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^3 \, dx}{6 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x)^2 \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{24} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {\left (c^2 \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x) \, dx}{6 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3 \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x)^2 \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {5}{24} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx}{36 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right )^2 \, dx}{36 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \cosh ^{-1}(a x)^2 \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {5}{24} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx}{48 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \, dx}{144 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right ) \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 a^2 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {5}{24} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{96 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\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^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (-x+a^2 x^3\right ) \, dx}{144 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{96 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int \left (-x+a^2 x^3\right ) \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{128 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=-\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{768 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {5}{24} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac {1}{6} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^4}{64 a \sqrt {-1+a x} \sqrt {1+a x}}\\ \end {align*}

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Mathematica [A]  time = 1.20, size = 189, normalized size = 0.31 \[ \frac {c^2 \sqrt {c-a^2 c x^2} \left (-4320 \cosh ^{-1}(a x)^4-72 \left (270 \cosh \left (2 \cosh ^{-1}(a x)\right )-27 \cosh \left (4 \cosh ^{-1}(a x)\right )+2 \cosh \left (6 \cosh ^{-1}(a x)\right )\right ) \cosh ^{-1}(a x)^2-9720 \cosh \left (2 \cosh ^{-1}(a x)\right )+243 \cosh \left (4 \cosh ^{-1}(a x)\right )-8 \cosh \left (6 \cosh ^{-1}(a x)\right )+288 \cosh ^{-1}(a x)^3 \left (45 \sinh \left (2 \cosh ^{-1}(a x)\right )-9 \sinh \left (4 \cosh ^{-1}(a x)\right )+\sinh \left (6 \cosh ^{-1}(a x)\right )\right )+12 \cosh ^{-1}(a x) \left (1620 \sinh \left (2 \cosh ^{-1}(a x)\right )-81 \sinh \left (4 \cosh ^{-1}(a x)\right )+4 \sinh \left (6 \cosh ^{-1}(a x)\right )\right )\right )}{55296 a \sqrt {\frac {a x-1}{a x+1}} (a x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(c^2*Sqrt[c - a^2*c*x^2]*(-4320*ArcCosh[a*x]^4 - 9720*Cosh[2*ArcCosh[a*x]] + 243*Cosh[4*ArcCosh[a*x]] - 8*Cosh
[6*ArcCosh[a*x]] - 72*ArcCosh[a*x]^2*(270*Cosh[2*ArcCosh[a*x]] - 27*Cosh[4*ArcCosh[a*x]] + 2*Cosh[6*ArcCosh[a*
x]]) + 288*ArcCosh[a*x]^3*(45*Sinh[2*ArcCosh[a*x]] - 9*Sinh[4*ArcCosh[a*x]] + Sinh[6*ArcCosh[a*x]]) + 12*ArcCo
sh[a*x]*(1620*Sinh[2*ArcCosh[a*x]] - 81*Sinh[4*ArcCosh[a*x]] + 4*Sinh[6*ArcCosh[a*x]])))/(55296*a*Sqrt[(-1 + a
*x)/(1 + a*x)]*(1 + a*x))

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fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{3}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^4*c^2*x^4 - 2*a^2*c^2*x^2 + c^2)*sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^3, x)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.51, size = 887, normalized size = 1.47 \[ -\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \mathrm {arccosh}\left (a x \right )^{4} c^{2}}{64 \sqrt {a x -1}\, \sqrt {a x +1}\, a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (32 x^{7} a^{7}-64 x^{5} a^{5}+32 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}+38 x^{3} a^{3}-48 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}-6 a x +18 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (36 \mathrm {arccosh}\left (a x \right )^{3}-18 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-1\right ) c^{2}}{13824 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}+8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \mathrm {arccosh}\left (a x \right )^{3}-24 \mathrm {arccosh}\left (a x \right )^{2}+12 \,\mathrm {arccosh}\left (a x \right )-3\right ) c^{2}}{4096 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {15 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}-6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )-3\right ) c^{2}}{512 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {15 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 x^{3} a^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \mathrm {arccosh}\left (a x \right )^{3}+6 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+3\right ) c^{2}}{512 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 x^{5} a^{5}-12 x^{3} a^{3}-8 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \mathrm {arccosh}\left (a x \right )^{3}+24 \mathrm {arccosh}\left (a x \right )^{2}+12 \,\mathrm {arccosh}\left (a x \right )+3\right ) c^{2}}{4096 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-32 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}+32 x^{7} a^{7}+48 \sqrt {a x +1}\, \sqrt {a x -1}\, x^{4} a^{4}-64 x^{5} a^{5}-18 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+38 x^{3} a^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-6 a x \right ) \left (36 \mathrm {arccosh}\left (a x \right )^{3}+18 \mathrm {arccosh}\left (a x \right )^{2}+6 \,\mathrm {arccosh}\left (a x \right )+1\right ) c^{2}}{13824 \left (a x -1\right ) \left (a x +1\right ) a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-a^2*c*x^2+c)^(5/2)*arccosh(a*x)^3,x)

[Out]

-5/64*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4*c^2+1/13824*(-c*(a^2*x^2-1))^(1/2)*(
32*x^7*a^7-64*x^5*a^5+32*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^6*x^6+38*x^3*a^3-48*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4
-6*a*x+18*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(36*arccosh(a*x)^3-18*arccosh(a*x)^
2+6*arccosh(a*x)-1)*c^2/(a*x-1)/(a*x+1)/a-3/4096*(-c*(a^2*x^2-1))^(1/2)*(8*x^5*a^5-12*x^3*a^3+8*(a*x+1)^(1/2)*
(a*x-1)^(1/2)*x^4*a^4+4*a*x-8*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(32*arccosh(a*x
)^3-24*arccosh(a*x)^2+12*arccosh(a*x)-3)*c^2/(a*x-1)/(a*x+1)/a+15/512*(-c*(a^2*x^2-1))^(1/2)*(2*x^3*a^3-2*a*x+
2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3-6*arccosh(a*x)^2+6*arccos
h(a*x)-3)*c^2/(a*x-1)/(a*x+1)/a+15/512*(-c*(a^2*x^2-1))^(1/2)*(2*x^3*a^3-2*a*x-2*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)
^(1/2)+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(4*arccosh(a*x)^3+6*arccosh(a*x)^2+6*arccosh(a*x)+3)*c^2/(a*x-1)/(a*x+1)/a
-3/4096*(-c*(a^2*x^2-1))^(1/2)*(8*x^5*a^5-12*x^3*a^3-8*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4+4*a*x+8*a^2*x^2*(a*
x-1)^(1/2)*(a*x+1)^(1/2)-(a*x-1)^(1/2)*(a*x+1)^(1/2))*(32*arccosh(a*x)^3+24*arccosh(a*x)^2+12*arccosh(a*x)+3)*
c^2/(a*x-1)/(a*x+1)/a+1/13824*(-c*(a^2*x^2-1))^(1/2)*(-32*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^6*x^6+32*x^7*a^7+48*(a
*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4-64*x^5*a^5-18*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+38*x^3*a^3+(a*x-1)^(1/2)*(
a*x+1)^(1/2)-6*a*x)*(36*arccosh(a*x)^3+18*arccosh(a*x)^2+6*arccosh(a*x)+1)*c^2/(a*x-1)/(a*x+1)/a

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{5/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^3*(c - a^2*c*x^2)^(5/2),x)

[Out]

int(acosh(a*x)^3*(c - a^2*c*x^2)^(5/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-a**2*c*x**2+c)**(5/2)*acosh(a*x)**3,x)

[Out]

Timed out

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