\(\int (c-a^2 c x^2)^{5/2} \text {arccosh}(a x)^3 \, dx\) [246]

Optimal result

Integrand size = 22, antiderivative size = 605 \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=-\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {245}{384} c^2 x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{768 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{12 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {5}{16} c^2 x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[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] (verified)

Time = 0.94 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5897, 5895, 5893, 5883, 5939, 30, 5912, 5914, 5898, 5896, 74, 14, 267} \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=-\frac {15 a c^2 x^2 \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{32 \sqrt {a x-1} \sqrt {a x+1}}+\frac {5}{16} c^2 x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}+\frac {245}{384} c^2 x \text {arccosh}(a x) \sqrt {c-a^2 c x^2}+\frac {1}{36} c^2 x (1-a x)^2 (a x+1)^2 \text {arccosh}(a x) \sqrt {c-a^2 c x^2}+\frac {65}{576} c^2 x (1-a x) (a x+1) \text {arccosh}(a x) \sqrt {c-a^2 c x^2}-\frac {5 c^2 \text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{64 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{12 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {5 c^2 \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {115 c^2 \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{768 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{6} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{5/2}+\frac {5}{24} c x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}-\frac {865 a c^2 x^2 \sqrt {c-a^2 c x^2}}{2304 \sqrt {a x-1} \sqrt {a x+1}}+\frac {c^2 \left (1-a^2 x^2\right )^3 \sqrt {c-a^2 c x^2}}{216 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {65 a^3 c^2 x^4 \sqrt {c-a^2 c x^2}}{2304 \sqrt {a x-1} \sqrt {a x+1}} \]

[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*x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3)/2
4 + (x*(c - a^2*c*x^2)^(5/2)*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 74

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[(a*c + b*
d*x^2)^m*(e + f*x)^p, x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[b*c + a*d, 0] && EqQ[n, m] && Integer
Q[m] && (NeQ[m, -1] || (EqQ[e, 0] && (EqQ[p, 1] ||  !IntegerQ[p])))

Rule 267

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 5883

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)/(Sqrt
[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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 5896

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[(1/2)*Simp[Sqrt[d1 + e1*x]/Sq
rt[1 + c*x]]*Simp[Sqrt[d2 + e2*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[d1 + e1*x]/Sqrt[1 + c*x]]*Simp[Sqrt[d2 + e2*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 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 5898

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/(2*p + 1))*Simp[(d1
+ e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*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, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] &&
GtQ[n, 0] && GtQ[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 5939

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3+\frac {1}{6} (5 c) \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx-\frac {\left (a c^2 \sqrt {c-a^2 c x^2}\right ) \int x (-1+a x)^2 (1+a x)^2 \text {arccosh}(a x)^2 \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx-\frac {\left (a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \left (-1+a^2 x^2\right )^2 \text {arccosh}(a x)^2 \, dx}{2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x (-1+a x) (1+a x) \text {arccosh}(a x)^2 \, dx}{8 \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} \text {arccosh}(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} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(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} \text {arccosh}(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 \frac {\text {arccosh}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \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 ) \text {arccosh}(a x)^2 \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (15 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x \text {arccosh}(a x)^2 \, dx}{16 \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} \text {arccosh}(a x)-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(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} \text {arccosh}(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 (-1+a x)^2 (1+a x)^2 \, dx}{36 \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 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {15}{32} c^2 x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x) \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (15 c^2 \sqrt {c-a^2 c x^2}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{32 \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 (5 a c^2 \sqrt {c-a^2 c x^2}\right ) \int x (-1+a x) (1+a x) \, 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 (-1+a x) (1+a x) \, 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}{32 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2}}{64 \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} \text {arccosh}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {15 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{64 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {15 a c^2 x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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 {\text {arccosh}(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 {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{128 \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 \, 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 x \left (-1+a^2 x^2\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 {245 a c^2 x^2 \sqrt {c-a^2 c x^2}}{768 \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} \text {arccosh}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{64 a \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 \left (-x+a^2 x^3\right ) \, dx}{64 \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} \text {arccosh}(a x)+\frac {65}{576} c^2 x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {1}{36} c^2 x (1-a x)^2 (1+a x)^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {115 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(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} \text {arccosh}(a x)^3+\frac {5}{24} c x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{6} x \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3-\frac {5 c^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{64 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.92 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.31 \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\frac {c^2 \sqrt {c-a^2 c x^2} \left (-4320 \text {arccosh}(a x)^4-9720 \cosh (2 \text {arccosh}(a x))+243 \cosh (4 \text {arccosh}(a x))-8 \cosh (6 \text {arccosh}(a x))-72 \text {arccosh}(a x)^2 (270 \cosh (2 \text {arccosh}(a x))-27 \cosh (4 \text {arccosh}(a x))+2 \cosh (6 \text {arccosh}(a x)))+288 \text {arccosh}(a x)^3 (45 \sinh (2 \text {arccosh}(a x))-9 \sinh (4 \text {arccosh}(a x))+\sinh (6 \text {arccosh}(a x)))+12 \text {arccosh}(a x) (1620 \sinh (2 \text {arccosh}(a x))-81 \sinh (4 \text {arccosh}(a x))+4 \sinh (6 \text {arccosh}(a x)))\right )}{55296 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]

[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))

Maple [A] (verified)

Time = 1.31 (sec) , antiderivative size = 887, normalized size of antiderivative = 1.47

method	result	size
default	\(-\frac {5 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {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 a^{7} x^{7}-64 a^{5} x^{5}+32 x^{6} a^{6} \sqrt {a x -1}\, \sqrt {a x +1}+38 a^{3} x^{3}-48 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{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 \operatorname {arccosh}\left (a x \right )^{3}-18 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {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 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{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 \operatorname {arccosh}\left (a x \right )^{3}-24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {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 a^{3} x^{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 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {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 a^{3} x^{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 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {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 a^{5} x^{5}-12 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{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 \operatorname {arccosh}\left (a x \right )^{3}+24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {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 x^{6} a^{6} \sqrt {a x -1}\, \sqrt {a x +1}+32 a^{7} x^{7}+48 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-64 a^{5} x^{5}-18 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+38 a^{3} x^{3}+\sqrt {a x -1}\, \sqrt {a x +1}-6 a x \right ) \left (36 \operatorname {arccosh}\left (a x \right )^{3}+18 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+1\right ) c^{2}}{13824 \left (a x -1\right ) \left (a x +1\right ) a}\)	\(887\)

[In]

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

[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*a^7*x^7-64*a^5*x^5+32*x^6*a^6*(a*x-1)^(1/2)*(a*x+1)^(1/2)+38*a^3*x^3-48*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^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*a^5*x^5-12*a^3*x^3+8*(a*x-1)^(1/2)*
(a*x+1)^(1/2)*a^4*x^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*a^3*x^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*a^3*x^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*a^5*x^5-12*a^3*x^3-8*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^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*x^6*a^6*(a*x-1)^(1/2)*(a*x+1)^(1/2)+32*a^7*x^7+48*(a
*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4-64*a^5*x^5-18*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+38*a^3*x^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

Fricas [F]

\[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arcosh}\left (a x\right )^{3} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

Exception generated. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]

[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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (c-a^2 c x^2\right )^{5/2} \text {arccosh}(a x)^3 \, dx=\int {\mathrm {acosh}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{5/2} \,d x \]

[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)