\(\int (c-a^2 c x^2)^{3/2} \text {arccosh}(a x)^3 \, dx\) [247]

Optimal result

Integrand size = 22, antiderivative size = 402 \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx=-\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 c \left (1-a^2 x^2\right )^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}} \]

[Out]

1/4*x*(-a^2*c*x^2+c)^(3/2)*arccosh(a*x)^3+45/64*c*x*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+3/32*c*x*(-a*x+1)*(a*x+1
)*arccosh(a*x)*(-a^2*c*x^2+c)^(1/2)+3/8*c*x*arccosh(a*x)^3*(-a^2*c*x^2+c)^(1/2)-51/128*a*c*x^2*(-a^2*c*x^2+c)^
(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/128*a^3*c*x^4*(-a^2*c*x^2+c)^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+27/128*c*ar
ccosh(a*x)^2*(-a^2*c*x^2+c)^(1/2)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)-9/16*a*c*x^2*arccosh(a*x)^2*(-a^2*c*x^2+c)^(1/
2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)+3/16*c*(-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)-3/32*c*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 = 0.55 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5897, 5895, 5893, 5883, 5939, 30, 5912, 5914, 5898, 5896, 74, 14} \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx=-\frac {9 a c x^2 \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{16 \sqrt {a x-1} \sqrt {a x+1}}+\frac {1}{4} x \text {arccosh}(a x)^3 \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{8} c x \text {arccosh}(a x)^3 \sqrt {c-a^2 c x^2}+\frac {45}{64} c x \text {arccosh}(a x) \sqrt {c-a^2 c x^2}+\frac {3}{32} c x (1-a x) (a x+1) \text {arccosh}(a x) \sqrt {c-a^2 c x^2}-\frac {3 c \text {arccosh}(a x)^4 \sqrt {c-a^2 c x^2}}{32 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 c \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{16 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {27 c \text {arccosh}(a x)^2 \sqrt {c-a^2 c x^2}}{128 a \sqrt {a x-1} \sqrt {a x+1}}-\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {a x-1} \sqrt {a x+1}}+\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {a x-1} \sqrt {a x+1}} \]

[In]

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

[Out]

(-51*a*c*x^2*Sqrt[c - a^2*c*x^2])/(128*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*a^3*c*x^4*Sqrt[c - a^2*c*x^2])/(128*
Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (45*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x])/64 + (3*c*x*(1 - a*x)*(1 + a*x)*Sqrt
[c - a^2*c*x^2]*ArcCosh[a*x])/32 + (27*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(128*a*Sqrt[-1 + a*x]*Sqrt[1 + a*
x]) - (9*a*c*x^2*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(16*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*(1 - a^2*x^2)^2*
Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^2)/(16*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (3*c*x*Sqrt[c - a^2*c*x^2]*ArcCosh[a
*x]^3)/8 + (x*(c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^3)/4 - (3*c*Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^4)/(32*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 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}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3 \, dx+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x (-1+a x) (1+a x) \text {arccosh}(a x)^2 \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\text {arccosh}(a x)^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \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 ) \text {arccosh}(a x)^2 \, dx}{4 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \text {arccosh}(a x)^2 \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \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} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x) \, dx}{8 \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 \text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{8 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = \frac {9}{16} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \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} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x (-1+a x) (1+a x) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{16 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {9 a c x^2 \sqrt {c-a^2 c x^2}}{32 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {9 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \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} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\text {arccosh}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{64 \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 ) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{64 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {45 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \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} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \left (-x+a^2 x^3\right ) \, dx}{32 \sqrt {-1+a x} \sqrt {1+a x}} \\ & = -\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {45}{64} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {3}{32} c x (1-a x) (1+a x) \sqrt {c-a^2 c x^2} \text {arccosh}(a x)+\frac {27 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{128 a \sqrt {-1+a x} \sqrt {1+a x}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^2}{16 \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} \text {arccosh}(a x)^2}{16 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3-\frac {3 c \sqrt {c-a^2 c x^2} \text {arccosh}(a x)^4}{32 a \sqrt {-1+a x} \sqrt {1+a x}} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.37 \[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx=-\frac {c \sqrt {c-a^2 c x^2} \left (96 \text {arccosh}(a x)^4-3 (-64 \cosh (2 \text {arccosh}(a x))+\cosh (4 \text {arccosh}(a x)))-24 \text {arccosh}(a x)^2 (-16 \cosh (2 \text {arccosh}(a x))+\cosh (4 \text {arccosh}(a x)))+12 \text {arccosh}(a x) (-32 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x)))+32 \text {arccosh}(a x)^3 (-8 \sinh (2 \text {arccosh}(a x))+\sinh (4 \text {arccosh}(a x)))\right )}{1024 a \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \]

[In]

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

[Out]

-1/1024*(c*Sqrt[c - a^2*c*x^2]*(96*ArcCosh[a*x]^4 - 3*(-64*Cosh[2*ArcCosh[a*x]] + Cosh[4*ArcCosh[a*x]]) - 24*A
rcCosh[a*x]^2*(-16*Cosh[2*ArcCosh[a*x]] + Cosh[4*ArcCosh[a*x]]) + 12*ArcCosh[a*x]*(-32*Sinh[2*ArcCosh[a*x]] +
Sinh[4*ArcCosh[a*x]]) + 32*ArcCosh[a*x]^3*(-8*Sinh[2*ArcCosh[a*x]] + Sinh[4*ArcCosh[a*x]])))/(a*Sqrt[(-1 + a*x
)/(1 + a*x)]*(1 + a*x))

Maple [A] (verified)

Time = 1.09 (sec) , antiderivative size = 536, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \operatorname {arccosh}\left (a x \right )^{4} c}{32 \sqrt {a x -1}\, \sqrt {a x +1}\, a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}+8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x -8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}-24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )-3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x +2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}-6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )-3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 a^{3} x^{3}-2 a x -2 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (4 \operatorname {arccosh}\left (a x \right )^{3}+6 \operatorname {arccosh}\left (a x \right )^{2}+6 \,\operatorname {arccosh}\left (a x \right )+3\right ) c}{32 \left (a x -1\right ) \left (a x +1\right ) a}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-12 a^{3} x^{3}-8 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+4 a x +8 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-\sqrt {a x -1}\, \sqrt {a x +1}\right ) \left (32 \operatorname {arccosh}\left (a x \right )^{3}+24 \operatorname {arccosh}\left (a x \right )^{2}+12 \,\operatorname {arccosh}\left (a x \right )+3\right ) c}{2048 \left (a x -1\right ) \left (a x +1\right ) a}\)	\(536\)

[In]

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

[Out]

-3/32*(-c*(a^2*x^2-1))^(1/2)/(a*x-1)^(1/2)/(a*x+1)^(1/2)/a*arccosh(a*x)^4*c-1/2048*(-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/(a*x-1)/(a*x+1)/a+1/32*(-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/(a*x-1)/(a*x+1)/a+1/32*(-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/(a*x-1)/(a*x+1)/a-1/2048*(-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/(a*x-1)/(a*x+1)/a

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \left (c-a^2 c x^2\right )^{3/2} \text {arccosh}(a x)^3 \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {acosh}^{3}{\left (a x \right )}\, dx \]

[In]

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

[Out]

Integral((-c*(a*x - 1)*(a*x + 1))**(3/2)*acosh(a*x)**3, x)

Maxima [F(-2)]

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

[In]

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

[In]

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

[In]

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

[Out]

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