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

Optimal. Leaf size=402 \[ \frac{3 a^3 c x^4 \sqrt{c-a^2 c x^2}}{128 \sqrt{a x-1} \sqrt{a x+1}}-\frac{51 a c x^2 \sqrt{c-a^2 c x^2}}{128 \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt{a x-1} \sqrt{a x+1}}+\frac{1}{4} x \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{45}{64} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{3}{32} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{27 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt{a x-1} \sqrt{a x+1}} \]

[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])

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Rubi [A]  time = 0.945539, antiderivative size = 414, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {5713, 5685, 5683, 5676, 5662, 5759, 30, 5716, 14} \[ \frac{3 a^3 c x^4 \sqrt{c-a^2 c x^2}}{128 \sqrt{a x-1} \sqrt{a x+1}}-\frac{51 a c x^2 \sqrt{c-a^2 c x^2}}{128 \sqrt{a x-1} \sqrt{a x+1}}-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 \sqrt{a x-1} \sqrt{a x+1}}+\frac{3}{8} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{1}{4} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3+\frac{45}{64} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{3}{32} c x (1-a x) (a x+1) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{3 c \left (1-a^2 x^2\right )^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{16 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{27 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^2}{128 a \sqrt{a x-1} \sqrt{a x+1}} \]

Antiderivative was successfully verified.

[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 + (c*x*(1 - a*x)*(1 + a*x)*Sqrt[c - a^2*c*x^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 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist[((-d)^IntPart[p]*(
d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rule 5685

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

Rule 5683

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

Rule 5676

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

Rule 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 5759

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

Rule 30

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

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 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]]

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^3 \, 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 \, 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+\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 \, 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 ) \cosh ^{-1}(a x)^2 \, dx}{4 \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} \cosh ^{-1}(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} \cosh ^{-1}(a x)^3+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{\left (3 c \sqrt{c-a^2 c x^2}\right ) \int (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x) \, dx}{8 \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}{\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 \cosh ^{-1}(a x)^2 \, dx}{8 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{3}{32} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(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} \cosh ^{-1}(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} \cosh ^{-1}(a x)^3+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(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} \cosh ^{-1}(a x) \, dx}{32 \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^2 c \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}{8 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{45}{64} c x \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{3}{32} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)-\frac{9 a c x^2 \sqrt{c-a^2 c x^2} \cosh ^{-1}(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} \cosh ^{-1}(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} \cosh ^{-1}(a x)^3+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(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{\cosh ^{-1}(a x)}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{64 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{\left (9 c \sqrt{c-a^2 c x^2}\right ) \int \frac{\cosh ^{-1}(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 \left (-x+a^2 x^3\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{\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{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} \cosh ^{-1}(a x)+\frac{3}{32} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)+\frac{27 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(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} \cosh ^{-1}(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} \cosh ^{-1}(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} \cosh ^{-1}(a x)^3+\frac{1}{4} c x (1-a x) (1+a x) \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^3-\frac{3 c \sqrt{c-a^2 c x^2} \cosh ^{-1}(a x)^4}{32 a \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}

Mathematica [A]  time = 0.493394, size = 148, normalized size = 0.37 \[ -\frac{c \sqrt{c-a^2 c x^2} \left (96 \cosh ^{-1}(a x)^4-24 \left (\cosh \left (4 \cosh ^{-1}(a x)\right )-16 \cosh \left (2 \cosh ^{-1}(a x)\right )\right ) \cosh ^{-1}(a x)^2-3 \left (\cosh \left (4 \cosh ^{-1}(a x)\right )-64 \cosh \left (2 \cosh ^{-1}(a x)\right )\right )+32 \cosh ^{-1}(a x)^3 \left (\sinh \left (4 \cosh ^{-1}(a x)\right )-8 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )+12 \cosh ^{-1}(a x) \left (\sinh \left (4 \cosh ^{-1}(a x)\right )-32 \sinh \left (2 \cosh ^{-1}(a x)\right )\right )\right )}{1024 a \sqrt{\frac{a x-1}{a x+1}} (a x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(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*ArcCosh[
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]])))/(1024*a*Sqrt[(-1 + a*x)/
(1 + a*x)]*(1 + a*x))

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Maple [A]  time = 0.174, size = 536, normalized size = 1.3 \begin{align*} -{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{4}c}{32\,a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\frac{1}{\sqrt{ax-1}}}{\frac{1}{\sqrt{ax+1}}}}-{\frac{ \left ( 32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-24\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+12\,{\rm arccosh} \left (ax\right )-3 \right ) c}{ \left ( 2048\,ax-2048 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 8\,{x}^{5}{a}^{5}-12\,{x}^{3}{a}^{3}+8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+4\,ax-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{ \left ( 4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}-6\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )-3 \right ) c}{ \left ( 32\,ax-32 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,{x}^{3}{a}^{3}-2\,ax+2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{ \left ( 4\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+6\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+6\,{\rm arccosh} \left (ax\right )+3 \right ) c}{ \left ( 32\,ax-32 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,{x}^{3}{a}^{3}-2\,ax-2\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{ \left ( 32\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}+24\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}+12\,{\rm arccosh} \left (ax\right )+3 \right ) c}{ \left ( 2048\,ax-2048 \right ) \left ( ax+1 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 8\,{x}^{5}{a}^{5}-12\,{x}^{3}{a}^{3}-8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+4\,ax+8\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-\sqrt{ax-1}\sqrt{ax+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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*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*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^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*x^3*a^3-2*a*x+2*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^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*x^3*a^3-2*a*x-2*(a*
x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^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*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*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (a^{2} c x^{2} - c\right )} \sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}} \operatorname{arcosh}\left (a x\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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