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

Optimal. Leaf size=505 \[ \frac{6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{16}{315} a c^3 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{976 c^3 \sqrt{a x-1} \sqrt{a x+1}}{315 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}+\frac{3 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)^2}{49 a}-\frac{18 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{8 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{48 c^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{35 a} \]

[Out]

(-976*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(315*a) + (16*a*c^3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/315 + (7104*c^3*
(1 - a^2*x^2))/(42875*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (1184*c^3*(1 - a^2*x^2)^2)/(42875*a*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]) + (2664*c^3*(1 - a^2*x^2)^3)/(214375*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (6*c^3*(1 - a^2*x^2)^4)/(24
01*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (4322*c^3*x*ArcCosh[a*x])/1225 - (1514*a^2*c^3*x^3*ArcCosh[a*x])/3675 + (
702*a^4*c^3*x^5*ArcCosh[a*x])/6125 - (6*a^6*c^3*x^7*ArcCosh[a*x])/343 - (48*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*A
rcCosh[a*x]^2)/(35*a) + (8*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(35*a) - (18*c^3*(-1 + a*x)^(5
/2)*(1 + a*x)^(5/2)*ArcCosh[a*x]^2)/(175*a) + (3*c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2)*ArcCosh[a*x]^2)/(49*a) +
(16*c^3*x*ArcCosh[a*x]^3)/35 + (8*c^3*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/35 + (6*c^3*x*(1 - a^2*x^2)^2*ArcCosh[a
*x]^3)/35 + (c^3*x*(1 - a^2*x^2)^3*ArcCosh[a*x]^3)/7

________________________________________________________________________________________

Rubi [A]  time = 1.40818, antiderivative size = 505, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5681, 5718, 194, 5680, 12, 1610, 1799, 1850, 520, 1247, 698, 460, 74, 5654} \[ \frac{6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt{a x-1} \sqrt{a x+1}}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{16}{315} a c^3 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{976 c^3 \sqrt{a x-1} \sqrt{a x+1}}{315 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}+\frac{3 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \cosh ^{-1}(a x)^2}{49 a}-\frac{18 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{8 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{48 c^3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{35 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-976*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(315*a) + (16*a*c^3*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/315 + (7104*c^3*
(1 - a^2*x^2))/(42875*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (1184*c^3*(1 - a^2*x^2)^2)/(42875*a*Sqrt[-1 + a*x]*Sqr
t[1 + a*x]) + (2664*c^3*(1 - a^2*x^2)^3)/(214375*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (6*c^3*(1 - a^2*x^2)^4)/(24
01*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (4322*c^3*x*ArcCosh[a*x])/1225 - (1514*a^2*c^3*x^3*ArcCosh[a*x])/3675 + (
702*a^4*c^3*x^5*ArcCosh[a*x])/6125 - (6*a^6*c^3*x^7*ArcCosh[a*x])/343 - (48*c^3*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*A
rcCosh[a*x]^2)/(35*a) + (8*c^3*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(35*a) - (18*c^3*(-1 + a*x)^(5
/2)*(1 + a*x)^(5/2)*ArcCosh[a*x]^2)/(175*a) + (3*c^3*(-1 + a*x)^(7/2)*(1 + a*x)^(7/2)*ArcCosh[a*x]^2)/(49*a) +
(16*c^3*x*ArcCosh[a*x]^3)/35 + (8*c^3*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/35 + (6*c^3*x*(1 - a^2*x^2)^2*ArcCosh[a
*x]^3)/35 + (c^3*x*(1 - a^2*x^2)^3*ArcCosh[a*x]^3)/7

Rule 5681

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[(b*c*n*(-d)^p)/(2*p + 1), Int[x*(-1 + c*x)^(p - 1/2)*(1 + c*x)^
(p - 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] + Dist[(2*d*p)/(2*p + 1), Int[(d + e*x^2)^(p - 1)*(a + b*ArcCos
h[c*x])^n, x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && IntegerQ[p]

Rule 5718

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

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5680

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 520

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[((a1 + b1*x^(n/2))^FracPart[p]*(a2 + b2*x^(n/2))^FracPart[p])/(a1
*a2 + b1*b2*x^n)^FracPart[p], Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 74

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

Rule 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rubi steps

\begin{align*} \int \left (c-a^2 c x^2\right )^3 \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \cosh ^{-1}(a x)^3 \, dx+\frac{1}{7} \left (3 a c^3\right ) \int x (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx-\frac{1}{49} \left (6 c^3\right ) \int \left (-1+a^2 x^2\right )^3 \cosh ^{-1}(a x) \, dx-\frac{1}{35} \left (18 a c^3\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac{6}{49} c^3 x \cosh ^{-1}(a x)-\frac{6}{49} a^2 c^3 x^3 \cosh ^{-1}(a x)+\frac{18}{245} a^4 c^3 x^5 \cosh ^{-1}(a x)-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{1}{175} \left (36 c^3\right ) \int \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x) \, dx+\frac{1}{35} \left (16 c^3\right ) \int \cosh ^{-1}(a x)^3 \, dx+\frac{1}{49} \left (6 a c^3\right ) \int \frac{x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right )}{35 \sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{1}{35} \left (24 a c^3\right ) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac{402 c^3 x \cosh ^{-1}(a x)}{1225}-\frac{318 a^2 c^3 x^3 \cosh ^{-1}(a x)}{1225}+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)+\frac{8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3-\frac{1}{35} \left (16 c^3\right ) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx+\frac{\left (6 a c^3\right ) \int \frac{x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx}{1715}-\frac{1}{175} \left (36 a c^3\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{15 \sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{1}{35} \left (48 a c^3\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{962 c^3 x \cosh ^{-1}(a x)}{1225}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac{48 c^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac{8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{1}{35} \left (96 c^3\right ) \int \cosh ^{-1}(a x) \, dx-\frac{1}{875} \left (12 a c^3\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{1}{35} \left (16 a c^3\right ) \int \frac{x \left (-3+a^2 x^2\right )}{3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{\left (6 a c^3 \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right )}{\sqrt{-1+a^2 x^2}} \, dx}{1715 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac{48 c^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac{8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3+\frac{1}{105} \left (16 a c^3\right ) \int \frac{x \left (-3+a^2 x^2\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{1}{35} \left (96 a c^3\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{\left (3 a c^3 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{-35+35 a^2 x-21 a^4 x^2+5 a^6 x^3}{\sqrt{-1+a^2 x}} \, dx,x,x^2\right )}{1715 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (12 a c^3 \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (15-10 a^2 x^2+3 a^4 x^4\right )}{\sqrt{-1+a^2 x^2}} \, dx}{875 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{96 c^3 \sqrt{-1+a x} \sqrt{1+a x}}{35 a}+\frac{16}{315} a c^3 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac{48 c^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac{8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3-\frac{1}{45} \left (16 a c^3\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx+\frac{\left (3 a c^3 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{16}{\sqrt{-1+a^2 x}}+8 \sqrt{-1+a^2 x}-6 \left (-1+a^2 x\right )^{3/2}+5 \left (-1+a^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )}{1715 \sqrt{-1+a x} \sqrt{1+a x}}-\frac{\left (6 a c^3 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{15-10 a^2 x+3 a^4 x^2}{\sqrt{-1+a^2 x}} \, dx,x,x^2\right )}{875 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{976 c^3 \sqrt{-1+a x} \sqrt{1+a x}}{315 a}+\frac{16}{315} a c^3 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{96 c^3 \left (1-a^2 x^2\right )}{1715 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{16 c^3 \left (1-a^2 x^2\right )^2}{1715 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{36 c^3 \left (1-a^2 x^2\right )^3}{8575 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac{48 c^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac{8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3-\frac{\left (6 a c^3 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{-1+a^2 x}}-4 \sqrt{-1+a^2 x}+3 \left (-1+a^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{875 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{976 c^3 \sqrt{-1+a x} \sqrt{1+a x}}{315 a}+\frac{16}{315} a c^3 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{4322 c^3 x \cosh ^{-1}(a x)}{1225}-\frac{1514 a^2 c^3 x^3 \cosh ^{-1}(a x)}{3675}+\frac{702 a^4 c^3 x^5 \cosh ^{-1}(a x)}{6125}-\frac{6}{343} a^6 c^3 x^7 \cosh ^{-1}(a x)-\frac{48 c^3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{35 a}+\frac{8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{35 a}-\frac{18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{175 a}+\frac{3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \cosh ^{-1}(a x)^2}{49 a}+\frac{16}{35} c^3 x \cosh ^{-1}(a x)^3+\frac{8}{35} c^3 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{6}{35} c^3 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{7} c^3 x \left (1-a^2 x^2\right )^3 \cosh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.41332, size = 179, normalized size = 0.35 \[ \frac{c^3 \left (2 \sqrt{a x-1} \sqrt{a x+1} \left (16875 a^6 x^6-134541 a^4 x^4+747937 a^2 x^2-22329151\right )-385875 a x \left (5 a^6 x^6-21 a^4 x^4+35 a^2 x^2-35\right ) \cosh ^{-1}(a x)^3+11025 \sqrt{a x-1} \sqrt{a x+1} \left (75 a^6 x^6-351 a^4 x^4+757 a^2 x^2-2161\right ) \cosh ^{-1}(a x)^2-210 a x \left (1125 a^6 x^6-7371 a^4 x^4+26495 a^2 x^2-226905\right ) \cosh ^{-1}(a x)\right )}{13505625 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-22329151 + 747937*a^2*x^2 - 134541*a^4*x^4 + 16875*a^6*x^6) - 210*a*x*(
-226905 + 26495*a^2*x^2 - 7371*a^4*x^4 + 1125*a^6*x^6)*ArcCosh[a*x] + 11025*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(-216
1 + 757*a^2*x^2 - 351*a^4*x^4 + 75*a^6*x^6)*ArcCosh[a*x]^2 - 385875*a*x*(-35 + 35*a^2*x^2 - 21*a^4*x^4 + 5*a^6
*x^6)*ArcCosh[a*x]^3))/(13505625*a)

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Maple [A]  time = 0.076, size = 294, normalized size = 0.6 \begin{align*} -{\frac{{c}^{3}}{13505625\,a} \left ( 1929375\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{7}{x}^{7}-826875\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{6}{x}^{6}-8103375\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{5}{x}^{5}+3869775\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{4}{x}^{4}+236250\,{\rm arccosh} \left (ax\right ){a}^{7}{x}^{7}-33750\,\sqrt{ax-1}\sqrt{ax+1}{a}^{6}{x}^{6}[phone]\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3}-8345925\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}-1547910\,{a}^{5}{x}^{5}{\rm arccosh} \left (ax\right )+269082\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}-13505625\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax+23825025\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}+5563950\,{\rm arccosh} \left (ax\right ){a}^{3}{x}^{3}-1495874\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}-47650050\,ax{\rm arccosh} \left (ax\right )[phone]\,\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*arccosh(a*x)^3,x)

[Out]

-1/13505625/a*c^3*(1929375*arccosh(a*x)^3*a^7*x^7-826875*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^6*x^6-81
03375*arccosh(a*x)^3*a^5*x^5+3869775*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4+236250*arccosh(a*x)*a^
7*x^7-33750*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^6*x^6+13505625*arccosh(a*x)^3*a^3*x^3-8345925*arccosh(a*x)^2*(a*x-1)
^(1/2)*(a*x+1)^(1/2)*a^2*x^2-1547910*a^5*x^5*arccosh(a*x)+269082*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^4*a^4-13505625*
arccosh(a*x)^3*a*x+23825025*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+5563950*arccosh(a*x)*a^3*x^3-1495874*(a
*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2-47650050*a*x*arccosh(a*x)[phone]*(a*x-1)^(1/2)*(a*x+1)^(1/2))

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Maxima [A]  time = 1.15399, size = 373, normalized size = 0.74 \begin{align*} \frac{1}{1225} \,{\left (75 \, \sqrt{a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 351 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 757 \, \sqrt{a^{2} x^{2} - 1} c^{3} x^{2} - \frac{2161 \, \sqrt{a^{2} x^{2} - 1} c^{3}}{a^{2}}\right )} a \operatorname{arcosh}\left (a x\right )^{2} - \frac{1}{35} \,{\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \operatorname{arcosh}\left (a x\right )^{3} + \frac{2}{13505625} \,{\left (16875 \, \sqrt{a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 134541 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt{a^{2} x^{2} - 1} c^{3} x^{2} - \frac{22329151 \, \sqrt{a^{2} x^{2} - 1} c^{3}}{a^{2}} - \frac{105 \,{\left (1125 \, a^{6} c^{3} x^{7} - 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} - 226905 \, c^{3} x\right )} \operatorname{arcosh}\left (a x\right )}{a}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1225*(75*sqrt(a^2*x^2 - 1)*a^4*c^3*x^6 - 351*sqrt(a^2*x^2 - 1)*a^2*c^3*x^4 + 757*sqrt(a^2*x^2 - 1)*c^3*x^2 -
2161*sqrt(a^2*x^2 - 1)*c^3/a^2)*a*arccosh(a*x)^2 - 1/35*(5*a^6*c^3*x^7 - 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 - 35
*c^3*x)*arccosh(a*x)^3 + 2/13505625*(16875*sqrt(a^2*x^2 - 1)*a^4*c^3*x^6 - 134541*sqrt(a^2*x^2 - 1)*a^2*c^3*x^
4 + 747937*sqrt(a^2*x^2 - 1)*c^3*x^2 - 22329151*sqrt(a^2*x^2 - 1)*c^3/a^2 - 105*(1125*a^6*c^3*x^7 - 7371*a^4*c
^3*x^5 + 26495*a^2*c^3*x^3 - 226905*c^3*x)*arccosh(a*x)/a)*a

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Fricas [A]  time = 2.17705, size = 605, normalized size = 1.2 \begin{align*} -\frac{385875 \,{\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 11025 \,{\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 210 \,{\left (1125 \, a^{7} c^{3} x^{7} - 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} - 226905 \, a c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (16875 \, a^{6} c^{3} x^{6} - 134541 \, a^{4} c^{3} x^{4} + 747937 \, a^{2} c^{3} x^{2} - 22329151 \, c^{3}\right )} \sqrt{a^{2} x^{2} - 1}}{13505625 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13505625*(385875*(5*a^7*c^3*x^7 - 21*a^5*c^3*x^5 + 35*a^3*c^3*x^3 - 35*a*c^3*x)*log(a*x + sqrt(a^2*x^2 - 1)
)^3 - 11025*(75*a^6*c^3*x^6 - 351*a^4*c^3*x^4 + 757*a^2*c^3*x^2 - 2161*c^3)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a
^2*x^2 - 1))^2 + 210*(1125*a^7*c^3*x^7 - 7371*a^5*c^3*x^5 + 26495*a^3*c^3*x^3 - 226905*a*c^3*x)*log(a*x + sqrt
(a^2*x^2 - 1)) - 2*(16875*a^6*c^3*x^6 - 134541*a^4*c^3*x^4 + 747937*a^2*c^3*x^2 - 22329151*c^3)*sqrt(a^2*x^2 -
1))/a

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Sympy [A]  time = 24.5367, size = 367, normalized size = 0.73 \begin{align*} \begin{cases} - \frac{a^{6} c^{3} x^{7} \operatorname{acosh}^{3}{\left (a x \right )}}{7} - \frac{6 a^{6} c^{3} x^{7} \operatorname{acosh}{\left (a x \right )}}{343} + \frac{3 a^{5} c^{3} x^{6} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{49} + \frac{6 a^{5} c^{3} x^{6} \sqrt{a^{2} x^{2} - 1}}{2401} + \frac{3 a^{4} c^{3} x^{5} \operatorname{acosh}^{3}{\left (a x \right )}}{5} + \frac{702 a^{4} c^{3} x^{5} \operatorname{acosh}{\left (a x \right )}}{6125} - \frac{351 a^{3} c^{3} x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{1225} - \frac{29898 a^{3} c^{3} x^{4} \sqrt{a^{2} x^{2} - 1}}{1500625} - a^{2} c^{3} x^{3} \operatorname{acosh}^{3}{\left (a x \right )} - \frac{1514 a^{2} c^{3} x^{3} \operatorname{acosh}{\left (a x \right )}}{3675} + \frac{757 a c^{3} x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{1225} + \frac{1495874 a c^{3} x^{2} \sqrt{a^{2} x^{2} - 1}}{13505625} + c^{3} x \operatorname{acosh}^{3}{\left (a x \right )} + \frac{4322 c^{3} x \operatorname{acosh}{\left (a x \right )}}{1225} - \frac{2161 c^{3} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{1225 a} - \frac{44658302 c^{3} \sqrt{a^{2} x^{2} - 1}}{13505625 a} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} c^{3} x}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**6*c**3*x**7*acosh(a*x)**3/7 - 6*a**6*c**3*x**7*acosh(a*x)/343 + 3*a**5*c**3*x**6*sqrt(a**2*x**2
- 1)*acosh(a*x)**2/49 + 6*a**5*c**3*x**6*sqrt(a**2*x**2 - 1)/2401 + 3*a**4*c**3*x**5*acosh(a*x)**3/5 + 702*a*
*4*c**3*x**5*acosh(a*x)/6125 - 351*a**3*c**3*x**4*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/1225 - 29898*a**3*c**3*x**
4*sqrt(a**2*x**2 - 1)/1500625 - a**2*c**3*x**3*acosh(a*x)**3 - 1514*a**2*c**3*x**3*acosh(a*x)/3675 + 757*a*c**
3*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/1225 + 1495874*a*c**3*x**2*sqrt(a**2*x**2 - 1)/13505625 + c**3*x*acos
h(a*x)**3 + 4322*c**3*x*acosh(a*x)/1225 - 2161*c**3*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(1225*a) - 44658302*c**3
*sqrt(a**2*x**2 - 1)/(13505625*a), Ne(a, 0)), (-I*pi**3*c**3*x/8, True))

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Giac [A]  time = 1.34547, size = 333, normalized size = 0.66 \begin{align*} -\frac{1}{13505625} \,{\left (210 \,{\left (1125 \, a^{6} x^{7} - 7371 \, a^{4} x^{5} + 26495 \, a^{2} x^{3} - 226905 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{11025 \,{\left (75 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{7}{2}} - 126 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 280 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 1680 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{a} - \frac{2 \,{\left (16875 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{7}{2}} - 83916 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} + 529480 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} - 21698880 \, \sqrt{a^{2} x^{2} - 1}\right )}}{a}\right )} c^{3} - \frac{1}{35} \,{\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13505625*(210*(1125*a^6*x^7 - 7371*a^4*x^5 + 26495*a^2*x^3 - 226905*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 11025
*(75*(a^2*x^2 - 1)^(7/2) - 126*(a^2*x^2 - 1)^(5/2) + 280*(a^2*x^2 - 1)^(3/2) - 1680*sqrt(a^2*x^2 - 1))*log(a*x
+ sqrt(a^2*x^2 - 1))^2/a - 2*(16875*(a^2*x^2 - 1)^(7/2) - 83916*(a^2*x^2 - 1)^(5/2) + 529480*(a^2*x^2 - 1)^(3
/2) - 21698880*sqrt(a^2*x^2 - 1))/a)*c^3 - 1/35*(5*a^6*c^3*x^7 - 21*a^4*c^3*x^5 + 35*a^2*c^3*x^3 - 35*c^3*x)*l
og(a*x + sqrt(a^2*x^2 - 1))^3