\(\int (c-a^2 c x^2)^3 \text {arccosh}(a x)^3 \, dx\) [240]

Optimal result

Integrand size = 20, antiderivative size = 505 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^3 \, dx=-\frac {976 c^3 \sqrt {-1+a x} \sqrt {1+a x}}{315 a}+\frac {16}{315} a c^3 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {4322 c^3 x \text {arccosh}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3 \]

[Out]

4322/1225*c^3*x*arccosh(a*x)-1514/3675*a^2*c^3*x^3*arccosh(a*x)+702/6125*a^4*c^3*x^5*arccosh(a*x)-6/343*a^6*c^
3*x^7*arccosh(a*x)+8/35*c^3*(a*x-1)^(3/2)*(a*x+1)^(3/2)*arccosh(a*x)^2/a-18/175*c^3*(a*x-1)^(5/2)*(a*x+1)^(5/2
)*arccosh(a*x)^2/a+3/49*c^3*(a*x-1)^(7/2)*(a*x+1)^(7/2)*arccosh(a*x)^2/a+16/35*c^3*x*arccosh(a*x)^3+8/35*c^3*x
*(-a^2*x^2+1)*arccosh(a*x)^3+6/35*c^3*x*(-a^2*x^2+1)^2*arccosh(a*x)^3+1/7*c^3*x*(-a^2*x^2+1)^3*arccosh(a*x)^3+
7104/42875*c^3*(-a^2*x^2+1)/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+1184/42875*c^3*(-a^2*x^2+1)^2/a/(a*x-1)^(1/2)/(a*x+1
)^(1/2)+2664/214375*c^3*(-a^2*x^2+1)^3/a/(a*x-1)^(1/2)/(a*x+1)^(1/2)+6/2401*c^3*(-a^2*x^2+1)^4/a/(a*x-1)^(1/2)
/(a*x+1)^(1/2)-976/315*c^3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a+16/315*a*c^3*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-48/35*c^
3*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5897, 5879, 5915, 75, 5889, 5894, 12, 471, 200, 534, 1261, 712, 1624, 1813, 1864} \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^3 \, dx=-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6 c^3 \left (1-a^2 x^2\right )^4}{2401 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {2664 c^3 \left (1-a^2 x^2\right )^3}{214375 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {1184 c^3 \left (1-a^2 x^2\right )^2}{42875 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {7104 c^3 \left (1-a^2 x^2\right )}{42875 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {4322 c^3 x \text {arccosh}(a x)}{1225}+\frac {3 c^3 (a x-1)^{7/2} (a x+1)^{7/2} \text {arccosh}(a x)^2}{49 a}-\frac {18 c^3 (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {8 c^3 (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {48 c^3 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{35 a}+\frac {16}{315} a c^3 x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {976 c^3 \sqrt {a x-1} \sqrt {a x+1}}{315 a} \]

[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 12

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

Rule 75

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 200

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 471

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 534

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 712

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 1261

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 1624

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 1813

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 1864

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 5879

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]

Rule 5889

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

Rule 5894

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 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 5915

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3+\frac {1}{7} (6 c) \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(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} \text {arccosh}(a x)^2 \, dx \\ & = \frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3+\frac {1}{35} \left (24 c^2\right ) \int \left (c-a^2 c x^2\right ) \text {arccosh}(a x)^3 \, dx-\frac {1}{49} \left (6 c^3\right ) \int (-1+a x)^3 (1+a x)^3 \text {arccosh}(a x) \, dx-\frac {1}{35} \left (18 a c^3\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2 \, dx \\ & = -\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3-\frac {1}{49} \left (6 c^3\right ) \int \left (-1+a^2 x^2\right )^3 \text {arccosh}(a x) \, dx+\frac {1}{175} \left (36 c^3\right ) \int (-1+a x)^2 (1+a x)^2 \text {arccosh}(a x) \, dx+\frac {1}{35} \left (16 c^3\right ) \int \text {arccosh}(a x)^3 \, dx+\frac {1}{35} \left (24 a c^3\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \, dx \\ & = \frac {6}{49} c^3 x \text {arccosh}(a x)-\frac {6}{49} a^2 c^3 x^3 \text {arccosh}(a x)+\frac {18}{245} a^4 c^3 x^5 \text {arccosh}(a x)-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3+\frac {1}{175} \left (36 c^3\right ) \int \left (-1+a^2 x^2\right )^2 \text {arccosh}(a x) \, dx-\frac {1}{35} \left (16 c^3\right ) \int (-1+a x) (1+a x) \text {arccosh}(a x) \, 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 (48 a c^3\right ) \int \frac {x \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {402 c^3 x \text {arccosh}(a x)}{1225}-\frac {318 a^2 c^3 x^3 \text {arccosh}(a x)}{1225}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3-\frac {1}{35} \left (16 c^3\right ) \int \left (-1+a^2 x^2\right ) \text {arccosh}(a x) \, dx+\frac {1}{35} \left (96 c^3\right ) \int \text {arccosh}(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 {4322 c^3 x \text {arccosh}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3-\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 {1}{35} \left (96 a c^3\right ) \int \frac {x}{\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 {96 c^3 \sqrt {-1+a x} \sqrt {1+a x}}{35 a}+\frac {4322 c^3 x \text {arccosh}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(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 {\left (3 a c^3 \sqrt {-1+a^2 x^2}\right ) \text {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 \text {arccosh}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(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 ) \text {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 ) \text {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 \text {arccosh}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3-\frac {\left (6 a c^3 \sqrt {-1+a^2 x^2}\right ) \text {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 \text {arccosh}(a x)}{1225}-\frac {1514 a^2 c^3 x^3 \text {arccosh}(a x)}{3675}+\frac {702 a^4 c^3 x^5 \text {arccosh}(a x)}{6125}-\frac {6}{343} a^6 c^3 x^7 \text {arccosh}(a x)-\frac {48 c^3 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{35 a}+\frac {8 c^3 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{35 a}-\frac {18 c^3 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{175 a}+\frac {3 c^3 (-1+a x)^{7/2} (1+a x)^{7/2} \text {arccosh}(a x)^2}{49 a}+\frac {16}{35} c^3 x \text {arccosh}(a x)^3+\frac {8}{35} c^3 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6}{35} c^3 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{7} c^3 x \left (1-a^2 x^2\right )^3 \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.35 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^3 \, dx=\frac {c^3 \left (2 \sqrt {-1+a x} \sqrt {1+a x} \left (-22329151+747937 a^2 x^2-134541 a^4 x^4+16875 a^6 x^6\right )-210 a x \left (-226905+26495 a^2 x^2-7371 a^4 x^4+1125 a^6 x^6\right ) \text {arccosh}(a x)+11025 \sqrt {-1+a x} \sqrt {1+a x} \left (-2161+757 a^2 x^2-351 a^4 x^4+75 a^6 x^6\right ) \text {arccosh}(a x)^2-385875 a x \left (-35+35 a^2 x^2-21 a^4 x^4+5 a^6 x^6\right ) \text {arccosh}(a x)^3\right )}{13505625 a} \]

[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)

Maple [A] (verified)

Time = 0.55 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.58

method	result	size
derivativedivides	\(-\frac {c^{3} \left (1929375 \operatorname {arccosh}\left (a x \right )^{3} a^{7} x^{7}-826875 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-8103375 \operatorname {arccosh}\left (a x \right )^{3} a^{5} x^{5}+3869775 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+236250 \,\operatorname {arccosh}\left (a x \right ) a^{7} x^{7}-33750 x^{6} a^{6} \sqrt {a x -1}\, \sqrt {a x +1}[phone] a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}-8345925 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-1547910 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )+269082 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-13505625 a x \operatorname {arccosh}\left (a x \right )^{3}[phone] \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+5563950 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-1495874 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-47650050 a x \,\operatorname {arccosh}\left (a x \right )[phone] \sqrt {a x -1}\, \sqrt {a x +1}\right )}{13505625 a}\)	\(294\)
default	\(-\frac {c^{3} \left (1929375 \operatorname {arccosh}\left (a x \right )^{3} a^{7} x^{7}-826875 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{6} x^{6}-8103375 \operatorname {arccosh}\left (a x \right )^{3} a^{5} x^{5}+3869775 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+236250 \,\operatorname {arccosh}\left (a x \right ) a^{7} x^{7}-33750 x^{6} a^{6} \sqrt {a x -1}\, \sqrt {a x +1}[phone] a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}-8345925 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-1547910 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )+269082 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-13505625 a x \operatorname {arccosh}\left (a x \right )^{3}[phone] \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+5563950 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )-1495874 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}-47650050 a x \,\operatorname {arccosh}\left (a x \right )[phone] \sqrt {a x -1}\, \sqrt {a x +1}\right )}{13505625 a}\)	\(294\)

[In]

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

[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*x^6*a^6*(a*x-1)^(1/2)*(a*x+1)^(1/2)[phone]*a^3*x^3*arccosh(a*x)^3-8345925*a^2*x^2*arccosh(a*x)^2
*(a*x-1)^(1/2)*(a*x+1)^(1/2)-1547910*a^5*x^5*arccosh(a*x)+269082*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4-13505625*
a*x*arccosh(a*x)^3+23825025*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+5563950*a^3*x^3*arccosh(a*x)-1495874*a^
2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-47650050*a*x*arccosh(a*x)[phone]*(a*x-1)^(1/2)*(a*x+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.49 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^3 \, dx=-\frac {385875 \, {\left (5 \, a^{7} c^{3} x^{7} - 21 \, a^{5} c^{3} x^{5} + 35 \, a^{3} c^{3} x^{3} - 35 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 11025 \, {\left (75 \, a^{6} c^{3} x^{6} - 351 \, a^{4} c^{3} x^{4} + 757 \, a^{2} c^{3} x^{2} - 2161 \, c^{3}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 210 \, {\left (1125 \, a^{7} c^{3} x^{7} - 7371 \, a^{5} c^{3} x^{5} + 26495 \, a^{3} c^{3} x^{3} - 226905 \, a c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (16875 \, a^{6} c^{3} x^{6} - 134541 \, a^{4} c^{3} x^{4} + 747937 \, a^{2} c^{3} x^{2} - 22329151 \, c^{3}\right )} \sqrt {a^{2} x^{2} - 1}}{13505625 \, a} \]

[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

Sympy [F]

\[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^3 \, dx=- c^{3} \left (\int 3 a^{2} x^{2} \operatorname {acosh}^{3}{\left (a x \right )}\, dx + \int \left (- 3 a^{4} x^{4} \operatorname {acosh}^{3}{\left (a x \right )}\right )\, dx + \int a^{6} x^{6} \operatorname {acosh}^{3}{\left (a x \right )}\, dx + \int \left (- \operatorname {acosh}^{3}{\left (a x \right )}\right )\, dx\right ) \]

[In]

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

[Out]

-c**3*(Integral(3*a**2*x**2*acosh(a*x)**3, x) + Integral(-3*a**4*x**4*acosh(a*x)**3, x) + Integral(a**6*x**6*a
cosh(a*x)**3, x) + Integral(-acosh(a*x)**3, x))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.55 \[ \int \left (c-a^2 c x^2\right )^3 \text {arccosh}(a x)^3 \, dx=\frac {1}{1225} \, {\left (75 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 351 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 757 \, \sqrt {a^{2} x^{2} - 1} c^{3} x^{2} - \frac {2161 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} - \frac {1}{35} \, {\left (5 \, a^{6} c^{3} x^{7} - 21 \, a^{4} c^{3} x^{5} + 35 \, a^{2} c^{3} x^{3} - 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right )^{3} + \frac {2}{13505625} \, {\left (16875 \, \sqrt {a^{2} x^{2} - 1} a^{4} c^{3} x^{6} - 134541 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{3} x^{4} + 747937 \, \sqrt {a^{2} x^{2} - 1} c^{3} x^{2} - \frac {22329151 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}} - \frac {105 \, {\left (1125 \, a^{6} c^{3} x^{7} - 7371 \, a^{4} c^{3} x^{5} + 26495 \, a^{2} c^{3} x^{3} - 226905 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right )}{a}\right )} a \]

[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

Giac [F(-2)]

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

[In]

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

[In]

int(acosh(a*x)^3*(c - a^2*c*x^2)^3,x)

[Out]

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