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

Optimal. Leaf size=388 \[ \frac{6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{8}{135} a c^2 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{488 c^2 \sqrt{a x-1} \sqrt{a x+1}}{135 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a} \]

[Out]

(-488*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(135*a) + (8*a*c^2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/135 + (16*c^2*(1
- a^2*x^2))/(125*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (8*c^2*(1 - a^2*x^2)^2)/(375*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]
) + (6*c^2*(1 - a^2*x^2)^3)/(625*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (298*c^2*x*ArcCosh[a*x])/75 - (76*a^2*c^2*x
^3*ArcCosh[a*x])/225 + (6*a^4*c^2*x^5*ArcCosh[a*x])/125 - (8*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/
(5*a) + (4*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(15*a) - (3*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^(5/
2)*ArcCosh[a*x]^2)/(25*a) + (8*c^2*x*ArcCosh[a*x]^3)/15 + (4*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/15 + (c^2*x*(
1 - a^2*x^2)^2*ArcCosh[a*x]^3)/5

________________________________________________________________________________________

Rubi [A]  time = 0.844015, antiderivative size = 388, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {5681, 5718, 194, 5680, 12, 520, 1247, 698, 460, 74, 5654} \[ \frac{6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt{a x-1} \sqrt{a x+1}}+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{8}{135} a c^2 x^2 \sqrt{a x-1} \sqrt{a x+1}-\frac{488 c^2 \sqrt{a x-1} \sqrt{a x+1}}{135 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{8 c^2 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2}{5 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-488*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(135*a) + (8*a*c^2*x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/135 + (16*c^2*(1
- a^2*x^2))/(125*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (8*c^2*(1 - a^2*x^2)^2)/(375*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]
) + (6*c^2*(1 - a^2*x^2)^3)/(625*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]) + (298*c^2*x*ArcCosh[a*x])/75 - (76*a^2*c^2*x
^3*ArcCosh[a*x])/225 + (6*a^4*c^2*x^5*ArcCosh[a*x])/125 - (8*c^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/
(5*a) + (4*c^2*(-1 + a*x)^(3/2)*(1 + a*x)^(3/2)*ArcCosh[a*x]^2)/(15*a) - (3*c^2*(-1 + a*x)^(5/2)*(1 + a*x)^(5/
2)*ArcCosh[a*x]^2)/(25*a) + (8*c^2*x*ArcCosh[a*x]^3)/15 + (4*c^2*x*(1 - a^2*x^2)*ArcCosh[a*x]^3)/15 + (c^2*x*(
1 - a^2*x^2)^2*ArcCosh[a*x]^3)/5

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 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 )^2 \cosh ^{-1}(a x)^3 \, dx &=\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \cosh ^{-1}(a x)^3 \, dx-\frac{1}{5} \left (3 a c^2\right ) \int x (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2 \, dx\\ &=-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{25} \left (6 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \cosh ^{-1}(a x) \, dx+\frac{1}{15} \left (8 c^2\right ) \int \cosh ^{-1}(a x)^3 \, dx+\frac{1}{5} \left (4 a c^2\right ) \int x \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \, dx\\ &=\frac{6}{25} c^2 x \cosh ^{-1}(a x)-\frac{4}{25} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac{1}{15} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \cosh ^{-1}(a x) \, dx-\frac{1}{25} \left (6 a c^2\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}{5} \left (8 a c^2\right ) \int \frac{x \cosh ^{-1}(a x)^2}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{58}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{5} \left (16 c^2\right ) \int \cosh ^{-1}(a x) \, dx-\frac{1}{125} \left (2 a c^2\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}{15} \left (8 a c^2\right ) \int \frac{x \left (-3+a^2 x^2\right )}{3 \sqrt{-1+a x} \sqrt{1+a x}} \, dx\\ &=\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3+\frac{1}{45} \left (8 a c^2\right ) \int \frac{x \left (-3+a^2 x^2\right )}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{1}{5} \left (16 a c^2\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (2 a c^2 \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}{125 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{16 c^2 \sqrt{-1+a x} \sqrt{1+a x}}{5 a}+\frac{8}{135} a c^2 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac{1}{135} \left (56 a c^2\right ) \int \frac{x}{\sqrt{-1+a x} \sqrt{1+a x}} \, dx-\frac{\left (a c^2 \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 )}{125 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{488 c^2 \sqrt{-1+a x} \sqrt{1+a x}}{135 a}+\frac{8}{135} a c^2 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3-\frac{\left (a c^2 \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 )}{125 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=-\frac{488 c^2 \sqrt{-1+a x} \sqrt{1+a x}}{135 a}+\frac{8}{135} a c^2 x^2 \sqrt{-1+a x} \sqrt{1+a x}+\frac{16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt{-1+a x} \sqrt{1+a x}}+\frac{298}{75} c^2 x \cosh ^{-1}(a x)-\frac{76}{225} a^2 c^2 x^3 \cosh ^{-1}(a x)+\frac{6}{125} a^4 c^2 x^5 \cosh ^{-1}(a x)-\frac{8 c^2 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2}{5 a}+\frac{4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^2}{15 a}-\frac{3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \cosh ^{-1}(a x)^2}{25 a}+\frac{8}{15} c^2 x \cosh ^{-1}(a x)^3+\frac{4}{15} c^2 x \left (1-a^2 x^2\right ) \cosh ^{-1}(a x)^3+\frac{1}{5} c^2 x \left (1-a^2 x^2\right )^2 \cosh ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.209834, size = 147, normalized size = 0.38 \[ \frac{c^2 \left (-2 \sqrt{a x-1} \sqrt{a x+1} \left (81 a^4 x^4-842 a^2 x^2+31841\right )+1125 a x \left (3 a^4 x^4-10 a^2 x^2+15\right ) \cosh ^{-1}(a x)^3-225 \sqrt{a x-1} \sqrt{a x+1} \left (9 a^4 x^4-38 a^2 x^2+149\right ) \cosh ^{-1}(a x)^2+30 a x \left (27 a^4 x^4-190 a^2 x^2+2235\right ) \cosh ^{-1}(a x)\right )}{16875 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^2*(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(31841 - 842*a^2*x^2 + 81*a^4*x^4) + 30*a*x*(2235 - 190*a^2*x^2 + 27*a^4
*x^4)*ArcCosh[a*x] - 225*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*(149 - 38*a^2*x^2 + 9*a^4*x^4)*ArcCosh[a*x]^2 + 1125*a*x
*(15 - 10*a^2*x^2 + 3*a^4*x^4)*ArcCosh[a*x]^3))/(16875*a)

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Maple [A]  time = 0.056, size = 218, normalized size = 0.6 \begin{align*}{\frac{{c}^{2}}{16875\,a} \left ( 3375\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{5}{x}^{5}-2025\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{4}{x}^{4}-11250\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}{a}^{3}{x}^{3}+8550\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}{a}^{2}{x}^{2}+810\,{a}^{5}{x}^{5}{\rm arccosh} \left (ax\right )-162\,\sqrt{ax+1}\sqrt{ax-1}{x}^{4}{a}^{4}+16875\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}ax-33525\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\sqrt{ax-1}\sqrt{ax+1}-5700\,{\rm arccosh} \left (ax\right ){a}^{3}{x}^{3}+1684\,\sqrt{ax+1}\sqrt{ax-1}{x}^{2}{a}^{2}+67050\,ax{\rm arccosh} \left (ax\right )-63682\,\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)^2*arccosh(a*x)^3,x)

[Out]

1/16875/a*c^2*(3375*arccosh(a*x)^3*a^5*x^5-2025*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^4*x^4-11250*arcco
sh(a*x)^3*a^3*x^3+8550*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)*a^2*x^2+810*a^5*x^5*arccosh(a*x)-162*(a*x+1)
^(1/2)*(a*x-1)^(1/2)*x^4*a^4+16875*arccosh(a*x)^3*a*x-33525*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-5700*ar
ccosh(a*x)*a^3*x^3+1684*(a*x+1)^(1/2)*(a*x-1)^(1/2)*x^2*a^2+67050*a*x*arccosh(a*x)-63682*(a*x-1)^(1/2)*(a*x+1)
^(1/2))

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Maxima [A]  time = 1.2006, size = 284, normalized size = 0.73 \begin{align*} -\frac{1}{75} \,{\left (9 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt{a^{2} x^{2} - 1} c^{2} x^{2} + \frac{149 \, \sqrt{a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname{arcosh}\left (a x\right )^{2} + \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname{arcosh}\left (a x\right )^{3} - \frac{2}{16875} \,{\left (81 \, \sqrt{a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 842 \, \sqrt{a^{2} x^{2} - 1} c^{2} x^{2} - \frac{15 \,{\left (27 \, a^{4} c^{2} x^{5} - 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname{arcosh}\left (a x\right )}{a} + \frac{31841 \, \sqrt{a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/75*(9*sqrt(a^2*x^2 - 1)*a^2*c^2*x^4 - 38*sqrt(a^2*x^2 - 1)*c^2*x^2 + 149*sqrt(a^2*x^2 - 1)*c^2/a^2)*a*arcco
sh(a*x)^2 + 1/15*(3*a^4*c^2*x^5 - 10*a^2*c^2*x^3 + 15*c^2*x)*arccosh(a*x)^3 - 2/16875*(81*sqrt(a^2*x^2 - 1)*a^
2*c^2*x^4 - 842*sqrt(a^2*x^2 - 1)*c^2*x^2 - 15*(27*a^4*c^2*x^5 - 190*a^2*c^2*x^3 + 2235*c^2*x)*arccosh(a*x)/a
+ 31841*sqrt(a^2*x^2 - 1)*c^2/a^2)*a

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Fricas [A]  time = 2.17825, size = 467, normalized size = 1.2 \begin{align*} \frac{1125 \,{\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} - 225 \,{\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt{a^{2} x^{2} - 1} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2} + 30 \,{\left (27 \, a^{5} c^{2} x^{5} - 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - 2 \,{\left (81 \, a^{4} c^{2} x^{4} - 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt{a^{2} x^{2} - 1}}{16875 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16875*(1125*(3*a^5*c^2*x^5 - 10*a^3*c^2*x^3 + 15*a*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1))^3 - 225*(9*a^4*c^2*x^
4 - 38*a^2*c^2*x^2 + 149*c^2)*sqrt(a^2*x^2 - 1)*log(a*x + sqrt(a^2*x^2 - 1))^2 + 30*(27*a^5*c^2*x^5 - 190*a^3*
c^2*x^3 + 2235*a*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 2*(81*a^4*c^2*x^4 - 842*a^2*c^2*x^2 + 31841*c^2)*sqrt(a
^2*x^2 - 1))/a

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Sympy [A]  time = 8.76748, size = 274, normalized size = 0.71 \begin{align*} \begin{cases} \frac{a^{4} c^{2} x^{5} \operatorname{acosh}^{3}{\left (a x \right )}}{5} + \frac{6 a^{4} c^{2} x^{5} \operatorname{acosh}{\left (a x \right )}}{125} - \frac{3 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{25} - \frac{6 a^{3} c^{2} x^{4} \sqrt{a^{2} x^{2} - 1}}{625} - \frac{2 a^{2} c^{2} x^{3} \operatorname{acosh}^{3}{\left (a x \right )}}{3} - \frac{76 a^{2} c^{2} x^{3} \operatorname{acosh}{\left (a x \right )}}{225} + \frac{38 a c^{2} x^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{75} + \frac{1684 a c^{2} x^{2} \sqrt{a^{2} x^{2} - 1}}{16875} + c^{2} x \operatorname{acosh}^{3}{\left (a x \right )} + \frac{298 c^{2} x \operatorname{acosh}{\left (a x \right )}}{75} - \frac{149 c^{2} \sqrt{a^{2} x^{2} - 1} \operatorname{acosh}^{2}{\left (a x \right )}}{75 a} - \frac{63682 c^{2} \sqrt{a^{2} x^{2} - 1}}{16875 a} & \text{for}\: a \neq 0 \\- \frac{i \pi ^{3} c^{2} 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)**2*acosh(a*x)**3,x)

[Out]

Piecewise((a**4*c**2*x**5*acosh(a*x)**3/5 + 6*a**4*c**2*x**5*acosh(a*x)/125 - 3*a**3*c**2*x**4*sqrt(a**2*x**2
- 1)*acosh(a*x)**2/25 - 6*a**3*c**2*x**4*sqrt(a**2*x**2 - 1)/625 - 2*a**2*c**2*x**3*acosh(a*x)**3/3 - 76*a**2*
c**2*x**3*acosh(a*x)/225 + 38*a*c**2*x**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/75 + 1684*a*c**2*x**2*sqrt(a**2*x*
*2 - 1)/16875 + c**2*x*acosh(a*x)**3 + 298*c**2*x*acosh(a*x)/75 - 149*c**2*sqrt(a**2*x**2 - 1)*acosh(a*x)**2/(
75*a) - 63682*c**2*sqrt(a**2*x**2 - 1)/(16875*a), Ne(a, 0)), (-I*pi**3*c**2*x/8, True))

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Giac [A]  time = 1.33745, size = 273, normalized size = 0.7 \begin{align*} \frac{1}{15} \,{\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{3} + \frac{1}{16875} \,{\left (30 \,{\left (27 \, a^{4} x^{5} - 190 \, a^{2} x^{3} + 2235 \, x\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right ) - \frac{225 \,{\left (9 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} - 20 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 120 \, \sqrt{a^{2} x^{2} - 1}\right )} \log \left (a x + \sqrt{a^{2} x^{2} - 1}\right )^{2}}{a} - \frac{2 \,{\left (81 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{5}{2}} - 680 \,{\left (a^{2} x^{2} - 1\right )}^{\frac{3}{2}} + 31080 \, \sqrt{a^{2} x^{2} - 1}\right )}}{a}\right )} c^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(3*a^4*c^2*x^5 - 10*a^2*c^2*x^3 + 15*c^2*x)*log(a*x + sqrt(a^2*x^2 - 1))^3 + 1/16875*(30*(27*a^4*x^5 - 19
0*a^2*x^3 + 2235*x)*log(a*x + sqrt(a^2*x^2 - 1)) - 225*(9*(a^2*x^2 - 1)^(5/2) - 20*(a^2*x^2 - 1)^(3/2) + 120*s
qrt(a^2*x^2 - 1))*log(a*x + sqrt(a^2*x^2 - 1))^2/a - 2*(81*(a^2*x^2 - 1)^(5/2) - 680*(a^2*x^2 - 1)^(3/2) + 310
80*sqrt(a^2*x^2 - 1))/a)*c^2