\(\int (c-a^2 c x^2)^2 \text {arccosh}(a x)^3 \, dx\) [241]

Optimal result

Integrand size = 20, antiderivative size = 388 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^3 \, dx=-\frac {488 c^2 \sqrt {-1+a x} \sqrt {1+a x}}{135 a}+\frac {8}{135} a c^2 x^2 \sqrt {-1+a x} \sqrt {1+a x}+\frac {16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt {-1+a x} \sqrt {1+a x}}+\frac {298}{75} c^2 x \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3 \]

[Out]

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

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5897, 5879, 5915, 75, 5889, 5894, 12, 471, 200, 534, 1261, 712} \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^3 \, dx=\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {6 c^2 \left (1-a^2 x^2\right )^3}{625 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {8 c^2 \left (1-a^2 x^2\right )^2}{375 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {16 c^2 \left (1-a^2 x^2\right )}{125 a \sqrt {a x-1} \sqrt {a x+1}}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {298}{75} c^2 x \text {arccosh}(a x)-\frac {3 c^2 (a x-1)^{5/2} (a x+1)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {4 c^2 (a x-1)^{3/2} (a x+1)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {8 c^2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^2}{5 a}+\frac {8}{135} a c^2 x^2 \sqrt {a x-1} \sqrt {a x+1}-\frac {488 c^2 \sqrt {a x-1} \sqrt {a x+1}}{135 a} \]

[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 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 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}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{5} (4 c) \int \left (c-a^2 c x^2\right ) \text {arccosh}(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} \text {arccosh}(a x)^2 \, dx \\ & = -\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{25} \left (6 c^2\right ) \int (-1+a x)^2 (1+a x)^2 \text {arccosh}(a x) \, dx+\frac {1}{15} \left (8 c^2\right ) \int \text {arccosh}(a x)^3 \, dx+\frac {1}{5} \left (4 a c^2\right ) \int x \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2 \, dx \\ & = \frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3+\frac {1}{25} \left (6 c^2\right ) \int \left (-1+a^2 x^2\right )^2 \text {arccosh}(a x) \, dx-\frac {1}{15} \left (8 c^2\right ) \int (-1+a x) (1+a x) \text {arccosh}(a x) \, dx-\frac {1}{5} \left (8 a c^2\right ) \int \frac {x \text {arccosh}(a x)^2}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = \frac {6}{25} c^2 x \text {arccosh}(a x)-\frac {4}{25} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3-\frac {1}{15} \left (8 c^2\right ) \int \left (-1+a^2 x^2\right ) \text {arccosh}(a x) \, dx+\frac {1}{5} \left (16 c^2\right ) \int \text {arccosh}(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 {298}{75} c^2 x \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3-\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 {1}{5} \left (16 a c^2\right ) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {16 c^2 \sqrt {-1+a x} \sqrt {1+a x}}{5 a}+\frac {298}{75} c^2 x \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(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 {\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 \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(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 ) \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 )}{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 \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3-\frac {\left (a c^2 \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 )}{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 \text {arccosh}(a x)-\frac {76}{225} a^2 c^2 x^3 \text {arccosh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arccosh}(a x)-\frac {8 c^2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^2}{5 a}+\frac {4 c^2 (-1+a x)^{3/2} (1+a x)^{3/2} \text {arccosh}(a x)^2}{15 a}-\frac {3 c^2 (-1+a x)^{5/2} (1+a x)^{5/2} \text {arccosh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arccosh}(a x)^3+\frac {4}{15} c^2 x \left (1-a^2 x^2\right ) \text {arccosh}(a x)^3+\frac {1}{5} c^2 x \left (1-a^2 x^2\right )^2 \text {arccosh}(a x)^3 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.38 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^3 \, dx=\frac {c^2 \left (-2 \sqrt {-1+a x} \sqrt {1+a x} \left (31841-842 a^2 x^2+81 a^4 x^4\right )+30 a x \left (2235-190 a^2 x^2+27 a^4 x^4\right ) \text {arccosh}(a x)-225 \sqrt {-1+a x} \sqrt {1+a x} \left (149-38 a^2 x^2+9 a^4 x^4\right ) \text {arccosh}(a x)^2+1125 a x \left (15-10 a^2 x^2+3 a^4 x^4\right ) \text {arccosh}(a x)^3\right )}{16875 a} \]

[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)

Maple [A] (verified)

Time = 0.50 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.56

method	result	size
derivativedivides	\(\frac {c^{2} \left (3375 \operatorname {arccosh}\left (a x \right )^{3} a^{5} x^{5}-2025 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-11250 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}+8550 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+810 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )-162 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+16875 a x \operatorname {arccosh}\left (a x \right )^{3}-33525 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5700 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )+1684 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+67050 a x \,\operatorname {arccosh}\left (a x \right )-63682 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{16875 a}\)	\(218\)
default	\(\frac {c^{2} \left (3375 \operatorname {arccosh}\left (a x \right )^{3} a^{5} x^{5}-2025 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}-11250 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )^{3}+8550 a^{2} x^{2} \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}+810 a^{5} x^{5} \operatorname {arccosh}\left (a x \right )-162 \sqrt {a x -1}\, \sqrt {a x +1}\, a^{4} x^{4}+16875 a x \operatorname {arccosh}\left (a x \right )^{3}-33525 \operatorname {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}-5700 a^{3} x^{3} \operatorname {arccosh}\left (a x \right )+1684 a^{2} x^{2} \sqrt {a x -1}\, \sqrt {a x +1}+67050 a x \,\operatorname {arccosh}\left (a x \right )-63682 \sqrt {a x -1}\, \sqrt {a x +1}\right )}{16875 a}\)	\(218\)

[In]

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

[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*a^3*x
^3*arccosh(a*x)^3+8550*a^2*x^2*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+810*a^5*x^5*arccosh(a*x)-162*(a*x-1)
^(1/2)*(a*x+1)^(1/2)*a^4*x^4+16875*a*x*arccosh(a*x)^3-33525*arccosh(a*x)^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)-5700*a^
3*x^3*arccosh(a*x)+1684*a^2*x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)+67050*a*x*arccosh(a*x)-63682*(a*x-1)^(1/2)*(a*x+1)
^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.53 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^3 \, dx=\frac {1125 \, {\left (3 \, a^{5} c^{2} x^{5} - 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{3} - 225 \, {\left (9 \, a^{4} c^{2} x^{4} - 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 30 \, {\left (27 \, a^{5} c^{2} x^{5} - 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 2 \, {\left (81 \, a^{4} c^{2} x^{4} - 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{16875 \, a} \]

[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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.54 \[ \int \left (c-a^2 c x^2\right )^2 \text {arccosh}(a x)^3 \, dx=-\frac {1}{75} \, {\left (9 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 38 \, \sqrt {a^{2} x^{2} - 1} c^{2} x^{2} + \frac {149 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \operatorname {arcosh}\left (a x\right )^{2} + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} - 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right )^{3} - \frac {2}{16875} \, {\left (81 \, \sqrt {a^{2} x^{2} - 1} a^{2} c^{2} x^{4} - 842 \, \sqrt {a^{2} x^{2} - 1} c^{2} x^{2} - \frac {15 \, {\left (27 \, a^{4} c^{2} x^{5} - 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname {arcosh}\left (a x\right )}{a} + \frac {31841 \, \sqrt {a^{2} x^{2} - 1} c^{2}}{a^{2}}\right )} a \]

[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

Giac [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

int(acosh(a*x)^3*(c - a^2*c*x^2)^2,x)

[Out]

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