\(\int \frac {x^2 (1-c^2 x^2)^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx\) [335]

Optimal result

Integrand size = 28, antiderivative size = 454 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}} \]

[Out]

1/16*cosh(2*a/b)*Shi(2*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)+1/8*cosh(4*a/b)*Shi(4*(a+b*a
rccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-3/16*cosh(6*a/b)*Shi(6*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/
2)/b^2/c^3/(c*x-1)^(1/2)+1/16*cosh(8*a/b)*Shi(8*(a+b*arccosh(c*x))/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-1/1
6*Chi(2*(a+b*arccosh(c*x))/b)*sinh(2*a/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-1/8*Chi(4*(a+b*arccosh(c*x))/b)
*sinh(4*a/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)+3/16*Chi(6*(a+b*arccosh(c*x))/b)*sinh(6*a/b)*(-c*x+1)^(1/2)/
b^2/c^3/(c*x-1)^(1/2)-1/16*Chi(8*(a+b*arccosh(c*x))/b)*sinh(8*a/b)*(-c*x+1)^(1/2)/b^2/c^3/(c*x-1)^(1/2)-x^2*(-
c^2*x^2+1)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(a+b*arccosh(c*x))

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5942, 5912, 5952, 5556, 3384, 3379, 3382} \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=-\frac {\sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b^2 c^3 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right ) \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right ) \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b^2 c^3 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {c x-1}}-\frac {x^2 \sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))} \]

[In]

Int[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

-((x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*(a + b*ArcCosh[c*x]))) - (Sqrt[1 - c*x]*CoshInte
gral[(2*(a + b*ArcCosh[c*x]))/b]*Sinh[(2*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(4*(
a + b*ArcCosh[c*x]))/b]*Sinh[(4*a)/b])/(8*b^2*c^3*Sqrt[-1 + c*x]) + (3*Sqrt[1 - c*x]*CoshIntegral[(6*(a + b*Ar
cCosh[c*x]))/b]*Sinh[(6*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*CoshIntegral[(8*(a + b*ArcCosh[c*x
]))/b]*Sinh[(8*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcCos
h[c*x]))/b])/(16*b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(4*a)/b]*SinhIntegral[(4*(a + b*ArcCosh[c*x]))/
b])/(8*b^2*c^3*Sqrt[-1 + c*x]) - (3*Sqrt[1 - c*x]*Cosh[(6*a)/b]*SinhIntegral[(6*(a + b*ArcCosh[c*x]))/b])/(16*
b^2*c^3*Sqrt[-1 + c*x]) + (Sqrt[1 - c*x]*Cosh[(8*a)/b]*SinhIntegral[(8*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c^3*S
qrt[-1 + c*x])

Rule 3379

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[I*(SinhIntegral[c*f*(fz/
d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 0]

Rule 5556

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 5912

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

Rule 5942

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^m*Simp[Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d + e*x^2)^p]*((a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (
Dist[f*(m/(b*c*(n + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m - 1)*(1 + c*x)^(p - 1/2)
*(-1 + c*x)^(p - 1/2)*(a + b*ArcCosh[c*x])^(n + 1), x], x] - Dist[c*((m + 2*p + 1)/(b*f*(n + 1)))*Simp[(d + e*
x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(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, f, m, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, -1] && IGtQ[2*p, 0
] && NeQ[m + 2*p + 1, 0] && IGtQ[m, -3]

Rule 5952

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Subst[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^
(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2
, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {x (-1+c x)^2 (1+c x)^2}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {-1+c x}}+\frac {\left (8 c \sqrt {1-c x}\right ) \int \frac {x^3 (-1+c x)^2 (1+c x)^2}{a+b \text {arccosh}(c x)} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (2 \sqrt {1-c x}\right ) \int \frac {x \left (-1+c^2 x^2\right )^2}{a+b \text {arccosh}(c x)} \, dx}{b c \sqrt {-1+c x}}+\frac {\left (8 c \sqrt {1-c x}\right ) \int \frac {x^3 \left (-1+c^2 x^2\right )^2}{a+b \text {arccosh}(c x)} \, dx}{b \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^5\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}}-\frac {\left (8 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\cosh ^3\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^5\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}+\frac {\left (2 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{32 x}-\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}+\frac {5 \sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{32 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}}-\frac {\left (8 \sqrt {1-c x}\right ) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{128 x}-\frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{64 x}-\frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{64 x}+\frac {3 \sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arccosh}(c x)\right )}{b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {8 a}{b}-\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {6 a}{b}-\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (5 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\left (5 \sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (3 \sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (5 \sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (3 \sqrt {1-c x} \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{4 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\left (\sqrt {1-c x} \sinh \left (\frac {6 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {6 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {\left (\sqrt {1-c x} \sinh \left (\frac {8 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {8 x}{b}\right )}{x} \, dx,x,a+b \text {arccosh}(c x)\right )}{16 b^2 c^3 \sqrt {-1+c x}} \\ & = -\frac {x^2 \sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c (a+b \text {arccosh}(c x))}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {2 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {4 a}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \text {Chi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {6 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \text {Chi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right ) \sinh \left (\frac {8 a}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 (a+b \text {arccosh}(c x))}{b}\right )}{8 b^2 c^3 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (\frac {6 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (\frac {8 (a+b \text {arccosh}(c x))}{b}\right )}{16 b^2 c^3 \sqrt {-1+c x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.33 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (-16 b c^2 x^2+48 b c^4 x^4-48 b c^6 x^6+16 b c^8 x^8+(a+b \text {arccosh}(c x)) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {2 a}{b}\right )+2 (a+b \text {arccosh}(c x)) \text {Chi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {4 a}{b}\right )-3 a \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )-3 b \text {arccosh}(c x) \text {Chi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {6 a}{b}\right )+a \text {Chi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {8 a}{b}\right )+b \text {arccosh}(c x) \text {Chi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right ) \sinh \left (\frac {8 a}{b}\right )-a \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-b \text {arccosh}(c x) \cosh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 a \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-2 b \text {arccosh}(c x) \cosh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 a \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )+3 b \text {arccosh}(c x) \cosh \left (\frac {6 a}{b}\right ) \text {Shi}\left (6 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-a \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )-b \text {arccosh}(c x) \cosh \left (\frac {8 a}{b}\right ) \text {Shi}\left (8 \left (\frac {a}{b}+\text {arccosh}(c x)\right )\right )\right )}{16 b^2 c^3 \sqrt {1-c^2 x^2} (a+b \text {arccosh}(c x))} \]

[In]

Integrate[(x^2*(1 - c^2*x^2)^(5/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b*c^2*x^2 + 48*b*c^4*x^4 - 48*b*c^6*x^6 + 16*b*c^8*x^8 + (a + b*ArcCosh[c*x
])*CoshIntegral[2*(a/b + ArcCosh[c*x])]*Sinh[(2*a)/b] + 2*(a + b*ArcCosh[c*x])*CoshIntegral[4*(a/b + ArcCosh[c
*x])]*Sinh[(4*a)/b] - 3*a*CoshIntegral[6*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] - 3*b*ArcCosh[c*x]*CoshIntegral[6
*(a/b + ArcCosh[c*x])]*Sinh[(6*a)/b] + a*CoshIntegral[8*(a/b + ArcCosh[c*x])]*Sinh[(8*a)/b] + b*ArcCosh[c*x]*C
oshIntegral[8*(a/b + ArcCosh[c*x])]*Sinh[(8*a)/b] - a*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - b*A
rcCosh[c*x]*Cosh[(2*a)/b]*SinhIntegral[2*(a/b + ArcCosh[c*x])] - 2*a*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCo
sh[c*x])] - 2*b*ArcCosh[c*x]*Cosh[(4*a)/b]*SinhIntegral[4*(a/b + ArcCosh[c*x])] + 3*a*Cosh[(6*a)/b]*SinhIntegr
al[6*(a/b + ArcCosh[c*x])] + 3*b*ArcCosh[c*x]*Cosh[(6*a)/b]*SinhIntegral[6*(a/b + ArcCosh[c*x])] - a*Cosh[(8*a
)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])] - b*ArcCosh[c*x]*Cosh[(8*a)/b]*SinhIntegral[8*(a/b + ArcCosh[c*x])])
)/(16*b^2*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.70

method	result	size
default	\(\text {Expression too large to display}\)	\(773\)

[In]

int(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/32*(-c^2*x^2+1)^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*(32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^8*x^8
-Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*b*arccosh(c*x)+arccosh(c*x)*b*Ei(1,-2*arccosh(c*x)-2*a
/b)*exp(-(-b*arccosh(c*x)+2*a)/b)+3*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)*b*arccosh(c*x)+arcc
osh(c*x)*b*Ei(1,-8*arccosh(c*x)-8*a/b)*exp(-(-b*arccosh(c*x)+8*a)/b)-Ei(1,8*arccosh(c*x)+8*a/b)*exp((b*arccosh
(c*x)+8*a)/b)*b*arccosh(c*x)+2*arccosh(c*x)*b*Ei(1,-4*arccosh(c*x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)-3*arcc
osh(c*x)*b*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-(-b*arccosh(c*x)+6*a)/b)-2*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arcco
sh(c*x)+4*a)/b)*b*arccosh(c*x)-96*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^6*x^6+96*x^4*c^4*b*(c*x-1)^(1/2)*(c*x+1)^(1/
2)-32*(c*x-1)^(1/2)*(c*x+1)^(1/2)*b*c^2*x^2+32*b*c^9*x^9-96*b*c^7*x^7-3*a*Ei(1,-6*arccosh(c*x)-6*a/b)*exp(-(-b
*arccosh(c*x)+6*a)/b)+3*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)*a+a*Ei(1,-2*arccosh(c*x)-2*a/b)
*exp(-(-b*arccosh(c*x)+2*a)/b)-Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)*a+2*a*Ei(1,-4*arccosh(c*
x)-4*a/b)*exp(-(-b*arccosh(c*x)+4*a)/b)-2*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)+4*a)/b)*a-Ei(1,8*arcc
osh(c*x)+8*a/b)*exp((b*arccosh(c*x)+8*a)/b)*a+a*Ei(1,-8*arccosh(c*x)-8*a/b)*exp(-(-b*arccosh(c*x)+8*a)/b)-32*b
*c^3*x^3+96*b*c^5*x^5)/(c*x+1)/c^3/(c*x-1)/b^2/(a+b*arccosh(c*x))

Fricas [F]

\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^6 - 2*c^2*x^4 + x^2)*sqrt(-c^2*x^2 + 1)/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\text {Timed out} \]

[In]

integrate(x**2*(-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x))**2,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^8 - 3*c^4*x^6 + 3*c^2*x^4 - x^2)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^9 - 3*c^5*x^7 + 3*c^3*x^5 - c*x^3)*
sqrt(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + sq
rt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((8*c^7*x^8 -
17*c^5*x^6 + 10*c^3*x^4 - c*x^2)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(8*c^8*x^9 - 22*c^6*x^7 + 21*c^4*x^5 - 8*c^2*x
^3 + x)*(c*x + 1)*sqrt(c*x - 1) + (8*c^9*x^10 - 27*c^7*x^8 + 33*c^5*x^6 - 17*c^3*x^4 + 3*c*x^2)*sqrt(c*x + 1))
*sqrt(-c*x + 1)/(a*b*c^5*x^4 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^2 - 2*a*b*c^3*x^2 + a*b*c + 2*(a*b*c^4*x^3 - a*b*
c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^4 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^2 - 2*b^2*c^3*x^2 + b^2*c +
2*(b^2*c^4*x^3 - b^2*c^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)

Giac [F]

\[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x^{2}}{{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(x^2*(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)*x^2/(b*arccosh(c*x) + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{(a+b \text {arccosh}(c x))^2} \, dx=\int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

[In]

int((x^2*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2,x)

[Out]

int((x^2*(1 - c^2*x^2)^(5/2))/(a + b*acosh(c*x))^2, x)