3.335 \(\int \frac{x^2 (1-c^2 x^2)^{5/2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=454 \[ -\frac{\sqrt{1-c x} \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3 \sqrt{c x-1}}+\frac{3 \sqrt{1-c x} \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \sinh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 b^2 c^3 \sqrt{c x-1}}-\frac{3 \sqrt{1-c x} \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}+\frac{\sqrt{1-c x} \cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^3 \sqrt{c x-1}}-\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )^{5/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.51466, antiderivative size = 565, normalized size of antiderivative = 1.24, number of steps used = 29, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5798, 5778, 5780, 5448, 3303, 3298, 3301} \[ -\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{2 a}{b}\right ) \text{Chi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{4 a}{b}\right ) \text{Chi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{3 \sqrt{1-c^2 x^2} \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{8 a}{b}\right ) \text{Chi}\left (\frac{8 a}{b}+8 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (\frac{2 a}{b}+2 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (\frac{4 a}{b}+4 \cosh ^{-1}(c x)\right )}{8 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3 \sqrt{1-c^2 x^2} \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (\frac{6 a}{b}+6 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (\frac{8 a}{b}+8 \cosh ^{-1}(c x)\right )}{16 b^2 c^3 \sqrt{c x-1} \sqrt{c x+1}}+\frac{x^2 (c x+1)^{5/2} \sqrt{1-c^2 x^2} (1-c x)^3}{b c \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^2*(1 - c*x)^3*(1 + c*x)^(5/2)*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) - (Sqrt[1 - c^2*
x^2]*CoshIntegral[(2*a)/b + 2*ArcCosh[c*x]]*Sinh[(2*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (Sqrt[1
- c^2*x^2]*CoshIntegral[(4*a)/b + 4*ArcCosh[c*x]]*Sinh[(4*a)/b])/(8*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
3*Sqrt[1 - c^2*x^2]*CoshIntegral[(6*a)/b + 6*ArcCosh[c*x]]*Sinh[(6*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]) - (Sqrt[1 - c^2*x^2]*CoshIntegral[(8*a)/b + 8*ArcCosh[c*x]]*Sinh[(8*a)/b])/(16*b^2*c^3*Sqrt[-1 + c*x]*Sq
rt[1 + c*x]) + (Sqrt[1 - c^2*x^2]*Cosh[(2*a)/b]*SinhIntegral[(2*a)/b + 2*ArcCosh[c*x]])/(16*b^2*c^3*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) + (Sqrt[1 - c^2*x^2]*Cosh[(4*a)/b]*SinhIntegral[(4*a)/b + 4*ArcCosh[c*x]])/(8*b^2*c^3*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - (3*Sqrt[1 - c^2*x^2]*Cosh[(6*a)/b]*SinhIntegral[(6*a)/b + 6*ArcCosh[c*x]])/(16*b^2
*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (Sqrt[1 - c^2*x^2]*Cosh[(8*a)/b]*SinhIntegral[(8*a)/b + 8*ArcCosh[c*x]])/
(16*b^2*c^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5798

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

Rule 5778

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[
c*x])^(n + 1))/(b*c*(n + 1)), x] + (Dist[(f*m*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPa
rt[p])/(b*c*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m - 1)*(-1 + c^2*x^2)^(p - 1/2)*
(a + b*ArcCosh[c*x])^(n + 1), x], x] - Dist[(c*(m + 2*p + 1)*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2
+ e2*x)^FracPart[p])/(b*f*(n + 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-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, f}, x] && EqQ[e1 - c*d
1, 0] && EqQ[e2 + c*d2, 0] && LtQ[n, -1] && IGtQ[m, -3] && IGtQ[p + 1/2, 0]

Rule 5780

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

Rule 5448

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 3303

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 3298

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 3301

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]

Rubi steps

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

Mathematica [A]  time = 1.7033, size = 446, normalized size = 0.98 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (\sinh \left (\frac{2 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 \sinh \left (\frac{4 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 a \sinh \left (\frac{6 a}{b}\right ) \text{Chi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-3 b \sinh \left (\frac{6 a}{b}\right ) \cosh ^{-1}(c x) \text{Chi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+a \sinh \left (\frac{8 a}{b}\right ) \text{Chi}\left (8 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+b \sinh \left (\frac{8 a}{b}\right ) \cosh ^{-1}(c x) \text{Chi}\left (8 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-a \cosh \left (\frac{2 a}{b}\right ) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-b \cosh \left (\frac{2 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (2 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-2 a \cosh \left (\frac{4 a}{b}\right ) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-2 b \cosh \left (\frac{4 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (4 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+3 a \cosh \left (\frac{6 a}{b}\right ) \text{Shi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+3 b \cosh \left (\frac{6 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (6 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-a \cosh \left (\frac{8 a}{b}\right ) \text{Shi}\left (8 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-b \cosh \left (\frac{8 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (8 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+16 b c^8 x^8-48 b c^6 x^6+48 b c^4 x^4-16 b c^2 x^2\right )}{16 b^2 c^3 \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Warning: Unable to verify antiderivative.

[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]))

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Maple [B]  time = 0.43, size = 1676, normalized size = 3.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256*(-c^2*x^2+1)^(1/2)*(-128*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^8*c^8+128*c^9*x^9+256*(c*x+1)^(1/2)*(c*x-1)^(1/2)
*x^6*c^6-320*c^7*x^7-160*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+272*c^5*x^5+32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^
2-88*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+8*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b-1/32*(-c^2*x^2+1)^(1/
2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,8*arccosh(c*x)+8*a/b)*exp((b*arccosh(c*x)+8*a)/b)/(c*x+1)
/(c*x-1)/c^3/b^2-1/256/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(128*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^7*b*c
^7+128*x^8*b*c^8-192*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*b*c^5-256*x^6*b*c^6+80*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*b*
c^3+160*x^4*b*c^4-8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*b*c-32*x^2*b*c^2+8*arccosh(c*x)*exp(-8*a/b)*Ei(1,-8*arccosh(
c*x)-8*a/b)*b+8*exp(-8*a/b)*Ei(1,-8*arccosh(c*x)-8*a/b)*a+b)/c^3/b^2/(a+b*arccosh(c*x))+5/128/(c*x+1)^(1/2)/(c
*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/(a+b*arccosh(c*x))/b-1/64*(-c^2*x^2+1)^(1/2)*(-32*(c*x+1)^(1/2)*(c*x-1)^(1/
2)*x^6*c^6+32*c^7*x^7+48*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4-64*c^5*x^5-18*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2
+38*c^3*x^3+(c*x-1)^(1/2)*(c*x+1)^(1/2)-6*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b+3/32*(-c^2*x^2+1)^(1/2
)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,6*arccosh(c*x)+6*a/b)*exp((b*arccosh(c*x)+6*a)/b)/(c*x+1)/
(c*x-1)/c^3/b^2+1/64*(-c^2*x^2+1)^(1/2)*(-8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4*c^4+8*c^5*x^5+8*(c*x+1)^(1/2)*(c*x
-1)^(1/2)*x^2*c^2-12*c^3*x^3-(c*x-1)^(1/2)*(c*x+1)^(1/2)+4*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b-1/16*
(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,4*arccosh(c*x)+4*a/b)*exp((b*arccosh(c*x)
+4*a)/b)/(c*x+1)/(c*x-1)/c^3/b^2+1/64*(-c^2*x^2+1)^(1/2)*(-2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*c^2+2*c^3*x^3+(c*
x-1)^(1/2)*(c*x+1)^(1/2)-2*c*x)/(c*x+1)/(c*x-1)/c^3/(a+b*arccosh(c*x))/b-1/32*(-c^2*x^2+1)^(1/2)*(-(c*x+1)^(1/
2)*(c*x-1)^(1/2)*x*c+c^2*x^2-1)*Ei(1,2*arccosh(c*x)+2*a/b)*exp((b*arccosh(c*x)+2*a)/b)/(c*x+1)/(c*x-1)/c^3/b^2
-1/64/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*b*c+2*x^2*b*c^2+2*arccos
h(c*x)*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*b+2*Ei(1,-2*arccosh(c*x)-2*a/b)*exp(-2*a/b)*a-b)/c^3/b^2/(a+b*a
rccosh(c*x))-1/64/(c*x+1)^(1/2)/(c*x-1)^(1/2)*(-c^2*x^2+1)^(1/2)*(8*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*b*c^3+8*x^
4*b*c^4-4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*b*c-8*x^2*b*c^2+4*arccosh(c*x)*exp(-4*a/b)*Ei(1,-4*arccosh(c*x)-4*a/b)
*b+4*exp(-4*a/b)*Ei(1,-4*arccosh(c*x)-4*a/b)*a+b)/c^3/b^2/(a+b*arccosh(c*x))+1/64/(c*x+1)^(1/2)/(c*x-1)^(1/2)*
(-c^2*x^2+1)^(1/2)*(32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*b*c^5+32*x^6*b*c^6-32*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*b
*c^3-48*x^4*b*c^4+6*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*b*c+18*x^2*b*c^2+6*arccosh(c*x)*exp(-6*a/b)*Ei(1,-6*arccosh(
c*x)-6*a/b)*b+6*exp(-6*a/b)*Ei(1,-6*arccosh(c*x)-6*a/b)*a-b)/c^3/b^2/(a+b*arccosh(c*x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{{\left ({\left (c^{6} x^{8} - 3 \, c^{4} x^{6} + 3 \, c^{2} x^{4} - x^{2}\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{7} x^{9} - 3 \, c^{5} x^{7} + 3 \, c^{3} x^{5} - c x^{3}\right )} \sqrt{c x + 1}\right )} \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 +{\left (b^{2} c^{3} x^{2} + \sqrt{c x + 1} \sqrt{c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )} + \int \frac{{\left ({\left (8 \, c^{7} x^{8} - 17 \, c^{5} x^{6} + 10 \, c^{3} x^{4} - c x^{2}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} + 2 \,{\left (8 \, c^{8} x^{9} - 22 \, c^{6} x^{7} + 21 \, c^{4} x^{5} - 8 \, c^{2} x^{3} + x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (8 \, c^{9} x^{10} - 27 \, c^{7} x^{8} + 33 \, c^{5} x^{6} - 17 \, c^{3} x^{4} + 3 \, c x^{2}\right )} \sqrt{c x + 1}\right )} \sqrt{-c x + 1}}{a b c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} a b c^{3} x^{2} - 2 \, a b c^{3} x^{2} + a b c + 2 \,{\left (a b c^{4} x^{3} - a b c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (b^{2} c^{5} x^{4} +{\left (c x + 1\right )}{\left (c x - 1\right )} b^{2} c^{3} x^{2} - 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \,{\left (b^{2} c^{4} x^{3} - b^{2} c^{2} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{4} x^{6} - 2 \, c^{2} x^{4} + x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)