3.329 \(\int \frac{x (1-c^2 x^2)^{3/2}}{(a+b \cosh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=348 \[ \frac{\sqrt{1-c x} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 c^2 \sqrt{c x-1}}-\frac{9 \sqrt{1-c x} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}+\frac{5 \sqrt{1-c x} \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}-\frac{\sqrt{1-c x} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a+b \cosh ^{-1}(c x)}{b}\right )}{8 b^2 c^2 \sqrt{c x-1}}+\frac{9 \sqrt{1-c x} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}-\frac{5 \sqrt{1-c x} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^2 \sqrt{c x-1}}-\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (1-c^2 x^2\right )^{3/2}}{b c \left (a+b \cosh ^{-1}(c x)\right )} \]

[Out]

-((x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcCosh[c*x]))) + (Sqrt[1 - c*x]*CoshIntegr
al[(a + b*ArcCosh[c*x])/b]*Sinh[a/b])/(8*b^2*c^2*Sqrt[-1 + c*x]) - (9*Sqrt[1 - c*x]*CoshIntegral[(3*(a + b*Arc
Cosh[c*x]))/b]*Sinh[(3*a)/b])/(16*b^2*c^2*Sqrt[-1 + c*x]) + (5*Sqrt[1 - c*x]*CoshIntegral[(5*(a + b*ArcCosh[c*
x]))/b]*Sinh[(5*a)/b])/(16*b^2*c^2*Sqrt[-1 + c*x]) - (Sqrt[1 - c*x]*Cosh[a/b]*SinhIntegral[(a + b*ArcCosh[c*x]
)/b])/(8*b^2*c^2*Sqrt[-1 + c*x]) + (9*Sqrt[1 - c*x]*Cosh[(3*a)/b]*SinhIntegral[(3*(a + b*ArcCosh[c*x]))/b])/(1
6*b^2*c^2*Sqrt[-1 + c*x]) - (5*Sqrt[1 - c*x]*Cosh[(5*a)/b]*SinhIntegral[(5*(a + b*ArcCosh[c*x]))/b])/(16*b^2*c
^2*Sqrt[-1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.05554, antiderivative size = 429, normalized size of antiderivative = 1.23, number of steps used = 23, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {5798, 5778, 5700, 3312, 3303, 3298, 3301, 5780, 5448} \[ \frac{\sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{9 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right ) \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{5 \sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{9 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1}}+\frac{x (c x+1)^{3/2} \sqrt{1-c^2 x^2} (1-c x)^2}{b c \sqrt{c x-1} \left (a+b \cosh ^{-1}(c x)\right )} \]

Antiderivative was successfully verified.

[In]

Int[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

(x*(1 - c*x)^2*(1 + c*x)^(3/2)*Sqrt[1 - c^2*x^2])/(b*c*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x])) + (Sqrt[1 - c^2*x^
2]*CoshIntegral[a/b + ArcCosh[c*x]]*Sinh[a/b])/(8*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (9*Sqrt[1 - c^2*x^2]
*CoshIntegral[(3*a)/b + 3*ArcCosh[c*x]]*Sinh[(3*a)/b])/(16*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*Sqrt[1 -
c^2*x^2]*CoshIntegral[(5*a)/b + 5*ArcCosh[c*x]]*Sinh[(5*a)/b])/(16*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (S
qrt[1 - c^2*x^2]*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]])/(8*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (9*Sqr
t[1 - c^2*x^2]*Cosh[(3*a)/b]*SinhIntegral[(3*a)/b + 3*ArcCosh[c*x]])/(16*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
- (5*Sqrt[1 - c^2*x^2]*Cosh[(5*a)/b]*SinhIntegral[(5*a)/b + 5*ArcCosh[c*x]])/(16*b^2*c^2*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 5700

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(-d)^p/c, Subst[I
nt[(a + b*x)^n*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]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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]

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]

Rubi steps

\begin{align*} \int \frac{x \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=-\frac{\sqrt{1-c^2 x^2} \int \frac{x (-1+c x)^{3/2} (1+c x)^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (1-c x)^2 (1+c x)^{3/2} \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} \int \frac{-1+c^2 x^2}{a+b \cosh ^{-1}(c x)} \, dx}{b c \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 c \sqrt{1-c^2 x^2}\right ) \int \frac{x^2 \left (-1+c^2 x^2\right )}{a+b \cosh ^{-1}(c x)} \, dx}{b \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (1-c x)^2 (1+c x)^{3/2} \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 ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cosh ^2(x) \sinh ^3(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (1-c x)^2 (1+c x)^{3/2} \sqrt{1-c^2 x^2}}{b c \sqrt{-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac{\left (i \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{3 i \sinh (x)}{4 (a+b x)}-\frac{i \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \left (-\frac{\sinh (x)}{8 (a+b x)}-\frac{\sinh (3 x)}{16 (a+b x)}+\frac{\sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (1-c x)^2 (1+c x)^{3/2} \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 (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (1-c x)^2 (1+c x)^{3/2} \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{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 \sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (\sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\sinh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{8 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 \sqrt{1-c^2 x^2} \sinh \left (\frac{a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (\sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{4 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{3 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (5 \sqrt{1-c^2 x^2} \sinh \left (\frac{5 a}{b}\right )\right ) \operatorname{Subst}\left (\int \frac{\cosh \left (\frac{5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{16 b c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{x (1-c x)^2 (1+c x)^{3/2} \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{a}{b}+\cosh ^{-1}(c x)\right ) \sinh \left (\frac{a}{b}\right )}{8 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{9 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{3 a}{b}\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{5 \sqrt{1-c^2 x^2} \text{Chi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right ) \sinh \left (\frac{5 a}{b}\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\sqrt{1-c^2 x^2} \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )}{8 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{9 \sqrt{1-c^2 x^2} \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (\frac{3 a}{b}+3 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{5 \sqrt{1-c^2 x^2} \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (\frac{5 a}{b}+5 \cosh ^{-1}(c x)\right )}{16 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 0.913682, size = 327, normalized size = 0.94 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (-2 \sinh \left (\frac{a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+9 \sinh \left (\frac{3 a}{b}\right ) \left (a+b \cosh ^{-1}(c x)\right ) \text{Chi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-5 a \sinh \left (\frac{5 a}{b}\right ) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-5 b \sinh \left (\frac{5 a}{b}\right ) \cosh ^{-1}(c x) \text{Chi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+2 a \cosh \left (\frac{a}{b}\right ) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )+2 b \cosh \left (\frac{a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (\frac{a}{b}+\cosh ^{-1}(c x)\right )-9 a \cosh \left (\frac{3 a}{b}\right ) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-9 b \cosh \left (\frac{3 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (3 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+5 a \cosh \left (\frac{5 a}{b}\right ) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )+5 b \cosh \left (\frac{5 a}{b}\right ) \cosh ^{-1}(c x) \text{Shi}\left (5 \left (\frac{a}{b}+\cosh ^{-1}(c x)\right )\right )-16 b c^5 x^5+32 b c^3 x^3-16 b c x\right )}{16 b^2 c^2 \sqrt{1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*(1 - c^2*x^2)^(3/2))/(a + b*ArcCosh[c*x])^2,x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-16*b*c*x + 32*b*c^3*x^3 - 16*b*c^5*x^5 - 2*(a + b*ArcCosh[c*x])*CoshIntegral[a
/b + ArcCosh[c*x]]*Sinh[a/b] + 9*(a + b*ArcCosh[c*x])*CoshIntegral[3*(a/b + ArcCosh[c*x])]*Sinh[(3*a)/b] - 5*a
*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Sinh[(5*a)/b] - 5*b*ArcCosh[c*x]*CoshIntegral[5*(a/b + ArcCosh[c*x])]*Si
nh[(5*a)/b] + 2*a*Cosh[a/b]*SinhIntegral[a/b + ArcCosh[c*x]] + 2*b*ArcCosh[c*x]*Cosh[a/b]*SinhIntegral[a/b + A
rcCosh[c*x]] - 9*a*Cosh[(3*a)/b]*SinhIntegral[3*(a/b + ArcCosh[c*x])] - 9*b*ArcCosh[c*x]*Cosh[(3*a)/b]*SinhInt
egral[3*(a/b + ArcCosh[c*x])] + 5*a*Cosh[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])] + 5*b*ArcCosh[c*x]*Cosh
[(5*a)/b]*SinhIntegral[5*(a/b + ArcCosh[c*x])]))/(16*b^2*c^2*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x]))

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Maple [B]  time = 0.276, size = 1029, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x)

[Out]

-1/32*(-c^2*x^2+1)^(1/2)*(-16*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^5*c^5+16*c^6*x^6+20*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^
3*c^3-28*c^4*x^4-5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x*c+13*c^2*x^2-1)/(c*x+1)/c^2/(c*x-1)/b/(a+b*arccosh(c*x))+5/32
*(-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(1,5*arccosh(c*x)+5*a/b)*exp((b*arccosh(c*x
)+5*a)/b)/(c*x+1)/c^2/(c*x-1)/b^2+1/32*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(16*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*x^4*b*c^4+16*x^5*b*c^5-12*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*b*c^2-20*x^3*b*c^3+5*arccosh(c*x)*exp(-5*a/b
)*Ei(1,-5*arccosh(c*x)-5*a/b)*b+5*exp(-5*a/b)*Ei(1,-5*arccosh(c*x)-5*a/b)*a+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+5*x*
b*c)/c^2/b^2/(a+b*arccosh(c*x))+3/32*(-c^2*x^2+1)^(1/2)*(-4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3*c^3+4*c^4*x^4+3*(c
*x+1)^(1/2)*(c*x-1)^(1/2)*x*c-5*c^2*x^2+1)/(c*x+1)/c^2/(c*x-1)/b/(a+b*arccosh(c*x))-9/32*(-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(1,3*arccosh(c*x)+3*a/b)*exp((b*arccosh(c*x)+3*a)/b)/(c*x+1)/c^2/
(c*x-1)/b^2+1/16*(-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(arccosh(c*x)*exp(-a/b)*Ei(1,-arccosh(c*x)-a/b
)*b+(c*x+1)^(1/2)*(c*x-1)^(1/2)*b+exp(-a/b)*Ei(1,-arccosh(c*x)-a/b)*a+x*b*c)/c^2/b^2/(a+b*arccosh(c*x))-3/32*(
-c^2*x^2+1)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(4*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2*b*c^2+4*x^3*b*c^3+3*arccosh(c
*x)*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b)*b+3*exp(-3*a/b)*Ei(1,-3*arccosh(c*x)-3*a/b)*a-(c*x+1)^(1/2)*(c*x-1
)^(1/2)*b-3*x*b*c)/c^2/b^2/(a+b*arccosh(c*x))-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)/(c*x+1)/c^2/(c*x-1)/b/(a+b*arccosh(c*x))+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,arccosh(c*x)+a/b)*exp((a+b*arccosh(c*x))/b)/(c*x+1)/c^2/(c*x-1)/b^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left ({\left (c^{4} x^{5} - 2 \, c^{2} x^{3} + x\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (c^{5} x^{6} - 2 \, c^{3} x^{4} + c x^{2}\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 (5 \,{\left (c^{5} x^{5} - c^{3} x^{3}\right )}{\left (c x + 1\right )}^{\frac{3}{2}}{\left (c x - 1\right )} +{\left (10 \, c^{6} x^{6} - 17 \, c^{4} x^{4} + 8 \, c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} \sqrt{c x - 1} +{\left (5 \, c^{7} x^{7} - 12 \, c^{5} x^{5} + 9 \, c^{3} x^{3} - 2 \, c x\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*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

((c^4*x^5 - 2*c^2*x^3 + x)*(c*x + 1)*sqrt(c*x - 1) + (c^5*x^6 - 2*c^3*x^4 + c*x^2)*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 + sqrt(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((5*(c^5*x^5 - c^3*x^3)*(c*x + 1)^(3/2)*(
c*x - 1) + (10*c^6*x^6 - 17*c^4*x^4 + 8*c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (5*c^7*x^7 - 12*c^5*x^5 + 9*c^3
*x^3 - 2*c*x)*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)*s
qrt(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^{2} x^{3} - x\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*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(-(c^2*x^3 - x)*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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}}}{\left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(-c**2*x**2+1)**(3/2)/(a+b*acosh(c*x))**2,x)

[Out]

Integral(x*(-(c*x - 1)*(c*x + 1))**(3/2)/(a + b*acosh(c*x))**2, x)

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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{3}{2}} x}{{\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*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)*x/(b*arccosh(c*x) + a)^2, x)