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

Optimal. Leaf size=453 \[ -\frac{b^2 c d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{c x-1} \sqrt{c x+1}}+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{2 b c d \sqrt{d-c^2 d x^2} \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-(b^2*c^2*d*x*Sqrt[d - c^2*d*x^2])/4 - (5*b^2*c*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(4*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]) + (3*b*c^3*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d*(1
- c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*c^2*d*x*Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 +
c*x]) - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x])^2)/x + (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^3)/(2
*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1 + E^(-2*ArcCosh[c*x
])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b^2*c*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(-2*ArcCosh[c*x])])/(Sqrt[-1
+ c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 0.965913, antiderivative size = 465, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.483, Rules used = {5798, 5740, 5683, 5676, 5662, 90, 52, 5727, 5660, 3718, 2190, 2279, 2391, 38} \[ \frac{b^2 c d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{c x-1} \sqrt{c x+1}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{c x-1} \sqrt{c x+1}}-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{2 b c d \sqrt{d-c^2 d x^2} \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt{c x-1} \sqrt{c x+1}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(b^2*c^2*d*x*Sqrt[d - c^2*d*x^2])/4 - (5*b^2*c*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(4*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]) + (3*b*c^3*d*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d*(1
- c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (3*c^2*d*x*Sqrt[d - c^2
*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 - (c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(Sqrt[-1 + c*x]*Sqrt[1 +
c*x]) - (d*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/x + (c*d*Sqrt[d - c^2*d*x^2]*(a + b
*ArcCosh[c*x])^3)/(2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (2*b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*Log[1
+ E^(2*ArcCosh[c*x])])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b^2*c*d*Sqrt[d - c^2*d*x^2]*PolyLog[2, -E^(2*ArcCosh
[c*x])])/(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 5740

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

Rule 5683

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)], x_Symbol] :
> Simp[(x*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/2, x] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2
*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(a + b*ArcCosh[c*x])^n/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dis
t[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(2*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[x*(a + b*ArcCosh[c*x])^(n - 1)
, x], x]) /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5676

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol]
:> Simp[(a + b*ArcCosh[c*x])^(n + 1)/(b*c*Sqrt[-(d1*d2)]*(n + 1)), x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n},
x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[d1, 0] && LtQ[d2, 0] && NeQ[n, -1]

Rule 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rule 5727

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

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(x*(a + b*x)^m*(c + d*x)^m)/(2*m + 1)
, x] + Dist[(2*a*c*m)/(2*m + 1), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^2} \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^2} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}-\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (3 b c^3 d \sqrt{d-c^2 d x^2}\right ) \int x \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{1}{2} b^2 c^2 d x \sqrt{d-c^2 d x^2}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b^2 c^4 d \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (4 b c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (3 b^2 c^2 d \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b^2 c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b^2 c d \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{1}{4} b^2 c^2 d x \sqrt{d-c^2 d x^2}-\frac{5 b^2 c d \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{3 b c^3 d x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{3}{2} c^2 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}}-\frac{d (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x}+\frac{c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^3}{2 b \sqrt{-1+c x} \sqrt{1+c x}}+\frac{2 b c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{b^2 c d \sqrt{d-c^2 d x^2} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{\sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [A]  time = 3.98464, size = 433, normalized size = 0.96 \[ \frac{-8 b^2 d \sqrt{d-c^2 d x^2} \left (3 c x \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x) \left (3 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+3\right )+6 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )\right )\right )+36 a^2 c d^{3/2} x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-12 a^2 d \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (c^2 x^2+2\right ) \sqrt{d-c^2 d x^2}-24 a b d \sqrt{d-c^2 d x^2} \left (2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (2 \log (c x)+\cosh ^{-1}(c x)^2\right )\right )+6 a b c d x \sqrt{d-c^2 d x^2} \left (\cosh \left (2 \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-\sinh \left (2 \cosh ^{-1}(c x)\right )\right )\right )+b^2 c d x \sqrt{d-c^2 d x^2} \left (4 \cosh ^{-1}(c x)^3+6 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \left (2 \cosh ^{-1}(c x)^2+1\right ) \sinh \left (2 \cosh ^{-1}(c x)\right )\right )}{24 x \sqrt{\frac{c x-1}{c x+1}} (c x+1)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-12*a^2*d*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(2 + c^2*x^2)*Sqrt[d - c^2*d*x^2] + 36*a^2*c*d^(3/2)*x*Sqrt[(-
1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] - 24*a*b*d*Sqrt[d - c
^2*d*x^2]*(2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]^2 + 2*Log[c*x])) - 8*b^2*d*
Sqrt[d - c^2*d*x^2]*(ArcCosh[c*x]*(3*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] - c*x*(ArcCosh[c*x]*(3
+ ArcCosh[c*x]) + 6*Log[1 + E^(-2*ArcCosh[c*x])])) + 3*c*x*PolyLog[2, -E^(-2*ArcCosh[c*x])]) + 6*a*b*c*d*x*Sqr
t[d - c^2*d*x^2]*(Cosh[2*ArcCosh[c*x]] + 2*ArcCosh[c*x]*(ArcCosh[c*x] - Sinh[2*ArcCosh[c*x]])) + b^2*c*d*x*Sqr
t[d - c^2*d*x^2]*(4*ArcCosh[c*x]^3 + 6*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*(1 + 2*ArcCosh[c*x]^2)*Sinh[2*Arc
Cosh[c*x]]))/(24*x*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))

________________________________________________________________________________________

Maple [B]  time = 0.329, size = 942, normalized size = 2.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/d/x*(-c^2*d*x^2+d)^(5/2)-a^2*c^2*x*(-c^2*d*x^2+d)^(3/2)-3/2*a^2*c^2*d*x*(-c^2*d*x^2+d)^(1/2)-3/2*a^2*c^2*
d^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*c^4*d/(c*x+1)/(c
*x-1)*x^3+1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*c^2*d/(c*x+1)/(c*x-1)*x+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^3*d/(c*x+1)^
(1/2)/(c*x-1)^(1/2)*arccosh(c*x)*x^2+1/2*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^3
*c*d+b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2)*c*
d-b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*c*d-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^4
*d/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x^3-1/2*b^2*(-d*(c^2*x^2-1))^(1/2)*c^2*d/(c*x+1)/(c*x-1)*arccosh(c*x)^2*x+b^
2*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)^2*d/(c*x+1)/(c*x-1)/x-1/4*b^2*(-d*(c^2*x^2-1))^(1/2)*c*d/(c*x+1)^(1/2)/(
c*x-1)^(1/2)*arccosh(c*x)+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)*ln((c*x+(c*x-1
)^(1/2)*(c*x+1)^(1/2))^2+1)*c*d+3/2*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*arccosh(c*x)^2*c*d-
a*b*(-d*(c^2*x^2-1))^(1/2)*c^4*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x^3+1/2*a*b*(-d*(c^2*x^2-1))^(1/2)*c^3*d/(c*x+1)
^(1/2)/(c*x-1)^(1/2)*x^2-2*a*b*(-d*(c^2*x^2-1))^(1/2)*c*d/(c*x+1)^(1/2)/(c*x-1)^(1/2)*arccosh(c*x)-a*b*(-d*(c^
2*x^2-1))^(1/2)*c^2*d/(c*x+1)/(c*x-1)*arccosh(c*x)*x-1/4*a*b*(-d*(c^2*x^2-1))^(1/2)*c*d/(c*x+1)^(1/2)/(c*x-1)^
(1/2)+2*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)*d/(c*x+1)/(c*x-1)/x+2*a*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)
/(c*x+1)^(1/2)*ln((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))^2+1)*c*d

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- d \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}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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