3.222 \(\int \frac{(a+b \cosh ^{-1}(c x))^2}{x^3 (d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=796 \[ -\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )^2 c^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )^2 c^2}{6 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{5 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right ) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right ) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) c}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \left (d-c^2 d x^2\right )^{3/2}} \]

[Out]

-(b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(d^2*x*(1 - c
^2*x^2)*Sqrt[d - c^2*d*x^2]) - (2*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c^2*x
^2)*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x])^2)/(6*d*(d - c^2*d*x^2)^(3/2)) - (a + b*ArcCosh[c*x])^2
/(2*d*x^2*(d - c^2*d*x^2)^(3/2)) + (5*c^2*(a + b*ArcCosh[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*Sqrt[-1
+ c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + (26*b*c^2*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (13*b^2*c^2*Sq
rt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) - ((5*I)*b*c^2*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) + ((5*I)*b*c
^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) -
(13*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*d^2*Sqrt[d - c^2*d*x^2]) + ((5*I)*b^2
*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - ((5*I)*b^2*c^2*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.93812, antiderivative size = 826, normalized size of antiderivative = 1.04, number of steps used = 39, number of rules used = 19, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.655, Rules used = {5798, 5748, 5756, 5761, 4180, 2531, 2282, 6589, 5694, 4182, 2279, 2391, 5689, 74, 5746, 104, 21, 92, 205} \[ -\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )^2 c^2}{6 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{5 \left (a+b \cosh ^{-1}(c x)\right )^2 c^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right ) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right ) c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right ) c^2}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right ) c}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCosh[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

-(b^2*c^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) + (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(d^2*x*(1 - c
^2*x^2)*Sqrt[d - c^2*d*x^2]) - (2*b*c^3*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*d^2*(1 - c^2*x
^2)*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[c*x])^2)/(2*d^2*Sqrt[d - c^2*d*x^2]) + (5*c^2*(a + b*ArcCosh[
c*x])^2)/(6*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(2*d^2*x^2*(1 - c*x)*(1 + c*
x)*Sqrt[d - c^2*d*x^2]) + (5*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(
d^2*Sqrt[d - c^2*d*x^2]) - (b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(d^2*Sq
rt[d - c^2*d*x^2]) + (26*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*d
^2*Sqrt[d - c^2*d*x^2]) + (13*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*d^2*Sqrt[d
- c^2*d*x^2]) - ((5*I)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]]
)/(d^2*Sqrt[d - c^2*d*x^2]) + ((5*I)*b*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^Ar
cCosh[c*x]])/(d^2*Sqrt[d - c^2*d*x^2]) - (13*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/
(3*d^2*Sqrt[d - c^2*d*x^2]) + ((5*I)*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d^
2*Sqrt[d - c^2*d*x^2]) - ((5*I)*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[3, I*E^ArcCosh[c*x]])/(d^2*Sqrt[d
- c^2*d*x^2])

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 5748

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 + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d1*
d2*f*(m + 1)), x] + (Dist[(c^2*(m + 2*p + 3))/(f^2*(m + 1)), Int[(f*x)^(m + 2)*(d1 + e1*x)^p*(d2 + e2*x)^p*(a
+ b*ArcCosh[c*x])^n, x], x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p
])/(f*(m + 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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 +
c*d2, 0] && GtQ[n, 0] && LtQ[m, -1] && IntegerQ[m] && IntegerQ[p + 1/2]

Rule 5756

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 + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*
d1*d2*f*(p + 1)), x] + (Dist[(m + 2*p + 3)/(2*d1*d2*(p + 1)), Int[(f*x)^m*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p +
1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^Fra
cPart[p])/(2*f*(p + 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, m}, x] && EqQ[e1 - c*d1, 0] &&
EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] || EqQ[n, 1]) && IntegerQ[p + 1/2]

Rule 5761

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

Rule 4180

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

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

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

Rule 5694

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

Rule 4182

Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, 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 5689

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

Rule 74

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 5746

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

Rule 104

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 92

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))), x_Symbol] :> Dist[b*f, Subst[I
nt[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sqrt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
EqQ[2*b*d*e - f*(b*c + a*d), 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\left (\sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^3 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x^2 \left (-1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\left (-1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{c+c^2 x}{x \sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 c^4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{\left (5 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{-1+c x} \sqrt{1+c x}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{c+c x^2} \, dx,x,\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (3 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (5 i b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (5 i b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 c^2}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{d^2 x \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{2 b c^3 x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{6 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2}}+\frac{5 c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \tan ^{-1}\left (\sqrt{-1+c x} \sqrt{1+c x}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{26 b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{13 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b c^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \text{Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{13 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt{d-c^2 d x^2}}+\frac{5 i b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}-\frac{5 i b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 97.9998, size = 1181, normalized size = 1.48 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*ArcCosh[c*x])^2/(x^3*(d - c^2*d*x^2)^(5/2)),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*(-a^2/(2*d^3*x^2) + (a^2*c^2)/(3*d^3*(-1 + c^2*x^2)^2) - (2*a^2*c^2)/(d^3*(-1 + c^2*
x^2))) + (5*a^2*c^2*Log[x])/(2*d^(5/2)) - (5*a^2*c^2*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/(2*d^(5/2)) +
(a*b*c^2*((6*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*x) + (6*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) +
26*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 - Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]*Coth[ArcCosh[c*x]/2]^2 - (30*I)*S
qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]
*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] - 26*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c
*x]/2]] - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)
/(1 + c*x)]*(1 + c*x)*PolyLog[2, I/E^ArcCosh[c*x]] - 26*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2 - Tanh[ArcCosh[c*x
]/2] - ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2]^2))/(6*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (b^2*c^2*Sqrt[d - c^2*d
*x^2]*((12*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) + 6*(1 - 1/(c^2*x^2))*ArcCosh[c*x]^2 - 24*
Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[Tanh[ArcCosh[c*x]/2]] - 4*Cosh[ArcCosh[c*x]/2]^2 + 26*ArcCosh[c*x]
^2*Cosh[ArcCosh[c*x]/2]^2 - 2*ArcCosh[c*x]*Coth[ArcCosh[c*x]/2] - ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2]^2 - 52*S
qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] - (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)
]*(1 + c*x)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] + (30*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x
]^2*Log[1 + I/E^ArcCosh[c*x]] + 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])
] - 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] - (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]
*(1 + c*x)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] + (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[
c*x]*PolyLog[2, I/E^ArcCosh[c*x]] + 52*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-ArcCosh[c*x])] - (6
0*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, (-I)/E^ArcCosh[c*x]] + (60*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*
(1 + c*x)*PolyLog[3, I/E^ArcCosh[c*x]] + 4*Sinh[ArcCosh[c*x]/2]^2 - 26*ArcCosh[c*x]^2*Sinh[ArcCosh[c*x]/2]^2 -
2*ArcCosh[c*x]*Tanh[ArcCosh[c*x]/2] - ArcCosh[c*x]^2*Tanh[ArcCosh[c*x]/2]^2))/(12*d^3*(-1 + c^2*x^2))

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Maple [F]  time = 0.449, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{2}}{{x}^{3}} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x)

[Out]

int((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x)

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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((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b^{2} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname{arcosh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{9} - 3 \, c^{4} d^{3} x^{7} + 3 \, c^{2} d^{3} x^{5} - d^{3} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2)/(c^6*d^3*x^9 - 3*c^4*d^3*x^7 +
3*c^2*d^3*x^5 - d^3*x^3), 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((a+b*acosh(c*x))**2/x**3/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arccosh(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*x^3), x)