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

Optimal. Leaf size=630 \[ -\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}} \]

[Out]

-2*b^2*c^2*d*Sqrt[d - c^2*d*x^2] + (3*a*b*c^3*d*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b^2
*c^3*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*
ArcCosh[c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) - (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 - ((d - c^2*d*x^2)^(3/2)*(a +
b*ArcCosh[c*x])^2)/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) + (b^2*c^2*d*Sqrt[d - c^2*d*x^2]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(Sqrt[-1 + c
*x]*Sqrt[1 + c*x]) - ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCos
h[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((3*I)*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]]
)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((3*I)*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1
+ c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 1.37413, antiderivative size = 642, normalized size of antiderivative = 1.02, number of steps used = 18, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {5798, 5740, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 14, 5731, 460, 92, 205} \[ -\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 i b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3 i b^2 c^2 d \sqrt{d-c^2 d x^2} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 a b c^3 d x \sqrt{d-c^2 d x^2}}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{b c^3 d x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{c x-1} \sqrt{c x+1}}-\frac{3}{2} c^2 d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac{b c d \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt{c x-1} \sqrt{c x+1}}-\frac{d (1-c x) (c x+1) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac{3 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{c x-1} \sqrt{c x+1}}-2 b^2 c^2 d \sqrt{d-c^2 d x^2}+\frac{b^2 c^2 d \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{\sqrt{c x-1} \sqrt{c x+1}}+\frac{3 b^2 c^3 d x \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b^2*c^2*d*Sqrt[d - c^2*d*x^2] + (3*a*b*c^3*d*x*Sqrt[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (3*b^2
*c^3*d*x*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d*Sqrt[d - c^2*d*x^2]*(a + b*
ArcCosh[c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(Sqrt[-
1 + c*x]*Sqrt[1 + c*x]) - (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 - (d*(1 - c*x)*(1 + c*x)*Sqrt
[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*x^2) + (3*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2*ArcTan[E
^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b^2*c^2*d*Sqrt[d - c^2*d*x^2]*ArcTan[Sqrt[-1 + c*x]*Sqrt[1 +
c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, (-
I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((3*I)*b*c^2*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])*P
olyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((3*I)*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3,
(-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((3*I)*b^2*c^2*d*Sqrt[d - c^2*d*x^2]*PolyLog[3, I*E^Arc
Cosh[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 5743

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*
(x_)], x_Symbol] :> Simp[((f*x)^(m + 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(f*(m + 2)), x
] + (-Dist[(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/((m + 2)*Sqrt[1 + c*x]*Sqrt[-1 + c*x]), Int[((f*x)^m*(a + b*ArcCo
sh[c*x])^n)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x] - Dist[(b*c*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(f*(m + 2)*S
qrt[1 + c*x]*Sqrt[-1 + c*x]), Int[(f*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d1, e
1, d2, e2, f, m}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] |
| EqQ[n, 1])

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 5654

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[(x*(a + b*ArcCosh[c*x])^(n - 1))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 5731

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 460

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[(d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*(a2 + b2*x^(n/2))^(p + 1))/(b1*b2*e*
(m + n*(p + 1) + 1)), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

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

Mathematica [A]  time = 168.74, size = 1129, normalized size = 1.79 \[ \frac{1}{2} d \sqrt{d-c^2 d x^2} \left (\frac{4 x^2 \cosh ^{-1}(c x) c^4}{(c x-1)^{3/2} \sqrt{c x+1}}-\frac{2 x \cosh ^{-1}(c x)^2 c^3}{c x-1}-\frac{4 x \cosh ^{-1}(c x) c^3}{(c x-1)^{3/2} \sqrt{c x+1}}-\frac{2 x \tan ^{-1}\left (\frac{1}{\sqrt{c^2 x^2-1}}\right ) c^3}{(c x-1) \sqrt{c^2 x^2-1}}-\frac{4 x c^3}{c x-1}+\frac{2 \cosh ^{-1}(c x)^2 c^2}{c x-1}-\frac{2 \cosh ^{-1}(c x) c^2}{(c x-1)^{3/2} \sqrt{c x+1}}+\frac{2 \tan ^{-1}\left (\frac{1}{\sqrt{c^2 x^2-1}}\right ) c^2}{(c x-1) \sqrt{c^2 x^2-1}}-\frac{3 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac{3 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}-\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}-\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac{6 i \sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right ) c^2}{c x-1}+\frac{4 c^2}{c x-1}+\frac{\cosh ^{-1}(c x)^2 c}{x-c x^2}+\frac{2 \cosh ^{-1}(c x) c}{x (c x-1)^{3/2} \sqrt{c x+1}}+\frac{\cosh ^{-1}(c x)^2}{x^2 (c x-1)}\right ) b^2-2 a c^2 d \sqrt{-d (c x-1) (c x+1)} \left (-\frac{c x}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\cosh ^{-1}(c x)+\frac{i \cosh ^{-1}(c x) \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}+\frac{i \left (\text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}\right ) b+\frac{i a c^2 d^2 \left (-\frac{i (c x-1) \cosh ^{-1}(c x) (c x+1)}{c^2 x^2}+\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)-\sqrt{\frac{c x-1}{c x+1}} \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)+\sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)-\sqrt{\frac{c x-1}{c x+1}} \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right ) (c x+1)-\frac{i \sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}\right ) b}{\sqrt{-d (c x-1) (c x+1)}}-\frac{3}{2} a^2 c^2 d^{3/2} \log (x)+\frac{3}{2} a^2 c^2 d^{3/2} \log \left (d+\sqrt{-d \left (c^2 x^2-1\right )} \sqrt{d}\right )+\left (-c^2 d a^2-\frac{d a^2}{2 x^2}\right ) \sqrt{-d \left (c^2 x^2-1\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-(a^2*c^2*d) - (a^2*d)/(2*x^2))*Sqrt[-(d*(-1 + c^2*x^2))] - (3*a^2*c^2*d^(3/2)*Log[x])/2 + (3*a^2*c^2*d^(3/2)
*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 2*a*b*c^2*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-((c*x)/(Sqrt[(-
1 + c*x)/(1 + c*x)]*(1 + c*x))) + ArcCosh[c*x] + (I*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcC
osh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcC
osh[c*x]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (I*a*b*c^2*d^2*(((-I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*
x))/(c*x) - (I*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x
]*Log[1 - I/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + Sq
rt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Poly
Log[2, I/E^ArcCosh[c*x]]))/Sqrt[-(d*(-1 + c*x)*(1 + c*x))] + (b^2*d*Sqrt[d - c^2*d*x^2]*((4*c^2)/(-1 + c*x) -
(4*c^3*x)/(-1 + c*x) - (2*c^2*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (2*c*ArcCosh[c*x])/(x*(-1 + c*x
)^(3/2)*Sqrt[1 + c*x]) - (4*c^3*x*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (4*c^4*x^2*ArcCosh[c*x])/((
-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (2*c^2*ArcCosh[c*x]^2)/(-1 + c*x) + ArcCosh[c*x]^2/(x^2*(-1 + c*x)) - (2*c^3*
x*ArcCosh[c*x]^2)/(-1 + c*x) + (c*ArcCosh[c*x]^2)/(x - c*x^2) + (2*c^2*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*
x)*Sqrt[-1 + c^2*x^2]) - (2*c^3*x*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - ((3*I)*c^2*S
qrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]])/(-1 + c*x) + ((3*I)*c^2*Sqrt[(-1 + c*x)/(1
+ c*x)]*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]])/(-1 + c*x) - ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[
c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) + ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[
2, I/E^ArcCosh[c*x]])/(-1 + c*x) - ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, (-I)/E^ArcCosh[c*x]])/(-1
+ c*x) + ((6*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, I/E^ArcCosh[c*x]])/(-1 + c*x)))/2

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Maple [F]  time = 0.35, 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{3}{2}}}}\, dx \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^3,x)

[Out]

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

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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^{3}}\, 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**3,x)

[Out]

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

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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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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