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

Optimal. Leaf size=650 \[ -\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{4 b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d \sqrt{d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.48649, antiderivative size = 650, normalized size of antiderivative = 1., number of steps used = 26, number of rules used = 15, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.517, Rules used = {5798, 5748, 5756, 5761, 4180, 2531, 2282, 6589, 5694, 4182, 2279, 2391, 5746, 92, 205} \[ -\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 i b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}+\frac{b c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{d x \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{4 b c^2 \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c^2 \sqrt{c x-1} \sqrt{c x+1} \tan ^{-1}\left (\sqrt{c x-1} \sqrt{c x+1}\right )}{d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(d*x*Sqrt[d - c^2*d*x^2]) + (3*c^2*(a + b*ArcCosh[c*x]
)^2)/(2*d*Sqrt[d - c^2*d*x^2]) - (a + b*ArcCosh[c*x])^2/(2*d*x^2*Sqrt[d - c^2*d*x^2]) + (3*c^2*Sqrt[-1 + c*x]*
Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(d*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*Sqrt[d - c^2*d*x^2]) + (4*b*c^2*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + (2*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[
1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - ((3*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*Sqrt[d - c^2*d*x^2]) + ((3*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*Sqrt[d - c^2*d*x^2]) - (2*b^2*c^2*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) + ((3*I)*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(d*Sqrt[d - c^2*d*x^2]) - ((3*I)*b^2*c^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Poly
Log[3, I*E^ArcCosh[c*x]])/(d*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 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 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 )^{3/2}} \, 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)^{3/2} (1+c x)^{3/2}} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \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 )} \, dx}{d \sqrt{d-c^2 d x^2}}-\frac{\left (3 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 \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 x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 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 \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 \sqrt{d-c^2 d x^2}}+\frac{\left (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 \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}{d \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 x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{\left (3 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 \sqrt{d-c^2 d x^2}}+\frac{\left (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 \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 )}{d \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 \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 x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 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 \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 \sqrt{d-c^2 d x^2}}+\frac{4 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 )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (3 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 \sqrt{d-c^2 d x^2}}+\frac{\left (3 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 \sqrt{d-c^2 d x^2}}-\frac{\left (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 \sqrt{d-c^2 d x^2}}+\frac{\left (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 \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 )}{d \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 )}{d \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 x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 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 \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 \sqrt{d-c^2 d x^2}}+\frac{4 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 )}{d \sqrt{d-c^2 d x^2}}-\frac{3 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 \sqrt{d-c^2 d x^2}}+\frac{3 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 \sqrt{d-c^2 d x^2}}+\frac{\left (3 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 \sqrt{d-c^2 d x^2}}-\frac{\left (3 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 \sqrt{d-c^2 d x^2}}-\frac{\left (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 \sqrt{d-c^2 d x^2}}+\frac{\left (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 \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 )}{d \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 )}{d \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 x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 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 \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 \sqrt{d-c^2 d x^2}}+\frac{4 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 )}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 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 \sqrt{d-c^2 d x^2}}+\frac{3 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 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (3 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 \sqrt{d-c^2 d x^2}}-\frac{\left (3 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 \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 x \sqrt{d-c^2 d x^2}}+\frac{3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{2 d x^2 \sqrt{d-c^2 d x^2}}+\frac{3 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 \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 \sqrt{d-c^2 d x^2}}+\frac{4 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 )}{d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{3 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 \sqrt{d-c^2 d x^2}}+\frac{3 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 \sqrt{d-c^2 d x^2}}-\frac{2 b^2 c^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{3 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 \sqrt{d-c^2 d x^2}}-\frac{3 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 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 91.3921, size = 979, normalized size = 1.51 \[ \frac{b^2 \sqrt{d-c^2 d x^2} \left (-2 \cosh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2+2 \sinh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)^2+\left (\frac{1}{c^2 x^2}-1\right ) \cosh ^{-1}(c x)^2+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)^2-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)^2-\frac{2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)}{c x}+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (1+e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)+6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)-6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right ) \cosh ^{-1}(c x)+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \tan ^{-1}\left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )+4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-4 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )+6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )-6 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right )\right ) c^2}{2 d^2 \left (c^2 x^2-1\right )}+\frac{a \left (3 a \sqrt{d} \log (x) c^2-3 a \sqrt{d} \log \left (d+\sqrt{d-c^2 d x^2} \sqrt{d}\right ) c^2-\frac{2 b d \left (-2 \cosh ^{-1}(c x) \cosh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )+2 \cosh ^{-1}(c x) \sinh ^2\left (\frac{1}{2} \cosh ^{-1}(c x)\right )-\frac{\sqrt{\frac{c x-1}{c x+1}} (c x+1)}{c x}+\left (\frac{1}{c^2 x^2}-1\right ) \cosh ^{-1}(c x)+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )+3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-3 i \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right ) c^2}{\sqrt{d-c^2 d x^2}}-\frac{a \left (3 c^2 x^2-1\right ) \sqrt{d-c^2 d x^2}}{x^2 \left (c^2 x^2-1\right )}\right )}{2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(b^2*c^2*Sqrt[d - c^2*d*x^2]*((-2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/(c*x) + (-1 + 1/(c^2*x^2)
)*ArcCosh[c*x]^2 + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcTan[Tanh[ArcCosh[c*x]/2]] - 2*ArcCosh[c*x]^2*Cosh
[ArcCosh[c*x]/2]^2 + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + (3*I)*Sq
rt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2*Log[1 - I/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]
*(1 + c*x)*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[
1 + E^(-ArcCosh[c*x])] + 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, -E^(-ArcCosh[c*x])] + (6*I)*Sqrt[(-
1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]] - (6*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*
(1 + c*x)*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]] - 4*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, E^(-Ar
cCosh[c*x])] + (6*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, (-I)/E^ArcCosh[c*x]] - (6*I)*Sqrt[(-1 + c
*x)/(1 + c*x)]*(1 + c*x)*PolyLog[3, I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]^2*Sinh[ArcCosh[c*x]/2]^2))/(2*d^2*(-1 +
c^2*x^2)) + (a*(-((a*(-1 + 3*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(x^2*(-1 + c^2*x^2))) + 3*a*c^2*Sqrt[d]*Log[x] - 3
*a*c^2*Sqrt[d]*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] - (2*b*c^2*d*(-((Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*
x)) + (-1 + 1/(c^2*x^2))*ArcCosh[c*x] - 2*ArcCosh[c*x]*Cosh[ArcCosh[c*x]/2]^2 + (3*I)*Sqrt[(-1 + c*x)/(1 + c*x
)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - I/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*
Log[1 + I/E^ArcCosh[c*x]] + 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + (3*I)*Sqrt[(-1
+ c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - (3*I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyL
og[2, I/E^ArcCosh[c*x]] + 2*ArcCosh[c*x]*Sinh[ArcCosh[c*x]/2]^2))/Sqrt[d - c^2*d*x^2]))/(2*d^2)

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Maple [F]  time = 0.372, 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((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),x)

[Out]

int((a+b*arccosh(c*x))^2/x^3/(-c^2*d*x^2+d)^(3/2),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)^(3/2),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^{4} d^{2} x^{7} - 2 \, c^{2} d^{2} x^{5} + d^{2} 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)^(3/2),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^4*d^2*x^7 - 2*c^2*d^2*x^5 + d
^2*x^3), x)

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

[Out]

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

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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{3}{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)^(3/2),x, algorithm="giac")

[Out]

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