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

Optimal. Leaf size=341 \[ -\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.936362, antiderivative size = 341, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.379, Rules used = {5798, 5748, 5688, 5715, 3716, 2190, 2279, 2391, 5721, 5461, 4182} \[ -\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + b*ArcCosh[c*x])^2/(d*x*Sqrt[d - c^2*d*x^2])) + (2*c^2*x*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2])
+ (2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(d*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[-1 + c*x]*S
qrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (4*b*c*Sqrt[-1 + c*x]
*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c*Sqrt[-1 + c*
x]*Sqrt[1 + c*x]*PolyLog[2, -E^(2*ArcCosh[c*x])])/(d*Sqrt[d - c^2*d*x^2]) - (b^2*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]*PolyLog[2, E^(2*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 5688

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(((d1_) + (e1_.)*(x_))^(3/2)*((d2_) + (e2_.)*(x_))^(3/2)), x_Sym
bol] :> Simp[(x*(a + b*ArcCosh[c*x])^n)/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), x] + Dist[(b*c*n*Sqrt[1 + c*x
]*Sqrt[-1 + c*x])/(d1*d2*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), Int[(x*(a + b*ArcCosh[c*x])^(n - 1))/(1 - c^2*x^2),
x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, -(c*d2)] && GtQ[n, 0]

Rule 5715

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

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && 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 5721

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

Rule 5461

Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]

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]

Rubi steps

\begin{align*} \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 \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^2 (-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}{d x \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 c^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{(-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}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \text{sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c^3 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{\left (4 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{b^2 c \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 c \sqrt{-1+c x} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}\\ &=-\frac{\left (a+b \cosh ^{-1}(c x)\right )^2}{d x \sqrt{d-c^2 d x^2}}+\frac{2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}+\frac{2 c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{4 b c \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}-\frac{b^2 c \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 1.63105, size = 315, normalized size = 0.92 \[ \frac{b^2 \left (c x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+c x \sqrt{\frac{c x-1}{c x+1}} (c x+1) \text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )+\cosh ^{-1}(c x) \left (c^2 x^2 \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-2 c x \left (\cosh ^{-1}(c x)+\log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )+\log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )\right )\right )\right )+a^2 \left (2 c^2 x^2-1\right )+2 a b \left (c^2 x^2 \cosh ^{-1}(c x)+\sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)-c x \left (\log (c x)+\log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )\right )\right )\right )}{d x \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.286, size = 826, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^2/d/x/(-c^2*d*x^2+d)^(1/2)+2*a^2*c^2/d*x/(-c^2*d*x^2+d)^(1/2)-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c
*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)^2*c-2*b^2*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)^2*x/(c^2*x^2-1)/d^2*c^2
+b^2*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)^2/x/(c^2*x^2-1)/d^2+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1
)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*c+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c+2*b^2*(-d*(c^2*x^2-1))^(1
/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c+2*b^2*(-d
*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))*c+
2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccosh(c*x)*ln((c*x+(c*x-1)^(1/2)*(c
*x+1)^(1/2))^2+1)*c+b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*polylog(2,-(c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))^2)*c-4*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*arccos
h(c*x)*c-4*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)*x/(c^2*x^2-1)/d^2*c^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(
c*x)/x/(c^2*x^2-1)/d^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*x^2-1)/d^2*ln((c*x+(c*x-1
)^(1/2)*(c*x+1)^(1/2))^4-1)*c

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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^2/(-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^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccosh(c*x))^2/x^2/(-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^6 - 2*c^2*d^2*x^4 + d
^2*x^2), 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^{2} \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**2/(-c**2*d*x**2+d)**(3/2),x)

[Out]

Integral((a + b*acosh(c*x))**2/(x**2*(-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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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