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

Optimal. Leaf size=389 \[ \frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac{b^2 (1-c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

-b^2/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*(1 - c*x))/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x]*ArcCosh[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos
h[c*x]))/(3*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x])^2)/(3*d*(d - c^2*d*x^2)^(3/2)
) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (2*b*Sqrt[-1 + c*x
]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[
-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.730808, antiderivative size = 404, normalized size of antiderivative = 1.04, number of steps used = 12, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {5798, 5724, 5750, 89, 12, 78, 52, 5715, 3716, 2190, 2279, 2391} \[ \frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b x^2 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{2 b \sqrt{c x-1} \sqrt{c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (1-c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2}{3 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^3 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-b^2/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*(1 - c*x))/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*Sqrt[-1 + c*x]*S
qrt[1 + c*x]*ArcCosh[c*x])/(3*c^3*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCos
h[c*x]))/(3*c*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]) + (x^3*(a + b*ArcCosh[c*x])^2)/(3*d^2*(1 - c*x)*(1 + c*x)
*Sqrt[d - c^2*d*x^2]) - (Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^2)/(3*c^3*d^2*Sqrt[d - c^2*d*x^2])
+ (2*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*Log[1 - E^(2*ArcCosh[c*x])])/(3*c^3*d^2*Sqrt[d - c^2*
d*x^2]) + (b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^(2*ArcCosh[c*x])])/(3*c^3*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 5724

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)/(d
1*d2*f*(m + 1)), 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, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5750

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

Rule 89

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 78

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

Rule 52

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

Rule 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]

Rubi steps

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

Mathematica [A]  time = 1.67831, size = 264, normalized size = 0.68 \[ \frac{b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (-\text{PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )-\frac{c x \left (c^2 x^2+c^2 x^2 \cosh ^{-1}(c x)^2-1\right )}{\left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3}+\cosh ^{-1}(c x) \left (\frac{1}{1-c^2 x^2}+\cosh ^{-1}(c x)+2 \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )\right )\right )+\frac{a^2 c^3 x^3}{1-c^2 x^2}+a b \left (\frac{\sqrt{\frac{c x-1}{c x+1}} \left (2 \left (c^2 x^2-1\right ) \log \left (\sqrt{\frac{c x-1}{c x+1}} (c x+1)\right )-1\right )}{c x-1}+\frac{2 c^3 x^3 \cosh ^{-1}(c x)}{1-c^2 x^2}\right )}{3 c^3 d^2 \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.352, size = 3445, normalized size = 8.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*x^3+1/3*b^2*(-d*(c^2*x^2-1))^(
1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*arccosh(c*x)^2*x^3+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8
*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*arccosh(c*x)*x^3+2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^
6+10*c^4*x^4-5*c^2*x^2+1)*c^4/d^3*x^7-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*
c^2/d^3*x^5-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*(c*x+1)*(c*x-1)*x^
3+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^4/d^3*arccosh(c*x)*x^7-b^2*(-d
*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*arccosh(c*x)^2*x^5-2/3*b^2*(-d*(c^2*x
^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*arccosh(c*x)*x^5+1/3*b^2*(-d*(c^2*x^2-1))^(1
/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c^3/d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+b^2*(-d*(c^2*x^2-1))^(1/2
)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^4/d^3*arccosh(c*x)^2*x^7-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^
8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-4/3*b^2*(-d*(c^2*x^
2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2+1/3*b^2*(-d*(c^
2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c^3/d^3*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2
)-2/3*b^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(
c*x+1)^(1/2))+2/3*b^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*arccosh(c*x)^2-2/
3*b^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x
+1)^(1/2))+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c^3/d^3*arccosh(c*x)*(c
*x+1)^(1/2)*(c*x-1)^(1/2)+2*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^4/d^3*ar
ccosh(c*x)*x^7-2*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*arccosh(c*x)*
x^5+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*(c*x+1)*(c*x-1)*x^3+1/3*a*
b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c^3/d^3*(c*x-1)^(1/2)*(c*x+1)^(1/2)+1/3*
b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*arccosh(c*x)*(c*x+1)*(c*x-1)*x^3+1
/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*(c*x+1)*(c*x-1)*x^5-b^2*(-d
*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^3/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^6+2*b^2
*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^4+1/3
*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c^2/d^3*(c*x+1)*(c*x-1)*x-1/3*a^2/c^2
/d^2*x/(-c^2*d*x^2+d)^(1/2)+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*x^
3+1/3*a^2/c^2/d*x/(-c^2*d*x^2+d)^(3/2)+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^
2+1)*c^4/d^3*x^7-2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*x^5+2/3*a
*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/d^3*arccosh(c*x)*x^3-2*a*b*(-d*(c^2*x^2
-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^3/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^6+4
*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x
-1)^(1/2)*x^4+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c/d^3*arccosh(c*x)*(c*x+
1)^(1/2)*(c*x-1)^(1/2)*x^4-4/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c/d^3*a
rccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2
*x^2+1)/c/d^3*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^2-2/3*b^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1)
)^(1/2)/d^3/c^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-2/3*b^2*(c*x+1)^(1/2)*(c*x-1)^(
1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-b^2*(-d*(c^
2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^3/d^3*arccosh(c*x)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x^6-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*arccosh(c*x)*(c*x+1)
*(c*x-1)*x^5+2*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c/d^3*arccosh(c*x)^2*(c
*x+1)^(1/2)*(c*x-1)^(1/2)*x^4+4/3*a*b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*a
rccosh(c*x)-1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c^2/d^3*(c*x+1)*(c*x-1
)*x^5+a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)*c/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x^4-a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c/d^3*(c*x+1)^(1/2)*(c*x-1)^(1/2
)*x^2+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^8*x^8-9*c^6*x^6+10*c^4*x^4-5*c^2*x^2+1)/c^3/d^3*arccosh(c*x)*(c*x+1)
^(1/2)*(c*x-1)^(1/2)-2/3*a*b*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^3/(c^2*x^2-1)*ln((c*x+(c
*x-1)^(1/2)*(c*x+1)^(1/2))^2-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a b c{\left (\frac{\sqrt{-d}}{c^{6} d^{3} x^{2} - c^{4} d^{3}} - \frac{\sqrt{-d} \log \left (c x + 1\right )}{c^{4} d^{3}} - \frac{\sqrt{-d} \log \left (c x - 1\right )}{c^{4} d^{3}}\right )} - \frac{2}{3} \, a b{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} \operatorname{arcosh}\left (c x\right ) - \frac{1}{3} \, a^{2}{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} + b^{2} \int \frac{x^{2} \log \left (c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*b*c*(sqrt(-d)/(c^6*d^3*x^2 - c^4*d^3) - sqrt(-d)*log(c*x + 1)/(c^4*d^3) - sqrt(-d)*log(c*x - 1)/(c^4*d^3
)) - 2/3*a*b*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d))*arccosh(c*x) - 1/3*a^2*(x/(
sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2)*c^2*d)) + b^2*integrate(x^2*log(c*x + sqrt(c*x + 1)*
sqrt(c*x - 1))^2/(-c^2*d*x^2 + d)^(5/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*acosh(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral(x**2*(a + b*acosh(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), 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} x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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