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

Optimal. Leaf size=556 \[ \frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (c x+1)}{27 c^4 d \sqrt{d-c^2 d x^2}}+\frac{94 b^2 (1-c x) (c x+1)}{27 c^6 d \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}} \]

[Out]

(16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) + (94*b^2*(1 - c*x)*(1 + c*x))/(27*c^6*d
*Sqrt[d - c^2*d*x^2]) + (2*b^2*x^2*(1 - c*x)*(1 + c*x))/(27*c^4*d*Sqrt[d - c^2*d*x^2]) + (16*b^2*x*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*Arc
Cosh[c*x]))/(c^5*d*Sqrt[d - c^2*d*x^2]) + (2*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3*d
*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (8*Sqrt[d - c^2*d*x^2]*(a +
b*ArcCosh[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(3*c^4*d^2) + (4*b*Sqrt[-
1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt
[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.36703, antiderivative size = 578, normalized size of antiderivative = 1.04, number of steps used = 23, number of rules used = 14, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.483, Rules used = {5798, 5752, 5759, 5718, 5654, 74, 5662, 100, 12, 5766, 5694, 4182, 2279, 2391} \[ \frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (c x+1)}{27 c^4 d \sqrt{d-c^2 d x^2}}+\frac{94 b^2 (1-c x) (c x+1)}{27 c^6 d \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) + (94*b^2*(1 - c*x)*(1 + c*x))/(27*c^6*d
*Sqrt[d - c^2*d*x^2]) + (2*b^2*x^2*(1 - c*x)*(1 + c*x))/(27*c^4*d*Sqrt[d - c^2*d*x^2]) + (16*b^2*x*Sqrt[-1 + c
*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/(3*c^5*d*Sqrt[d - c^2*d*x^2]) - (2*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*Arc
Cosh[c*x]))/(c^5*d*Sqrt[d - c^2*d*x^2]) + (2*b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(9*c^3*d
*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (8*(1 - c*x)*(1 + c*x)*(a +
b*ArcCosh[c*x])^2)/(3*c^6*d*Sqrt[d - c^2*d*x^2]) + (4*x^2*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x])^2)/(3*c^4*
d*Sqrt[d - c^2*d*x^2]) + (4*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(c^6*
d*Sqrt[d - c^2*d*x^2]) + (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(c^6*d*Sqrt[d - c^2*
d*x^2]) - (2*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(c^6*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 5752

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_
))^(p_), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2
*e1*e2*(p + 1)), x] + (-Dist[(f^2*(m - 1))/(2*e1*e2*(p + 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*f*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*
x)^FracPart[p])/(2*c*(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}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && LtQ[p, -1] && GtQ[m, 1] && IntegerQ[p + 1/2]

Rule 5759

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_
.)*(x_)]), x_Symbol] :> Simp[(f*(f*x)^(m - 1)*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n)/(e1*e2*m
), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m - 2)*(a + b*ArcCosh[c*x])^n)/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*
x]), x], x] + Dist[(b*f*n*Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x])/(c*d1*d2*m*Sqrt[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, e1, d2, e2, f}, x] && EqQ[e1 - c*d1, 0]
&& EqQ[e2 + c*d2, 0] && GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-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, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

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 5662

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
osh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCosh[c*x])^(n - 1))/(Sqr
t[-1 + c*x]*Sqrt[1 + c*x]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 100

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

Rule 12

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

Rule 5766

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)/(e*(m + 2*p + 1)), x] + (-Dist[(b*f*n*(-d)^p)/(c*
(m + 2*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))/(c^2*(m + 2*p + 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] && GtQ[m, 1] && NeQ[m + 2*p + 1, 0] && Inte
gerQ[p] && IntegerQ[m]

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]

Rubi steps

\begin{align*} \int \frac{x^5 \left (a+b \cosh ^{-1}(c x)\right )^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{x^5 \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{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (4 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^2 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (8 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x \left (a+b \cosh ^{-1}(c x)\right )^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{-1+c^2 x^2} \, dx}{c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (8 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int x^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^3 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c^2 d \sqrt{d-c^2 d x^2}}\\ &=-\frac{2 b^2 x^2 (1-c x) (1+c x)}{9 c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (2 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{\left (16 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (2 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (8 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x^3}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c^2 d \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 (1-c x) (1+c x)}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \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) \text{csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \cosh ^{-1}(c x) \, dx}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{\left (8 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{2 x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 c^4 d \sqrt{d-c^2 d x^2}}+\frac{\left (4 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{9 c^4 d \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{22 b^2 (1-c x) (1+c x)}{9 c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \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 )}{c^6 d \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^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d \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^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{27 c^4 d \sqrt{d-c^2 d x^2}}-\frac{\left (16 b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{3 c^4 d \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \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 )}{c^6 d \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 \frac{\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d \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 \frac{\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}\\ &=\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d \sqrt{d-c^2 d x^2}}+\frac{94 b^2 (1-c x) (1+c x)}{27 c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x^2 (1-c x) (1+c x)}{27 c^4 d \sqrt{d-c^2 d x^2}}+\frac{16 b^2 x \sqrt{-1+c x} \sqrt{1+c x} \cosh ^{-1}(c x)}{3 c^5 d \sqrt{d-c^2 d x^2}}-\frac{2 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d \sqrt{d-c^2 d x^2}}+\frac{2 b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{9 c^3 d \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{c^2 d \sqrt{d-c^2 d x^2}}+\frac{8 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d \sqrt{d-c^2 d x^2}}+\frac{4 x^2 (1-c x) (1+c x) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d \sqrt{d-c^2 d x^2}}+\frac{4 b \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 )}{c^6 d \sqrt{d-c^2 d x^2}}+\frac{2 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}-\frac{2 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^6 d \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 3.78422, size = 358, normalized size = 0.64 \[ \frac{-b^2 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \left (216 \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-216 \text{PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )+378 \sqrt{\frac{c x-1}{c x+1}} (c x+1)+189 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2-378 c x \cosh ^{-1}(c x)-6 \cosh \left (3 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+216 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )-216 \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+9 \cosh ^{-1}(c x)^2 \sinh \left (3 \cosh ^{-1}(c x)\right )+2 \sinh \left (3 \cosh ^{-1}(c x)\right )+54 \cosh ^{-1}(c x)^2 \tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )-54 \cosh ^{-1}(c x)^2 \coth \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )-36 a^2 \left (c^4 x^4+4 c^2 x^2-8\right )+3 a b \left (-60 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)-3 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+135 \cosh ^{-1}(c x)+62 \sinh \left (2 \cosh ^{-1}(c x)\right )+\sinh \left (4 \cosh ^{-1}(c x)\right )-72 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{108 c^6 d \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-36*a^2*(-8 + 4*c^2*x^2 + c^4*x^4) + 3*a*b*(135*ArcCosh[c*x] - 60*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 3*ArcCo
sh[c*x]*Cosh[4*ArcCosh[c*x]] - 72*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + 62*Sinh[2*A
rcCosh[c*x]] + Sinh[4*ArcCosh[c*x]]) - b^2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(378*Sqrt[(-1 + c*x)/(1 + c*x)
]*(1 + c*x) - 378*c*x*ArcCosh[c*x] + 189*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]^2 - 6*ArcCosh[c*x]*
Cosh[3*ArcCosh[c*x]] - 54*ArcCosh[c*x]^2*Coth[ArcCosh[c*x]/2] + 216*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] -
216*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 216*PolyLog[2, -E^(-ArcCosh[c*x])] - 216*PolyLog[2, E^(-ArcCosh[
c*x])] + 2*Sinh[3*ArcCosh[c*x]] + 9*ArcCosh[c*x]^2*Sinh[3*ArcCosh[c*x]] + 54*ArcCosh[c*x]^2*Tanh[ArcCosh[c*x]/
2]))/(108*c^6*d*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.486, size = 1099, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a^2*x^4/c^2/d/(-c^2*d*x^2+d)^(1/2)-4/3*a^2/c^4*x^2/d/(-c^2*d*x^2+d)^(1/2)+8/3*a^2/c^6/d/(-c^2*d*x^2+d)^(1
/2)-8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/c^6/d^2/(c^2*x^2-1)*arccosh(c*x)^2-2/9*b^2*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/(
c^2*x^2-1)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3-10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/c^5/d^2/(c^2*x^2-1)*ar
ccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(c^2
*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^
(1/2)/c^6/d^2/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x
-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+2/27*b^2*(-d*(c^2*x^2-1
))^(1/2)/c^2/d^2/(c^2*x^2-1)*x^4+92/27*b^2*(-d*(c^2*x^2-1))^(1/2)/c^4/d^2/(c^2*x^2-1)*x^2-94/27*b^2*(-d*(c^2*x
^2-1))^(1/2)/c^6/d^2/(c^2*x^2-1)+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arccosh(c*x)^2*x^4+4/3*b^2
*(-d*(c^2*x^2-1))^(1/2)/c^4/d^2/(c^2*x^2-1)*arccosh(c*x)^2*x^2-2*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x
+1)^(1/2)/c^6/d^2/(c^2*x^2-1)*polylog(2,-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)
^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(c^2*x^2-1)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))+2*a*b*(-d*(c^2*x^2-1))^(1/2)*(c
*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^2/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)-16/3*a*b*(-d*(c^2*x^2-1))^
(1/2)/c^6/d^2/(c^2*x^2-1)*arccosh(c*x)+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/c^2/d^2/(c^2*x^2-1)*arccosh(c*x)*x^4+8/3
*a*b*(-d*(c^2*x^2-1))^(1/2)/c^4/d^2/(c^2*x^2-1)*arccosh(c*x)*x^2-2/9*a*b*(-d*(c^2*x^2-1))^(1/2)/c^3/d^2/(c^2*x
^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x^3-10/3*a*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^2/(c^2*x^2-1)*(c*x+1)^(1/2)*(c*x-1
)^(1/2)*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(x^5*(a+b*arccosh(c*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{{\left (b^{2} x^{5} \operatorname{arcosh}\left (c x\right )^{2} + 2 \, a b x^{5} \operatorname{arcosh}\left (c x\right ) + a^{2} x^{5}\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^5*arccosh(c*x)^2 + 2*a*b*x^5*arccosh(c*x) + a^2*x^5)*sqrt(-c^2*d*x^2 + d)/(c^4*d^2*x^4 - 2*c^2
*d^2*x^2 + d^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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