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

Optimal. Leaf size=568 \[ -\frac{11 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^3}-\frac{22 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}} \]

[Out]

-(b^2*x^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2*d
*x^2]) - (7*b^2*(1 - c*x)*(1 + c*x))/(3*c^6*d^2*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^2*Sqrt[d - c^2*d*x^2]) + (11*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*
c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2*x
^2)*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCosh[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) - (4*x^2*(a + b*ArcCos
h[c*x])^2)/(3*c^4*d^2*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^3) - (22*
b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) -
(11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) + (11*b^2*S
qrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.47623, antiderivative size = 594, normalized size of antiderivative = 1.05, number of steps used = 27, number of rules used = 13, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.448, Rules used = {5798, 5752, 5718, 5654, 74, 5766, 5694, 4182, 2279, 2391, 5750, 98, 21} \[ -\frac{11 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b^2 \sqrt{c x-1} \sqrt{c x+1} \text{PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{c x-1} \sqrt{c x+1}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (c x+1) \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{c x-1} \sqrt{c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{8 (1-c x) (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{22 b \sqrt{c x-1} \sqrt{c x+1} \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b^2 (1-c x) (c x+1)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 b^2 x \sqrt{c x-1} \sqrt{c x+1} \cosh ^{-1}(c x)}{3 c^5 d^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*x^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (16*a*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(3*c^5*d^2*Sqrt[d - c^2*d
*x^2]) - (7*b^2*(1 - c*x)*(1 + c*x))/(3*c^6*d^2*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^2*Sqrt[d - c^2*d*x^2]) + (11*b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*
c^5*d^2*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(3*c^3*d^2*(1 - c^2*x
^2)*Sqrt[d - c^2*d*x^2]) - (4*x^2*(a + b*ArcCosh[c*x])^2)/(3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (x^4*(a + b*ArcCos
h[c*x])^2)/(3*c^2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]) - (8*(1 - c*x)*(1 + c*x)*(a + b*ArcCosh[c*x])^2
)/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (22*b*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])*ArcTanh[E^ArcCosh[
c*x]])/(3*c^6*d^2*Sqrt[d - c^2*d*x^2]) - (11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, -E^ArcCosh[c*x]])/(3*
c^6*d^2*Sqrt[d - c^2*d*x^2]) + (11*b^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*PolyLog[2, E^ArcCosh[c*x]])/(3*c^6*d^2*Sqr
t[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 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 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]

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 98

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 )^{5/2}} \, 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)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 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^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 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}{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^3}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 c^2 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 b \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{-1+c^2 x^2} \, dx}{3 c^5 d^2 \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^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 \sqrt{-1+c x} \sqrt{1+c x}\right ) \int \frac{x (-2-2 c x)}{\sqrt{-1+c x} (1+c x)^{3/2}} \, dx}{3 c^4 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} \sqrt{1+c x}} \, dx}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (8 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^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (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^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 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 )}{3 c^6 d^2 \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^2 \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}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{3 b^2 (1-c x) (1+c x)}{c^6 d^2 \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^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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^2 \sqrt{d-c^2 d x^2}}-\frac{22 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 )}{3 c^6 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 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 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 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (8 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 )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 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 )}{3 c^6 d^2 \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^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \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^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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^2 \sqrt{d-c^2 d x^2}}-\frac{22 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 )}{3 c^6 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^{\cosh ^{-1}(c x)}\right )}{c^6 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^{\cosh ^{-1}(c x)}\right )}{c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (8 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 )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (8 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 )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{b^2 x^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{16 a b x \sqrt{-1+c x} \sqrt{1+c x}}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}-\frac{7 b^2 (1-c x) (1+c x)}{3 c^6 d^2 \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^2 \sqrt{d-c^2 d x^2}}+\frac{11 b x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^5 d^2 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 c^3 d^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}-\frac{4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{x^4 \left (a+b \cosh ^{-1}(c x)\right )^2}{3 c^2 d^2 (1-c x) (1+c x) \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^2 \sqrt{d-c^2 d x^2}}-\frac{22 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 )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}-\frac{11 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}+\frac{11 b^2 \sqrt{-1+c x} \sqrt{1+c x} \text{Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{3 c^6 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 5.56397, size = 437, normalized size = 0.77 \[ -\frac{b^2 \left (88 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(c x)}\right )-88 \left (\frac{c x-1}{c x+1}\right )^{3/2} (c x+1)^3 \text{PolyLog}\left (2,e^{-\cosh ^{-1}(c x)}\right )+25 \cosh ^{-1}(c x)^2-4 \left (9 \cosh ^{-1}(c x)^2+7\right ) \cosh \left (2 \cosh ^{-1}(c x)\right )+3 \left (\cosh ^{-1}(c x)^2+2\right ) \cosh \left (4 \cosh ^{-1}(c x)\right )-66 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right )+66 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+8 \cosh ^{-1}(c x) \sinh \left (2 \cosh ^{-1}(c x)\right )-6 \cosh ^{-1}(c x) \sinh \left (4 \cosh ^{-1}(c x)\right )+22 \cosh ^{-1}(c x) \log \left (1-e^{-\cosh ^{-1}(c x)}\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )-22 \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right ) \sinh \left (3 \cosh ^{-1}(c x)\right )+22\right )+8 a^2 \left (3 c^4 x^4-12 c^2 x^2+8\right )+2 a b \left (-36 \cosh \left (2 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+3 \cosh \left (4 \cosh ^{-1}(c x)\right ) \cosh ^{-1}(c x)+25 \cosh ^{-1}(c x)+4 \sinh \left (2 \cosh ^{-1}(c x)\right )-3 \sinh \left (4 \cosh ^{-1}(c x)\right )-33 \sqrt{\frac{c x-1}{c x+1}} (c x+1) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )+11 \sinh \left (3 \cosh ^{-1}(c x)\right ) \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(c x)\right )\right )\right )}{24 c^6 d \left (d-c^2 d x^2\right )^{3/2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(8*a^2*(8 - 12*c^2*x^2 + 3*c^4*x^4) + 2*a*b*(25*ArcCosh[c*x] - 36*ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] + 3*ArcCo
sh[c*x]*Cosh[4*ArcCosh[c*x]] - 33*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Tanh[ArcCosh[c*x]/2]] + 4*Sinh[2*Ar
cCosh[c*x]] + 11*Log[Tanh[ArcCosh[c*x]/2]]*Sinh[3*ArcCosh[c*x]] - 3*Sinh[4*ArcCosh[c*x]]) + b^2*(22 + 25*ArcCo
sh[c*x]^2 - 4*(7 + 9*ArcCosh[c*x]^2)*Cosh[2*ArcCosh[c*x]] + 3*(2 + ArcCosh[c*x]^2)*Cosh[4*ArcCosh[c*x]] - 66*S
qrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])] + 66*Sqrt[(-1 + c*x)/(1 + c*x)]*(1
+ c*x)*ArcCosh[c*x]*Log[1 + E^(-ArcCosh[c*x])] + 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, -E^(-
ArcCosh[c*x])] - 88*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*PolyLog[2, E^(-ArcCosh[c*x])] + 8*ArcCosh[c*x]*Si
nh[2*ArcCosh[c*x]] + 22*ArcCosh[c*x]*Log[1 - E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 22*ArcCosh[c*x]*Log[1 +
E^(-ArcCosh[c*x])]*Sinh[3*ArcCosh[c*x]] - 6*ArcCosh[c*x]*Sinh[4*ArcCosh[c*x]]))/(24*c^6*d*(d - c^2*d*x^2)^(3/
2))

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Maple [B]  time = 0.479, size = 1211, normalized size = 2.1 \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)^(5/2),x)

[Out]

-1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6+b^2*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c
*x)^2+11/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^3/(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^4/d^3/(c^2*x^2-1)*arccosh(c*x)*x^2+4*a*b*(-d*(c^2*x^2-1))^(
1/2)/d^3/(c^2*x^2-1)^2/c^4*arccosh(c*x)*x^2-11/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c^6/d^
3/(c^2*x^2-1)*polylog(2,c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6*
arccosh(c*x)^2-2*b^2*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*x^2-8/3*a^2/c^6/d/(-c^2*d*x^2+d)^(3/2)+2*b^2*(
-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)-a^2*x^4/c^2/d/(-c^2*d*x^2+d)^(3/2)+4*a^2/c^4*x^2/d/(-c^2*d*x^2+d)^(3
/2)-b^2*(-d*(c^2*x^2-1))^(1/2)/c^4/d^3/(c^2*x^2-1)*arccosh(c*x)^2*x^2+2*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^
2-1)^2/c^4*arccosh(c*x)^2*x^2+2*a*b*(-d*(c^2*x^2-1))^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*x)-10/3*a*b*(-d*(c^2*
x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^6*arccosh(c*x)+11/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c
^6/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))-11/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^
(1/2)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*arccosh(c*x)*ln(1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))+1/3*a*b*(-d*(c^2*x^
2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^5*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x-11/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*
(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)-1)+11/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1
)^(1/2)*(c*x+1)^(1/2)/c^6/d^3/(c^2*x^2-1)*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
^5/d^3/(c^2*x^2-1)*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2
/c^5*arccosh(c*x)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+2*a*b*(-d*(c^2*x^2-1))^(1/2)/c^5/d^3/(c^2*x^2-1)*(c*x+1)^(1/2)
*(c*x-1)^(1/2)*x+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(c^2*x^2-1)^2/c^4*x^2

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

[Out]

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