3.2.92 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^2}{x^3} \, dx\) [192]

Optimal. Leaf size=890 \[ -\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{(1-c x) (1+c x)}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]

[Out]

-5/6*c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))^2-1/2*(-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^2/x^2-170/27*
b^2*c^2*d^2*(-c^2*d*x^2+d)^(1/2)+5/27*b^2*c^4*d^2*x^2*(-c^2*d*x^2+d)^(1/2)+5/3*b^2*c^2*d^2*(-c^2*x^2+1)*(-c^2*
d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)+1/9*b^2*c^2*d^2*(-c^2*x^2+1)^2*(-c^2*d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)-5/2*c^2*d
^2*(a+b*arccosh(c*x))^2*(-c^2*d*x^2+d)^(1/2)+5*a*b*c^3*d^2*x*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+
5*b^2*c^3*d^2*x*arccosh(c*x)*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b*c*d^2*(a+b*arccosh(c*x))*(-c^2
*d*x^2+d)^(1/2)/x/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/3*b*c^3*d^2*x*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^
(1/2)/(c*x+1)^(1/2)-2/9*b*c^5*d^2*x^3*(a+b*arccosh(c*x))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5*c^
2*d^2*(a+b*arccosh(c*x))^2*arctan(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^
(1/2)-5*I*b*c^2*d^2*(a+b*arccosh(c*x))*polylog(2,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/(c
*x-1)^(1/2)/(c*x+1)^(1/2)+5*I*b^2*c^2*d^2*polylog(3,-I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/2)
/(c*x-1)^(1/2)/(c*x+1)^(1/2)-5*I*b^2*c^2*d^2*polylog(3,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))*(-c^2*d*x^2+d)^(1/
2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)+5*I*b*c^2*d^2*(a+b*arccosh(c*x))*polylog(2,I*(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2)))
*(-c^2*d*x^2+d)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)-b^2*c^2*d^2*arctan((c^2*x^2-1)^(1/2))*(c^2*x^2-1)^(1/2)*(-c^
2*d*x^2+d)^(1/2)/(-c*x+1)/(c*x+1)

________________________________________________________________________________________

Rubi [A]
time = 0.92, antiderivative size = 890, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 22, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.759, Rules used = {5928, 5930, 5926, 5947, 4265, 2611, 2320, 6724, 5879, 75, 5889, 5894, 12, 471, 5912, 276, 5921, 534, 1265, 911, 1167, 211} \begin {gather*} -\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^5}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{27} b^2 d^2 x^2 \sqrt {d-c^2 d x^2} c^4+\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 a b d^2 x \sqrt {d-c^2 d x^2} c^3}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{6} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2-\frac {5}{2} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 d^2 \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2} \text {ArcTan}\left (\sqrt {c^2 x^2-1}\right ) c^2}{(1-c x) (c x+1)}-\frac {5 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {170}{27} b^2 d^2 \sqrt {d-c^2 d x^2} c^2+\frac {b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} c^2}{9 (1-c x) (c x+1)}+\frac {5 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} c^2}{3 (1-c x) (c x+1)}-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-170*b^2*c^2*d^2*Sqrt[d - c^2*d*x^2])/27 + (5*b^2*c^4*d^2*x^2*Sqrt[d - c^2*d*x^2])/27 + (5*a*b*c^3*d^2*x*Sqrt
[d - c^2*d*x^2])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (5*b^2*c^2*d^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2])/(3*(1 - c*
x)*(1 + c*x)) + (b^2*c^2*d^2*(1 - c^2*x^2)^2*Sqrt[d - c^2*d*x^2])/(9*(1 - c*x)*(1 + c*x)) + (5*b^2*c^3*d^2*x*S
qrt[d - c^2*d*x^2]*ArcCosh[c*x])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[
c*x]))/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*Sqrt[-1 +
c*x]*Sqrt[1 + c*x]) - (2*b*c^5*d^2*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(9*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) - (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/2 - (5*c^2*d*(d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh
[c*x])^2)/6 - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2)/(2*x^2) + (5*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x])^2*ArcTan[E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b^2*c^2*d^2*Sqrt[-1 + c^2*x^2]*Sqrt[d
- c^2*d*x^2]*ArcTan[Sqrt[-1 + c^2*x^2]])/((1 - c*x)*(1 + c*x)) - ((5*I)*b*c^2*d^2*Sqrt[d - c^2*d*x^2]*(a + b*
ArcCosh[c*x])*PolyLog[2, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((5*I)*b*c^2*d^2*Sqrt[d - c^2*
d*x^2]*(a + b*ArcCosh[c*x])*PolyLog[2, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + ((5*I)*b^2*c^2*d^2*
Sqrt[d - c^2*d*x^2]*PolyLog[3, (-I)*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - ((5*I)*b^2*c^2*d^2*Sqrt[
d - c^2*d*x^2]*PolyLog[3, I*E^ArcCosh[c*x]])/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 12

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

Rule 75

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rule 534

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.) + (e_.)*(x_)^(n2_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b
2_.)*(x_)^(non2_.))^(p_.), x_Symbol] :> Dist[(a1 + b1*x^(n/2))^FracPart[p]*((a2 + b2*x^(n/2))^FracPart[p]/(a1*
a2 + b1*b2*x^n)^FracPart[p]), Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n + e*x^(2*n))^q, x], x] /; FreeQ[{a1, b1,
a2, b2, c, d, e, n, p, q}, x] && EqQ[non2, n/2] && EqQ[n2, 2*n] && EqQ[a2*b1 + a1*b2, 0]

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 2320

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 2611

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 4265

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 5879

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 5889

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_Symbo
l] :> Int[(d1*d2 + e1*e2*x^2)^p*(a + b*ArcCosh[c*x])^n, x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2
*e1 + d1*e2, 0] && IntegerQ[p]

Rule 5894

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x
], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5912

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

Rule 5921

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcCosh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(Sqrt[1
+ c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 5926

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

Rule 5928

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*((a + b*ArcCosh[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 1)))*Simp[(d + e*x^2)^p/((1
+ c*x)^p*(-1 + c*x)^p)], 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]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 5930

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*((a + b*ArcCosh[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int
[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcCosh[c*x])^n, x], x] - Dist[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^
p/((1 + c*x)^p*(-1 + c*x)^p)], 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]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m
, -1]

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rule 6724

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]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x^3} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}-\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{3 x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^3 d^2 \sqrt {d-c^2 d x^2}\right ) \int \cosh ^{-1}(c x) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c^2 x^2}} \, dx}{3 (-1+c x) (1+c x)}\\ &=\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (35 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-3-6 c^2 x+c^4 x^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 (-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {-8-4 x^2+x^4}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 (-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (-5 c^2+c^2 x^2-\frac {3}{\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 (-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{(-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2}+\frac {5 c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{(1-c x) (1+c x)}-\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 78.80, size = 1384, normalized size = 1.56 \begin {gather*} \sqrt {-d \left (-1+c^2 x^2\right )} \left (-\frac {7}{3} a^2 c^2 d^2-\frac {a^2 d^2}{2 x^2}+\frac {1}{3} a^2 c^4 d^2 x^2\right )-\frac {a b c^2 d^2 \sqrt {-d (-1+c x) (1+c x)} \left (-9 c x-12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \cosh ^{-1}(c x)+\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{18 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {5}{2} a^2 c^2 d^{5/2} \log (x)+\frac {5}{2} a^2 c^2 d^{5/2} \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )-4 a b c^2 d^2 \sqrt {-d (-1+c x) (1+c x)} \left (-\frac {c x}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\cosh ^{-1}(c x)+\frac {i \cosh ^{-1}(c x) \left (\log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\log \left (1+i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+\frac {i \left (\text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )+\frac {i a b c^2 d^3 \left (-\frac {i \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}{c x}-\frac {i (-1+c x) (1+c x) \cosh ^{-1}(c x)}{c^2 x^2}+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )\right )}{\sqrt {-d (-1+c x) (1+c x)}}+\frac {1}{54} b^2 d^2 \sqrt {d-c^2 d x^2} \left (\frac {244 c^2}{-1+c x}-\frac {244 c^3 x}{-1+c x}-\frac {4 c^4 x^2}{-1+c x}+\frac {4 c^5 x^3}{-1+c x}-\frac {54 c^2 \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}}+\frac {54 c \cosh ^{-1}(c x)}{x (-1+c x)^{3/2} \sqrt {1+c x}}-\frac {252 c^3 x \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}}+\frac {252 c^4 x^2 \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}}+\frac {12 c^5 x^3 \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}}-\frac {12 c^6 x^4 \cosh ^{-1}(c x)}{(-1+c x)^{3/2} \sqrt {1+c x}}+\frac {126 c^2 \cosh ^{-1}(c x)^2}{-1+c x}+\frac {27 \cosh ^{-1}(c x)^2}{x^2 (-1+c x)}-\frac {126 c^3 x \cosh ^{-1}(c x)^2}{-1+c x}-\frac {18 c^4 x^2 \cosh ^{-1}(c x)^2}{-1+c x}+\frac {18 c^5 x^3 \cosh ^{-1}(c x)^2}{-1+c x}+\frac {27 c \cosh ^{-1}(c x)^2}{x-c x^2}+\frac {54 c^2 \text {ArcTan}\left (\frac {1}{\sqrt {-1+c^2 x^2}}\right )}{(-1+c x) \sqrt {-1+c^2 x^2}}-\frac {54 c^3 x \text {ArcTan}\left (\frac {1}{\sqrt {-1+c^2 x^2}}\right )}{(-1+c x) \sqrt {-1+c^2 x^2}}-\frac {135 i c^2 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )}{-1+c x}+\frac {135 i c^2 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )}{-1+c x}-\frac {270 i c^2 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )}{-1+c x}+\frac {270 i c^2 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )}{-1+c x}-\frac {270 i c^2 \sqrt {\frac {-1+c x}{1+c x}} \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )}{-1+c x}+\frac {270 i c^2 \sqrt {\frac {-1+c x}{1+c x}} \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(c x)}\right )}{-1+c x}\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((-7*a^2*c^2*d^2)/3 - (a^2*d^2)/(2*x^2) + (a^2*c^4*d^2*x^2)/3) - (a*b*c^2*d^2*Sqrt[-
(d*(-1 + c*x)*(1 + c*x))]*(-9*c*x - 12*((-1 + c*x)/(1 + c*x))^(3/2)*(1 + c*x)^3*ArcCosh[c*x] + Cosh[3*ArcCosh[
c*x]]))/(18*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) - (5*a^2*c^2*d^(5/2)*Log[x])/2 + (5*a^2*c^2*d^(5/2)*Log[d +
Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^2))]])/2 - 4*a*b*c^2*d^2*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*(-((c*x)/(Sqrt[(-1 + c*x
)/(1 + c*x)]*(1 + c*x))) + ArcCosh[c*x] + (I*ArcCosh[c*x]*(Log[1 - I/E^ArcCosh[c*x]] - Log[1 + I/E^ArcCosh[c*x
]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + (I*(PolyLog[2, (-I)/E^ArcCosh[c*x]] - PolyLog[2, I/E^ArcCosh[c*x
]]))/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))) + (I*a*b*c^2*d^3*(((-I)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))/(c*
x) - (I*(-1 + c*x)*(1 + c*x)*ArcCosh[c*x])/(c^2*x^2) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1
- I/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x]*Log[1 + I/E^ArcCosh[c*x]] + Sqrt[(-1
+ c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2, (-I)/E^ArcCosh[c*x]] - Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*PolyLog[2,
I/E^ArcCosh[c*x]]))/Sqrt[-(d*(-1 + c*x)*(1 + c*x))] + (b^2*d^2*Sqrt[d - c^2*d*x^2]*((244*c^2)/(-1 + c*x) - (24
4*c^3*x)/(-1 + c*x) - (4*c^4*x^2)/(-1 + c*x) + (4*c^5*x^3)/(-1 + c*x) - (54*c^2*ArcCosh[c*x])/((-1 + c*x)^(3/2
)*Sqrt[1 + c*x]) + (54*c*ArcCosh[c*x])/(x*(-1 + c*x)^(3/2)*Sqrt[1 + c*x]) - (252*c^3*x*ArcCosh[c*x])/((-1 + c*
x)^(3/2)*Sqrt[1 + c*x]) + (252*c^4*x^2*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (12*c^5*x^3*ArcCosh[c*
x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) - (12*c^6*x^4*ArcCosh[c*x])/((-1 + c*x)^(3/2)*Sqrt[1 + c*x]) + (126*c^2*A
rcCosh[c*x]^2)/(-1 + c*x) + (27*ArcCosh[c*x]^2)/(x^2*(-1 + c*x)) - (126*c^3*x*ArcCosh[c*x]^2)/(-1 + c*x) - (18
*c^4*x^2*ArcCosh[c*x]^2)/(-1 + c*x) + (18*c^5*x^3*ArcCosh[c*x]^2)/(-1 + c*x) + (27*c*ArcCosh[c*x]^2)/(x - c*x^
2) + (54*c^2*ArcTan[1/Sqrt[-1 + c^2*x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - (54*c^3*x*ArcTan[1/Sqrt[-1 + c^2*
x^2]])/((-1 + c*x)*Sqrt[-1 + c^2*x^2]) - ((135*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 - I/E^Ar
cCosh[c*x]])/(-1 + c*x) + ((135*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]^2*Log[1 + I/E^ArcCosh[c*x]])/(-
1 + c*x) - ((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[2, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) +
((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]*PolyLog[2, I/E^ArcCosh[c*x]])/(-1 + c*x) - ((270*I)*c^2*S
qrt[(-1 + c*x)/(1 + c*x)]*PolyLog[3, (-I)/E^ArcCosh[c*x]])/(-1 + c*x) + ((270*I)*c^2*Sqrt[(-1 + c*x)/(1 + c*x)
]*PolyLog[3, I/E^ArcCosh[c*x]])/(-1 + c*x)))/54

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(15*c^2*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - 3*(-c^2*d*x^2 + d)^(5/2)*c^2 - 5
*(-c^2*d*x^2 + d)^(3/2)*c^2*d - 15*sqrt(-c^2*d*x^2 + d)*c^2*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)/(d*x^2))*a^2 + inte
grate((-c^2*d*x^2 + d)^(5/2)*b^2*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))^2/x^3 + 2*(-c^2*d*x^2 + d)^(5/2)*a*b*l
og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/x^3, x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acosh(c*x))**2/x**3, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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