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

Optimal. Leaf size=1744 \[ \text{result too large to display} \]

[Out]

(2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d -
c^2*d*x^2])/(3*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c
^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*x^3*Sqrt
[d - c^2*d*x^2])/(45*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(
9*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a*d^2*(c^2*f^2 - g^2)^2*(1 -
c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g^5*(1 - c*x)*(1 + c*x)) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCosh
[c*x])/g^5 + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4
*g^2) - (2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*g) - (d^2*(c^2*f^2 - 2*g^2)*(
1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*g^3) - (c^2*d^2*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*g) + (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*g^2*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*g^4*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*g^5
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^6
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (d^2*(c^2*f^2 - g^2)^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[-1 + c^2*
x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^6*(1 - c*x)*(1 +
c*x)) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c
^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh
[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(c^
2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/
(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [A]  time = 4.77926, antiderivative size = 1744, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 31, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {5836, 5826, 5683, 5676, 30, 5718, 5743, 5759, 100, 12, 74, 5733, 5824, 683, 5816, 6742, 93, 208, 1610, 1654, 725, 206, 5860, 5858, 8, 5832, 3320, 2264, 2190, 2279, 2391} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*c*d^2*x*Sqrt[d - c^2*d*x^2])/(15*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c*d^2*(c^2*f^2 - 2*g^2)*x*Sqrt[d -
c^2*d*x^2])/(3*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2])/(g^5*Sqrt
[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*f*x^2*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c
^3*d^2*f*(c^2*f^2 - 2*g^2)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^3*d^2*x^3*Sqrt
[d - c^2*d*x^2])/(45*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^3*d^2*(c^2*f^2 - 2*g^2)*x^3*Sqrt[d - c^2*d*x^2])/(
9*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*c^5*d^2*f*x^4*Sqrt[d - c^2*d*x^2])/(16*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) - (b*c^5*d^2*x^5*Sqrt[d - c^2*d*x^2])/(25*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (a*d^2*(c^2*f^2 - g^2)^2*(1 -
c^2*x^2)*Sqrt[d - c^2*d*x^2])/(g^5*(1 - c*x)*(1 + c*x)) + (b*d^2*(c^2*f^2 - g^2)^2*Sqrt[d - c^2*d*x^2]*ArcCosh
[c*x])/g^5 + (c^2*d^2*f*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(8*g^2) - (c^2*d^2*f*(c^2*f^2 - 2*g^2)*x*S
qrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^4) - (c^4*d^2*f*x^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(4
*g^2) - (2*d^2*(1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(15*g) - (d^2*(c^2*f^2 - 2*g^2)*(
1 - c*x)*(1 + c*x)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(3*g^3) - (c^2*d^2*x^2*(1 - c*x)*(1 + c*x)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(5*g) + (c*d^2*f*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(16*b*g^2*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) + (c*d^2*f*(c^2*f^2 - 2*g^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(4*b*g^4*S
qrt[-1 + c*x]*Sqrt[1 + c*x]) - (c*d^2*(c^2*f^2 - g^2)^2*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*g^5
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (d^2*(c^2*f^2 - g^2)^3*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^6
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (d^2*(c^2*f^2 - g^2)^2*(1 - c^2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Cosh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (a*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[-1 + c^2*
x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^6*(1 - c*x)*(1 +
c*x)) + (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c
^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*ArcCosh
[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d^2*(c^
2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]) - (b*d^2*(c^2*f^2 - g^2)^(5/2)*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/
(c*f + Sqrt[c^2*f^2 - g^2]))])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rule 5836

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(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 + g*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, g, n}, x] && EqQ[c^2*d
+ e, 0] && IntegerQ[m] && IntegerQ[p - 1/2]

Rule 5826

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

Rule 5683

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

Rule 5676

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 5743

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

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 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 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 5733

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Sym
bol] :> With[{u = IntHide[x^m*(1 + c*x)^p*(-1 + c*x)^p, x]}, Dist[(-(d1*d2))^p*(a + b*ArcCosh[c*x]), u, x] - D
ist[b*c*(-(d1*d2))^p, Int[SimplifyIntegrand[u/(Sqrt[1 + c*x]*Sqrt[-1 + c*x]), x], x], x]] /; FreeQ[{a, b, c, d
1, e1, d2, e2}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p - 1/2] && (IGtQ[(m + 1)/2, 0] || IL
tQ[(m + 2*p + 3)/2, 0]) && NeQ[p, -2^(-1)] && GtQ[d1, 0] && LtQ[d2, 0]

Rule 5824

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

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 5816

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

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 208

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

Rule 1610

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Dist[((
a + b*x)^FracPart[m]*(c + d*x)^FracPart[m])/(a*c + b*d*x^2)^FracPart[m], Int[Px*(a*c + b*d*x^2)^m*(e + f*x)^p,
x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && EqQ[b*c + a*d, 0] && EqQ[m, n] &&  !Intege
rQ[m]

Rule 1654

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[(f*(d + e*x)^(m + q - 1)*(a + c*x^2)^(p + 1))/(c*e^(q - 1)*(m + q + 2*p + 1)), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rule 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 5860

Int[(ArcCosh[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_S
ymbol] :> Int[ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p, RFx*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b,
c, d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] &&
IntegerQ[p - 1/2]

Rule 5858

Int[ArcCosh[(c_.)*(x_)]^(n_.)*(RFx_)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_), x_Symbol] :> With[
{u = ExpandIntegrand[(d1 + e1*x)^p*(d2 + e2*x)^p*ArcCosh[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c,
d1, e1, d2, e2}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && In
tegerQ[p - 1/2]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5832

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_
) + (e2_.)*(x_)]), x_Symbol] :> Dist[1/(c^(m + 1)*Sqrt[-(d1*d2)]), Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m,
x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2
, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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{\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, 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 )}{f+g x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\left (d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{c^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^4}+\frac{c^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^3}-\frac{c^4 f x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^2}+\frac{c^4 x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g}+\frac{\left (-c^2 f^2+g^2\right )^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^4 (f+g x)}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (c^4 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x^2 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^4 d^2 \sqrt{d-c^2 d x^2}\right ) \int x^3 \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 d^2 \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int x \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^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 )}{f+g x} \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (c^4 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^5 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x^3 \, dx}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c^5 d^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{-2-c^2 x^2+3 c^4 x^4}{15 c^4} \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) \, dx}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (c^2 d^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c^3 d^2 f \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{8 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c d^2 \sqrt{d-c^2 d x^2}\right ) \int \left (-2-c^2 x^2+3 c^4 x^4\right ) \, dx}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (\frac{1}{f+g x}-\frac{c^2 \left (g x+\frac{f^2}{f+g x}\right )}{g^2}\right ) \left (-a-b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 f^2-g^2\right ) \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{g^6 (-1+c x) (1+c x)}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (b c^2 d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{c^4 f^2 g^2-c^2 g^4}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{c^2 g^8 (-1+c x) (1+c x)}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (b c d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{g^6 (-1+c x) (1+c x)}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 f^2-g^2-x^2} \, dx,x,\frac{-g-c^2 f x}{\sqrt{-1+c^2 x^2}}\right )}{g^6 (-1+c x) (1+c x)}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^6 (1-c x) (1+c x)}+\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^6 (1-c x) (1+c x)}+\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^6 (1-c x) (1+c x)}+\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{2 b c d^2 x \sqrt{d-c^2 d x^2}}{15 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c d^2 \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2}}{3 g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2}}{g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 f x^2 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 f \left (c^2 f^2-2 g^2\right ) x^2 \sqrt{d-c^2 d x^2}}{4 g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^3 d^2 x^3 \sqrt{d-c^2 d x^2}}{45 g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^3 d^2 \left (c^2 f^2-2 g^2\right ) x^3 \sqrt{d-c^2 d x^2}}{9 g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b c^5 d^2 f x^4 \sqrt{d-c^2 d x^2}}{16 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^5 d^2 x^5 \sqrt{d-c^2 d x^2}}{25 g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{a d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^5 (1-c x) (1+c x)}+\frac{b d^2 \left (c^2 f^2-g^2\right )^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^5}+\frac{c^2 d^2 f x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{8 g^2}-\frac{c^2 d^2 f \left (c^2 f^2-2 g^2\right ) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^4}-\frac{c^4 d^2 f x^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{4 g^2}-\frac{2 d^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 g}-\frac{d^2 \left (c^2 f^2-2 g^2\right ) (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 g^3}-\frac{c^2 d^2 x^2 (1-c x) (1+c x) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{5 g}+\frac{c d^2 f \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{16 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d^2 f \left (c^2 f^2-2 g^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{c d^2 \left (c^2 f^2-g^2\right )^2 x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^5 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{d^2 \left (c^2 f^2-g^2\right )^3 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^6 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{d^2 \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{a d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^6 (1-c x) (1+c x)}+\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d^2 (c f-g) (c f+g) \left (c^2 f^2-g^2\right )^{3/2} \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^6 \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [C]  time = 20.71, size = 6244, normalized size = 3.58 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.368, size = 4234, normalized size = 2.4 \begin{align*} \text{output too large to display} \end{align*}

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))/(g*x+f),x)

[Out]

b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*dilog((-(c*x+(c*x-1)^(1/2)*(c
*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)
^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*
f+(c^2*f^2-g^2)^(1/2)))-1/25*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g*x^5*c^5+11/45*b*(-d*(c
^2*x^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g*x^3*c^3-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c
*x+1)^(1/2)/g*c*x+33/128*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2-1/8*b*(-d*(c^2*x^2-1
))^(1/2)*f^3*d^2*c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^6/(c*x-1)/(c*x+1)/g^
2*arccosh(c*x)*x^5+11/8*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^4/(c*x-1)/(c*x+1)/g^2*arccosh(c*x)*x^3-9/8*b*(-d*(c^2
*x^2-1))^(1/2)*f*d^2*c^2/(c*x-1)/(c*x+1)/g^2*arccosh(c*x)*x+1/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g
^3*arccosh(c*x)*x^4*c^6*f^2-8/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g^3*arccosh(c*x)*x^2*c^4*f^2-1/2*
b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^6/(c*x-1)/(c*x+1)/g^4*arccosh(c*x)*x^3+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2
*c^4/(c*x-1)/(c*x+1)/g^4*arccosh(c*x)*x+b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g^5*arccosh(c*x)*x^2*c^6*
f^4+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*dilog((-(c*x+(c*x-1)^(1/2
)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(
c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2
)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4-2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(
c*x+1)^(1/2)/g^4*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2
)))*c^2*f^2+2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*dilog(((c*x+(c*
x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*c^2*f^2+1/5*a/g*(-(x+f/g)^2*c^
2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(5/2)+1/3*a/g*d*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-
g^2)/g^2)^(3/2)+a/g*d^2*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)-b*(-d*(c^2*x^2-1))^(1
/2)*d^2/(c*x-1)/(c*x+1)/g^5*arccosh(c*x)*c^4*f^4-15/16*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f*
arccosh(c*x)^2*d^2*c/g^2+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g*arccosh(c*x)*x^6*c^6-14/15*b*(-d*(
c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g*arccosh(c*x)*x^4*c^4+1/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^5/(c*x-1)^(
1/2)/(c*x+1)^(1/2)/g^2*x^4-9/16*b*(-d*(c^2*x^2-1))^(1/2)*f*d^2*c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*x^2-1/9*b*(
-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^3*x^3*c^5*f^2+7/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)
^(1/2)/(c*x+1)^(1/2)/g^3*x*c^3*f^2+1/4*b*(-d*(c^2*x^2-1))^(1/2)*f^3*d^2*c^5/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*x^
2-b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^5*x*c^5*f^4+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2
-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*arccosh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2
-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^4*f^4-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2
)/(c*x+1)^(1/2)/g^6*arccosh(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^
2-g^2)^(1/2)))*c^4*f^4-2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*arcc
osh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))*c^2*f^2
+2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*arccosh(c*x)*ln(((c*x+(c*x
-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))*c^2*f^2-a/g*d^3/(-d*(c^2*f^2-g^
2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+
2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))-1/3*a/g^3*d*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(
c^2*f^2-g^2)/g^2)^(3/2)*c^2*f^2+a/g^5*d^2*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*c^4
*f^4-2*a/g^3*d^2*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*c^2*f^2+34/15*b*(-d*(c^2*x^2
-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g*arccosh(c*x)*x^2*c^2+7/3*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g^3*arc
cosh(c*x)*c^2*f^2-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f^5*arccosh(c*x)^2*d^2*c^5/g^6+5/4*
b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f^3*arccosh(c*x)^2*d^2*c^3/g^4+b*d^2*(-d*(c^2*x^2-1))^(1/
2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*arccosh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c
*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(
1/2)/(c*x+1)^(1/2)/g^2*arccosh(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2
*f^2-g^2)^(1/2)))+a/g^7*d^3/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*
(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))*c^6*f^6-3*
a/g^5*d^3/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)
^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))*c^4*f^4+3*a/g^3*d^3/(-d*(c^2
*f^2-g^2)/g^2)^(1/2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2
*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))*c^2*f^2+15/8*a/g^2*c^2*d^3*f/(c^2*d)^(1/2)*arc
tan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))-1/2*a/g^4*d^2*c^4*f^3*(-
(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*x-5/2*a/g^4*d^3*c^4*f^3/(c^2*d)^(1/2)*arctan((c
^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+a/g^6*d^3*c^6*f^5/(c^2*d)^(1/2
)*arctan((c^2*d)^(1/2)*x/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+1/4*a/g^2*c^2*d*f*(
-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)*x+7/8*a/g^2*c^2*d^2*f*(-(x+f/g)^2*c^2*d+2*c^2*
d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*x-23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c*x-1)/(c*x+1)/g*arccosh(c*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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),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 (a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} +{\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{g x + f}, x\right ) \end{align*}

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))/(g*x+f),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sq
rt(-c^2*d*x^2 + d)/(g*x + f), 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((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/(g*x+f),x)

[Out]

Timed out

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

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))/(g*x+f),x, algorithm="giac")

[Out]

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