3.5 \(\int \frac{\sqrt{d-c^2 d x^2} (a+b \cos ^{-1}(c x))}{(f+g x)^2} \, dx\)

Optimal. Leaf size=851 \[ \frac{b f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{a f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b (f+g x)^2 c}-\frac{\left (f x c^2+g\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2} c} \]

[Out]

-((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/(g*(f + g*x)) + (b*c^3*f^2*Sqrt
[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) - ((g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2
]*(a + b*ArcCos[c*x])^2)/(2*b*c*(c^2*f^2 - g^2)*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (Sqrt[1 - c^2*x^2]*Sqrt[d - c
^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*(f + g*x)^2) - (a*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^2*(c^2*f
^2 - g^2)*Sqrt[1 - c^2*x^2]) + (a*c^2*f*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 -
c^2*x^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (I*b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 +
(E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (I*b*c^2*f*S
qrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 -
g^2]*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 - c^2*x^2]) + (b*c^2*f*Sqrt[d -
c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 -
c^2*x^2]) - (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g
^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.69367, antiderivative size = 851, normalized size of antiderivative = 1., number of steps used = 35, number of rules used = 22, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.71, Rules used = {4778, 4766, 37, 4756, 12, 1651, 844, 216, 725, 204, 4800, 4798, 4642, 4774, 3324, 3321, 2264, 2190, 2279, 2391, 2668, 31} \[ \frac{b f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{a f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}+1\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b (f+g x)^2 c}-\frac{\left (f x c^2+g\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2} c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d - c^2*d*x^2]*ArcCos[c*x])/(g*(f + g*x)) + (b*c^3*f^2*Sqrt
[d - c^2*d*x^2]*ArcCos[c*x]^2)/(2*g^2*(c^2*f^2 - g^2)*Sqrt[1 - c^2*x^2]) - ((g + c^2*f*x)^2*Sqrt[d - c^2*d*x^2
]*(a + b*ArcCos[c*x])^2)/(2*b*c*(c^2*f^2 - g^2)*(f + g*x)^2*Sqrt[1 - c^2*x^2]) - (Sqrt[1 - c^2*x^2]*Sqrt[d - c
^2*d*x^2]*(a + b*ArcCos[c*x])^2)/(2*b*c*(f + g*x)^2) - (a*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g^2*(c^2*f
^2 - g^2)*Sqrt[1 - c^2*x^2]) + (a*c^2*f*Sqrt[d - c^2*d*x^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 -
c^2*x^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) + (I*b*c^2*f*Sqrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 +
(E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2]) - (I*b*c^2*f*S
qrt[d - c^2*d*x^2]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(g^2*Sqrt[c^2*f^2 -
g^2]*Sqrt[1 - c^2*x^2]) - (b*c*Sqrt[d - c^2*d*x^2]*Log[f + g*x])/(g^2*Sqrt[1 - c^2*x^2]) + (b*c^2*f*Sqrt[d -
c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 -
c^2*x^2]) - (b*c^2*f*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(g
^2*Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])

Rule 4778

Int[((a_.) + ArcCos[(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^2*x^2)^FracPart[p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a +
b*ArcCos[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] &&  !GtQ[d, 0]

Rule 4766

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 4756

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(p_.), x_Symbol] :>
With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcCos[c*x])^n, u, x] + Dist[b*c*n, Int[SimplifyIn
tegrand[(u*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && I
GtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]

Rule 12

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

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 216

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

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 204

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

Rule 4800

Int[(ArcCos[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^2)^p, RFx*(a + b*ArcCos[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] &
& IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4798

Int[ArcCos[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcCos[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4642

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> -Simp[(a + b*ArcCos[c*x])
^(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4774

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

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3321

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

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )}{(f+g x)^2} \, dx &=\frac{\sqrt{d-c^2 d x^2} \int \frac{\sqrt{1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )}{(f+g x)^2} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (-2 g-2 c^2 f x\right ) \left (a+b \cos ^{-1}(c x)\right )^2}{(f+g x)^3} \, dx}{2 b c \sqrt{1-c^2 x^2}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (g+c^2 f x\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{\left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{\sqrt{1-c^2 x^2}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{\sqrt{d-c^2 d x^2} \int \frac{\left (g+c^2 f x\right )^2 \left (a+b \cos ^{-1}(c x)\right )}{(f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{\sqrt{d-c^2 d x^2} \int \left (\frac{a \left (g+c^2 f x\right )^2}{(f+g x)^2 \sqrt{1-c^2 x^2}}+\frac{b \left (g+c^2 f x\right )^2 \cos ^{-1}(c x)}{(f+g x)^2 \sqrt{1-c^2 x^2}}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{\left (a \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (g+c^2 f x\right )^2}{(f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (g+c^2 f x\right )^2 \cos ^{-1}(c x)}{(f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{\left (a \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f \left (c^2 f^2-g^2\right )+c^4 f^2 \left (\frac{c^2 f^2}{g}-g\right ) x}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2}}-\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^4 f^2 \cos ^{-1}(c x)}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (-c^2 f^2+g^2\right )^2 \cos ^{-1}(c x)}{g^2 (f+g x)^2 \sqrt{1-c^2 x^2}}+\frac{2 c^2 f \left (-c^2 f^2+g^2\right ) \cos ^{-1}(c x)}{g^2 (f+g x) \sqrt{1-c^2 x^2}}\right ) \, dx}{\left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}+\frac{\left (a c^2 f \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (a c^4 f^2 \left (1-\frac{c^2 f^2}{g^2}\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{\left (c^2 f^2-g^2\right )^2 \sqrt{1-c^2 x^2}}-\frac{\left (b c^4 f^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\cos ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (b \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{\cos ^{-1}(c x)}{(f+g x)^2 \sqrt{1-c^2 x^2}} \, dx}{g^2 \sqrt{1-c^2 x^2}}-\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \int \frac{\cos ^{-1}(c x)}{(f+g x) \sqrt{1-c^2 x^2}} \, dx}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (a c^2 f \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^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c \left (c^2 f^2-g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{(c f+g \cos (x))^2} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a c^2 f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\sin (x)}{c f+g \cos (x)} \, dx,x,\cos ^{-1}(c x)\right )}{g \sqrt{1-c^2 x^2}}+\frac{\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a c^2 f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c f+x} \, dx,x,c g x\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c e^{i x} f+g+e^{2 i x} g} \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f+2 e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}-\frac{\left (4 b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f+2 e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a c^2 f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{2 i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (2 b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f+2 e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (2 b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f+2 e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cos ^{-1}(c x)\right )}{g \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 i b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}-\frac{\left (2 i b c^2 f \left (-c^2 f^2+g^2\right ) \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a c^2 f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (i b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (i b c^2 f \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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^{i \cos ^{-1}(c x)}\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}-\frac{\left (2 b c^2 f \left (-c^2 f^2+g^2\right ) \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^{i \cos ^{-1}(c x)}\right )}{g^2 \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a c^2 f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{1-c^2 x^2}}+\frac{2 b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{2 b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{\left (b c^2 f \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^{i \cos ^{-1}(c x)}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{\left (b c^2 f \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^{i \cos ^{-1}(c x)}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ &=-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}-\frac{b \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)}{g (f+g x)}+\frac{b c^3 f^2 \sqrt{d-c^2 d x^2} \cos ^{-1}(c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{\left (g+c^2 f x\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2}}-\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c (f+g x)^2}-\frac{a c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a c^2 f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b c^2 f \sqrt{d-c^2 d x^2} \cos ^{-1}(c x) \log \left (1+\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt{1-c^2 x^2}}+\frac{b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b c^2 f \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{i \cos ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 9.48203, size = 1130, normalized size = 1.33 \[ \frac{a \sqrt{d} f \log (f+g x) c^2}{g^2 \sqrt{g^2-c^2 f^2}}-\frac{a \sqrt{d} f \log \left (d f x c^2+d g+\sqrt{d} \sqrt{g^2-c^2 f^2} \sqrt{-d \left (c^2 x^2-1\right )}\right ) c^2}{g^2 \sqrt{g^2-c^2 f^2}}+\frac{a \sqrt{d} \tan ^{-1}\left (\frac{c x \sqrt{-d \left (c^2 x^2-1\right )}}{\sqrt{d} \left (c^2 x^2-1\right )}\right ) c}{g^2}-\frac{b \sqrt{d \left (1-c^2 x^2\right )} \left (-\frac{\cos ^{-1}(c x)^2}{\sqrt{1-c^2 x^2}}+\frac{2 g \cos ^{-1}(c x)}{c f+c g x}+\frac{2 \log \left (\frac{g x}{f}+1\right )}{\sqrt{1-c^2 x^2}}+\frac{2 c f \left (2 \cos ^{-1}(c x) \tanh ^{-1}\left (\frac{(c f+g) \cot \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )-2 \cos ^{-1}\left (-\frac{c f}{g}\right ) \tanh ^{-1}\left (\frac{(g-c f) \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+\left (\cos ^{-1}\left (-\frac{c f}{g}\right )-2 i \tanh ^{-1}\left (\frac{(c f+g) \cot \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )+2 i \tanh ^{-1}\left (\frac{(g-c f) \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{e^{-\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c f+c g x}}\right )+\left (\cos ^{-1}\left (-\frac{c f}{g}\right )+2 i \left (\tanh ^{-1}\left (\frac{(c f+g) \cot \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )-\tanh ^{-1}\left (\frac{(g-c f) \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right )\right ) \log \left (\frac{e^{\frac{1}{2} i \cos ^{-1}(c x)} \sqrt{g^2-c^2 f^2}}{\sqrt{2} \sqrt{g} \sqrt{c f+c g x}}\right )-\left (\cos ^{-1}\left (-\frac{c f}{g}\right )-2 i \tanh ^{-1}\left (\frac{(g-c f) \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{(c f+g) \left (-i c f+i g+\sqrt{g^2-c^2 f^2}\right ) \left (\tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )-i\right )}{g \left (c f+g+\sqrt{g^2-c^2 f^2} \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )\right )}\right )-\left (\cos ^{-1}\left (-\frac{c f}{g}\right )+2 i \tanh ^{-1}\left (\frac{(g-c f) \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )}{\sqrt{g^2-c^2 f^2}}\right )\right ) \log \left (\frac{(c f+g) \left (i c f-i g+\sqrt{g^2-c^2 f^2}\right ) \left (\tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )+i\right )}{g \left (c f+g+\sqrt{g^2-c^2 f^2} \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )\right )}\right )+i \left (\text{PolyLog}\left (2,\frac{\left (c f-i \sqrt{g^2-c^2 f^2}\right ) \left (c f+g-\sqrt{g^2-c^2 f^2} \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )\right )}{g \left (c f+g+\sqrt{g^2-c^2 f^2} \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )\right )}\right )-\text{PolyLog}\left (2,\frac{\left (c f+i \sqrt{g^2-c^2 f^2}\right ) \left (c f+g-\sqrt{g^2-c^2 f^2} \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )\right )}{g \left (c f+g+\sqrt{g^2-c^2 f^2} \tan \left (\frac{1}{2} \cos ^{-1}(c x)\right )\right )}\right )\right )\right )}{\sqrt{g^2-c^2 f^2} \sqrt{1-c^2 x^2}}\right ) c}{2 g^2}-\frac{a \sqrt{-d \left (c^2 x^2-1\right )}}{g (f+g x)} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-((a*Sqrt[-(d*(-1 + c^2*x^2))])/(g*(f + g*x))) + (a*c*Sqrt[d]*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*
(-1 + c^2*x^2))])/g^2 + (a*c^2*Sqrt[d]*f*Log[f + g*x])/(g^2*Sqrt[-(c^2*f^2) + g^2]) - (a*c^2*Sqrt[d]*f*Log[d*g
+ c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(g^2*Sqrt[-(c^2*f^2) + g^2]) - (b*c*
Sqrt[d*(1 - c^2*x^2)]*((2*g*ArcCos[c*x])/(c*f + c*g*x) - ArcCos[c*x]^2/Sqrt[1 - c^2*x^2] + (2*Log[1 + (g*x)/f]
)/Sqrt[1 - c^2*x^2] + (2*c*f*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2
*ArcCos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] -
(2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[
c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f +
c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcT
anh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g
^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/
2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2])*(-I + Tan[ArcCos[c*x]/2]
))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) +
g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Ta
n[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-
(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*
Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[Arc
Cos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))])))/(Sqrt[-(c^2*f^2) + g^2]*Sqrt[1 - c
^2*x^2])))/(2*g^2)

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Maple [C]  time = 0.476, size = 1573, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x)

[Out]

a/d/(c^2*f^2-g^2)/(x+f/g)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(3/2)-a/g*c^2*f/(c^2*f^2-
g^2)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)-a/g^2*c^4*f^2/(c^2*f^2-g^2)*d/(c^2*d)^(1
/2)*arctan((c^2*d)^(1/2)*x/(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))-a/g^3*c^4*f^3/(c^
2*f^2-g^2)*d/(-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)*(-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))+a/g*c^2*f/(c^2*f^2-g^2)*d
/(-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)*(-
d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))+a*c^2/(c^2*f^2-g^2)*(-d*c^2*(x+f/g)^2
+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2)*x+a*c^2/(c^2*f^2-g^2)*d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(
-d*c^2*(x+f/g)^2+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))+b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(
1/2)/(c^2*x^2-1)*arccos(c*x)^2*c/g^2-(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*arccos(c*x)*(
c^2*f*x+g+I*(-c^2*x^2+1)^(1/2)*c*f)/(c^2*x^2-1)/g^2/(g*x+f)-c*(I*arccos(c*x)*ln((-(c*x+I*(-c^2*x^2+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-g^2)^(1/2)*c*f-I*arccos(c*x)*ln(((c*x+I*(-c^2*x
^2+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-g^2)^(1/2)*c*f-2*Im(arccos(c*x))*c
^2*f^2+2*ln(exp(I*Re(arccos(c*x))))*c^2*f^2-ln((c*x+I*(-c^2*x^2+1)^(1/2))^2*g+2*c*f*(c*x+I*(-c^2*x^2+1)^(1/2))
+g)*c^2*f^2+dilog(-1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c*x+I*(-c^2*x^2+1)^(1/2))*g-1/(-c*f+(c^2*f^2-g^2)^(1/2))*c*f+
1/(-c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2)*c*f-dilog((c*x+I*(-c^2*x^2+1)^(1/2))*g/(
c*f+(c^2*f^2-g^2)^(1/2))+1/(c*f+(c^2*f^2-g^2)^(1/2))*c*f+1/(c*f+(c^2*f^2-g^2)^(1/2))*(c^2*f^2-g^2)^(1/2))*(c^2
*f^2-g^2)^(1/2)*c*f+2*Im(arccos(c*x))*g^2-2*ln(exp(I*Re(arccos(c*x))))*g^2+ln((c*x+I*(-c^2*x^2+1)^(1/2))^2*g+2
*c*f*(c*x+I*(-c^2*x^2+1)^(1/2))+g)*g^2)*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)/(c^2*x^2-1)/g^
2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(g^2*x^2 + 2*f*g*x + f^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acos(c*x))*(-c**2*d*x**2+d)**(1/2)/(g*x+f)**2,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*acos(c*x))/(f + g*x)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x, algorithm="giac")

[Out]

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