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

Optimal. Leaf size=860 \[ -\frac{b f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{a f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b (f+g x)^2 c}+\frac{\left (f x c^2+g\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2} c} \]

[Out]

-((a*Sqrt[d - c^2*d*x^2])/(g*(f + g*x))) - (b*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(g*(f + g*x)) - (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]) - (b*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[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*ArcSin[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*ArcSin[
c*x])^2)/(2*b*c*(f + g*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]*ArcSin[c*x]*Log[1 -
(I*E^(I*ArcSin[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
*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[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, (I*E^(I*ArcSin[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, (I*E^(I*ArcSin[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 = 2.71488, antiderivative size = 860, 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 = {4777, 4765, 37, 4755, 12, 1651, 844, 216, 725, 204, 4799, 4797, 4641, 4773, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{b f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}-\frac{a f^2 \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt{1-c^2 x^2}}+\frac{a f \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{f x c^2+g}{\sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{i b f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{i b f \sqrt{d-c^2 d x^2} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}-\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b f \sqrt{d-c^2 d x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt{c^2 f^2-g^2} \sqrt{1-c^2 x^2}}+\frac{b \sqrt{d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt{1-c^2 x^2}}-\frac{b \sqrt{d-c^2 d x^2} \sin ^{-1}(c x)}{g (f+g x)}-\frac{a \sqrt{d-c^2 d x^2}}{g (f+g x)}+\frac{\sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b (f+g x)^2 c}+\frac{\left (f x c^2+g\right )^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt{1-c^2 x^2} c} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[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]*ArcSin[c*x])/(g*(f + g*x)) - (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]) - (b*c^3*f^2*Sqrt[d - c^2*d*x^2]*ArcSin[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*ArcSin[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*ArcSin[
c*x])^2)/(2*b*c*(f + g*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]*ArcSin[c*x]*Log[1 -
(I*E^(I*ArcSin[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
*Sqrt[d - c^2*d*x^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[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, (I*E^(I*ArcSin[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, (I*E^(I*ArcSin[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 4777

Int[((a_.) + ArcSin[(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*ArcSin[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 4765

Int[((a_.) + ArcSin[(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*ArcSin[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*ArcSin[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 4755

Int[((a_.) + ArcSin[(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*ArcSin[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIn
tegrand[(u*(a + b*ArcSin[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 4799

Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^2)^p, RFx*(a + b*ArcSin[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 4797

Int[ArcSin[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcSin[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 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[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 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[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 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (-c^2 f^2+g^2\right )^2 \sin ^{-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 ) \sin ^{-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 \sin ^{-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 \sin ^{-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{\sin ^{-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{\sin ^{-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{\sin ^{-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{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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-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 ) \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 \sin (x))^2} \, dx,x,\sin ^{-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 \sin (x)} \, dx,x,\sin ^{-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} \sin ^{-1}(c x)}{g (f+g x)}-\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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )^2}{2 b c (f+g 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 \sin (x)} \, dx,x,\sin ^{-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{\cos (x)}{c f+g \sin (x)} \, dx,x,\sin ^{-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+i g-i e^{2 i x} g} \, dx,x,\sin ^{-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} \sin ^{-1}(c x)}{g (f+g x)}-\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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )^2}{2 b c (f+g 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+i g-i e^{2 i x} g} \, dx,x,\sin ^{-1}(c x)\right )}{g^2 \sqrt{1-c^2 x^2}}+\frac{\left (4 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 \frac{e^{i x} x}{2 c f-2 i e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-1}(c x)\right )}{g \left (c^2 f^2-g^2\right )^{3/2} \sqrt{1-c^2 x^2}}-\frac{\left (4 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 \frac{e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-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} \sin ^{-1}(c x)}{g (f+g x)}-\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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )^2}{2 b c (f+g 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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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 i 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 i e^{i x} g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-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 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{2 c f-2 i e^{i x} g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\sin ^{-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 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-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 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-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} \sin ^{-1}(c x)}{g (f+g x)}-\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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )^2}{2 b c (f+g 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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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 i e^{i x} g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-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 i e^{i x} g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\sin ^{-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 i g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-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 i g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-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} \sin ^{-1}(c x)}{g (f+g x)}-\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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )^2}{2 b c (f+g 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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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 i g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-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 i g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{i \sin ^{-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} \sin ^{-1}(c x)}{g (f+g x)}-\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{b c^3 f^2 \sqrt{d-c^2 d x^2} \sin ^{-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 \sin ^{-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 \sin ^{-1}(c x)\right )^2}{2 b c (f+g 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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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} \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-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{i e^{i \sin ^{-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{i e^{i \sin ^{-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 = 2.6496, size = 600, normalized size = 0.7 \[ \frac{\sqrt{d-c^2 d x^2} \left (\frac{2 b c^2 \left (\frac{c f \left (b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )-b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )+i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )\right )}{\sqrt{c^2 f^2-g^2}}-\frac{g \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c f+c g x}+b \log (f+g x)\right )}{g^2}+\frac{4 b c^3 f \left (-b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{c f-\sqrt{c^2 f^2-g^2}}\right )+b \text{PolyLog}\left (2,\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}-c f}\right )-\log \left (1-\frac{i g e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 f^2-g^2}+c f}\right )\right )\right )}{g^2 \sqrt{c^2 f^2-g^2}}+\frac{\left (c^2 f^2-g^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{g^2 (f+g x)^2}-\frac{2 c^2 f \left (a+b \sin ^{-1}(c x)\right )^2}{g^2 (f+g x)}+\frac{\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{(f+g x)^2}\right )}{2 b c \sqrt{1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2)/(g^2*(f + g*x)^2) - (2*c^2*f*(a + b*ArcSin[c*x])
^2)/(g^2*(f + g*x)) + ((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(f + g*x)^2 + (4*b*c^3*f*((-I)*(a + b*ArcSin[c*x])
*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqr
t[c^2*f^2 - g^2])]) - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + b*PolyLog[2, (I*E^(I
*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/(g^2*Sqrt[c^2*f^2 - g^2]) + (2*b*c^2*(-((g*Sqrt[1 - c^2*x^2]*(
a + b*ArcSin[c*x]))/(c*f + c*g*x)) + b*Log[f + g*x] + (c*f*(I*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x]
)*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) + b*PolyL
og[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[
c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2]))/g^2))/(2*b*c*Sqrt[1 - c^2*x^2])

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Maple [C]  time = 0.49, size = 1572, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(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)*arcsin(c*x)^2*c/g^2-(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*arcsin(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)+(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(arcsin(c
*x)*ln((-I*c*f-(I*c*x+(-c^2*x^2+1)^(1/2))*g+(-c^2*f^2+g^2)^(1/2))/(-I*c*f+(-c^2*f^2+g^2)^(1/2)))*(-c^2*f^2+g^2
)^(1/2)*c*f-arcsin(c*x)*ln((I*c*f+(I*c*x+(-c^2*x^2+1)^(1/2))*g+(-c^2*f^2+g^2)^(1/2))/(I*c*f+(-c^2*f^2+g^2)^(1/
2)))*(-c^2*f^2+g^2)^(1/2)*c*f-ln((I*c*x+(-c^2*x^2+1)^(1/2))^2*g+2*I*c*f*(I*c*x+(-c^2*x^2+1)^(1/2))-g)*c^2*f^2-
2*Im(arcsin(c*x))*c^2*f^2+2*ln(exp(I*Re(arcsin(c*x))))*c^2*f^2-I*dilog(-I/(-I*c*f+(-c^2*f^2+g^2)^(1/2))*c*f-1/
(-I*c*f+(-c^2*f^2+g^2)^(1/2))*(I*c*x+(-c^2*x^2+1)^(1/2))*g+1/(-I*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+I*dilog(I/(I*c*f+(-c^2*f^2+g^2)^(1/2))*c*f+1/(I*c*f+(-c^2*f^2+g^2)^(1/2))*(I*c*x+(
-c^2*x^2+1)^(1/2))*g+1/(I*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+ln((I*c*x+(
-c^2*x^2+1)^(1/2))^2*g+2*I*c*f*(I*c*x+(-c^2*x^2+1)^(1/2))-g)*g^2+2*Im(arcsin(c*x))*g^2-2*ln(exp(I*Re(arcsin(c*
x))))*g^2)*c/(c^2*x^2-1)/g^2/(c^2*f^2-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*arcsin(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 \arcsin \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*arcsin(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*arcsin(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{asin}{\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*asin(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*asin(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 \arcsin \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*arcsin(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*arcsin(c*x) + a)/(g*x + f)^2, x)