\(\int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx\) [35]

Optimal result

Integrand size = 31, antiderivative size = 860 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \]

[Out]

-a*(-c^2*d*x^2+d)^(1/2)/g/(g*x+f)-b*arcsin(c*x)*(-c^2*d*x^2+d)^(1/2)/g/(g*x+f)-a*c^3*f^2*arcsin(c*x)*(-c^2*d*x
^2+d)^(1/2)/g^2/(c^2*f^2-g^2)/(-c^2*x^2+1)^(1/2)-1/2*b*c^3*f^2*arcsin(c*x)^2*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2
-g^2)/(-c^2*x^2+1)^(1/2)+1/2*(c^2*f*x+g)^2*(a+b*arcsin(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/(c^2*f^2-g^2)/(g*x+f)^
2/(-c^2*x^2+1)^(1/2)+b*c*ln(g*x+f)*(-c^2*d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)+a*c^2*f*arctan((c^2*f*x+g)/(c^2
*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b*c^2*f*
arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g
^2)^(1/2)/(-c^2*x^2+1)^(1/2)+I*b*c^2*f*arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2
)))*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)-b*c^2*f*polylog(2,I*(I*c*x+(-c^2*x^2+1)^(1
/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)+b*c^2*f*poly
log(2,I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)/(
-c^2*x^2+1)^(1/2)+1/2*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/b/c/(g*x+f)^2

Rubi [A] (verified)

Time = 1.81 (sec) , antiderivative size = 860, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.710, Rules used = {4861, 4849, 37, 4839, 12, 1665, 858, 222, 739, 210, 4883, 4881, 4737, 4857, 3405, 3404, 2296, 2221, 2317, 2438, 2747, 31} \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=-\frac {b f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)^2 c^3}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {a f^2 \sqrt {d-c^2 d x^2} \arcsin (c x) c^3}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a f \sqrt {d-c^2 d x^2} \arctan \left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b f \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b f \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right ) c^2}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b \sqrt {d-c^2 d x^2} \log (f+g x) c}{g^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {d-c^2 d x^2} \arcsin (c x)}{g (f+g x)}-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b (f+g x)^2 c}+\frac {\left (f x c^2+g\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2} c} \]

[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 12

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

Rule 31

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

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 210

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

Rule 222

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 739

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 858

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 1665

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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2296

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 2317

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 2438

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 2747

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 3404

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 3405

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 4737

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

Rule 4839

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 4849

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 4857

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 4861

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Dist[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; F
reeQ[{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 4881

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 4883

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(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} (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 (a+b \arcsin (c x))}{\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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 (a+b \arcsin (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 {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 \arcsin (c x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (-c^2 f^2+g^2\right )^2 \arcsin (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 ) \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^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 {\arcsin (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 {\arcsin (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 {\arcsin (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} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}-\frac {\left (a c^2 f \sqrt {d-c^2 d x^2}\right ) \text {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 ) \text {Subst}\left (\int \frac {x}{(c f+g \sin (x))^2} \, dx,x,\arcsin (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 ) \text {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\arcsin (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} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \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 ) \text {Subst}\left (\int \frac {x}{c f+g \sin (x)} \, dx,x,\arcsin (c x)\right )}{g^2 \sqrt {1-c^2 x^2}}+\frac {\left (b c \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{c f+g \sin (x)} \, dx,x,\arcsin (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 ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (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} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \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 ) \text {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 ) \text {Subst}\left (\int \frac {e^{i x} x}{2 c e^{i x} f+i g-i e^{2 i x} g} \, dx,x,\arcsin (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 ) \text {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,\arcsin (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 ) \text {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,\arcsin (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} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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 ) \text {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,\arcsin (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 ) \text {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,\arcsin (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 ) \text {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,\arcsin (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 ) \text {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,\arcsin (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} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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 ) \text {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,\arcsin (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 ) \text {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,\arcsin (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 ) \text {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 \arcsin (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 ) \text {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 \arcsin (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} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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 ) \text {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 \arcsin (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 ) \text {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 \arcsin (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} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (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} \arcsin (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} (a+b \arcsin (c x))^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} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (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] (verified)

Time = 2.37 (sec) , antiderivative size = 600, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2}{g^2 (f+g x)^2}-\frac {2 c^2 f (a+b \arcsin (c x))^2}{g^2 (f+g x)}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(f+g x)^2}+\frac {4 b c^3 f \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g^2 \sqrt {c^2 f^2-g^2}}+\frac {2 b c^2 \left (-\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}+b \log (f+g x)+\frac {c f \left (i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2}}\right )}{g^2}\right )}{2 b c \sqrt {1-c^2 x^2}} \]

[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])

Maple [A] (warning: unable to verify)

Time = 0.66 (sec) , antiderivative size = 1352, normalized size of antiderivative = 1.57

method	result	size
default	\(\text {Expression too large to display}\)	\(1352\)
parts	\(\text {Expression too large to display}\)	\(1352\)

[In]

int((a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/(g*x+f)^2,x,method=_RETURNVERBOSE)

[Out]

a/g^2*(1/d/(c^2*f^2-g^2)*g^2/(x+f/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)^(3/2)-c^2*f*g/
(c^2*f^2-g^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*d*f/g/(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))+d*(c^2*f^2-g^2)/g^2/(-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)))+2*c^2/(c^2*f^2-g^2)*g^2*(-1/4*(-2*(x+f/g)*c^
2*d+2*c^2*d*f/g)/c^2/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)^(1/2)-1/8*(4*c^2*d^2*(c^2*f^
2-g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d/(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))))+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)*(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)))*arcsin(c*x)*(-c^2*f^2+g^2)^(1/2)*c*f-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)))*arcsin(c*x)*(-c^2*f^2+g^2)^(1/2)*c*f-I*dilo
g((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+I*dilog((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+2*ln(I*c*x+(-c^2*x^2+1)^(1/2))*c^2*f^2-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*ln(I*c*x+(-c^2*x^2+1)^(1/2))*g^2+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)*c/(c^2*x^2-1)/g^2/(c^2*f^2-g^2))

Fricas [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \]

[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)

Sympy [F]

\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]

[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)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\text {Exception raised: ValueError} \]

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for m
ore details)

Giac [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{{\left (f+g\,x\right )}^2} \,d x \]

[In]

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

[Out]

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