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

Optimal. Leaf size=1067 \[ \text{result too large to display} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.94796, antiderivative size = 1067, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 23, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.821, Rules used = {4759, 698, 4753, 12, 6742, 780, 216, 2404, 4741, 4519, 2190, 2279, 2391, 4619, 4677, 8, 4627, 4707, 4641, 30, 2531, 2282, 6589} \[ -\frac{i b^2 \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i a b \left (f g^2-e h g+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}+\frac{b^2 \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{2 a b \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}+\frac{2 a b \left (f g^2-e h g+d h^2\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}-\frac{2 i b^2 \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}-\frac{2 i b^2 \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \sin ^{-1}(c x)}{h^3}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 \left (f g^2-e h g+d h^2\right ) \log (g+h x)}{h^3}-\frac{2 i a b \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i a b \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e h g+d h^2\right ) \text{PolyLog}\left (3,\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x + f*x^2)*(a + b*ArcSin[c*x])^2)/(g + h*x),x]

[Out]

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

Rule 4759

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(Px_)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
Px*(d + e*x)^m*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[Px, x] && IGtQ[n, 0]
&& IntegerQ[m]

Rule 698

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

Rule 4753

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rule 12

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

Rule 6742

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

Rule 780

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*(a + c*x^2)^(p + 1))/(2*c*(p + 1)*(2*p + 3)), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

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 2404

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4741

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Subst[Int[((a + b*x)^n*Cos[x])/
(c*d + e*Sin[x]), x], x, ArcSin[c*x]] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[n, 0]

Rule 4519

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

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 4619

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

Rule 4677

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

Rule 8

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

Rule 4627

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

Rule 4707

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

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 30

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

Rule 2531

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

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (d+e x+f x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{g+h x} \, dx &=\int \left (\frac{a^2 \left (d+e x+f x^2\right )}{g+h x}+\frac{2 a b \left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{g+h x}+\frac{b^2 \left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{g+h x}\right ) \, dx\\ &=a^2 \int \frac{d+e x+f x^2}{g+h x} \, dx+(2 a b) \int \frac{\left (d+e x+f x^2\right ) \sin ^{-1}(c x)}{g+h x} \, dx+b^2 \int \frac{\left (d+e x+f x^2\right ) \sin ^{-1}(c x)^2}{g+h x} \, dx\\ &=-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}+a^2 \int \left (\frac{-f g+e h}{h^2}+\frac{f x}{h}+\frac{f g^2-e g h+d h^2}{h^2 (g+h x)}\right ) \, dx+b^2 \int \left (\frac{(-f g+e h) \sin ^{-1}(c x)^2}{h^2}+\frac{f x \sin ^{-1}(c x)^2}{h}+\frac{\left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^2 (g+h x)}\right ) \, dx-(2 a b c) \int \frac{h x (-2 f g+2 e h+f h x)+2 \left (f g^2+h (-e g+d h)\right ) \log (g+h x)}{2 h^3 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac{(a b c) \int \frac{h x (-2 f g+2 e h+f h x)+2 \left (f g^2+h (-e g+d h)\right ) \log (g+h x)}{\sqrt{1-c^2 x^2}} \, dx}{h^3}+\frac{\left (b^2 f\right ) \int x \sin ^{-1}(c x)^2 \, dx}{h}-\frac{\left (b^2 (f g-e h)\right ) \int \sin ^{-1}(c x)^2 \, dx}{h^2}+\frac{\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \int \frac{\sin ^{-1}(c x)^2}{g+h x} \, dx}{h^2}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac{(a b c) \int \left (\frac{h x (-2 f g+2 e h+f h x)}{\sqrt{1-c^2 x^2}}+\frac{2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{\sqrt{1-c^2 x^2}}\right ) \, dx}{h^3}-\frac{\left (b^2 c f\right ) \int \frac{x^2 \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{h}+\frac{\left (2 b^2 c (f g-e h)\right ) \int \frac{x \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{h^2}+\frac{\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2 \cos (x)}{c g+h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log (g+h x)}{h^3}-\frac{(a b c) \int \frac{x (-2 f g+2 e h+f h x)}{\sqrt{1-c^2 x^2}} \, dx}{h^2}-\frac{\left (b^2 f\right ) \int x \, dx}{2 h}-\frac{\left (b^2 f\right ) \int \frac{\sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 c h}+\frac{\left (2 b^2 (f g-e h)\right ) \int 1 \, dx}{h^2}-\frac{\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac{\log (g+h x)}{\sqrt{1-c^2 x^2}} \, dx}{h^3}+\frac{\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{c g-i e^{i x} h-\sqrt{c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac{\left (b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x^2}{c g-i e^{i x} h+\sqrt{c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac{(a b f) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 c h}-\frac{\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int x \log \left (1-\frac{i e^{i x} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac{\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int x \log \left (1-\frac{i e^{i x} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac{\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \int \frac{\sin ^{-1}(c x)}{c g+c h x} \, dx}{h^2}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{\left (2 i b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{i e^{i x} h}{c g-\sqrt{c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac{\left (2 i b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (\frac{i e^{i x} h}{c g+\sqrt{c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}+\frac{\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{x \cos (x)}{c^2 g+c h \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac{i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i h x}{c g-\sqrt{c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}+\frac{\left (2 b^2 \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{i h x}{c g+\sqrt{c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}+\frac{\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 g-i c e^{i x} h-c \sqrt{c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}+\frac{\left (2 a b c \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{e^{i x} x}{c^2 g-i c e^{i x} h+c \sqrt{c^2 g^2-h^2}} \, dx,x,\sin ^{-1}(c x)\right )}{h^2}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac{i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e g h+d h^2\right ) \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e g h+d h^2\right ) \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{\left (2 a b \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{i c e^{i x} h}{c^2 g-c \sqrt{c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}-\frac{\left (2 a b \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \log \left (1-\frac{i c e^{i x} h}{c^2 g+c \sqrt{c^2 g^2-h^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{h^3}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac{i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e g h+d h^2\right ) \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e g h+d h^2\right ) \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{\left (2 i a b \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c h x}{c^2 g-c \sqrt{c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}+\frac{\left (2 i a b \left (f g^2-e g h+d h^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i c h x}{c^2 g+c \sqrt{c^2 g^2-h^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{h^3}\\ &=-\frac{a^2 (f g-e h) x}{h^2}+\frac{2 b^2 (f g-e h) x}{h^2}+\frac{a^2 f x^2}{2 h}-\frac{b^2 f x^2}{4 h}-\frac{a b (4 (f g-e h)-f h x) \sqrt{1-c^2 x^2}}{2 c h^2}-\frac{a b f \sin ^{-1}(c x)}{2 c^2 h}-\frac{2 a b (f g-e h) x \sin ^{-1}(c x)}{h^2}+\frac{a b f x^2 \sin ^{-1}(c x)}{h}-\frac{2 b^2 (f g-e h) \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c h^2}+\frac{b^2 f x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{2 c h}-\frac{b^2 f \sin ^{-1}(c x)^2}{4 c^2 h}-\frac{i a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2}{h^3}-\frac{b^2 (f g-e h) x \sin ^{-1}(c x)^2}{h^2}+\frac{b^2 f x^2 \sin ^{-1}(c x)^2}{2 h}-\frac{i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^3}{3 h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 a b \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x)^2 \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{a^2 \left (f g^2-e g h+d h^2\right ) \log (g+h x)}{h^3}-\frac{2 i a b \left (f g^2-e g h+d h^2\right ) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i a b \left (f g^2-e g h+d h^2\right ) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}-\frac{2 i b^2 \left (f g^2-e g h+d h^2\right ) \sin ^{-1}(c x) \text{Li}_2\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e g h+d h^2\right ) \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g-\sqrt{c^2 g^2-h^2}}\right )}{h^3}+\frac{2 b^2 \left (f g^2-e g h+d h^2\right ) \text{Li}_3\left (\frac{i e^{i \sin ^{-1}(c x)} h}{c g+\sqrt{c^2 g^2-h^2}}\right )}{h^3}\\ \end{align*}

Mathematica [A]  time = 0.677825, size = 556, normalized size = 0.52 \[ \frac{-24 b \left (h (d h-e g)+f g^2\right ) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i h e^{i \sin ^{-1}(c x)}}{c g-\sqrt{c^2 g^2-h^2}}\right )-b \text{PolyLog}\left (3,\frac{i h e^{i \sin ^{-1}(c x)}}{c g-\sqrt{c^2 g^2-h^2}}\right )\right )-24 b \left (h (d h-e g)+f g^2\right ) \left (i \left (a+b \sin ^{-1}(c x)\right ) \text{PolyLog}\left (2,\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )-b \text{PolyLog}\left (3,\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )\right )+12 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h-e g)+f g^2\right ) \log \left (1+\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}-c g}\right )+12 \left (a+b \sin ^{-1}(c x)\right )^2 \left (h (d h-e g)+f g^2\right ) \log \left (1-\frac{i h e^{i \sin ^{-1}(c x)}}{\sqrt{c^2 g^2-h^2}+c g}\right )+24 b h (f g-e h) \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-3 b f h^2 \left (-\frac{2 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b c^2}+b x^2\right )-\frac{4 i \left (a+b \sin ^{-1}(c x)\right )^3 \left (h (d h-e g)+f g^2\right )}{b}+12 h x (e h-f g) \left (a+b \sin ^{-1}(c x)\right )^2+6 f h^2 x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{12 h^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(12*h*(-(f*g) + e*h)*x*(a + b*ArcSin[c*x])^2 + 6*f*h^2*x^2*(a + b*ArcSin[c*x])^2 - ((4*I)*(f*g^2 + h*(-(e*g) +
d*h))*(a + b*ArcSin[c*x])^3)/b + 24*b*h*(f*g - e*h)*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c) - 3*b*f
*h^2*(b*x^2 - (2*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (a + b*ArcSin[c*x])^2/(b*c^2)) + 12*(f*g^2 + h*(
-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 + (I*E^(I*ArcSin[c*x])*h)/(-(c*g) + Sqrt[c^2*g^2 - h^2])] + 12*(f*g
^2 + h*(-(e*g) + d*h))*(a + b*ArcSin[c*x])^2*Log[1 - (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - 24
*b*(f*g^2 + h*(-(e*g) + d*h))*(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 -
h^2])] - b*PolyLog[3, (I*E^(I*ArcSin[c*x])*h)/(c*g - Sqrt[c^2*g^2 - h^2])]) - 24*b*(f*g^2 + h*(-(e*g) + d*h))*
(I*(a + b*ArcSin[c*x])*PolyLog[2, (I*E^(I*ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])] - b*PolyLog[3, (I*E^(I*
ArcSin[c*x])*h)/(c*g + Sqrt[c^2*g^2 - h^2])]))/(12*h^3)

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Maple [F]  time = 2.546, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( f{x}^{2}+ex+d \right ) \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}}{hx+g}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x)

[Out]

int((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} e{\left (\frac{x}{h} - \frac{g \log \left (h x + g\right )}{h^{2}}\right )} + \frac{1}{2} \, a^{2} f{\left (\frac{2 \, g^{2} \log \left (h x + g\right )}{h^{3}} + \frac{h x^{2} - 2 \, g x}{h^{2}}\right )} + \frac{a^{2} d \log \left (h x + g\right )}{h} + \int \frac{{\left (b^{2} f x^{2} + b^{2} e x + b^{2} d\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (a b f x^{2} + a b e x + a b d\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{h x + g}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x, algorithm="maxima")

[Out]

a^2*e*(x/h - g*log(h*x + g)/h^2) + 1/2*a^2*f*(2*g^2*log(h*x + g)/h^3 + (h*x^2 - 2*g*x)/h^2) + a^2*d*log(h*x +
g)/h + integrate(((b^2*f*x^2 + b^2*e*x + b^2*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*f*x^2 +
a*b*e*x + a*b*d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))/(h*x + g), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^2+e*x+d)*(a+b*arcsin(c*x))^2/(h*x+g),x, algorithm="fricas")

[Out]

integral((a^2*f*x^2 + a^2*e*x + a^2*d + (b^2*f*x^2 + b^2*e*x + b^2*d)*arcsin(c*x)^2 + 2*(a*b*f*x^2 + a*b*e*x +
a*b*d)*arcsin(c*x))/(h*x + g), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**2+e*x+d)*(a+b*asin(c*x))**2/(h*x+g),x)

[Out]

Integral((a + b*asin(c*x))**2*(d + e*x + f*x**2)/(g + h*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((f*x^2 + e*x + d)*(b*arcsin(c*x) + a)^2/(h*x + g), x)