3.79 \(\int \frac{(f+g x)^3 (a+b \sin ^{-1}(c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\)

Optimal. Leaf size=1589 \[ \text{result too large to display} \]

[Out]

((-I/12)*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/4)*(c*f - 2*
g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/12)*(c*f + g)^3*Sq
rt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((I/4)*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 -
c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Cot[Pi/4 +
ArcSin[c*x]/2])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi
/4 + ArcSin[c*x]/2])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcS
in[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*
(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*c^4*d^2*Sqrt[d - c^2*d*x^2])
+ (b*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt
[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c^4*d
^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(
3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^
(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*PolyLog[2,
I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/3)*b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I/E^
(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((I/3)*b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*A
rcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^
(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/
4 + ArcSin[c*x]/2]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + ArcSin[
c*x]/2])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Ta
n[Pi/4 + ArcSin[c*x]/2])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^
2*Tan[Pi/4 + ArcSin[c*x]/2])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c
*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(24*c^4*d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.11152, antiderivative size = 1589, normalized size of antiderivative = 1., number of steps used = 37, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4777, 4775, 4773, 3318, 4186, 3767, 8, 4184, 3717, 2190, 2279, 2391} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

((-I/12)*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/4)*(c*f - 2*
g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/12)*(c*f + g)^3*Sq
rt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((I/4)*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 -
c^2*x^2]*(a + b*ArcSin[c*x])^2)/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*Cot[Pi/4 +
ArcSin[c*x]/2])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Cot[Pi
/4 + ArcSin[c*x]/2])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcS
in[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((c*f - g)^3*Sqrt[1 - c^2*x^2]*
(a + b*ArcSin[c*x])^2*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*c^4*d^2*Sqrt[d - c^2*d*x^2])
+ (b*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt
[d - c^2*d*x^2]) + (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I/E^(I*ArcSin[c*x])])/(3*c^4*d
^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^(I*ArcSin[c*x])])/(
3*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - I*E^
(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + (I*b^2*(c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*PolyLog[2,
I/E^(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((I/3)*b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I/E^
(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - ((I/3)*b^2*(c*f - g)^3*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^(I*A
rcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (I*b^2*(c*f - g)^2*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*PolyLog[2, I*E^
(I*ArcSin[c*x])])/(c^4*d^2*Sqrt[d - c^2*d*x^2]) - (b*(c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Sec[Pi/
4 + ArcSin[c*x]/2]^2)/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + (b^2*(c*f + g)^3*Sqrt[1 - c^2*x^2]*Tan[Pi/4 + ArcSin[
c*x]/2])/(6*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f - 2*g)*(c*f + g)^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2*Ta
n[Pi/4 + ArcSin[c*x]/2])/(4*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^
2*Tan[Pi/4 + ArcSin[c*x]/2])/(12*c^4*d^2*Sqrt[d - c^2*d*x^2]) + ((c*f + g)^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c
*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(24*c^4*d^2*Sqrt[d - c^2*d*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 4775

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n/Sqrt[d + e*x^2], (f + g*x)^m*(d + e*x^2)^(p + 1/2), x], x] /; Free
Q[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d, 0] && IGtQ[n, 0]

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 3318

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

Rule 4186

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(b^2*(c + d*x)^m*Cot[e
+ f*x]*(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*d^2*m*(m - 1))/(f^2*(n - 1)*(n - 2)), Int[(c + d
*x)^(m - 2)*(b*Csc[e + f*x])^(n - 2), x], x] + Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)^m*(b*Csc[e + f*x])^(n
- 2), x], x] - Simp[(b^2*d*m*(c + d*x)^(m - 1)*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[
{b, c, d, e, f}, x] && GtQ[n, 1] && NeQ[n, 2] && GtQ[m, 1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(f+g x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{(c f+g)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (-1+c x)^2 \sqrt{1-c^2 x^2}}-\frac{(c f-2 g) (c f+g)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (-1+c x) \sqrt{1-c^2 x^2}}+\frac{(c f-g)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (1+c x)^2 \sqrt{1-c^2 x^2}}+\frac{(c f-g)^2 (c f+2 g) \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^3 (1+c x) \sqrt{1-c^2 x^2}}\right ) \, dx}{d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left ((c f-g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x)^2 \sqrt{1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left ((c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x) \sqrt{1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f+g)^3 \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(-1+c x)^2 \sqrt{1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \int \frac{\left (a+b \sin ^{-1}(c x)\right )^2}{(1+c x) \sqrt{1-c^2 x^2}} \, dx}{4 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left ((c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(c+c \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left ((c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{-c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(-c+c \sin (x))^2} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^2 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^2}{c+c \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{4 c^3 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{\left ((c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^4\left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{16 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{8 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{(c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \csc ^2\left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left ((c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x)^2 \csc ^2\left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \csc ^2\left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{2 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=\frac{i (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}+\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-i x} (a+b x)}{1-i e^{-i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \cot \left (\frac{\pi }{4}-\frac{x}{2}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b^2 (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int 1 \, dx,x,\cot \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{i (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 (c f-g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (b (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{i x} (a+b x)}{1-i e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{-i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{-i x} (a+b x)}{1-i e^{-i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{i (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 (c f-g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (i b^2 (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (b^2 (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-i e^{-i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (i b^2 (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{i (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 (c f-g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{\left (i b^2 (c f-g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{\left (i b^2 (c f+g)^3 \sqrt{1-c^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ &=-\frac{i (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b^2 (c f-g)^3 \sqrt{1-c^2 x^2} \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{6 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{(c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \cot \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \csc ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 (c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{i b^2 (c f+g)^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{-i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 (c f-g)^3 \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{i b^2 (c f-g)^2 (c f+2 g) \sqrt{1-c^2 x^2} \text{Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{c^4 d^2 \sqrt{d-c^2 d x^2}}-\frac{b (c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f-2 g) (c f+g)^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{4 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{12 c^4 d^2 \sqrt{d-c^2 d x^2}}+\frac{(c f+g)^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \sec ^2\left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right ) \tan \left (\frac{\pi }{4}+\frac{1}{2} \sin ^{-1}(c x)\right )}{24 c^4 d^2 \sqrt{d-c^2 d x^2}}\\ \end{align*}

Mathematica [A]  time = 6.25324, size = 715, normalized size = 0.45 \[ \frac{\sqrt{1-c^2 x^2} \left (-\frac{(c f-g)^3 \left (-2 \left (-\tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )+2 b \sec ^2\left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )+\tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+4 b^2 \tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right )\right )}{24 c^4}-\frac{(c f+g)^3 \left (2 \left (-\tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )+i \log \left (1+e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )\right )\right )\right )+2 b \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )-\tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2-4 b^2 \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )\right )}{24 c^4}+\frac{(c f+2 g) (c f-g)^2 \left (-\tan \left (\frac{\pi }{4}-\frac{1}{2} \sin ^{-1}(c x)\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}-4 \left (i \log \left (1+e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right ) \left (a+b \sin ^{-1}(c x)\right )-b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (\pi -2 \sin ^{-1}(c x)\right )}\right )\right )\right )\right )}{4 c^4}-\frac{(c f-2 g) (c f+g)^2 \left (-\tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \left (a+b \sin ^{-1}(c x)\right )^2+i b \left (\frac{\left (a+b \sin ^{-1}(c x)\right )^2}{b}+4 \left (b \text{PolyLog}\left (2,-e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right )+i \log \left (1+e^{\frac{1}{2} i \left (2 \sin ^{-1}(c x)+\pi \right )}\right ) \left (a+b \sin ^{-1}(c x)\right )\right )\right )\right )}{4 c^4}\right )}{d^2 \sqrt{d-c^2 d x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x)^3*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]

[Out]

(Sqrt[1 - c^2*x^2]*(((c*f - g)^2*(c*f + 2*g)*(I*b*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 +
E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[
Pi/4 - ArcSin[c*x]/2]))/(4*c^4) - ((c*f - g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 - ArcSin[c*x]/2]^2 + 4*b^2*Ta
n[Pi/4 - ArcSin[c*x]/2] + (a + b*ArcSin[c*x])^2*Sec[Pi/4 - ArcSin[c*x]/2]^2*Tan[Pi/4 - ArcSin[c*x]/2] - 2*(I*b
*((a + b*ArcSin[c*x])^2/b - 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi - 2*ArcSin[c*x]))] - b*PolyLog[2, -E
^((I/2)*(Pi - 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 - ArcSin[c*x]/2])))/(24*c^4) - ((c*f - 2*g)*
(c*f + g)^2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] +
b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))])) - (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2]))/(4*c^4) -
((c*f + g)^3*(2*b*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2 - 4*b^2*Tan[Pi/4 + ArcSin[c*x]/2] - (a + b*
ArcSin[c*x])^2*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2] + 2*(I*b*((a + b*ArcSin[c*x])^2/b + 4*(I*
(a + b*ArcSin[c*x])*Log[1 + E^((I/2)*(Pi + 2*ArcSin[c*x]))] + b*PolyLog[2, -E^((I/2)*(Pi + 2*ArcSin[c*x]))]))
- (a + b*ArcSin[c*x])^2*Tan[Pi/4 + ArcSin[c*x]/2])))/(24*c^4)))/(d^2*Sqrt[d - c^2*d*x^2])

________________________________________________________________________________________

Maple [B]  time = 0.802, size = 13136, normalized size = 8.3 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a b c f^{3}{\left (\frac{1}{c^{4} d^{\frac{5}{2}} x^{2} - c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x + 1\right )}{c^{2} d^{\frac{5}{2}}} + \frac{2 \, \log \left (c x - 1\right )}{c^{2} d^{\frac{5}{2}}}\right )} + \frac{2}{3} \, a b f^{3}{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} \arcsin \left (c x\right ) + \frac{1}{3} \, a^{2} f^{3}{\left (\frac{2 \, x}{\sqrt{-c^{2} d x^{2} + d} d^{2}} + \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} d}\right )} + \frac{1}{3} \, a^{2} g^{3}{\left (\frac{3 \, x^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} - \frac{2}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{4} d}\right )} - a^{2} f g^{2}{\left (\frac{x}{\sqrt{-c^{2} d x^{2} + d} c^{2} d^{2}} - \frac{x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d}\right )} + \sqrt{d} \int \frac{{\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + 2 \,{\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{{\left (c^{4} d^{3} x^{4} - 2 \, c^{2} d^{3} x^{2} + d^{3}\right )} \sqrt{c x + 1} \sqrt{-c x + 1}}\,{d x} + \frac{a^{2} f^{2} g}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*a*b*c*f^3*(1/(c^4*d^(5/2)*x^2 - c^2*d^(5/2)) + 2*log(c*x + 1)/(c^2*d^(5/2)) + 2*log(c*x - 1)/(c^2*d^(5/2))
) + 2/3*a*b*f^3*(2*x/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d))*arcsin(c*x) + 1/3*a^2*f^3*(2*x
/(sqrt(-c^2*d*x^2 + d)*d^2) + x/((-c^2*d*x^2 + d)^(3/2)*d)) + 1/3*a^2*g^3*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d
) - 2/((-c^2*d*x^2 + d)^(3/2)*c^4*d)) - a^2*f*g^2*(x/(sqrt(-c^2*d*x^2 + d)*c^2*d^2) - x/((-c^2*d*x^2 + d)^(3/2
)*c^2*d)) + sqrt(d)*integrate(((b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^2*g*x + b^2*f^3)*arctan2(c*x, sqrt(c*x
+ 1)*sqrt(-c*x + 1))^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c
*x + 1)))/((c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x) + a^2*f^2*g/((-c^2*d*x^2 + d)
^(3/2)*c^2*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a^{2} g^{3} x^{3} + 3 \, a^{2} f g^{2} x^{2} + 3 \, a^{2} f^{2} g x + a^{2} f^{3} +{\left (b^{2} g^{3} x^{3} + 3 \, b^{2} f g^{2} x^{2} + 3 \, b^{2} f^{2} g x + b^{2} f^{3}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b g^{3} x^{3} + 3 \, a b f g^{2} x^{2} + 3 \, a b f^{2} g x + a b f^{3}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="fricas")

[Out]

integral(-(a^2*g^3*x^3 + 3*a^2*f*g^2*x^2 + 3*a^2*f^2*g*x + a^2*f^3 + (b^2*g^3*x^3 + 3*b^2*f*g^2*x^2 + 3*b^2*f^
2*g*x + b^2*f^3)*arcsin(c*x)^2 + 2*(a*b*g^3*x^3 + 3*a*b*f*g^2*x^2 + 3*a*b*f^2*g*x + a*b*f^3)*arcsin(c*x))*sqrt
(-c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)**3*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (g x + f\right )}^{3}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)^3*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((g*x + f)^3*(b*arcsin(c*x) + a)^2/(-c^2*d*x^2 + d)^(5/2), x)