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

Optimal. Leaf size=1300 \[ \text{result too large to display} \]

[Out]

-((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2
*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*
x^2]) - (b*Sqrt[1 - c^2*x^2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c
^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f - g)*Sqrt[d -
c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2
- g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*S
qrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + ArcSin[c*x]/2]])/(6*d^2*(c*f + g
)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + ArcSin[c*x]/2]])/(2*d^2*(c*f + g)^2*S
qrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + ArcSin[c*x]/2]])/(2*d^2*(c*f - g)^2*Sqrt
[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + ArcSin[c*x]/2]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2])
- (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(
c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])
*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*S
qrt[1 - c^2*x^2]*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])*Tan[Pi/4 + ArcSin[c*x]/2])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*
x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + ArcSin[c*x]/2])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(24*d^2*(c*f + g)*Sqrt[d - c^2*d
*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.76735, antiderivative size = 1300, normalized size of antiderivative = 1., number of steps used = 30, number of rules used = 12, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.387, Rules used = {4777, 4775, 4773, 3318, 4185, 4184, 3475, 3323, 2264, 2190, 2279, 2391} \[ -\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}+\frac{i \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right ) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \text{PolyLog}\left (2,\frac{i e^{i \sin ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right ) g^4}{d^2 (c f-g)^2 (c f+g)^2 \sqrt{c^2 f^2-g^2} \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{24 d^2 (c f-g) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \csc ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{24 d^2 (c f-g) \sqrt{d-c^2 d x^2}}-\frac{b \sqrt{1-c^2 x^2} \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{24 d^2 (c f+g) \sqrt{d-c^2 d x^2}}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{12 d^2 (c f-g) \sqrt{d-c^2 d x^2}}-\frac{(c f-2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \cot \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{4 d^2 (c f-g)^2 \sqrt{d-c^2 d x^2}}+\frac{b (c f+2 g) \sqrt{1-c^2 x^2} \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )\right )}{2 d^2 (c f+g)^2 \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (\cos \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )\right )}{6 d^2 (c f+g) \sqrt{d-c^2 d x^2}}+\frac{b \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )\right )}{6 d^2 (c f-g) \sqrt{d-c^2 d x^2}}+\frac{b (c f-2 g) \sqrt{1-c^2 x^2} \log \left (\sin \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )\right )}{2 d^2 (c f-g)^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \sec ^2\left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right ) \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{24 d^2 (c f+g) \sqrt{d-c^2 d x^2}}+\frac{(c f+2 g) \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{4 d^2 (c f+g)^2 \sqrt{d-c^2 d x^2}}+\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan \left (\frac{1}{2} \sin ^{-1}(c x)+\frac{\pi }{4}\right )}{12 d^2 (c f+g) \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*f - 2*g)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2])/(4*d^2*(c*f - g)^2*Sqrt[d - c^2
*d*x^2]) - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2])/(12*d^2*(c*f - g)*Sqrt[d - c^2*d*
x^2]) - (b*Sqrt[1 - c^2*x^2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2]) - (Sqrt[1 - c
^2*x^2]*(a + b*ArcSin[c*x])*Cot[Pi/4 + ArcSin[c*x]/2]*Csc[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f - g)*Sqrt[d -
c^2*d*x^2]) - (I*g^4*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2
- g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (I*g^4*Sqrt[1 - c^2*x^2]*(a
+ b*ArcSin[c*x])*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*S
qrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + ArcSin[c*x]/2]])/(6*d^2*(c*f + g
)*Sqrt[d - c^2*d*x^2]) + (b*(c*f + 2*g)*Sqrt[1 - c^2*x^2]*Log[Cos[Pi/4 + ArcSin[c*x]/2]])/(2*d^2*(c*f + g)^2*S
qrt[d - c^2*d*x^2]) + (b*(c*f - 2*g)*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + ArcSin[c*x]/2]])/(2*d^2*(c*f - g)^2*Sqrt
[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*Log[Sin[Pi/4 + ArcSin[c*x]/2]])/(6*d^2*(c*f - g)*Sqrt[d - c^2*d*x^2])
- (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(
c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) + (b*g^4*Sqrt[1 - c^2*x^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])
*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(d^2*(c*f - g)^2*(c*f + g)^2*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]) - (b*S
qrt[1 - c^2*x^2]*Sec[Pi/4 + ArcSin[c*x]/2]^2)/(24*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])*Tan[Pi/4 + ArcSin[c*x]/2])/(12*d^2*(c*f + g)*Sqrt[d - c^2*d*x^2]) + ((c*f + 2*g)*Sqrt[1 - c^2*
x^2]*(a + b*ArcSin[c*x])*Tan[Pi/4 + ArcSin[c*x]/2])/(4*d^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^
2]*(a + b*ArcSin[c*x])*Sec[Pi/4 + ArcSin[c*x]/2]^2*Tan[Pi/4 + ArcSin[c*x]/2])/(24*d^2*(c*f + g)*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 4185

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

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3323

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

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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

Rule 2279

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

Rule 2391

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

Rubi steps

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

Mathematica [A]  time = 12.8867, size = 2078, normalized size = 1.6 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((a*g - a*c^2*f*x)/(3*d^3*(-(c^2*f^2) + g^2)*(-1 + c^2*x^2)^2) + (-3*a*g^3 - 2*a*c^4
*f^3*x + 5*a*c^2*f*g^2*x)/(3*d^3*(-(c^2*f^2) + g^2)^2*(-1 + c^2*x^2))) + (a*g^4*Log[f + g*x])/(d^(5/2)*(-(c*f)
+ g)^2*(c*f + g)^2*Sqrt[-(c^2*f^2) + g^2]) - (a*g^4*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt
[-(d*(-1 + c^2*x^2))]])/(d^(5/2)*(-(c*f) + g)^2*(c*f + g)^2*Sqrt[-(c^2*f^2) + g^2]) + (b*((g*(-(c^2*f^2) + 7*g
^2)*(1 - c^2*x^2)^(3/2)*ArcSin[c*x])/(6*(-(c^2*f^2) + g^2)^2*(d*(1 - c^2*x^2))^(3/2)) + ((4*c*f + 7*g)*(1 - c^
2*x^2)^(3/2)*Log[Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]/2]])/(6*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)) + ((4*c*f -
7*g)*(1 - c^2*x^2)^(3/2)*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2]])/(6*(c*f - g)^2*(d*(1 - c^2*x^2))^(3/2)
) + (g^4*(1 - c^2*x^2)^(3/2)*((Pi*ArcTan[(g + c*f*Tan[ArcSin[c*x]/2])/Sqrt[c^2*f^2 - g^2]])/Sqrt[c^2*f^2 - g^2
] + (2*(Pi/2 - ArcSin[c*x])*ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos
[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)]
- (2*I)*(ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[(
Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*(Pi/2 - ArcSin[
c*x]))*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[(Pi/2 - ArcSin[c*x])/
2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log
[(E^((I/2)*(Pi/2 - ArcSin[c*x]))*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f
)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - ArcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[1 - ((c*f - I*Sq
rt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2
*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] + (-ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[(Pi/2 - A
rcSin[c*x])/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[1 - ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2)
+ g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] + I*(
PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(
g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^
2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[(Pi/2 - ArcSin[c*x])/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[(
Pi/2 - ArcSin[c*x])/2]))]))/Sqrt[-(c^2*f^2) + g^2]))/((-(c*f) + g)^2*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)) + ((
1 - c^2*x^2)^(3/2)*(-1 + ArcSin[c*x]))/(12*(c*f + g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[
c*x]/2])^2) + ((1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(6*(c*f + g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[A
rcSin[c*x]/2] - Sin[ArcSin[c*x]/2])^3) + ((1 - c^2*x^2)^(3/2)*ArcSin[c*x]*Sin[ArcSin[c*x]/2])/(6*(c*f - g)*(d*
(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^3) + ((1 - c^2*x^2)^(3/2)*(-1 - ArcSin[c*x]))/(
12*(c*f - g)*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])^2) + ((1 - c^2*x^2)^(3/2)*(4*c*
f*ArcSin[c*x]*Sin[ArcSin[c*x]/2] - 7*g*ArcSin[c*x]*Sin[ArcSin[c*x]/2]))/(6*(c*f - g)^2*(d*(1 - c^2*x^2))^(3/2)
*(Cos[ArcSin[c*x]/2] + Sin[ArcSin[c*x]/2])) + ((1 - c^2*x^2)^(3/2)*(4*c*f*ArcSin[c*x]*Sin[ArcSin[c*x]/2] + 7*g
*ArcSin[c*x]*Sin[ArcSin[c*x]/2]))/(6*(c*f + g)^2*(d*(1 - c^2*x^2))^(3/2)*(Cos[ArcSin[c*x]/2] - Sin[ArcSin[c*x]
/2]))))/d

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Maple [B]  time = 0.483, size = 7977, normalized size = 6.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{c^{6} d^{3} g x^{7} + c^{6} d^{3} f x^{6} - 3 \, c^{4} d^{3} g x^{5} - 3 \, c^{4} d^{3} f x^{4} + 3 \, c^{2} d^{3} g x^{3} + 3 \, c^{2} d^{3} f x^{2} - d^{3} g x - d^{3} f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(c^6*d^3*g*x^7 + c^6*d^3*f*x^6 - 3*c^4*d^3*g*x^5 - 3*c^4*d^
3*f*x^4 + 3*c^2*d^3*g*x^3 + 3*c^2*d^3*f*x^2 - d^3*g*x - d^3*f), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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