3.301 \(\int \frac{\sin ^{-1}(a x)^3}{(c-a^2 c x^2)^{7/2}} \, dx\)

Optimal. Leaf size=547 \[ -\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]

[Out]

-1/(20*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x])/(c^3*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*
x])/(10*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - (3*ArcSin[a*x]^2)/(20*a*c^3*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*
c*x^2]) - (2*ArcSin[a*x]^2)/(5*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^3)/(5*c*(c - a^2*
c*x^2)^(5/2)) + (4*x*ArcSin[a*x]^3)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x]^3)/(15*c^3*Sqrt[c - a^2*
c*x^2]) - (((8*I)/15)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(a*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[1 - a^2*x^2]*ArcS
in[a*x]^2*Log[1 + E^((2*I)*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2])
/(2*a*c^3*Sqrt[c - a^2*c*x^2]) - (((8*I)/5)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/
(a*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^((2*I)*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c
*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.48107, antiderivative size = 547, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {4655, 4653, 4675, 3719, 2190, 2531, 2282, 6589, 4677, 4651, 260, 261} \[ -\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^3/(c - a^2*c*x^2)^(7/2),x]

[Out]

-1/(20*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x])/(c^3*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*
x])/(10*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - (3*ArcSin[a*x]^2)/(20*a*c^3*(1 - a^2*x^2)^(3/2)*Sqrt[c - a^2*
c*x^2]) - (2*ArcSin[a*x]^2)/(5*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin[a*x]^3)/(5*c*(c - a^2*
c*x^2)^(5/2)) + (4*x*ArcSin[a*x]^3)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x]^3)/(15*c^3*Sqrt[c - a^2*
c*x^2]) - (((8*I)/15)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(a*c^3*Sqrt[c - a^2*c*x^2]) + (8*Sqrt[1 - a^2*x^2]*ArcS
in[a*x]^2*Log[1 + E^((2*I)*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c*x^2]) + (Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2])
/(2*a*c^3*Sqrt[c - a^2*c*x^2]) - (((8*I)/5)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/
(a*c^3*Sqrt[c - a^2*c*x^2]) + (4*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^((2*I)*ArcSin[a*x])])/(5*a*c^3*Sqrt[c - a^2*c
*x^2])

Rule 4655

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(x*(d + e*x^2)^(p
+ 1)*(a + b*ArcSin[c*x])^n)/(2*d*(p + 1)), x] + (Dist[(2*p + 3)/(2*d*(p + 1)), Int[(d + e*x^2)^(p + 1)*(a + b*
ArcSin[c*x])^n, x], x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*(p + 1)*(1 - c^2*x^2)^FracPart[p
]), Int[x*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2
*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 4653

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSin[c
*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n*Sqrt[1 - c^2*x^2])/(d*Sqrt[d + e*x^2]), Int[(x*(a + b*ArcSin[c*x
])^(n - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0]

Rule 4675

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[e^(-1), Subst[In
t[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 3719

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x)))/(1 + E^(2*I*(e + f*x))), x], x] /; FreeQ[{c, d, e, f}, x] &&
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 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]

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 4651

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(x*(a + b*ArcSin[c
*x])^n)/(d*Sqrt[d + e*x^2]), x] - Dist[(b*c*n)/Sqrt[d], Int[(x*(a + b*ArcSin[c*x])^(n - 1))/(d + e*x^2), x], x
] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[d, 0]

Rule 260

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

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac{\left (3 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 \int \frac{\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}+\frac{\left (3 \sqrt{1-a^2 x^2}\right ) \int \frac{\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (4 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{1-a^2 x^2}\right ) \int \frac{\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{\left (1-a^2 x^2\right )^2} \, dx}{10 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (8 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (8 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int x^2 \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (4 a \sqrt{1-a^2 x^2}\right ) \int \frac{x}{1-a^2 x^2} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (16 i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x^2}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (16 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int x \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (8 i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{1}{20 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)}{10 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{3 \sin ^{-1}(a x)^2}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{2 \sin ^{-1}(a x)^2}{5 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^3}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{8 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )}{2 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 \sqrt{1-a^2 x^2} \text{Li}_3\left (-e^{2 i \sin ^{-1}(a x)}\right )}{5 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.726761, size = 319, normalized size = 0.58 \[ \frac{-96 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )+48 \sqrt{1-a^2 x^2} \text{PolyLog}\left (3,-e^{2 i \sin ^{-1}(a x)}\right )-\frac{3}{\sqrt{1-a^2 x^2}}+30 \sqrt{1-a^2 x^2} \log \left (1-a^2 x^2\right )-32 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^3+\frac{16 a x \sin ^{-1}(a x)^3}{1-a^2 x^2}+\frac{12 a x \sin ^{-1}(a x)^3}{\left (a^2 x^2-1\right )^2}-\frac{24 \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}-\frac{9 \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}+\frac{6 a x \sin ^{-1}(a x)}{1-a^2 x^2}+96 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2 \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )+32 a x \sin ^{-1}(a x)^3+60 a x \sin ^{-1}(a x)}{60 a c^3 \sqrt{c-a^2 c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^3/(c - a^2*c*x^2)^(7/2),x]

[Out]

(-3/Sqrt[1 - a^2*x^2] + 60*a*x*ArcSin[a*x] + (6*a*x*ArcSin[a*x])/(1 - a^2*x^2) - (9*ArcSin[a*x]^2)/(1 - a^2*x^
2)^(3/2) - (24*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + 32*a*x*ArcSin[a*x]^3 + (16*a*x*ArcSin[a*x]^3)/(1 - a^2*x^2)
- (32*I)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3 + (12*a*x*ArcSin[a*x]^3)/(-1 + a^2*x^2)^2 + 96*Sqrt[1 - a^2*x^2]*ArcS
in[a*x]^2*Log[1 + E^((2*I)*ArcSin[a*x])] + 30*Sqrt[1 - a^2*x^2]*Log[1 - a^2*x^2] - (96*I)*Sqrt[1 - a^2*x^2]*Ar
cSin[a*x]*PolyLog[2, -E^((2*I)*ArcSin[a*x])] + 48*Sqrt[1 - a^2*x^2]*PolyLog[3, -E^((2*I)*ArcSin[a*x])])/(60*a*
c^3*Sqrt[c - a^2*c*x^2])

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Maple [A]  time = 0.283, size = 1017, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^3/(-a^2*c*x^2+c)^(7/2),x)

[Out]

-1/60*(-c*(a^2*x^2-1))^(1/2)*(8*a^5*x^5-20*a^3*x^3+8*I*(-a^2*x^2+1)^(1/2)*x^4*a^4+15*a*x-16*I*(-a^2*x^2+1)^(1/
2)*x^2*a^2+8*I*(-a^2*x^2+1)^(1/2))*(1590*a^4*x^4*arcsin(a*x)-1410*a^2*x^2*arcsin(a*x)+105*a^3*x^3*(-a^2*x^2+1)
^(1/2)-45*a*x*(-a^2*x^2+1)^(1/2)+160*a^4*x^4*arcsin(a*x)^3+24*I*x^8*a^8+24*(-a^2*x^2+1)^(1/2)*x^7*a^7-84*(-a^2
*x^2+1)^(1/2)*a^5*x^5+24*I+256*arcsin(a*x)^3+480*arcsin(a*x)+756*I*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x^5*a^5+264*
I*arcsin(a*x)^2+1020*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x^3*a^3-495*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x*a-936*I*a
rcsin(a*x)*(-a^2*x^2+1)^(1/2)*x^3*a^3+372*I*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x*a-192*I*arcsin(a*x)*(-a^2*x^2+1)^
(1/2)*x^7*a^7-96*I*x^6*a^6+144*I*x^4*a^4-96*I*a^2*x^2-380*arcsin(a*x)^3*x^2*a^2+192*arcsin(a*x)*x^8*a^8-852*ar
csin(a*x)*x^6*a^6+192*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x^7*a^7-744*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*x^5*a^5+19
2*I*arcsin(a*x)^2*x^8*a^8-840*I*arcsin(a*x)^2*x^6*a^6+1368*I*arcsin(a*x)^2*x^4*a^4-984*I*arcsin(a*x)^2*x^2*a^2
)/c^4/(40*a^10*x^10-215*a^8*x^8+469*a^6*x^6-517*a^4*x^4+287*a^2*x^2-64)/a-(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^
(1/2)/a/c^4/(a^2*x^2-1)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+2*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^4/(
a^2*x^2-1)*ln(I*a*x+(-a^2*x^2+1)^(1/2))+16/15*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/a/c^4/(a^2*x^2-1)*ar
csin(a*x)^3-8/5*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^4/(a^2*x^2-1)*arcsin(a*x)^2*ln(1+(I*a*x+(-a^2*x^
2+1)^(1/2))^2)+8/5*I*(-a^2*x^2+1)^(1/2)*(-c*(a^2*x^2-1))^(1/2)/a/c^4/(a^2*x^2-1)*arcsin(a*x)*polylog(2,-(I*a*x
+(-a^2*x^2+1)^(1/2))^2)-4/5*(-c*(a^2*x^2-1))^(1/2)*(-a^2*x^2+1)^(1/2)/a/c^4/(a^2*x^2-1)*polylog(3,-(I*a*x+(-a^
2*x^2+1)^(1/2))^2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxima")

[Out]

integrate(arcsin(a*x)^3/(-a^2*c*x^2 + c)^(7/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{-a^{2} c x^{2} + c} \arcsin \left (a x\right )^{3}}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3/(-a^2*c*x^2+c)^(7/2),x, algorithm="fricas")

[Out]

integral(sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^3/(a^8*c^4*x^8 - 4*a^6*c^4*x^6 + 6*a^4*c^4*x^4 - 4*a^2*c^4*x^2 + c^4
), 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(asin(a*x)**3/(-a**2*c*x**2+c)**(7/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arcsin \left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^3/(-a^2*c*x^2 + c)^(7/2), x)