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

Optimal. Leaf size=390 \[ -\frac{8 i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]

[Out]

x/(3*c^3*Sqrt[c - a^2*c*x^2]) + x/(30*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - ArcSin[a*x]/(10*a*c^3*(1 - a^2*
x^2)^(3/2)*Sqrt[c - a^2*c*x^2]) - (4*ArcSin[a*x])/(15*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin
[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcSin[a*x]^2)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x]^2
)/(15*c^3*Sqrt[c - a^2*c*x^2]) - (((8*I)/15)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(a*c^3*Sqrt[c - a^2*c*x^2]) + (1
6*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[1 + E^((2*I)*ArcSin[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (((8*I)/15)*S
qrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/(a*c^3*Sqrt[c - a^2*c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.328618, antiderivative size = 390, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4655, 4653, 4675, 3719, 2190, 2279, 2391, 4677, 191, 192} \[ -\frac{8 i \sqrt{1-a^2 x^2} \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/(3*c^3*Sqrt[c - a^2*c*x^2]) + x/(30*c^3*(1 - a^2*x^2)*Sqrt[c - a^2*c*x^2]) - ArcSin[a*x]/(10*a*c^3*(1 - a^2*
x^2)^(3/2)*Sqrt[c - a^2*c*x^2]) - (4*ArcSin[a*x])/(15*a*c^3*Sqrt[1 - a^2*x^2]*Sqrt[c - a^2*c*x^2]) + (x*ArcSin
[a*x]^2)/(5*c*(c - a^2*c*x^2)^(5/2)) + (4*x*ArcSin[a*x]^2)/(15*c^2*(c - a^2*c*x^2)^(3/2)) + (8*x*ArcSin[a*x]^2
)/(15*c^3*Sqrt[c - a^2*c*x^2]) - (((8*I)/15)*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/(a*c^3*Sqrt[c - a^2*c*x^2]) + (1
6*Sqrt[1 - a^2*x^2]*ArcSin[a*x]*Log[1 + E^((2*I)*ArcSin[a*x])])/(15*a*c^3*Sqrt[c - a^2*c*x^2]) - (((8*I)/15)*S
qrt[1 - a^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[a*x])])/(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 2279

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

Rule 2391

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

Rule 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 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 192

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

Rubi steps

\begin{align*} \int \frac{\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 \int \frac{\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac{\left (2 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt{c-a^2 c x^2}}\\ &=-\frac{\sin ^{-1}(a x)}{10 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)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 \int \frac{\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}+\frac{\sqrt{1-a^2 x^2} \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, 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)}{\left (1-a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\sqrt{1-a^2 x^2} \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (4 \sqrt{1-a^2 x^2}\right ) \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (16 a \sqrt{1-a^2 x^2}\right ) \int \frac{x \sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{15 c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (16 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{\left (32 i \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{\left (16 \sqrt{1-a^2 x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 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 \frac{\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ &=\frac{x}{3 c^3 \sqrt{c-a^2 c x^2}}+\frac{x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt{c-a^2 c x^2}}-\frac{\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt{c-a^2 c x^2}}-\frac{4 \sin ^{-1}(a x)}{15 a c^3 \sqrt{1-a^2 x^2} \sqrt{c-a^2 c x^2}}+\frac{x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac{4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac{8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt{c-a^2 c x^2}}+\frac{16 \sqrt{1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}-\frac{8 i \sqrt{1-a^2 x^2} \text{Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt{c-a^2 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.809866, size = 234, normalized size = 0.6 \[ \frac{\sqrt{1-a^2 x^2} \left (-16 i \text{PolyLog}\left (2,-e^{2 i \sin ^{-1}(a x)}\right )+\frac{a^3 x^3}{\left (1-a^2 x^2\right )^{3/2}}+\frac{11 a x}{\sqrt{1-a^2 x^2}}+\frac{16 a x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}}+\frac{8 \sin ^{-1}(a x) \left (\frac{a x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-1\right )}{1-a^2 x^2}+\frac{3 \sin ^{-1}(a x) \left (\frac{2 a x \sin ^{-1}(a x)}{\sqrt{1-a^2 x^2}}-1\right )}{\left (1-a^2 x^2\right )^2}-16 i \sin ^{-1}(a x)^2+32 \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )\right )}{30 a c^3 \sqrt{c \left (1-a^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - a^2*x^2]*((a^3*x^3)/(1 - a^2*x^2)^(3/2) + (11*a*x)/Sqrt[1 - a^2*x^2] - (16*I)*ArcSin[a*x]^2 + (16*a*
x*ArcSin[a*x]^2)/Sqrt[1 - a^2*x^2] + (8*ArcSin[a*x]*(-1 + (a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2]))/(1 - a^2*x^2)
+ (3*ArcSin[a*x]*(-1 + (2*a*x*ArcSin[a*x])/Sqrt[1 - a^2*x^2]))/(1 - a^2*x^2)^2 + 32*ArcSin[a*x]*Log[1 + E^((2*
I)*ArcSin[a*x])] - (16*I)*PolyLog[2, -E^((2*I)*ArcSin[a*x])]))/(30*a*c^3*Sqrt[c*(1 - a^2*x^2)])

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Maple [A]  time = 0.207, size = 556, normalized size = 1.4 \begin{align*} -{\frac{1}{30\,{c}^{4} \left ( 40\,{a}^{10}{x}^{10}-215\,{x}^{8}{a}^{8}+469\,{x}^{6}{a}^{6}-517\,{a}^{4}{x}^{4}+287\,{a}^{2}{x}^{2}-64 \right ) a}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 8\,{a}^{5}{x}^{5}-20\,{a}^{3}{x}^{3}+8\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{4}{a}^{4}+15\,ax-16\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{2}{a}^{2}+8\,i\sqrt{-{a}^{2}{x}^{2}+1} \right ) \left ( 126\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{5}{a}^{5}+64\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{7}{a}^{7}-32\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{7}{a}^{7}+32\,{x}^{8}{a}^{8}+456\,i\arcsin \left ( ax \right ){x}^{4}{a}^{4}-248\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{5}{a}^{5}+62\,i\sqrt{-{a}^{2}{x}^{2}+1}xa-142\,{x}^{6}{a}^{6}+80\,{a}^{4}{x}^{4} \left ( \arcsin \left ( ax \right ) \right ) ^{2}+64\,i\arcsin \left ( ax \right ){x}^{8}{a}^{8}+340\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}-156\,i\sqrt{-{a}^{2}{x}^{2}+1}{x}^{3}{a}^{3}+265\,{a}^{4}{x}^{4}-190\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}-280\,i\arcsin \left ( ax \right ){x}^{6}{a}^{6}-165\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}xa-328\,i\arcsin \left ( ax \right ){x}^{2}{a}^{2}-235\,{a}^{2}{x}^{2}+128\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}+88\,i\arcsin \left ( ax \right ) +80 \right ) }+{\frac{{\frac{8\,i}{15}}}{a{c}^{4} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-{a}^{2}{x}^{2}+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( 2\,i\arcsin \left ( ax \right ) \ln \left ( 1+ \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) +2\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}+{\it polylog} \left ( 2,- \left ( iax+\sqrt{-{a}^{2}{x}^{2}+1} \right ) ^{2} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(-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))*(126*I*(-a^2*x^2+1)^(1/2)*x^5*a^5+64*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x^7*a^7
-32*I*(-a^2*x^2+1)^(1/2)*x^7*a^7+32*x^8*a^8+456*I*arcsin(a*x)*x^4*a^4-248*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x^5*a
^5+62*I*(-a^2*x^2+1)^(1/2)*x*a-142*x^6*a^6+80*a^4*x^4*arcsin(a*x)^2+64*I*arcsin(a*x)*x^8*a^8+340*arcsin(a*x)*(
-a^2*x^2+1)^(1/2)*x^3*a^3-156*I*(-a^2*x^2+1)^(1/2)*x^3*a^3+265*a^4*x^4-190*arcsin(a*x)^2*x^2*a^2-280*I*arcsin(
a*x)*x^6*a^6-165*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*x*a-328*I*arcsin(a*x)*x^2*a^2-235*a^2*x^2+128*arcsin(a*x)^2+88
*I*arcsin(a*x)+80)/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+8/15*I*(-a^2*x^2+1)
^(1/2)*(-c*(a^2*x^2-1))^(1/2)*(2*I*arcsin(a*x)*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)+2*arcsin(a*x)^2+polylog(2,-(
I*a*x+(-a^2*x^2+1)^(1/2))^2))/a/c^4/(a^2*x^2-1)

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

[Out]

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

[Out]

integral(sqrt(-a^2*c*x^2 + c)*arcsin(a*x)^2/(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)**2/(-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 )^{2}}{{\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)^2/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac")

[Out]

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