3.4.1 \(\int \frac {\text {ArcSin}(a x)^3}{(c-a^2 c x^2)^{7/2}} \, dx\) [301]

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

[Out]

1/5*x*arcsin(a*x)^3/c/(-a^2*c*x^2+c)^(5/2)+4/15*x*arcsin(a*x)^3/c^2/(-a^2*c*x^2+c)^(3/2)+x*arcsin(a*x)/c^3/(-a
^2*c*x^2+c)^(1/2)+1/10*x*arcsin(a*x)/c^3/(-a^2*x^2+1)/(-a^2*c*x^2+c)^(1/2)-3/20*arcsin(a*x)^2/a/c^3/(-a^2*x^2+
1)^(3/2)/(-a^2*c*x^2+c)^(1/2)+8/15*x*arcsin(a*x)^3/c^3/(-a^2*c*x^2+c)^(1/2)-1/20/a/c^3/(-a^2*x^2+1)^(1/2)/(-a^
2*c*x^2+c)^(1/2)-2/5*arcsin(a*x)^2/a/c^3/(-a^2*x^2+1)^(1/2)/(-a^2*c*x^2+c)^(1/2)-8/15*I*arcsin(a*x)^3*(-a^2*x^
2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)+8/5*arcsin(a*x)^2*ln(1+(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/
a/c^3/(-a^2*c*x^2+c)^(1/2)+1/2*ln(-a^2*x^2+1)*(-a^2*x^2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)-8/5*I*arcsin(a*x)*
polylog(2,-(I*a*x+(-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)+4/5*polylog(3,-(I*a*x+(
-a^2*x^2+1)^(1/2))^2)*(-a^2*x^2+1)^(1/2)/a/c^3/(-a^2*c*x^2+c)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.35, antiderivative size = 547, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4747, 4745, 4765, 3800, 2221, 2611, 2320, 6724, 4767, 266, 267} \begin {gather*} -\frac {8 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \text {Li}_2\left (-e^{2 i \text {ArcSin}(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 \text {ArcSin}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \text {ArcSin}(a x)^3}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {2 \text {ArcSin}(a x)^2}{5 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}-\frac {3 \text {ArcSin}(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 \text {ArcSin}(a x)}{c^3 \sqrt {c-a^2 c x^2}}+\frac {x \text {ArcSin}(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} \text {ArcSin}(a x)^2 \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{5 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \text {ArcSin}(a x)^3}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \text {ArcSin}(a x)^3}{5 c \left (c-a^2 c x^2\right )^{5/2}}-\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/20*1/(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]*Ar
cSin[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 266

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 267

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]

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 3800

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 4745

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/d)*Simp[Sqrt[1 - c^2*x^2]/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 4747

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/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4765

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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 6724

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

Rubi steps

\begin {align*} \int \frac {\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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.53, size = 319, normalized size = 0.58 \begin {gather*} \frac {-\frac {3}{\sqrt {1-a^2 x^2}}+60 a x \text {ArcSin}(a x)+\frac {6 a x \text {ArcSin}(a x)}{1-a^2 x^2}-\frac {9 \text {ArcSin}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}}-\frac {24 \text {ArcSin}(a x)^2}{\sqrt {1-a^2 x^2}}+32 a x \text {ArcSin}(a x)^3+\frac {16 a x \text {ArcSin}(a x)^3}{1-a^2 x^2}-32 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^3+\frac {12 a x \text {ArcSin}(a x)^3}{\left (-1+a^2 x^2\right )^2}+96 \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2 \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )+30 \sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )-96 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )+48 \sqrt {1-a^2 x^2} \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(a x)}\right )}{60 a c^3 \sqrt {c-a^2 c x^2}} \end {gather*}

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 = 0.28, size = 1017, normalized size = 1.86

method	result	size
default	\(-\frac {\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}\, a^{4} x^{4}+15 a x -16 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 i \sqrt {-a^{2} x^{2}+1}\right ) \left (160 a^{4} x^{4} \arcsin \left (a x \right )^{3}+1590 a^{4} x^{4} \arcsin \left (a x \right )+105 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-1410 a^{2} x^{2} \arcsin \left (a x \right )+192 i \arcsin \left (a x \right )^{2} a^{8} x^{8}-840 i \arcsin \left (a x \right )^{2} a^{6} x^{6}+1368 i \arcsin \left (a x \right )^{2} a^{4} x^{4}-984 i \arcsin \left (a x \right )^{2} a^{2} x^{2}-45 a x \sqrt {-a^{2} x^{2}+1}+264 i \arcsin \left (a x \right )^{2}+144 i a^{4} x^{4}+256 \arcsin \left (a x \right )^{3}+480 \arcsin \left (a x \right )+24 \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+372 i \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a x -192 i \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+756 i \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-936 i \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+24 i+192 \arcsin \left (a x \right ) a^{8} x^{8}-852 \arcsin \left (a x \right ) a^{6} x^{6}-380 \arcsin \left (a x \right )^{3} a^{2} x^{2}+1020 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-495 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a x -84 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+192 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}-744 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-96 i a^{2} x^{2}-96 i a^{6} x^{6}+24 i a^{8} x^{8}\right )}{60 c^{4} \left (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\right ) a}+\frac {2 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )}{a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{a \,c^{4} \left (a^{2} x^{2}-1\right )}+\frac {16 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \arcsin \left (a x \right )^{3}}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {8 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{2} \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{5 a \,c^{4} \left (a^{2} x^{2}-1\right )}+\frac {8 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \arcsin \left (a x \right ) \polylog \left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{5 a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {4 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \polylog \left (3, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{5 a \,c^{4} \left (a^{2} x^{2}-1\right )}\)	\(1017\)

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,method=_RETURNVERBOSE)

[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)*a^4*x^4+15*a*x-16*I*(-a^2*x^2+1)^(1/
2)*a^2*x^2+8*I*(-a^2*x^2+1)^(1/2))*(160*a^4*x^4*arcsin(a*x)^3+1590*a^4*x^4*arcsin(a*x)+105*a^3*x^3*(-a^2*x^2+1
)^(1/2)-1410*a^2*x^2*arcsin(a*x)+24*I-45*a*x*(-a^2*x^2+1)^(1/2)+256*arcsin(a*x)^3+480*arcsin(a*x)+24*(-a^2*x^2
+1)^(1/2)*a^7*x^7+372*I*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a*x-192*I*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^7*x^7+756*I*
arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^5*x^5-936*I*arcsin(a*x)*(-a^2*x^2+1)^(1/2)*a^3*x^3+264*I*arcsin(a*x)^2+192*ar
csin(a*x)*a^8*x^8-852*arcsin(a*x)*a^6*x^6-380*arcsin(a*x)^3*a^2*x^2+1020*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^3*
x^3-495*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a*x-84*(-a^2*x^2+1)^(1/2)*a^5*x^5+192*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2
)*a^7*x^7-744*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^5*x^5+192*I*arcsin(a*x)^2*a^8*x^8-840*I*arcsin(a*x)^2*a^6*x^6
+1368*I*arcsin(a*x)^2*a^4*x^4-984*I*arcsin(a*x)^2*a^2*x^2+144*I*a^4*x^4-96*I*a^2*x^2+24*I*a^8*x^8-96*I*a^6*x^6
)/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+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))-(-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)+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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \end {gather*}

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]

Integral(asin(a*x)**3/(-c*(a*x - 1)*(a*x + 1))**(7/2), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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