3.3.94 \(\int \frac {\text {ArcSin}(a x)^3}{(c-a^2 c x^2)^3} \, dx\) [294]

Optimal. Leaf size=455 \[ -\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \text {ArcSin}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\text {ArcSin}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \text {ArcSin}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \text {ArcSin}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \text {ArcSin}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 i \text {ArcSin}(a x) \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{a c^3}-\frac {3 i \text {ArcSin}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}+\frac {5 i \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(a x)}\right )}{2 a c^3}+\frac {9 i \text {ArcSin}(a x)^2 \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(a x)}\right )}{8 a c^3}-\frac {5 i \text {PolyLog}\left (2,i e^{i \text {ArcSin}(a x)}\right )}{2 a c^3}-\frac {9 i \text {ArcSin}(a x)^2 \text {PolyLog}\left (2,i e^{i \text {ArcSin}(a x)}\right )}{8 a c^3}-\frac {9 \text {ArcSin}(a x) \text {PolyLog}\left (3,-i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}+\frac {9 \text {ArcSin}(a x) \text {PolyLog}\left (3,i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}-\frac {9 i \text {PolyLog}\left (4,-i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}+\frac {9 i \text {PolyLog}\left (4,i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3} \]

[Out]

1/4*x*arcsin(a*x)/c^3/(-a^2*x^2+1)-1/4*arcsin(a*x)^2/a/c^3/(-a^2*x^2+1)^(3/2)+1/4*x*arcsin(a*x)^3/c^3/(-a^2*x^
2+1)^2+3/8*x*arcsin(a*x)^3/c^3/(-a^2*x^2+1)-9/8*I*arcsin(a*x)^2*polylog(2,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-
5/2*I*polylog(2,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-3/4*I*arcsin(a*x)^3*arctan(I*a*x+(-a^2*x^2+1)^(1/2))/a/c^3
-9/4*I*polylog(4,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3+9/8*I*arcsin(a*x)^2*polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/
2)))/a/c^3-5*I*arcsin(a*x)*arctan(I*a*x+(-a^2*x^2+1)^(1/2))/a/c^3-9/4*arcsin(a*x)*polylog(3,-I*(I*a*x+(-a^2*x^
2+1)^(1/2)))/a/c^3+9/4*arcsin(a*x)*polylog(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3+5/2*I*polylog(2,-I*(I*a*x+(-a
^2*x^2+1)^(1/2)))/a/c^3+9/4*I*polylog(4,I*(I*a*x+(-a^2*x^2+1)^(1/2)))/a/c^3-1/4/a/c^3/(-a^2*x^2+1)^(1/2)-9/8*a
rcsin(a*x)^2/a/c^3/(-a^2*x^2+1)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.36, antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {4747, 4749, 4266, 2611, 6744, 2320, 6724, 4767, 2317, 2438, 267} \begin {gather*} \frac {3 x \text {ArcSin}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {x \text {ArcSin}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {9 \text {ArcSin}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}-\frac {\text {ArcSin}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x \text {ArcSin}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}-\frac {3 i \text {ArcSin}(a x)^3 \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}-\frac {5 i \text {ArcSin}(a x) \text {ArcTan}\left (e^{i \text {ArcSin}(a x)}\right )}{a c^3}+\frac {9 i \text {ArcSin}(a x)^2 \text {Li}_2\left (-i e^{i \text {ArcSin}(a x)}\right )}{8 a c^3}-\frac {9 i \text {ArcSin}(a x)^2 \text {Li}_2\left (i e^{i \text {ArcSin}(a x)}\right )}{8 a c^3}-\frac {9 \text {ArcSin}(a x) \text {Li}_3\left (-i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}+\frac {9 \text {ArcSin}(a x) \text {Li}_3\left (i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}+\frac {5 i \text {Li}_2\left (-i e^{i \text {ArcSin}(a x)}\right )}{2 a c^3}-\frac {5 i \text {Li}_2\left (i e^{i \text {ArcSin}(a x)}\right )}{2 a c^3}-\frac {9 i \text {Li}_4\left (-i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3}+\frac {9 i \text {Li}_4\left (i e^{i \text {ArcSin}(a x)}\right )}{4 a c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*1/(a*c^3*Sqrt[1 - a^2*x^2]) + (x*ArcSin[a*x])/(4*c^3*(1 - a^2*x^2)) - ArcSin[a*x]^2/(4*a*c^3*(1 - a^2*x^2
)^(3/2)) - (9*ArcSin[a*x]^2)/(8*a*c^3*Sqrt[1 - a^2*x^2]) + (x*ArcSin[a*x]^3)/(4*c^3*(1 - a^2*x^2)^2) + (3*x*Ar
cSin[a*x]^3)/(8*c^3*(1 - a^2*x^2)) - ((5*I)*ArcSin[a*x]*ArcTan[E^(I*ArcSin[a*x])])/(a*c^3) - (((3*I)/4)*ArcSin
[a*x]^3*ArcTan[E^(I*ArcSin[a*x])])/(a*c^3) + (((5*I)/2)*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(a*c^3) + (((9*I)/
8)*ArcSin[a*x]^2*PolyLog[2, (-I)*E^(I*ArcSin[a*x])])/(a*c^3) - (((5*I)/2)*PolyLog[2, I*E^(I*ArcSin[a*x])])/(a*
c^3) - (((9*I)/8)*ArcSin[a*x]^2*PolyLog[2, I*E^(I*ArcSin[a*x])])/(a*c^3) - (9*ArcSin[a*x]*PolyLog[3, (-I)*E^(I
*ArcSin[a*x])])/(4*a*c^3) + (9*ArcSin[a*x]*PolyLog[3, I*E^(I*ArcSin[a*x])])/(4*a*c^3) - (((9*I)/4)*PolyLog[4,
(-I)*E^(I*ArcSin[a*x])])/(a*c^3) + (((9*I)/4)*PolyLog[4, I*E^(I*ArcSin[a*x])])/(a*c^3)

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 2317

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 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 2438

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 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 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 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 4749

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/(c*d), Subst[Int[(a +
b*x)^n*Sec[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]

Rule 6744

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

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac {(3 a) \int \frac {x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{4 c^3}+\frac {3 \int \frac {\sin ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {\int \frac {\sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{2 c^3}-\frac {(9 a) \int \frac {x \sin ^{-1}(a x)^2}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{8 c^3}+\frac {3 \int \frac {\sin ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=\frac {x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}+\frac {\int \frac {\sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{4 c^3}+\frac {9 \int \frac {\sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{4 c^3}+\frac {3 \text {Subst}\left (\int x^3 \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{8 a c^3}-\frac {a \int \frac {x}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{4 c^3}\\ &=-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {\text {Subst}\left (\int x \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a c^3}+\frac {9 \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{8 a c^3}+\frac {9 \text {Subst}\left (\int x \sec (x) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac {3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {(9 i) \text {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}+\frac {(9 i) \text {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac {\text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}+\frac {\text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac {3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \sin ^{-1}(a x) \text {Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \sin ^{-1}(a x) \text {Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {i \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac {i \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {(9 i) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac {(9 i) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \text {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}-\frac {9 \text {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac {3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {5 i \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}+\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {5 i \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \sin ^{-1}(a x) \text {Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \sin ^{-1}(a x) \text {Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac {(9 i) \text {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {(9 i) \text {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}\\ &=-\frac {1}{4 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)}{4 c^3 \left (1-a^2 x^2\right )}-\frac {\sin ^{-1}(a x)^2}{4 a c^3 \left (1-a^2 x^2\right )^{3/2}}-\frac {9 \sin ^{-1}(a x)^2}{8 a c^3 \sqrt {1-a^2 x^2}}+\frac {x \sin ^{-1}(a x)^3}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac {3 x \sin ^{-1}(a x)^3}{8 c^3 \left (1-a^2 x^2\right )}-\frac {5 i \sin ^{-1}(a x) \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{a c^3}-\frac {3 i \sin ^{-1}(a x)^3 \tan ^{-1}\left (e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {5 i \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}+\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {5 i \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{2 a c^3}-\frac {9 i \sin ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \sin ^{-1}(a x)}\right )}{8 a c^3}-\frac {9 \sin ^{-1}(a x) \text {Li}_3\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 \sin ^{-1}(a x) \text {Li}_3\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}-\frac {9 i \text {Li}_4\left (-i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}+\frac {9 i \text {Li}_4\left (i e^{i \sin ^{-1}(a x)}\right )}{4 a c^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1544\) vs. \(2(455)=910\).
time = 12.12, size = 1544, normalized size = 3.39 \begin {gather*} \text {Too large to display} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((1 + 5*ArcSin[a*x]^2)/4 - (5*(ArcSin[a*x]*(Log[1 - I*E^(I*ArcSin[a*x])] - Log[1 + I*E^(I*ArcSin[a*x])]) + I
*(PolyLog[2, (-I)*E^(I*ArcSin[a*x])] - PolyLog[2, I*E^(I*ArcSin[a*x])])))/2 - (3*((Pi^3*Log[Cot[(Pi/2 - ArcSin
[a*x])/2]])/8 + (3*Pi^2*((Pi/2 - ArcSin[a*x])*(Log[1 - E^(I*(Pi/2 - ArcSin[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcS
in[a*x]))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcSin[a*x]))])))/4 - (3*Pi
*((Pi/2 - ArcSin[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcSin[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcSin[a*x]))]) + (2*I)*(
Pi/2 - ArcSin[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcSin[a*x]))]) + 2*(-P
olyLog[3, -E^(I*(Pi/2 - ArcSin[a*x]))] + PolyLog[3, E^(I*(Pi/2 - ArcSin[a*x]))])))/2 + 8*((I/64)*(Pi/2 - ArcSi
n[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^4 - ((Pi/2 - ArcSin[a*x])^3*Log[1 + E^(I*(Pi/2 - ArcSin[a
*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))
]))/8 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + ((3*I)/8)
*(Pi/2 - ArcSin[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - ArcSin[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcSin[a*x
])/2)^2 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + (I/2)*Pol
yLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))]))/4 + ((3*I)/2)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^2*Po
lyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] - (3*(Pi/2 - ArcSin[a*x])*PolyLog[3, -E^(I*(Pi/2 - Arc
Sin[a*x]))])/4 - (3*Pi*((I/3)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^3 - (Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)^2*Log[
1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] + I*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2)*PolyLog[2, -E^((2*I)*
(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))] - PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))]/2))/2 - (3*(P
i/2 + (-1/2*Pi + ArcSin[a*x])/2)*PolyLog[3, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))])/2 - ((3*I)/4)*Poly
Log[4, -E^(I*(Pi/2 - ArcSin[a*x]))] - ((3*I)/4)*PolyLog[4, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcSin[a*x])/2))])))/8
- ArcSin[a*x]^3/(16*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])^4) - (2*ArcSin[a*x] - ArcSin[a*x]^2 + 3*ArcSin[
a*x]^3)/(16*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])^2) + (ArcSin[a*x]^2*Sin[ArcSin[a*x]/2])/(8*(Cos[ArcSin[a
*x]/2] - Sin[ArcSin[a*x]/2])^3) + ArcSin[a*x]^3/(16*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])^4) - (ArcSin[a*x
]^2*Sin[ArcSin[a*x]/2])/(8*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])^3) - (-2*ArcSin[a*x] - ArcSin[a*x]^2 - 3*
ArcSin[a*x]^3)/(16*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])^2) - (-Sin[ArcSin[a*x]/2] - 5*ArcSin[a*x]^2*Sin[A
rcSin[a*x]/2])/(4*(Cos[ArcSin[a*x]/2] - Sin[ArcSin[a*x]/2])) - (Sin[ArcSin[a*x]/2] + 5*ArcSin[a*x]^2*Sin[ArcSi
n[a*x]/2])/(4*(Cos[ArcSin[a*x]/2] + Sin[ArcSin[a*x]/2])))/(a*c^3))

________________________________________________________________________________________

Maple [A]
time = 0.31, size = 543, normalized size = 1.19

method	result	size
derivativedivides	\(\frac {-\frac {3 a^{3} x^{3} \arcsin \left (a x \right )^{3}-9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-5 a x \arcsin \left (a x \right )^{3}+2 a^{3} x^{3} \arcsin \left (a x \right )+11 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 a x \arcsin \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}+\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 i \arcsin \left (a x \right )^{2} \polylog \left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 \arcsin \left (a x \right ) \polylog \left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}+\frac {9 i \polylog \left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 i \arcsin \left (a x \right )^{2} \polylog \left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 \arcsin \left (a x \right ) \polylog \left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {9 i \polylog \left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {5 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 i \dilog \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}-\frac {5 i \dilog \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}}{a}\)	\(543\)
default	\(\frac {-\frac {3 a^{3} x^{3} \arcsin \left (a x \right )^{3}-9 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-5 a x \arcsin \left (a x \right )^{3}+2 a^{3} x^{3} \arcsin \left (a x \right )+11 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}-2 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}-2 a x \arcsin \left (a x \right )+2 \sqrt {-a^{2} x^{2}+1}}{8 \left (a^{4} x^{4}-2 a^{2} x^{2}+1\right ) c^{3}}+\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 i \arcsin \left (a x \right )^{2} \polylog \left (2, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 \arcsin \left (a x \right ) \polylog \left (3, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}+\frac {9 i \polylog \left (4, i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {3 \arcsin \left (a x \right )^{3} \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}+\frac {9 i \arcsin \left (a x \right )^{2} \polylog \left (2, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{8 c^{3}}-\frac {9 \arcsin \left (a x \right ) \polylog \left (3, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {9 i \polylog \left (4, -i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{4 c^{3}}-\frac {5 \arcsin \left (a x \right ) \ln \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 \arcsin \left (a x \right ) \ln \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}+\frac {5 i \dilog \left (1+i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}-\frac {5 i \dilog \left (1-i \left (i a x +\sqrt {-a^{2} x^{2}+1}\right )\right )}{2 c^{3}}}{a}\)	\(543\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^3/(-a^2*c*x^2+c)^3,x,method=_RETURNVERBOSE)

[Out]

1/a*(-1/8*(3*a^3*x^3*arcsin(a*x)^3-9*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)*a^2*x^2-5*a*x*arcsin(a*x)^3+2*a^3*x^3*ar
csin(a*x)+11*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-2*a^2*x^2*(-a^2*x^2+1)^(1/2)-2*a*x*arcsin(a*x)+2*(-a^2*x^2+1)^(1
/2))/(a^4*x^4-2*a^2*x^2+1)/c^3+3/8/c^3*arcsin(a*x)^3*ln(1-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-9/8*I/c^3*arcsin(a*x)^
2*polylog(2,I*(I*a*x+(-a^2*x^2+1)^(1/2)))+9/4/c^3*arcsin(a*x)*polylog(3,I*(I*a*x+(-a^2*x^2+1)^(1/2)))+9/4*I/c^
3*polylog(4,I*(I*a*x+(-a^2*x^2+1)^(1/2)))-3/8/c^3*arcsin(a*x)^3*ln(1+I*(I*a*x+(-a^2*x^2+1)^(1/2)))+9/8*I/c^3*a
rcsin(a*x)^2*polylog(2,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-9/4/c^3*arcsin(a*x)*polylog(3,-I*(I*a*x+(-a^2*x^2+1)^(1/
2)))-9/4*I/c^3*polylog(4,-I*(I*a*x+(-a^2*x^2+1)^(1/2)))-5/2/c^3*arcsin(a*x)*ln(1+I*(I*a*x+(-a^2*x^2+1)^(1/2)))
+5/2/c^3*arcsin(a*x)*ln(1-I*(I*a*x+(-a^2*x^2+1)^(1/2)))+5/2*I/c^3*dilog(1+I*(I*a*x+(-a^2*x^2+1)^(1/2)))-5/2*I/
c^3*dilog(1-I*(I*a*x+(-a^2*x^2+1)^(1/2))))

________________________________________________________________________________________

Maxima [A]
time = 0.83, size = 78, normalized size = 0.17 \begin {gather*} -\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, a^{2} x^{3} - 5 \, x\right )}}{a^{4} c^{3} x^{4} - 2 \, a^{2} c^{3} x^{2} + c^{3}} - \frac {3 \, \log \left (a x + 1\right )}{a c^{3}} + \frac {3 \, \log \left (a x - 1\right )}{a c^{3}}\right )} \arcsin \left (a x\right )^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(2*(3*a^2*x^3 - 5*x)/(a^4*c^3*x^4 - 2*a^2*c^3*x^2 + c^3) - 3*log(a*x + 1)/(a*c^3) + 3*log(a*x - 1)/(a*c^
3))*arcsin(a*x)^3

________________________________________________________________________________________

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)^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**3/(-a**2*c*x**2+c)**3,x)

[Out]

-Integral(asin(a*x)**3/(a**6*x**6 - 3*a**4*x**4 + 3*a**2*x**2 - 1), x)/c**3

________________________________________________________________________________________

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)^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

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 )}^3} \,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)^3,x)

[Out]

int(asin(a*x)^3/(c - a^2*c*x^2)^3, x)

________________________________________________________________________________________