3.309 \(\int \frac {\sin ^{-1}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=99 \[ -\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-3 i a \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {3}{2} a \text {Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )-i a \sin ^{-1}(a x)^3+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]

[Out]

-I*a*arcsin(a*x)^3+3*a*arcsin(a*x)^2*ln(1-(I*a*x+(-a^2*x^2+1)^(1/2))^2)-3*I*a*arcsin(a*x)*polylog(2,(I*a*x+(-a
^2*x^2+1)^(1/2))^2)+3/2*a*polylog(3,(I*a*x+(-a^2*x^2+1)^(1/2))^2)-arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/x

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {4681, 4625, 3717, 2190, 2531, 2282, 6589} \[ -3 i a \sin ^{-1}(a x) \text {PolyLog}\left (2,e^{2 i \sin ^{-1}(a x)}\right )+\frac {3}{2} a \text {PolyLog}\left (3,e^{2 i \sin ^{-1}(a x)}\right )-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-i a \sin ^{-1}(a x)^3+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

(-I)*a*ArcSin[a*x]^3 - (Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/x + 3*a*ArcSin[a*x]^2*Log[1 - E^((2*I)*ArcSin[a*x])]
- (3*I)*a*ArcSin[a*x]*PolyLog[2, E^((2*I)*ArcSin[a*x])] + (3*a*PolyLog[3, E^((2*I)*ArcSin[a*x])])/2

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 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 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 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4681

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

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]

Rubi steps

\begin {align*} \int \frac {\sin ^{-1}(a x)^3}{x^2 \sqrt {1-a^2 x^2}} \, dx &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+(3 a) \int \frac {\sin ^{-1}(a x)^2}{x} \, dx\\ &=-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+(3 a) \operatorname {Subst}\left (\int x^2 \cot (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}-(6 i a) \operatorname {Subst}\left (\int \frac {e^{2 i x} x^2}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-(6 a) \operatorname {Subst}\left (\int x \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 i a \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+(3 i a) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 i a \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {1}{2} (3 a) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )\\ &=-i a \sin ^{-1}(a x)^3-\frac {\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{x}+3 a \sin ^{-1}(a x)^2 \log \left (1-e^{2 i \sin ^{-1}(a x)}\right )-3 i a \sin ^{-1}(a x) \text {Li}_2\left (e^{2 i \sin ^{-1}(a x)}\right )+\frac {3}{2} a \text {Li}_3\left (e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.24, size = 108, normalized size = 1.09 \[ \frac {1}{8} a \left (-\frac {8 \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3}{a x}+24 i \sin ^{-1}(a x) \text {Li}_2\left (e^{-2 i \sin ^{-1}(a x)}\right )+12 \text {Li}_3\left (e^{-2 i \sin ^{-1}(a x)}\right )+8 i \sin ^{-1}(a x)^3+24 \sin ^{-1}(a x)^2 \log \left (1-e^{-2 i \sin ^{-1}(a x)}\right )-i \pi ^3\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcSin[a*x]^3/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

(a*((-I)*Pi^3 + (8*I)*ArcSin[a*x]^3 - (8*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/(a*x) + 24*ArcSin[a*x]^2*Log[1 - E^(
(-2*I)*ArcSin[a*x])] + (24*I)*ArcSin[a*x]*PolyLog[2, E^((-2*I)*ArcSin[a*x])] + 12*PolyLog[3, E^((-2*I)*ArcSin[
a*x])]))/8

________________________________________________________________________________________

fricas [F]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{3}}{a^{2} x^{4} - x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arcsin(a*x)^3/(a^2*x^4 - x^2), x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arcsin \left (a x\right )^{3}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arcsin(a*x)^3/(sqrt(-a^2*x^2 + 1)*x^2), x)

________________________________________________________________________________________

maple [A]  time = 0.23, size = 208, normalized size = 2.10 \[ \frac {\left (i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{3}}{x}-2 i \arcsin \left (a x \right )^{3} a +3 \ln \left (1-i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{2} a -6 i \polylog \left (2, i a x +\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right ) a +3 \ln \left (1+i a x +\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right )^{2} a -6 i \polylog \left (2, -i a x -\sqrt {-a^{2} x^{2}+1}\right ) \arcsin \left (a x \right ) a +6 \polylog \left (3, -i a x -\sqrt {-a^{2} x^{2}+1}\right ) a +6 \polylog \left (3, i a x +\sqrt {-a^{2} x^{2}+1}\right ) a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x)

[Out]

(I*a*x-(-a^2*x^2+1)^(1/2))/x*arcsin(a*x)^3-2*I*arcsin(a*x)^3*a+3*ln(1-I*a*x-(-a^2*x^2+1)^(1/2))*arcsin(a*x)^2*
a-6*I*polylog(2,I*a*x+(-a^2*x^2+1)^(1/2))*arcsin(a*x)*a+3*ln(1+I*a*x+(-a^2*x^2+1)^(1/2))*arcsin(a*x)^2*a-6*I*p
olylog(2,-I*a*x-(-a^2*x^2+1)^(1/2))*arcsin(a*x)*a+6*polylog(3,-I*a*x-(-a^2*x^2+1)^(1/2))*a+6*polylog(3,I*a*x+(
-a^2*x^2+1)^(1/2))*a

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {3}{8} \, {\left (x^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2} + 8 \, \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} a x^{2} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) + 3 \, {\left (a^{2} x^{3} - x\right )} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{2}}{4 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x}\right )} a^{3} x - \sqrt {a x + 1} \sqrt {-a x + 1} \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

(3*a^3*x*integrate(x*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2, x) - sqrt(a*x + 1)*sqrt(-a*x + 1)*arctan2(a
*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^3)/x

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {asin}\left (a\,x\right )}^3}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asin(a*x)^3/(x^2*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(asin(a*x)^3/(x^2*(1 - a^2*x^2)^(1/2)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asin}^{3}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asin(a*x)**3/x**2/(-a**2*x**2+1)**(1/2),x)

[Out]

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

________________________________________________________________________________________