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

Optimal. Leaf size=13 \[ -\frac {1}{2 a \sin ^{-1}(a x)^2} \]

[Out]

-1/2/a/arcsin(a*x)^2

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {4641} \[ -\frac {1}{2 a \sin ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2*a*ArcSin[a*x]^2)

Rule 4641

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^3} \, dx &=-\frac {1}{2 a \sin ^{-1}(a x)^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 13, normalized size = 1.00 \[ -\frac {1}{2 a \sin ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*1/(a*ArcSin[a*x]^2)

________________________________________________________________________________________

fricas [A]  time = 0.51, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a*arcsin(a*x)^2)

________________________________________________________________________________________

giac [A]  time = 0.64, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a*arcsin(a*x)^2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 12, normalized size = 0.92 \[ -\frac {1}{2 a \arcsin \left (a x \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/a/arcsin(a*x)^2

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 11, normalized size = 0.85 \[ -\frac {1}{2 \, a \arcsin \left (a x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/(a*arcsin(a*x)^2)

________________________________________________________________________________________

mupad [B]  time = 0.12, size = 11, normalized size = 0.85 \[ -\frac {1}{2\,a\,{\mathrm {asin}\left (a\,x\right )}^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a*asin(a*x)^2)

________________________________________________________________________________________

sympy [A]  time = 0.85, size = 12, normalized size = 0.92 \[ - \frac {1}{2 a \operatorname {asin}^{2}{\left (a x \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/(2*a*asin(a*x)**2)

________________________________________________________________________________________