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)

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Rubi [A]  time = 0.0309994, antiderivative size = 13, normalized size of antiderivative = 1., 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 = 0.0052446, size = 13, normalized size = 1. \[ -\frac{1}{2 a \sin ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.01, size = 12, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,a \left ( \arcsin \left ( ax \right ) \right ) ^{2}}} \end{align*}

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

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Maxima [A]  time = 1.43998, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2 \, a \arcsin \left (a x\right )^{2}} \end{align*}

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)

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Fricas [A]  time = 2.0407, size = 32, normalized size = 2.46 \begin{align*} -\frac{1}{2 \, a \arcsin \left (a x\right )^{2}} \end{align*}

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)

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Sympy [A]  time = 1.15787, size = 12, normalized size = 0.92 \begin{align*} - \frac{1}{2 a \operatorname{asin}^{2}{\left (a x \right )}} \end{align*}

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)

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Giac [A]  time = 1.43421, size = 15, normalized size = 1.15 \begin{align*} -\frac{1}{2 \, a \arcsin \left (a x\right )^{2}} \end{align*}

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)