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

Optimal. Leaf size=13 \[ \frac{\sin ^{-1}(a x)^4}{4 a} \]

[Out]

ArcSin[a*x]^4/(4*a)

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Rubi [A]  time = 0.0295044, 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} \[ \frac{\sin ^{-1}(a x)^4}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]^4/(4*a)

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{\sin ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx &=\frac{\sin ^{-1}(a x)^4}{4 a}\\ \end{align*}

Mathematica [A]  time = 0.0039839, size = 13, normalized size = 1. \[ \frac{\sin ^{-1}(a x)^4}{4 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSin[a*x]^4/(4*a)

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Maple [A]  time = 0.005, size = 12, normalized size = 0.9 \begin{align*}{\frac{ \left ( \arcsin \left ( ax \right ) \right ) ^{4}}{4\,a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arcsin(a*x)^4/a

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Maxima [A]  time = 1.47312, size = 15, normalized size = 1.15 \begin{align*} \frac{\arcsin \left (a x\right )^{4}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arcsin(a*x)^4/a

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Fricas [A]  time = 1.8194, size = 28, normalized size = 2.15 \begin{align*} \frac{\arcsin \left (a x\right )^{4}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arcsin(a*x)^4/a

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Sympy [A]  time = 0.8623, size = 10, normalized size = 0.77 \begin{align*} \begin{cases} \frac{\operatorname{asin}^{4}{\left (a x \right )}}{4 a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((asin(a*x)**4/(4*a), Ne(a, 0)), (0, True))

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Giac [A]  time = 1.38032, size = 15, normalized size = 1.15 \begin{align*} \frac{\arcsin \left (a x\right )^{4}}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*arcsin(a*x)^4/a