3.26 \(\int \sin ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=60 \[ -\frac{6 \sqrt{1-a^2 x^2}}{a}+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]

[Out]

(-6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcSin[a*x] + (3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + x*ArcSin[a*x]^3

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Rubi [A]  time = 0.0801758, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4619, 4677, 261} \[ -\frac{6 \sqrt{1-a^2 x^2}}{a}+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^3,x]

[Out]

(-6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcSin[a*x] + (3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + x*ArcSin[a*x]^3

Rule 4619

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

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 261

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]

Rubi steps

\begin{align*} \int \sin ^{-1}(a x)^3 \, dx &=x \sin ^{-1}(a x)^3-(3 a) \int \frac{x \sin ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 \int \sin ^{-1}(a x) \, dx\\ &=-6 x \sin ^{-1}(a x)+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3+(6 a) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{6 \sqrt{1-a^2 x^2}}{a}-6 x \sin ^{-1}(a x)+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0112293, size = 60, normalized size = 1. \[ -\frac{6 \sqrt{1-a^2 x^2}}{a}+\frac{3 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)^2}{a}+x \sin ^{-1}(a x)^3-6 x \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^3,x]

[Out]

(-6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcSin[a*x] + (3*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^2)/a + x*ArcSin[a*x]^3

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Maple [A]  time = 0.024, size = 57, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arcsin \left ( ax \right ) \right ) ^{3}+3\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}-6\,\sqrt{-{a}^{2}{x}^{2}+1}-6\,ax\arcsin \left ( ax \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^3,x)

[Out]

1/a*(a*x*arcsin(a*x)^3+3*arcsin(a*x)^2*(-a^2*x^2+1)^(1/2)-6*(-a^2*x^2+1)^(1/2)-6*a*x*arcsin(a*x))

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Maxima [A]  time = 1.68067, size = 77, normalized size = 1.28 \begin{align*} x \arcsin \left (a x\right )^{3} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac{6 \,{\left (a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3,x, algorithm="maxima")

[Out]

x*arcsin(a*x)^3 + 3*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^2/a - 6*(a*x*arcsin(a*x) + sqrt(-a^2*x^2 + 1))/a

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Fricas [A]  time = 2.04743, size = 116, normalized size = 1.93 \begin{align*} \frac{a x \arcsin \left (a x\right )^{3} - 6 \, a x \arcsin \left (a x\right ) + 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (\arcsin \left (a x\right )^{2} - 2\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3,x, algorithm="fricas")

[Out]

(a*x*arcsin(a*x)^3 - 6*a*x*arcsin(a*x) + 3*sqrt(-a^2*x^2 + 1)*(arcsin(a*x)^2 - 2))/a

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Sympy [A]  time = 0.832359, size = 54, normalized size = 0.9 \begin{align*} \begin{cases} x \operatorname{asin}^{3}{\left (a x \right )} - 6 x \operatorname{asin}{\left (a x \right )} + \frac{3 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a x \right )}}{a} - \frac{6 \sqrt{- a^{2} x^{2} + 1}}{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,x)

[Out]

Piecewise((x*asin(a*x)**3 - 6*x*asin(a*x) + 3*sqrt(-a**2*x**2 + 1)*asin(a*x)**2/a - 6*sqrt(-a**2*x**2 + 1)/a,
Ne(a, 0)), (0, True))

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Giac [A]  time = 1.36035, size = 76, normalized size = 1.27 \begin{align*} x \arcsin \left (a x\right )^{3} - 6 \, x \arcsin \left (a x\right ) + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )^{2}}{a} - \frac{6 \, \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^3,x, algorithm="giac")

[Out]

x*arcsin(a*x)^3 - 6*x*arcsin(a*x) + 3*sqrt(-a^2*x^2 + 1)*arcsin(a*x)^2/a - 6*sqrt(-a^2*x^2 + 1)/a