3.16 \(\int \sin ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=35 \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]

[Out]

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

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Rubi [A]  time = 0.0449068, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4619, 4677, 8} \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^2,x]

[Out]

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

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0109879, size = 35, normalized size = 1. \[ \frac{2 \sqrt{1-a^2 x^2} \sin ^{-1}(a x)}{a}+x \sin ^{-1}(a x)^2-2 x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^2,x]

[Out]

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

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Maple [A]  time = 0.021, size = 37, normalized size = 1.1 \begin{align*}{\frac{1}{a} \left ( ax \left ( \arcsin \left ( ax \right ) \right ) ^{2}-2\,ax+2\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsin(a*x)^2,x)

[Out]

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

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Maxima [A]  time = 1.82134, size = 45, normalized size = 1.29 \begin{align*} x \arcsin \left (a x\right )^{2} - 2 \, x + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.35201, size = 89, normalized size = 2.54 \begin{align*} \frac{a x \arcsin \left (a x\right )^{2} - 2 \, a x + 2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.23858, size = 32, normalized size = 0.91 \begin{align*} \begin{cases} x \operatorname{asin}^{2}{\left (a x \right )} - 2 x + \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{asin}{\left (a x \right )}}{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)**2,x)

[Out]

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

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Giac [A]  time = 1.34824, size = 45, normalized size = 1.29 \begin{align*} x \arcsin \left (a x\right )^{2} - 2 \, x + \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsin(a*x)^2,x, algorithm="giac")

[Out]

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