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

Optimal. Leaf size=25 \[ \frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x) \]

[Out]

Sqrt[1 - a^2*x^2]/a + x*ArcSin[a*x]

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Rubi [A]  time = 0.0075506, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4619, 261} \[ \frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x],x]

[Out]

Sqrt[1 - a^2*x^2]/a + x*ArcSin[a*x]

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

Mathematica [A]  time = 0.0075927, size = 25, normalized size = 1. \[ \frac{\sqrt{1-a^2 x^2}}{a}+x \sin ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x],x]

[Out]

Sqrt[1 - a^2*x^2]/a + x*ArcSin[a*x]

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Maple [A]  time = 0.001, size = 25, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax\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),x)

[Out]

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

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Maxima [A]  time = 1.68877, size = 32, normalized size = 1.28 \begin{align*} \frac{a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.10614, size = 57, normalized size = 2.28 \begin{align*} \frac{a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.618542, size = 20, normalized size = 0.8 \begin{align*} \begin{cases} x \operatorname{asin}{\left (a x \right )} + \frac{\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),x)

[Out]

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

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Giac [A]  time = 1.33051, size = 32, normalized size = 1.28 \begin{align*} \frac{a x \arcsin \left (a x\right ) + \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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