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

Optimal. Leaf size=9 \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a^2} \]

[Out]

SinIntegral[ArcSin[a*x]]/a^2

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Rubi [A]  time = 0.0750753, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {4723, 3299} \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinIntegral[ArcSin[a*x]]/a^2

Rule 4723

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[d^p/c^(
m + 1), Subst[Int[(a + b*x)^n*Sin[x]^m*Cos[x]^(2*p + 1), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n},
x] && EqQ[c^2*d + e, 0] && IntegerQ[2*p] && GtQ[p, -1] && IGtQ[m, 0] && (IntegerQ[p] || GtQ[d, 0])

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^2}\\ &=\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a^2}\\ \end{align*}

Mathematica [A]  time = 0.0482187, size = 9, normalized size = 1. \[ \frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinIntegral[ArcSin[a*x]]/a^2

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Maple [A]  time = 0.037, size = 10, normalized size = 1.1 \begin{align*}{\frac{{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{{a}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Si(arcsin(a*x))/a^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{-a^{2} x^{2} + 1} \arcsin \left (a x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-a^{2} x^{2} + 1} x}{{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname{asin}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(sqrt(-(a*x - 1)*(a*x + 1))*asin(a*x)), x)

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Giac [A]  time = 1.37371, size = 12, normalized size = 1.33 \begin{align*} \frac{\operatorname{Si}\left (\arcsin \left (a x\right )\right )}{a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin_integral(arcsin(a*x))/a^2