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

Optimal. Leaf size=55 \[ -\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}-\frac{x^2 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

[Out]

-((x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/(4*a^3) + (3*SinIntegral[3*ArcSin[a*x]])
/(4*a^3)

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Rubi [A]  time = 0.0439644, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4631, 3299} \[ -\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}-\frac{x^2 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((x^2*Sqrt[1 - a^2*x^2])/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/(4*a^3) + (3*SinIntegral[3*ArcSin[a*x]])
/(4*a^3)

Rule 4631

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[(x^m*Sqrt[1 - c^2*x^2]*(a + b*ArcSin
[c*x])^(n + 1))/(b*c*(n + 1)), x] - Dist[1/(b*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[(a + b*x)^(n + 1)
, Sin[x]^(m - 1)*(m - (m + 1)*Sin[x]^2), x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && G
eQ[n, -2] && LtQ[n, -1]

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^2}{\sin ^{-1}(a x)^2} \, dx &=-\frac{x^2 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\sin (x)}{4 x}+\frac{3 \sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^3}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^3}\\ &=-\frac{x^2 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{4 a^3}+\frac{3 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3}\\ \end{align*}

Mathematica [A]  time = 0.164837, size = 50, normalized size = 0.91 \[ -\frac{\frac{4 a^2 x^2 \sqrt{1-a^2 x^2}}{\sin ^{-1}(a x)}+\text{Si}\left (\sin ^{-1}(a x)\right )-3 \text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((4*a^2*x^2*Sqrt[1 - a^2*x^2])/ArcSin[a*x] + SinIntegral[ArcSin[a*x]] - 3*SinIntegral[3*ArcSin[a*x]])/(4*a^3)

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Maple [A]  time = 0.023, size = 57, normalized size = 1. \begin{align*}{\frac{1}{{a}^{3}} \left ( -{\frac{1}{4\,\arcsin \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{4}}+{\frac{\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) }{4\,\arcsin \left ( ax \right ) }}+{\frac{3\,{\it Si} \left ( 3\,\arcsin \left ( ax \right ) \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^3*(-1/4/arcsin(a*x)*(-a^2*x^2+1)^(1/2)-1/4*Si(arcsin(a*x))+1/4/arcsin(a*x)*cos(3*arcsin(a*x))+3/4*Si(3*arc
sin(a*x)))

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^2/arcsin(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/asin(a*x)**2,x)

[Out]

Integral(x**2/asin(a*x)**2, x)

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Giac [A]  time = 1.35974, size = 92, normalized size = 1.67 \begin{align*} \frac{3 \, \operatorname{Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a^{3}} - \frac{\operatorname{Si}\left (\arcsin \left (a x\right )\right )}{4 \, a^{3}} + \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{a^{3} \arcsin \left (a x\right )} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a^{3} \arcsin \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*sin_integral(3*arcsin(a*x))/a^3 - 1/4*sin_integral(arcsin(a*x))/a^3 + (-a^2*x^2 + 1)^(3/2)/(a^3*arcsin(a*x
)) - sqrt(-a^2*x^2 + 1)/(a^3*arcsin(a*x))