3.342 \(\int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx\)

Optimal. Leaf size=27 \[ \frac{3 \text{Si}\left (\sin ^{-1}(a x)\right )}{4 a^4}-\frac{\text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^4} \]

[Out]

(3*SinIntegral[ArcSin[a*x]])/(4*a^4) - SinIntegral[3*ArcSin[a*x]]/(4*a^4)

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Rubi [A]  time = 0.145394, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {4723, 3312, 3299} \[ \frac{3 \text{Si}\left (\sin ^{-1}(a x)\right )}{4 a^4}-\frac{\text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]

[Out]

(3*SinIntegral[ArcSin[a*x]])/(4*a^4) - SinIntegral[3*ArcSin[a*x]]/(4*a^4)

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 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 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^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sin ^3(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 x}-\frac{\sin (3 x)}{4 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^4}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^4}\\ &=\frac{3 \text{Si}\left (\sin ^{-1}(a x)\right )}{4 a^4}-\frac{\text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0604065, size = 24, normalized size = 0.89 \[ \frac{3 \text{Si}\left (\sin ^{-1}(a x)\right )-\text{Si}\left (3 \sin ^{-1}(a x)\right )}{4 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(Sqrt[1 - a^2*x^2]*ArcSin[a*x]),x]

[Out]

(3*SinIntegral[ArcSin[a*x]] - SinIntegral[3*ArcSin[a*x]])/(4*a^4)

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Maple [A]  time = 0.049, size = 21, normalized size = 0.8 \begin{align*} -{\frac{{\it Si} \left ( 3\,\arcsin \left ( ax \right ) \right ) -3\,{\it Si} \left ( \arcsin \left ( ax \right ) \right ) }{4\,{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(Si(3*arcsin(a*x))-3*Si(arcsin(a*x)))/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\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^3/arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(x^3/(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^{3}}{{\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^3/arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*x^3/((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^{3}}{\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**3/asin(a*x)/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(x**3/(sqrt(-(a*x - 1)*(a*x + 1))*asin(a*x)), x)

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Giac [A]  time = 1.35669, size = 31, normalized size = 1.15 \begin{align*} -\frac{\operatorname{Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a^{4}} + \frac{3 \, \operatorname{Si}\left (\arcsin \left (a x\right )\right )}{4 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arcsin(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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