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

Optimal. Leaf size=57 \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{x^3 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0486051, antiderivative size = 57, 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, 3302} \[ \frac{\text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{\text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )}{2 a^4}-\frac{x^3 \sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[x^3/ArcSin[a*x]^2,x]

[Out]

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

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 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rubi steps

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

Mathematica [A]  time = 0.0168856, size = 56, normalized size = 0.98 \[ \frac{4 \sin ^{-1}(a x) \text{CosIntegral}\left (2 \sin ^{-1}(a x)\right )-4 \sin ^{-1}(a x) \text{CosIntegral}\left (4 \sin ^{-1}(a x)\right )-2 \sin \left (2 \sin ^{-1}(a x)\right )+\sin \left (4 \sin ^{-1}(a x)\right )}{8 a^4 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/ArcSin[a*x]^2,x]

[Out]

(4*ArcSin[a*x]*CosIntegral[2*ArcSin[a*x]] - 4*ArcSin[a*x]*CosIntegral[4*ArcSin[a*x]] - 2*Sin[2*ArcSin[a*x]] +
Sin[4*ArcSin[a*x]])/(8*a^4*ArcSin[a*x])

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Maple [A]  time = 0.021, size = 54, normalized size = 1. \begin{align*}{\frac{1}{{a}^{4}} \left ( -{\frac{\sin \left ( 2\,\arcsin \left ( ax \right ) \right ) }{4\,\arcsin \left ( ax \right ) }}+{\frac{{\it Ci} \left ( 2\,\arcsin \left ( ax \right ) \right ) }{2}}+{\frac{\sin \left ( 4\,\arcsin \left ( ax \right ) \right ) }{8\,\arcsin \left ( ax \right ) }}-{\frac{{\it Ci} \left ( 4\,\arcsin \left ( ax \right ) \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^4*(-1/4/arcsin(a*x)*sin(2*arcsin(a*x))+1/2*Ci(2*arcsin(a*x))+1/8/arcsin(a*x)*sin(4*arcsin(a*x))-1/2*Ci(4*a
rcsin(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^3/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^{3}}{\arcsin \left (a x\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(x^3/arcsin(a*x)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/asin(a*x)**2,x)

[Out]

Integral(x**3/asin(a*x)**2, x)

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Giac [A]  time = 1.38474, size = 97, normalized size = 1.7 \begin{align*} \frac{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x}{a^{3} \arcsin \left (a x\right )} - \frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{3} \arcsin \left (a x\right )} - \frac{\operatorname{Ci}\left (4 \, \arcsin \left (a x\right )\right )}{2 \, a^{4}} + \frac{\operatorname{Ci}\left (2 \, \arcsin \left (a x\right )\right )}{2 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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