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

Optimal. Leaf size=51 \[ -\frac{\sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{2 a}+\frac{x}{2 \sin ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0845015, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4621, 4719, 4623, 3302} \[ -\frac{\sqrt{1-a^2 x^2}}{2 a \sin ^{-1}(a x)^2}-\frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{2 a}+\frac{x}{2 \sin ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^(-3),x]

[Out]

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

Rule 4621

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^(n + 1))
/(b*c*(n + 1)), x] + Dist[c/(b*(n + 1)), Int[(x*(a + b*ArcSin[c*x])^(n + 1))/Sqrt[1 - c^2*x^2], x], x] /; Free
Q[{a, b, c}, x] && LtQ[n, -1]

Rule 4719

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
((f*x)^m*(a + b*ArcSin[c*x])^(n + 1))/(b*c*Sqrt[d]*(n + 1)), x] - Dist[(f*m)/(b*c*Sqrt[d]*(n + 1)), Int[(f*x)^
(m - 1)*(a + b*ArcSin[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && LtQ[n,
-1] && GtQ[d, 0]

Rule 4623

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[a/b - x/b], x], x, a
+ b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

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

Mathematica [A]  time = 0.0224016, size = 48, normalized size = 0.94 \[ -\frac{\sqrt{1-a^2 x^2}+\sin ^{-1}(a x)^2 \text{CosIntegral}\left (\sin ^{-1}(a x)\right )-a x \sin ^{-1}(a x)}{2 a \sin ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^(-3),x]

[Out]

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

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Maple [A]  time = 0.021, size = 43, normalized size = 0.8 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{2\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{ax}{2\,\arcsin \left ( ax \right ) }}-{\frac{{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin(a*x)^3,x, algorithm="maxima")

[Out]

-1/2*(a*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^2*integrate(1/arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)), x
) - a*x*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) + sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*arctan2(a*x, sqrt(a*x +
1)*sqrt(-a*x + 1))^2)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\arcsin \left (a x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arcsin(a*x)^3,x, algorithm="fricas")

[Out]

integral(arcsin(a*x)^(-3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{asin}^{3}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/asin(a*x)**3,x)

[Out]

Integral(asin(a*x)**(-3), x)

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Giac [A]  time = 1.33887, size = 58, normalized size = 1.14 \begin{align*} \frac{x}{2 \, \arcsin \left (a x\right )} - \frac{\operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{2 \, a} - \frac{\sqrt{-a^{2} x^{2} + 1}}{2 \, a \arcsin \left (a x\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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