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

Optimal. Leaf size=36 \[ -\frac{\sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0779915, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4621, 4723, 3299} \[ -\frac{\sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/(a*ArcSin[a*x])) - SinIntegral[ArcSin[a*x]]/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 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{1}{\sin ^{-1}(a x)^2} \, dx &=-\frac{\sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-a \int \frac{x}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)} \, dx\\ &=-\frac{\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 )}{a}\\ &=-\frac{\sqrt{1-a^2 x^2}}{a \sin ^{-1}(a x)}-\frac{\text{Si}\left (\sin ^{-1}(a x)\right )}{a}\\ \end{align*}

Mathematica [A]  time = 0.0590502, size = 32, normalized size = 0.89 \[ -\frac{\frac{\sqrt{1-a^2 x^2}}{\sin ^{-1}(a x)}+\text{Si}\left (\sin ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.019, size = 33, normalized size = 0.9 \begin{align*}{\frac{1}{a} \left ( -{\frac{1}{\arcsin \left ( ax \right ) }\sqrt{-{a}^{2}{x}^{2}+1}}-{\it Si} \left ( \arcsin \left ( ax \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arcsin(a*x)^(-2), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\operatorname{asin}^{2}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asin(a*x)**(-2), x)

________________________________________________________________________________________

Giac [A]  time = 1.33019, size = 46, normalized size = 1.28 \begin{align*} -\frac{\operatorname{Si}\left (\arcsin \left (a x\right )\right )}{a} - \frac{\sqrt{-a^{2} x^{2} + 1}}{a \arcsin \left (a x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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