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

Optimal. Leaf size=9 \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{a} \]

[Out]

CosIntegral[ArcSin[a*x]]/a

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Rubi [A]  time = 0.0168823, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4623, 3302} \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcSin[a*x]^(-1),x]

[Out]

CosIntegral[ArcSin[a*x]]/a

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

Mathematica [A]  time = 0.0094762, size = 9, normalized size = 1. \[ \frac{\text{CosIntegral}\left (\sin ^{-1}(a x)\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcSin[a*x]^(-1),x]

[Out]

CosIntegral[ArcSin[a*x]]/a

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Maple [A]  time = 0.017, size = 10, normalized size = 1.1 \begin{align*}{\frac{{\it Ci} \left ( \arcsin \left ( ax \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Ci(arcsin(a*x))/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/arcsin(a*x), x)

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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 )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/arcsin(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/asin(a*x), x)

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Giac [A]  time = 1.35593, size = 12, normalized size = 1.33 \begin{align*} \frac{\operatorname{Ci}\left (\arcsin \left (a x\right )\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cos_integral(arcsin(a*x))/a