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

Optimal. Leaf size=30 \[ \frac{\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a} \]

[Out]

(Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/a

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Rubi [A]  time = 0.0244487, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4623, 3304, 3352} \[ \frac{\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[ArcSin[a*x]],x]

[Out]

(Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[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 3304

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

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rubi steps

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

Mathematica [C]  time = 0.0256741, size = 69, normalized size = 2.3 \[ -\frac{i \left (\sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )-\sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )\right )}{2 a \sqrt{\sin ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[ArcSin[a*x]],x]

[Out]

((-I/2)*(Sqrt[(-I)*ArcSin[a*x]]*Gamma[1/2, (-I)*ArcSin[a*x]] - Sqrt[I*ArcSin[a*x]]*Gamma[1/2, I*ArcSin[a*x]]))
/(a*Sqrt[ArcSin[a*x]])

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Maple [A]  time = 0.028, size = 25, normalized size = 0.8 \begin{align*}{\frac{\sqrt{2}\sqrt{\pi }}{a}{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( ax \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))*2^(1/2)*Pi^(1/2)/a

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\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)**(1/2),x)

[Out]

Integral(1/sqrt(asin(a*x)), x)

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Giac [C]  time = 1.36933, size = 63, normalized size = 2.1 \begin{align*} -\frac{\left (i + 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{4 \, a} + \frac{\left (i - 1\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (a x\right )}\right )}{4 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1/4*I + 1/4)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a + (1/4*I - 1/4)*sqrt(2)*sqrt(pi
)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a