3.464 \(\int \frac{x}{\sqrt{1-x^2} \sqrt{\sin ^{-1}(x)}} \, dx\)

Optimal. Leaf size=25 \[ \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(x)}\right ) \]

[Out]

Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[x]]]

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Rubi [A]  time = 0.0607728, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {4723, 3305, 3351} \[ \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - x^2]*Sqrt[ArcSin[x]]),x]

[Out]

Sqrt[2*Pi]*FresnelS[Sqrt[2/Pi]*Sqrt[ArcSin[x]]]

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 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(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 3351

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelS[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{x}{\sqrt{1-x^2} \sqrt{\sin ^{-1}(x)}} \, dx &=\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(x)\right )\\ &=2 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(x)}\right )\\ &=\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(x)}\right )\\ \end{align*}

Mathematica [C]  time = 0.0762037, size = 53, normalized size = 2.12 \[ -\frac{\sqrt{-i \sin ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(x)\right )+\sqrt{i \sin ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(x)\right )}{2 \sqrt{\sin ^{-1}(x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x/(Sqrt[1 - x^2]*Sqrt[ArcSin[x]]),x]

[Out]

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

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Maple [A]  time = 0.067, size = 20, normalized size = 0.8 \begin{align*}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arcsin \left ( x \right ) }} \right ) \sqrt{2}\sqrt{\pi } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+1)^(1/2)/arcsin(x)^(1/2),x)

[Out]

FresnelS(2^(1/2)/Pi^(1/2)*arcsin(x)^(1/2))*2^(1/2)*Pi^(1/2)

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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(x/(-x^2+1)^(1/2)/arcsin(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(x/(-x^2+1)^(1/2)/arcsin(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{x}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{\operatorname{asin}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+1)**(1/2)/asin(x)**(1/2),x)

[Out]

Integral(x/(sqrt(-(x - 1)*(x + 1))*sqrt(asin(x))), x)

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Giac [C]  time = 1.35562, size = 50, normalized size = 2. \begin{align*} \left (\frac{1}{4} i - \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\left (\frac{1}{2} i - \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (x\right )}\right ) - \left (\frac{1}{4} i + \frac{1}{4}\right ) \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\left (\frac{1}{2} i + \frac{1}{2}\right ) \, \sqrt{2} \sqrt{\arcsin \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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