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]

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

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Rubi [A]  time = 0.06, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [C]  time = 0.09, size = 53, normalized size = 2.12 \[ -\frac {\sqrt {-i \sin ^{-1}(x)} \Gamma \left (\frac {1}{2},-i \sin ^{-1}(x)\right )+\sqrt {i \sin ^{-1}(x)} \Gamma \left (\frac {1}{2},i \sin ^{-1}(x)\right )}{2 \sqrt {\sin ^{-1}(x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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giac [C]  time = 0.77, size = 37, normalized size = 1.48 \[ \left (\frac {1}{4} i - \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \relax (x)}\right ) - \left (\frac {1}{4} i + \frac {1}{4}\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arcsin \relax (x)}\right ) \]

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)))

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maple [A]  time = 0.13, size = 20, normalized size = 0.80 \[ \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \relax (x )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\pi } \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

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 >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x}{\sqrt {\mathrm {asin}\relax (x)}\,\sqrt {1-x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(asin(x)^(1/2)*(1 - x^2)^(1/2)),x)

[Out]

int(x/(asin(x)^(1/2)*(1 - x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )} \sqrt {\operatorname {asin}{\relax (x )}}}\, dx \]

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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