3.115 \(\int \frac{x}{\sqrt{1-x^2} \sqrt{\cos ^{-1}(x)}} \, dx\)

Optimal. Leaf size=26 \[ -\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(x)}\right ) \]

[Out]

-(Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[x]]])

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Rubi [A]  time = 0.0706919, antiderivative size = 26, 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 = {4724, 3304, 3352} \[ -\sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(x)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcCos[x]]])

Rule 4724

Int[((a_.) + ArcCos[(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*Cos[x]^m*Sin[x]^(2*p + 1), x], x, ArcCos[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 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{x}{\sqrt{1-x^2} \sqrt{\cos ^{-1}(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(x)}\right )\right )\\ &=-\sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(x)}\right )\\ \end{align*}

Mathematica [C]  time = 0.0848823, size = 56, normalized size = 2.15 \[ \frac{i \left (\sqrt{-i \cos ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(x)\right )-\sqrt{i \cos ^{-1}(x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(x)\right )\right )}{2 \sqrt{\cos ^{-1}(x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((I/2)*(Sqrt[(-I)*ArcCos[x]]*Gamma[1/2, (-I)*ArcCos[x]] - Sqrt[I*ArcCos[x]]*Gamma[1/2, I*ArcCos[x]]))/Sqrt[Arc
Cos[x]]

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Maple [A]  time = 0.106, size = 21, normalized size = 0.8 \begin{align*} -{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \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)/arccos(x)^(1/2),x)

[Out]

-FresnelC(2^(1/2)/Pi^(1/2)*arccos(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)/arccos(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)/arccos(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{acos}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.39764, size = 78, normalized size = 3. \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (x\right )}}{i - 1}\right )}{2 \,{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (x\right )}}{i - 1}\right )}{2 \,{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+1)^(1/2)/arccos(x)^(1/2),x, algorithm="giac")

[Out]

1/2*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(x))/(i - 1))/(i - 1) - 1/2*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt
(arccos(x))/(i - 1))/(i - 1)