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

Optimal. Leaf size=31 \[ -\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.022666, antiderivative size = 31, 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 = {4624, 3305, 3351} \[ -\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4624

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Sin[a/b - x/b], x], x, a
+ b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, n}, 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]

Rubi steps

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

Mathematica [C]  time = 0.0242184, size = 68, normalized size = 2.19 \[ -\frac{-\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )-\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )}{2 a \sqrt{\cos ^{-1}(a x)}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 26, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{a}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acos(a*x)**(1/2),x)

[Out]

Integral(1/sqrt(acos(a*x)), x)

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Giac [B]  time = 1.24571, size = 92, normalized size = 2.97 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{2 \, a{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{2 \, a{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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