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

Optimal. Leaf size=33 \[ -\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b} \]

[Out]

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

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Rubi [A]  time = 0.0297642, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4804, 4624, 3305, 3351} \[ -\frac{\sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a+b x)}\right )}{b} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[ArcCos[a + b*x]],x]

[Out]

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

Rule 4804

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

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

Mathematica [C]  time = 0.0316839, size = 78, normalized size = 2.36 \[ -\frac{-\sqrt{-i \cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a+b x)\right )-\sqrt{i \cos ^{-1}(a+b x)} \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a+b x)\right )}{2 b \sqrt{\cos ^{-1}(a+b x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/Sqrt[ArcCos[a + b*x]],x]

[Out]

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

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Maple [A]  time = 0.055, size = 28, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}\sqrt{\pi }}{b}{\it FresnelS} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( bx+a \right ) }} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-FresnelS(2^(1/2)/Pi^(1/2)*arccos(b*x+a)^(1/2))*2^(1/2)*Pi^(1/2)/b

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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(b*x+a)^(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(b*x+a)^(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 + b x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(acos(a + b*x)), x)

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Giac [B]  time = 1.57481, size = 97, normalized size = 2.94 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{2 \, b{\left (i - 1\right )}} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (b x + a\right )}}{i - 1}\right )}{2 \, b{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(2)*sqrt(pi)*i*erf(-sqrt(2)*i*sqrt(arccos(b*x + a))/(i - 1))/(b*(i - 1)) - 1/2*sqrt(2)*sqrt(pi)*erf(sq
rt(2)*sqrt(arccos(b*x + a))/(i - 1))/(b*(i - 1))