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

Optimal. Leaf size=44 \[ x \sqrt{\cos ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0900672, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4724, 3304, 3352} \[ x \sqrt{\cos ^{-1}(a x)}-\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4620

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCos[c*x])^n, x] + Dist[b*c*n, Int[
(x*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

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

Mathematica [C]  time = 0.0339111, size = 76, normalized size = 1.73 \[ \frac{\sqrt{\cos ^{-1}(a x)} \left (\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},-i \cos ^{-1}(a x)\right )+\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{3}{2},i \cos ^{-1}(a x)\right )\right )}{2 a \sqrt{\cos ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[ArcCos[a*x]]*(Sqrt[I*ArcCos[a*x]]*Gamma[3/2, (-I)*ArcCos[a*x]] + Sqrt[(-I)*ArcCos[a*x]]*Gamma[3/2, I*Arc
Cos[a*x]]))/(2*a*Sqrt[ArcCos[a*x]^2])

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Maple [A]  time = 0.065, size = 49, normalized size = 1.1 \begin{align*}{\frac{1}{2\,a} \left ( -\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}}{\sqrt{\pi }}\sqrt{\arccos \left ( ax \right ) }} \right ) +2\,ax\arccos \left ( ax \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(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(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 \sqrt{\operatorname{acos}{\left (a x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(acos(a*x)), x)

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Giac [B]  time = 1.24525, size = 142, normalized size = 3.23 \begin{align*} \frac{\sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (\frac{\sqrt{2} \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{4 \, a{\left (i - 1\right )}} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (i \arccos \left (a x\right )\right )}}{2 \, a} + \frac{\sqrt{\arccos \left (a x\right )} e^{\left (-i \arccos \left (a x\right )\right )}}{2 \, a} - \frac{\sqrt{2} \sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{2} i \sqrt{\arccos \left (a x\right )}}{i - 1}\right )}{4 \, a{\left (i - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(2)*sqrt(pi)*i*erf(sqrt(2)*sqrt(arccos(a*x))/(i - 1))/(a*(i - 1)) + 1/2*sqrt(arccos(a*x))*e^(i*arccos(
a*x))/a + 1/2*sqrt(arccos(a*x))*e^(-i*arccos(a*x))/a - 1/4*sqrt(2)*sqrt(pi)*erf(-sqrt(2)*i*sqrt(arccos(a*x))/(
i - 1))/(a*(i - 1))