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

Optimal. Leaf size=25 \[ 2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )-2 \sqrt{1-x} \]

[Out]

-2*Sqrt[1 - x] + 2*Sqrt[x]*ArcCos[Sqrt[x]]

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Rubi [A]  time = 0.0187569, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6715, 4620, 261} \[ 2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )-2 \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[Sqrt[x]]/Sqrt[x],x]

[Out]

-2*Sqrt[1 - x] + 2*Sqrt[x]*ArcCos[Sqrt[x]]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0078178, size = 25, normalized size = 1. \[ 2 \sqrt{x} \cos ^{-1}\left (\sqrt{x}\right )-2 \sqrt{1-x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[Sqrt[x]]/Sqrt[x],x]

[Out]

-2*Sqrt[1 - x] + 2*Sqrt[x]*ArcCos[Sqrt[x]]

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Maple [A]  time = 0.003, size = 20, normalized size = 0.8 \begin{align*} -2\,\sqrt{1-x}+2\,\arccos \left ( \sqrt{x} \right ) \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(1-x)^(1/2)+2*arccos(x^(1/2))*x^(1/2)

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Maxima [A]  time = 1.48198, size = 26, normalized size = 1.04 \begin{align*} 2 \, \sqrt{x} \arccos \left (\sqrt{x}\right ) - 2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x^(1/2))/x^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(x)*arccos(sqrt(x)) - 2*sqrt(-x + 1)

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Fricas [A]  time = 2.53294, size = 59, normalized size = 2.36 \begin{align*} 2 \, \sqrt{x} \arccos \left (\sqrt{x}\right ) - 2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x^(1/2))/x^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(x)*arccos(sqrt(x)) - 2*sqrt(-x + 1)

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Sympy [A]  time = 0.365094, size = 20, normalized size = 0.8 \begin{align*} 2 \sqrt{x} \operatorname{acos}{\left (\sqrt{x} \right )} - 2 \sqrt{1 - x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*acos(sqrt(x)) - 2*sqrt(1 - x)

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Giac [A]  time = 1.2737, size = 26, normalized size = 1.04 \begin{align*} 2 \, \sqrt{x} \arccos \left (\sqrt{x}\right ) - 2 \, \sqrt{-x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(x)*arccos(sqrt(x)) - 2*sqrt(-x + 1)