3.1.68 \(\int \frac {\text {ArcCos}(\sqrt {x})}{\sqrt {x}} \, dx\) [68]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6847, 4716, 267} \begin {gather*} 2 \sqrt {x} \text {ArcCos}\left (\sqrt {x}\right )-2 \sqrt {1-x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 267

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]

Rule 4716

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 6847

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]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} -2 \sqrt {1-x}+2 \sqrt {x} \text {ArcCos}\left (\sqrt {x}\right ) \end {gather*}

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 = 0.00, size = 20, normalized size = 0.80

method	result	size
derivativedivides	\(-2 \sqrt {1-x}+2 \arccos \left (\sqrt {x}\right ) \sqrt {x}\)	\(20\)
default	\(-2 \sqrt {1-x}+2 \arccos \left (\sqrt {x}\right ) \sqrt {x}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(x^(1/2))/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.46, size = 19, normalized size = 0.76 \begin {gather*} 2 \, \sqrt {x} \arccos \left (\sqrt {x}\right ) - 2 \, \sqrt {-x + 1} \end {gather*}

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 = 1.24, size = 19, normalized size = 0.76 \begin {gather*} 2 \, \sqrt {x} \arccos \left (\sqrt {x}\right ) - 2 \, \sqrt {-x + 1} \end {gather*}

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 = 0.10, size = 20, normalized size = 0.80 \begin {gather*} 2 \sqrt {x} \operatorname {acos}{\left (\sqrt {x} \right )} - 2 \sqrt {1 - x} \end {gather*}

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 = 0.42, size = 19, normalized size = 0.76 \begin {gather*} 2 \, \sqrt {x} \arccos \left (\sqrt {x}\right ) - 2 \, \sqrt {-x + 1} \end {gather*}

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)

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Mupad [B]
time = 0.40, size = 19, normalized size = 0.76 \begin {gather*} 2\,\sqrt {x}\,\mathrm {acos}\left (\sqrt {x}\right )-2\,\sqrt {1-x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x^(1/2)*acos(x^(1/2)) - 2*(1 - x)^(1/2)

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