3.1.40 \(\int \cos ^{-1}(x) \, dx\) [40]

Optimal. Leaf size=18 \[ -\sqrt {1-x^2}+x \cos ^{-1}(x) \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4716, 267} \begin {gather*} x \cos ^{-1}(x)-\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 18, normalized size = 1.00 \begin {gather*} -\sqrt {1-x^2}+x \cos ^{-1}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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Mathics [A]
time = 1.78, size = 16, normalized size = 0.89 \begin {gather*} x \text {ArcCos}\left [x\right ]-\sqrt {1-x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[ArcCos[x],x]')

[Out]

x ArcCos[x] - Sqrt[1 - x ^ 2]

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Maple [A]
time = 0.00, size = 17, normalized size = 0.94

method	result	size
lookup	\(x \arccos \left (x \right )-\sqrt {-x^{2}+1}\)	\(17\)
default	\(x \arccos \left (x \right )-\sqrt {-x^{2}+1}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.34, size = 16, normalized size = 0.89 \begin {gather*} x \arccos \left (x\right ) - \sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x),x, algorithm="maxima")

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

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Fricas [A]
time = 0.34, size = 16, normalized size = 0.89 \begin {gather*} x \arccos \left (x\right ) - \sqrt {-x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x),x, algorithm="fricas")

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

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Sympy [A]
time = 0.06, size = 12, normalized size = 0.67 \begin {gather*} x \operatorname {acos}{\left (x \right )} - \sqrt {1 - x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(x),x)

[Out]

x*acos(x) - sqrt(1 - x**2)

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Giac [A]
time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} x \arccos x-\sqrt {1-x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(x),x)

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

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Mupad [B]
time = 0.17, size = 16, normalized size = 0.89 \begin {gather*} x\,\mathrm {acos}\left (x\right )-\sqrt {1-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(x),x)

[Out]

x*acos(x) - (1 - x^2)^(1/2)

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