3.40 \(\int \cos ^{-1}(x) \, dx\)

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

[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 = {4620, 261} \[ x \cos ^{-1}(x)-\sqrt {1-x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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]

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]

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 \[ x \cos ^{-1}(x)-\sqrt {1-x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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fricas [A]  time = 0.42, size = 16, normalized size = 0.89 \[ x \arccos \relax (x) - \sqrt {-x^{2} + 1} \]

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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giac [A]  time = 0.82, size = 16, normalized size = 0.89 \[ x \arccos \relax (x) - \sqrt {-x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

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maple [A]  time = 0.00, size = 17, normalized size = 0.94 \[ x \arccos \relax (x )-\sqrt {-x^{2}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(x),x)

[Out]

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

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maxima [A]  time = 0.98, size = 16, normalized size = 0.89 \[ x \arccos \relax (x) - \sqrt {-x^{2} + 1} \]

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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mupad [B]  time = 0.17, size = 16, normalized size = 0.89 \[ x\,\mathrm {acos}\relax (x)-\sqrt {1-x^2} \]

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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sympy [A]  time = 0.13, size = 12, normalized size = 0.67 \[ x \operatorname {acos}{\relax (x )} - \sqrt {1 - x^{2}} \]

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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