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

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

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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Rubi [A]  time = 0.0041468, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 2, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 1., 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 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 \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*}

Mathematica [A]  time = 0.0024326, size = 18, normalized size = 1. $x \cos ^{-1}(x)-\sqrt{1-x^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

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Maple [A]  time = 0.001, size = 17, normalized size = 0.9 \begin{align*} x\arccos \left ( x \right ) -\sqrt{-{x}^{2}+1} \end{align*}

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 = 1.40595, size = 22, normalized size = 1.22 \begin{align*} x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \end{align*}

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 = 2.39014, size = 41, normalized size = 2.28 \begin{align*} x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \end{align*}

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 = 0.117869, size = 12, normalized size = 0.67 \begin{align*} x \operatorname{acos}{\left (x \right )} - \sqrt{1 - x^{2}} \end{align*}

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 = 1.04753, size = 22, normalized size = 1.22 \begin{align*} x \arccos \left (x\right ) - \sqrt{-x^{2} + 1} \end{align*}

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)