\(\int \arccos (x) \, dx\) [40]

Optimal result

Integrand size = 2, antiderivative size = 18 \[ \int \arccos (x) \, dx=-\sqrt {1-x^2}+x \arccos (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {4716, 267} \[ \int \arccos (x) \, dx=x \arccos (x)-\sqrt {1-x^2} \]

[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*} \text {integral}& = x \arccos (x)+\int \frac {x}{\sqrt {1-x^2}} \, dx \\ & = -\sqrt {1-x^2}+x \arccos (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \arccos (x) \, dx=-\sqrt {1-x^2}+x \arccos (x) \]

[In]

Integrate[ArcCos[x],x]

[Out]

-Sqrt[1 - x^2] + x*ArcCos[x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \arccos (x) \, dx=x \arccos \left (x\right ) - \sqrt {-x^{2} + 1} \]

[In]

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.67 \[ \int \arccos (x) \, dx=x \operatorname {acos}{\left (x \right )} - \sqrt {1 - x^{2}} \]

[In]

integrate(acos(x),x)

[Out]

x*acos(x) - sqrt(1 - x**2)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \arccos (x) \, dx=x \arccos \left (x\right ) - \sqrt {-x^{2} + 1} \]

[In]

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \arccos (x) \, dx=x \arccos \left (x\right ) - \sqrt {-x^{2} + 1} \]

[In]

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

[Out]

x*arccos(x) - sqrt(-x^2 + 1)

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \arccos (x) \, dx=x\,\mathrm {acos}\left (x\right )-\sqrt {1-x^2} \]

[In]

int(acos(x),x)

[Out]

x*acos(x) - (1 - x^2)^(1/2)