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

Optimal. Leaf size=26 \[ x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a} \]

[Out]

-(Sqrt[1 - a^2*x^2]/a) + x*ArcCos[a*x]

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Rubi [A]  time = 0.0075295, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 261} \[ x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x],x]

[Out]

-(Sqrt[1 - a^2*x^2]/a) + x*ArcCos[a*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}(a x) \, dx &=x \cos ^{-1}(a x)+a \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\sqrt{1-a^2 x^2}}{a}+x \cos ^{-1}(a x)\\ \end{align*}

Mathematica [A]  time = 0.0072581, size = 26, normalized size = 1. \[ x \cos ^{-1}(a x)-\frac{\sqrt{1-a^2 x^2}}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x],x]

[Out]

-(Sqrt[1 - a^2*x^2]/a) + x*ArcCos[a*x]

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Maple [A]  time = 0.002, size = 27, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax\arccos \left ( ax \right ) -\sqrt{-{a}^{2}{x}^{2}+1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x),x)

[Out]

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

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Maxima [A]  time = 1.44063, size = 35, normalized size = 1.35 \begin{align*} \frac{a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.91552, size = 57, normalized size = 2.19 \begin{align*} \frac{a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.150393, size = 24, normalized size = 0.92 \begin{align*} \begin{cases} x \operatorname{acos}{\left (a x \right )} - \frac{\sqrt{- a^{2} x^{2} + 1}}{a} & \text{for}\: a \neq 0 \\\frac{\pi x}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x),x)

[Out]

Piecewise((x*acos(a*x) - sqrt(-a**2*x**2 + 1)/a, Ne(a, 0)), (pi*x/2, True))

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Giac [A]  time = 1.15976, size = 35, normalized size = 1.35 \begin{align*} \frac{a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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