3.16 \(\int \cos ^{-1}(a x)^2 \, dx\)

Optimal. Leaf size=35 \[ -\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x \]

[Out]

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

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Rubi [A]  time = 0.0468431, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4678, 8} \[ -\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^2,x]

[Out]

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

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 4678

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcCos[c*x])^n)/(2*e*(p + 1)), x] - Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

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

Mathematica [A]  time = 0.0161168, size = 35, normalized size = 1. \[ -\frac{2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^2-2 x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^2,x]

[Out]

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

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Maple [A]  time = 0.046, size = 37, normalized size = 1.1 \begin{align*}{\frac{1}{a} \left ( ax \left ( \arccos \left ( ax \right ) \right ) ^{2}-2\,ax-2\,\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)^2,x)

[Out]

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

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Maxima [A]  time = 1.43443, size = 45, normalized size = 1.29 \begin{align*} x \arccos \left (a x\right )^{2} - 2 \, x - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.87394, size = 89, normalized size = 2.54 \begin{align*} \frac{a x \arccos \left (a x\right )^{2} - 2 \, a x - 2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.238852, size = 37, normalized size = 1.06 \begin{align*} \begin{cases} x \operatorname{acos}^{2}{\left (a x \right )} - 2 x - \frac{2 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**2,x)

[Out]

Piecewise((x*acos(a*x)**2 - 2*x - 2*sqrt(-a**2*x**2 + 1)*acos(a*x)/a, Ne(a, 0)), (pi**2*x/4, True))

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Giac [A]  time = 1.13932, size = 45, normalized size = 1.29 \begin{align*} x \arccos \left (a x\right )^{2} - 2 \, x - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^2,x, algorithm="giac")

[Out]

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