3.26 \(\int \cos ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=60 \[ \frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x) \]

[Out]

(6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcCos[a*x] - (3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a + x*ArcCos[a*x]^3

________________________________________________________________________________________

Rubi [A]  time = 0.0810922, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4678, 261} \[ \frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^3,x]

[Out]

(6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcCos[a*x] - (3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a + x*ArcCos[a*x]^3

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 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)^3 \, dx &=x \cos ^{-1}(a x)^3+(3 a) \int \frac{x \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 \int \cos ^{-1}(a x) \, dx\\ &=-6 x \cos ^{-1}(a x)-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-(6 a) \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{6 \sqrt{1-a^2 x^2}}{a}-6 x \cos ^{-1}(a x)-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0186135, size = 60, normalized size = 1. \[ \frac{6 \sqrt{1-a^2 x^2}}{a}-\frac{3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{a}+x \cos ^{-1}(a x)^3-6 x \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^3,x]

[Out]

(6*Sqrt[1 - a^2*x^2])/a - 6*x*ArcCos[a*x] - (3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/a + x*ArcCos[a*x]^3

________________________________________________________________________________________

Maple [A]  time = 0.049, size = 57, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arccos \left ( ax \right ) \right ) ^{3}-3\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\sqrt{-{a}^{2}{x}^{2}+1}+6\,\sqrt{-{a}^{2}{x}^{2}+1}-6\,ax\arccos \left ( ax \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^3,x)

[Out]

1/a*(a*x*arccos(a*x)^3-3*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)+6*(-a^2*x^2+1)^(1/2)-6*a*x*arccos(a*x))

________________________________________________________________________________________

Maxima [A]  time = 1.4488, size = 80, normalized size = 1.33 \begin{align*} x \arccos \left (a x\right )^{3} - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{a} - \frac{6 \,{\left (a x \arccos \left (a x\right ) - \sqrt{-a^{2} x^{2} + 1}\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^3,x, algorithm="maxima")

[Out]

x*arccos(a*x)^3 - 3*sqrt(-a^2*x^2 + 1)*arccos(a*x)^2/a - 6*(a*x*arccos(a*x) - sqrt(-a^2*x^2 + 1))/a

________________________________________________________________________________________

Fricas [A]  time = 2.26834, size = 116, normalized size = 1.93 \begin{align*} \frac{a x \arccos \left (a x\right )^{3} - 6 \, a x \arccos \left (a x\right ) - 3 \, \sqrt{-a^{2} x^{2} + 1}{\left (\arccos \left (a x\right )^{2} - 2\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^3,x, algorithm="fricas")

[Out]

(a*x*arccos(a*x)^3 - 6*a*x*arccos(a*x) - 3*sqrt(-a^2*x^2 + 1)*(arccos(a*x)^2 - 2))/a

________________________________________________________________________________________

Sympy [A]  time = 0.572868, size = 60, normalized size = 1. \begin{align*} \begin{cases} x \operatorname{acos}^{3}{\left (a x \right )} - 6 x \operatorname{acos}{\left (a x \right )} - \frac{3 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{a} + \frac{6 \sqrt{- a^{2} x^{2} + 1}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{3} x}{8} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**3,x)

[Out]

Piecewise((x*acos(a*x)**3 - 6*x*acos(a*x) - 3*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/a + 6*sqrt(-a**2*x**2 + 1)/a,
Ne(a, 0)), (pi**3*x/8, True))

________________________________________________________________________________________

Giac [A]  time = 1.11909, size = 76, normalized size = 1.27 \begin{align*} x \arccos \left (a x\right )^{3} - 6 \, x \arccos \left (a x\right ) - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{a} + \frac{6 \, \sqrt{-a^{2} x^{2} + 1}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^3,x, algorithm="giac")

[Out]

x*arccos(a*x)^3 - 6*x*arccos(a*x) - 3*sqrt(-a^2*x^2 + 1)*arccos(a*x)^2/a + 6*sqrt(-a^2*x^2 + 1)/a