3.1.26 \(\int \text {ArcCos}(a x)^3 \, dx\) [26]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4716, 4768, 267} \begin {gather*} -\frac {3 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{a}+\frac {6 \sqrt {1-a^2 x^2}}{a}+x \text {ArcCos}(a x)^3-6 x \text {ArcCos}(a x) \end {gather*}

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 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]

Rule 4768

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]

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*}

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Mathematica [A]
time = 0.02, size = 60, normalized size = 1.00 \begin {gather*} \frac {6 \sqrt {1-a^2 x^2}}{a}-6 x \text {ArcCos}(a x)-\frac {3 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{a}+x \text {ArcCos}(a x)^3 \end {gather*}

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

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Maple [A]
time = 0.08, size = 57, normalized size = 0.95

method	result	size
derivativedivides	\(\frac {a x \arccos \left (a x \right )^{3}-3 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+6 \sqrt {-a^{2} x^{2}+1}-6 a x \arccos \left (a x \right )}{a}\)	\(57\)
default	\(\frac {a x \arccos \left (a x \right )^{3}-3 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+6 \sqrt {-a^{2} x^{2}+1}-6 a x \arccos \left (a x \right )}{a}\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^3,x,method=_RETURNVERBOSE)

[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))

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Maxima [A]
time = 0.47, size = 59, normalized size = 0.98 \begin {gather*} 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 {gather*}

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

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Fricas [A]
time = 2.95, size = 44, normalized size = 0.73 \begin {gather*} \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 {gather*}

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

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Sympy [A]
time = 0.12, size = 60, normalized size = 1.00 \begin {gather*} \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 {gather*}

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))

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Giac [A]
time = 0.42, size = 56, normalized size = 0.93 \begin {gather*} 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 {gather*}

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

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Mupad [B]
time = 0.31, size = 59, normalized size = 0.98 \begin {gather*} \left \{\begin {array}{cl} \frac {x\,\pi ^3}{8} & \text {\ if\ \ }a=0\\ -x\,\left (6\,\mathrm {acos}\left (a\,x\right )-{\mathrm {acos}\left (a\,x\right )}^3\right )-\frac {\sqrt {1-a^2\,x^2}\,\left (3\,{\mathrm {acos}\left (a\,x\right )}^2-6\right )}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

piecewise(a == 0, (x*pi^3)/8, a ~= 0, - x*(6*acos(a*x) - acos(a*x)^3) - ((- a^2*x^2 + 1)^(1/2)*(3*acos(a*x)^2
- 6))/a)

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