3.37 \(\int \cos ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=69 \[ -\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]

[Out]

24*x + (24*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - 12*x*ArcCos[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a + x*A
rcCos[a*x]^4

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Rubi [A]  time = 0.120369, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4620, 4678, 8} \[ -\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^4,x]

[Out]

24*x + (24*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - 12*x*ArcCos[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a + x*A
rcCos[a*x]^4

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)^4 \, dx &=x \cos ^{-1}(a x)^4+(4 a) \int \frac{x \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4-12 \int \cos ^{-1}(a x)^2 \, dx\\ &=-12 x \cos ^{-1}(a x)^2-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4-(24 a) \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}-12 x \cos ^{-1}(a x)^2-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4+24 \int 1 \, dx\\ &=24 x+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}-12 x \cos ^{-1}(a x)^2-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+x \cos ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.0216089, size = 69, normalized size = 1. \[ -\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac{24 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^4,x]

[Out]

24*x + (24*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/a - 12*x*ArcCos[a*x]^2 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/a + x*A
rcCos[a*x]^4

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Maple [A]  time = 0.047, size = 67, normalized size = 1. \begin{align*}{\frac{1}{a} \left ( ax \left ( \arccos \left ( ax \right ) \right ) ^{4}-4\, \left ( \arccos \left ( ax \right ) \right ) ^{3}\sqrt{-{a}^{2}{x}^{2}+1}-12\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}+24\,ax+24\,\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)^4,x)

[Out]

1/a*(a*x*arccos(a*x)^4-4*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)-12*a*x*arccos(a*x)^2+24*a*x+24*arccos(a*x)*(-a^2*x^2
+1)^(1/2))

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Maxima [A]  time = 1.48534, size = 100, normalized size = 1.45 \begin{align*} x \arccos \left (a x\right )^{4} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} - 12 \,{\left (\frac{x \arccos \left (a x\right )^{2}}{a} - \frac{2 \,{\left (x + \frac{\sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4,x, algorithm="maxima")

[Out]

x*arccos(a*x)^4 - 4*sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a - 12*(x*arccos(a*x)^2/a - 2*(x + sqrt(-a^2*x^2 + 1)*arc
cos(a*x)/a)/a)*a

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Fricas [A]  time = 2.38187, size = 149, normalized size = 2.16 \begin{align*} \frac{a x \arccos \left (a x\right )^{4} - 12 \, a x \arccos \left (a x\right )^{2} + 24 \, a x - 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^4,x, algorithm="fricas")

[Out]

(a*x*arccos(a*x)^4 - 12*a*x*arccos(a*x)^2 + 24*a*x - 4*sqrt(-a^2*x^2 + 1)*(arccos(a*x)^3 - 6*arccos(a*x)))/a

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Sympy [A]  time = 1.10413, size = 70, normalized size = 1.01 \begin{align*} \begin{cases} x \operatorname{acos}^{4}{\left (a x \right )} - 12 x \operatorname{acos}^{2}{\left (a x \right )} + 24 x - \frac{4 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{a} + \frac{24 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{a} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**4,x)

[Out]

Piecewise((x*acos(a*x)**4 - 12*x*acos(a*x)**2 + 24*x - 4*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/a + 24*sqrt(-a**2*x
**2 + 1)*acos(a*x)/a, Ne(a, 0)), (pi**4*x/16, True))

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Giac [A]  time = 1.14171, size = 88, normalized size = 1.28 \begin{align*} x \arccos \left (a x\right )^{4} - 12 \, x \arccos \left (a x\right )^{2} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} + 24 \, x + \frac{24 \, \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)^4,x, algorithm="giac")

[Out]

x*arccos(a*x)^4 - 12*x*arccos(a*x)^2 - 4*sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a + 24*x + 24*sqrt(-a^2*x^2 + 1)*arc
cos(a*x)/a