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

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 8

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

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 = 0.03, size = 69, normalized size = 1.00 \[ -\frac {4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{a}+\frac {24 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{a}+x \cos ^{-1}(a x)^4-12 x \cos ^{-1}(a x)^2+24 x \]

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

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 55, normalized size = 0.80 \[ \frac {a x \arccos \left (a x\right )^{4} - 12 \, a x \arccos \left (a x\right )^{2} + 24 \, a x - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (\arccos \left (a x\right )^{3} - 6 \, \arccos \left (a x\right )\right )}}{a} \]

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

________________________________________________________________________________________

giac [A]  time = 0.18, size = 65, normalized size = 0.94 \[ x \arccos \left (a x\right )^{4} - 12 \, x \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} + 24 \, x + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a} \]

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

________________________________________________________________________________________

maple [A]  time = 0.07, size = 67, normalized size = 0.97 \[ \frac {a x \arccos \left (a x \right )^{4}-4 \arccos \left (a x \right )^{3} \sqrt {-a^{2} x^{2}+1}-12 a x \arccos \left (a x \right )^{2}+24 a x +24 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{a} \]

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))

________________________________________________________________________________________

maxima [A]  time = 0.52, size = 74, normalized size = 1.07 \[ x \arccos \left (a x\right )^{4} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{a} - 12 \, {\left (\frac {x \arccos \left (a x\right )^{2}}{a} - \frac {2 \, {\left (x + \frac {\sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{a}\right )}}{a}\right )} a \]

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

________________________________________________________________________________________

mupad [B]  time = 0.31, size = 63, normalized size = 0.91 \[ \left \{\begin {array}{cl} \frac {x\,\pi ^4}{16} & \text {\ if\ \ }a=0\\ x\,\left ({\mathrm {acos}\left (a\,x\right )}^4-12\,{\mathrm {acos}\left (a\,x\right )}^2+24\right )+\frac {\sqrt {1-a^2\,x^2}\,\left (24\,\mathrm {acos}\left (a\,x\right )-4\,{\mathrm {acos}\left (a\,x\right )}^3\right )}{a} & \text {\ if\ \ }a\neq 0 \end {array}\right . \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)^4,x)

[Out]

piecewise(a == 0, (x*pi^4)/16, a ~= 0, x*(- 12*acos(a*x)^2 + acos(a*x)^4 + 24) + ((- a^2*x^2 + 1)^(1/2)*(24*ac
os(a*x) - 4*acos(a*x)^3))/a)

________________________________________________________________________________________

sympy [A]  time = 0.81, size = 70, normalized size = 1.01 \[ \begin {cases} x \operatorname {acos}^{4}{\left (a x \right )} - 12 x \operatorname {acos}^{2}{\left (a x \right )} + 24 x - \frac {4 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{a} + \frac {24 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x}{16} & \text {otherwise} \end {cases} \]

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))

________________________________________________________________________________________