3.35 \(\int x^2 \cos ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=166 \[ -\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{160 x}{27 a^2}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{9} x^3 \cos ^{-1}(a x)^2+\frac{8 x^3}{81} \]

[Out]

(160*x)/(27*a^2) + (8*x^3)/81 + (160*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a^3) + (8*x^2*Sqrt[1 - a^2*x^2]*ArcCos
[a*x])/(27*a) - (8*x*ArcCos[a*x]^2)/(3*a^2) - (4*x^3*ArcCos[a*x]^2)/9 - (8*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9
*a^3) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9*a) + (x^3*ArcCos[a*x]^4)/3

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Rubi [A]  time = 0.365734, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4628, 4708, 4678, 4620, 8, 30} \[ -\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{160 x}{27 a^2}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{9} x^3 \cos ^{-1}(a x)^2+\frac{8 x^3}{81} \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCos[a*x]^4,x]

[Out]

(160*x)/(27*a^2) + (8*x^3)/81 + (160*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(27*a^3) + (8*x^2*Sqrt[1 - a^2*x^2]*ArcCos
[a*x])/(27*a) - (8*x*ArcCos[a*x]^2)/(3*a^2) - (4*x^3*ArcCos[a*x]^2)/9 - (8*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9
*a^3) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(9*a) + (x^3*ArcCos[a*x]^4)/3

Rule 4628

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
s[c*x])^n)/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCos[c*x])^(n - 1))/Sqrt[1
- c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4708

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcCos[c*x])^n)/Sqrt[d + e*x^2], x], x] - Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

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 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 8

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^2 \cos ^{-1}(a x)^4 \, dx &=\frac{1}{3} x^3 \cos ^{-1}(a x)^4+\frac{1}{3} (4 a) \int \frac{x^3 \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{4}{3} \int x^2 \cos ^{-1}(a x)^2 \, dx+\frac{8 \int \frac{x \cos ^{-1}(a x)^3}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4-\frac{8 \int \cos ^{-1}(a x)^2 \, dx}{3 a^2}-\frac{1}{9} (8 a) \int \frac{x^3 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4+\frac{8 \int x^2 \, dx}{27}-\frac{16 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{27 a}-\frac{16 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{3 a}\\ &=\frac{8 x^3}{81}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4+\frac{16 \int 1 \, dx}{27 a^2}+\frac{16 \int 1 \, dx}{3 a^2}\\ &=\frac{160 x}{27 a^2}+\frac{8 x^3}{81}+\frac{160 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a^3}+\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{27 a}-\frac{8 x \cos ^{-1}(a x)^2}{3 a^2}-\frac{4}{9} x^3 \cos ^{-1}(a x)^2-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a^3}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^4\\ \end{align*}

Mathematica [A]  time = 0.0722178, size = 114, normalized size = 0.69 \[ \frac{8 a x \left (a^2 x^2+60\right )+27 a^3 x^3 \cos ^{-1}(a x)^4-36 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \cos ^{-1}(a x)^3-36 a x \left (a^2 x^2+6\right ) \cos ^{-1}(a x)^2+24 \sqrt{1-a^2 x^2} \left (a^2 x^2+20\right ) \cos ^{-1}(a x)}{81 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCos[a*x]^4,x]

[Out]

(8*a*x*(60 + a^2*x^2) + 24*Sqrt[1 - a^2*x^2]*(20 + a^2*x^2)*ArcCos[a*x] - 36*a*x*(6 + a^2*x^2)*ArcCos[a*x]^2 -
36*Sqrt[1 - a^2*x^2]*(2 + a^2*x^2)*ArcCos[a*x]^3 + 27*a^3*x^3*ArcCos[a*x]^4)/(81*a^3)

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Maple [A]  time = 0.052, size = 130, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{4}}{3}}-{\frac{4\, \left ( \arccos \left ( ax \right ) \right ) ^{3} \left ({a}^{2}{x}^{2}+2 \right ) }{9}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{8\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}}{3}}+{\frac{160\,ax}{27}}+{\frac{16\,\arccos \left ( ax \right ) }{3}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{4\,{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{9}}+{\frac{8\,\arccos \left ( ax \right ) \left ({a}^{2}{x}^{2}+2 \right ) }{27}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{8\,{a}^{3}{x}^{3}}{81}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccos(a*x)^4,x)

[Out]

1/a^3*(1/3*a^3*x^3*arccos(a*x)^4-4/9*arccos(a*x)^3*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2)-8/3*a*x*arccos(a*x)^2+160/27
*a*x+16/3*arccos(a*x)*(-a^2*x^2+1)^(1/2)-4/9*a^3*x^3*arccos(a*x)^2+8/27*arccos(a*x)*(a^2*x^2+2)*(-a^2*x^2+1)^(
1/2)+8/81*a^3*x^3)

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Maxima [A]  time = 1.52923, size = 197, normalized size = 1.19 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac{4}{9} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{3} + \frac{4}{81} \,{\left (2 \, a{\left (\frac{3 \,{\left (\sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{20 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{3}} + \frac{a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac{9 \,{\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )^{2}}{a^{3}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^4,x, algorithm="maxima")

[Out]

1/3*x^3*arccos(a*x)^4 - 4/9*a*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccos(a*x)^3 + 4/81*(2*
a*(3*(sqrt(-a^2*x^2 + 1)*x^2 + 20*sqrt(-a^2*x^2 + 1)/a^2)*arccos(a*x)/a^3 + (a^2*x^3 + 60*x)/a^4) - 9*(a^2*x^3
+ 6*x)*arccos(a*x)^2/a^3)*a

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Fricas [A]  time = 2.42694, size = 247, normalized size = 1.49 \begin{align*} \frac{27 \, a^{3} x^{3} \arccos \left (a x\right )^{4} + 8 \, a^{3} x^{3} - 36 \,{\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right )^{2} + 480 \, a x - 12 \, \sqrt{-a^{2} x^{2} + 1}{\left (3 \,{\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{3} - 2 \,{\left (a^{2} x^{2} + 20\right )} \arccos \left (a x\right )\right )}}{81 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^4,x, algorithm="fricas")

[Out]

1/81*(27*a^3*x^3*arccos(a*x)^4 + 8*a^3*x^3 - 36*(a^3*x^3 + 6*a*x)*arccos(a*x)^2 + 480*a*x - 12*sqrt(-a^2*x^2 +
1)*(3*(a^2*x^2 + 2)*arccos(a*x)^3 - 2*(a^2*x^2 + 20)*arccos(a*x)))/a^3

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Sympy [A]  time = 4.37318, size = 165, normalized size = 0.99 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acos}^{4}{\left (a x \right )}}{3} - \frac{4 x^{3} \operatorname{acos}^{2}{\left (a x \right )}}{9} + \frac{8 x^{3}}{81} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{9 a} + \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{27 a} - \frac{8 x \operatorname{acos}^{2}{\left (a x \right )}}{3 a^{2}} + \frac{160 x}{27 a^{2}} - \frac{8 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{3}{\left (a x \right )}}{9 a^{3}} + \frac{160 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{27 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi ^{4} x^{3}}{48} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*acos(a*x)**4,x)

[Out]

Piecewise((x**3*acos(a*x)**4/3 - 4*x**3*acos(a*x)**2/9 + 8*x**3/81 - 4*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/
(9*a) + 8*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)/(27*a) - 8*x*acos(a*x)**2/(3*a**2) + 160*x/(27*a**2) - 8*sqrt(-a
**2*x**2 + 1)*acos(a*x)**3/(9*a**3) + 160*sqrt(-a**2*x**2 + 1)*acos(a*x)/(27*a**3), Ne(a, 0)), (pi**4*x**3/48,
True))

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Giac [A]  time = 1.17803, size = 189, normalized size = 1.14 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac{4}{9} \, x^{3} \arccos \left (a x\right )^{2} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{9 \, a} + \frac{8}{81} \, x^{3} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{27 \, a} - \frac{8 \, x \arccos \left (a x\right )^{2}}{3 \, a^{2}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{9 \, a^{3}} + \frac{160 \, x}{27 \, a^{2}} + \frac{160 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccos(a*x)^4,x, algorithm="giac")

[Out]

1/3*x^3*arccos(a*x)^4 - 4/9*x^3*arccos(a*x)^2 - 4/9*sqrt(-a^2*x^2 + 1)*x^2*arccos(a*x)^3/a + 8/81*x^3 + 8/27*s
qrt(-a^2*x^2 + 1)*x^2*arccos(a*x)/a - 8/3*x*arccos(a*x)^2/a^2 - 8/9*sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a^3 + 160
/27*x/a^2 + 160/27*sqrt(-a^2*x^2 + 1)*arccos(a*x)/a^3