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

Optimal. Leaf size=82 \[ -\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{2 x^3}{27} \]

[Out]

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

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Rubi [A]  time = 0.125599, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 4708, 4678, 8, 30} \[ -\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*x)/(9*a^2) - (2*x^3)/27 - (4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(9*a^3) - (2*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]
)/(9*a) + (x^3*ArcCos[a*x]^2)/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 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)^2 \, dx &=\frac{1}{3} x^3 \cos ^{-1}(a x)^2+\frac{1}{3} (2 a) \int \frac{x^3 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{2 \int x^2 \, dx}{9}+\frac{4 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac{2 x^3}{27}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{4 \int 1 \, dx}{9 a^2}\\ &=-\frac{4 x}{9 a^2}-\frac{2 x^3}{27}-\frac{4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a^3}-\frac{2 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{9 a}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.0463105, size = 63, normalized size = 0.77 \[ -\frac{2 \sqrt{1-a^2 x^2} \left (a^2 x^2+2\right ) \cos ^{-1}(a x)}{9 a^3}-\frac{4 x}{9 a^2}+\frac{1}{3} x^3 \cos ^{-1}(a x)^2-\frac{2 x^3}{27} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.048, size = 59, normalized size = 0.7 \begin{align*}{\frac{1}{{a}^{3}} \left ({\frac{{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{3}}-{\frac{2\,\arccos \left ( ax \right ) \left ({a}^{2}{x}^{2}+2 \right ) }{9}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{3}{x}^{3}}{27}}-{\frac{4\,ax}{9}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46577, size = 97, normalized size = 1.18 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{2} - \frac{2}{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 ) - \frac{2 \,{\left (a^{2} x^{3} + 6 \, x\right )}}{27 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccos(a*x)^2 - 2/9*a*(sqrt(-a^2*x^2 + 1)*x^2/a^2 + 2*sqrt(-a^2*x^2 + 1)/a^4)*arccos(a*x) - 2/27*(a^2*
x^3 + 6*x)/a^2

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Fricas [A]  time = 1.97583, size = 143, normalized size = 1.74 \begin{align*} \frac{9 \, a^{3} x^{3} \arccos \left (a x\right )^{2} - 2 \, a^{3} x^{3} - 6 \,{\left (a^{2} x^{2} + 2\right )} \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 12 \, a x}{27 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*(9*a^3*x^3*arccos(a*x)^2 - 2*a^3*x^3 - 6*(a^2*x^2 + 2)*sqrt(-a^2*x^2 + 1)*arccos(a*x) - 12*a*x)/a^3

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Sympy [A]  time = 1.15989, size = 83, normalized size = 1.01 \begin{align*} \begin{cases} \frac{x^{3} \operatorname{acos}^{2}{\left (a x \right )}}{3} - \frac{2 x^{3}}{27} - \frac{2 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{9 a} - \frac{4 x}{9 a^{2}} - \frac{4 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{9 a^{3}} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x^{3}}{12} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**3*acos(a*x)**2/3 - 2*x**3/27 - 2*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)/(9*a) - 4*x/(9*a**2) - 4*sq
rt(-a**2*x**2 + 1)*acos(a*x)/(9*a**3), Ne(a, 0)), (pi**2*x**3/12, True))

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Giac [A]  time = 1.16626, size = 92, normalized size = 1.12 \begin{align*} \frac{1}{3} \, x^{3} \arccos \left (a x\right )^{2} - \frac{2}{27} \, x^{3} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{9 \, a} - \frac{4 \, x}{9 \, a^{2}} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{9 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*arccos(a*x)^2 - 2/27*x^3 - 2/9*sqrt(-a^2*x^2 + 1)*x^2*arccos(a*x)/a - 4/9*x/a^2 - 4/9*sqrt(-a^2*x^2 +
1)*arccos(a*x)/a^3