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

Optimal. Leaf size=120 \[ -\frac{8 x^3}{225 a^2}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125} \]

[Out]

(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (16*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(75*a^5) - (8*x^2*Sqrt
[1 - a^2*x^2]*ArcCos[a*x])/(75*a^3) - (2*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(25*a) + (x^5*ArcCos[a*x]^2)/5

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Rubi [A]  time = 0.195771, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{8 x^3}{225 a^2}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (16*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(75*a^5) - (8*x^2*Sqrt
[1 - a^2*x^2]*ArcCos[a*x])/(75*a^3) - (2*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(25*a) + (x^5*ArcCos[a*x]^2)/5

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^4 \cos ^{-1}(a x)^2 \, dx &=\frac{1}{5} x^5 \cos ^{-1}(a x)^2+\frac{1}{5} (2 a) \int \frac{x^5 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 \int x^4 \, dx}{25}+\frac{8 \int \frac{x^3 \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac{2 x^5}{125}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2+\frac{16 \int \frac{x \cos ^{-1}(a x)}{\sqrt{1-a^2 x^2}} \, dx}{75 a^3}-\frac{8 \int x^2 \, dx}{75 a^2}\\ &=-\frac{8 x^3}{225 a^2}-\frac{2 x^5}{125}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{16 \int 1 \, dx}{75 a^4}\\ &=-\frac{16 x}{75 a^4}-\frac{8 x^3}{225 a^2}-\frac{2 x^5}{125}-\frac{16 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac{8 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac{2 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2\\ \end{align*}

Mathematica [A]  time = 0.0561761, size = 82, normalized size = 0.68 \[ -\frac{8 x^3}{225 a^2}-\frac{2 \sqrt{1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cos ^{-1}(a x)}{75 a^5}-\frac{16 x}{75 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^2-\frac{2 x^5}{125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (2*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a
*x])/(75*a^5) + (x^5*ArcCos[a*x]^2)/5

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Maple [A]  time = 0.05, size = 76, normalized size = 0.6 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{2}{a}^{5}{x}^{5}}{5}}-{\frac{2\,\arccos \left ( ax \right ) \left ( 3\,{a}^{4}{x}^{4}+4\,{a}^{2}{x}^{2}+8 \right ) }{75}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{2\,{a}^{5}{x}^{5}}{125}}-{\frac{8\,{a}^{3}{x}^{3}}{225}}-{\frac{16\,ax}{75}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a^5*(1/5*arccos(a*x)^2*a^5*x^5-2/75*arccos(a*x)*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)-2/125*a^5*x^5-8/2
25*a^3*x^3-16/75*a*x)

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Maxima [A]  time = 1.49582, size = 138, normalized size = 1.15 \begin{align*} \frac{1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac{2}{75} \,{\left (\frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac{8 \, \sqrt{-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right ) - \frac{2 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccos(a*x)^2 - 2/75*(3*sqrt(-a^2*x^2 + 1)*x^4/a^2 + 4*sqrt(-a^2*x^2 + 1)*x^2/a^4 + 8*sqrt(-a^2*x^2 +
1)/a^6)*a*arccos(a*x) - 2/1125*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)/a^4

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Fricas [A]  time = 2.00245, size = 189, normalized size = 1.58 \begin{align*} \frac{225 \, a^{5} x^{5} \arccos \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 30 \,{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1125*(225*a^5*x^5*arccos(a*x)^2 - 18*a^5*x^5 - 40*a^3*x^3 - 30*(3*a^4*x^4 + 4*a^2*x^2 + 8)*sqrt(-a^2*x^2 + 1
)*arccos(a*x) - 240*a*x)/a^5

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Sympy [A]  time = 4.14903, size = 121, normalized size = 1.01 \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acos}^{2}{\left (a x \right )}}{5} - \frac{2 x^{5}}{125} - \frac{2 x^{4} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{25 a} - \frac{8 x^{3}}{225 a^{2}} - \frac{8 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{75 a^{3}} - \frac{16 x}{75 a^{4}} - \frac{16 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}{\left (a x \right )}}{75 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi ^{2} x^{5}}{20} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*acos(a*x)**2/5 - 2*x**5/125 - 2*x**4*sqrt(-a**2*x**2 + 1)*acos(a*x)/(25*a) - 8*x**3/(225*a**2)
- 8*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)/(75*a**3) - 16*x/(75*a**4) - 16*sqrt(-a**2*x**2 + 1)*acos(a*x)/(75*a*
*5), Ne(a, 0)), (pi**2*x**5/20, True))

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Giac [A]  time = 1.17294, size = 135, normalized size = 1.12 \begin{align*} \frac{1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac{2}{125} \, x^{5} - \frac{2 \, \sqrt{-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{25 \, a} - \frac{8 \, x^{3}}{225 \, a^{2}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{75 \, a^{3}} - \frac{16 \, x}{75 \, a^{4}} - \frac{16 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )}{75 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccos(a*x)^2 - 2/125*x^5 - 2/25*sqrt(-a^2*x^2 + 1)*x^4*arccos(a*x)/a - 8/225*x^3/a^2 - 8/75*sqrt(-a^2
*x^2 + 1)*x^2*arccos(a*x)/a^3 - 16/75*x/a^4 - 16/75*sqrt(-a^2*x^2 + 1)*arccos(a*x)/a^5