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

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

[Out]

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

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Rubi [A]  time = 0.37, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, 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 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]

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 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 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 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]

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*}

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Mathematica [A]  time = 0.08, size = 114, normalized size = 0.69 \[ \frac {27 a^3 x^3 \cos ^{-1}(a x)^4+8 a x \left (a^2 x^2+60\right )-36 \sqrt {1-a^2 x^2} \left (a^2 x^2+2\right ) \cos ^{-1}(a x)^3-36 a x \left (a^2 x^2+6\right ) \cos ^{-1}(a x)^2+24 \sqrt {1-a^2 x^2} \left (a^2 x^2+20\right ) \cos ^{-1}(a x)}{81 a^3} \]

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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fricas [A]  time = 0.44, size = 99, normalized size = 0.60 \[ \frac {27 \, a^{3} x^{3} \arccos \left (a x\right )^{4} + 8 \, a^{3} x^{3} - 36 \, {\left (a^{3} x^{3} + 6 \, a x\right )} \arccos \left (a x\right )^{2} + 480 \, a x - 12 \, \sqrt {-a^{2} x^{2} + 1} {\left (3 \, {\left (a^{2} x^{2} + 2\right )} \arccos \left (a x\right )^{3} - 2 \, {\left (a^{2} x^{2} + 20\right )} \arccos \left (a x\right )\right )}}{81 \, a^{3}} \]

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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giac [A]  time = 0.20, size = 140, normalized size = 0.84 \[ \frac {1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac {4}{9} \, x^{3} \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{9 \, a} + \frac {8}{81} \, x^{3} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{27 \, a} - \frac {8 \, x \arccos \left (a x\right )^{2}}{3 \, a^{2}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{9 \, a^{3}} + \frac {160 \, x}{27 \, a^{2}} + \frac {160 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{27 \, a^{3}} \]

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

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maple [A]  time = 0.05, size = 130, normalized size = 0.78 \[ \frac {\frac {a^{3} x^{3} \arccos \left (a x \right )^{4}}{3}-\frac {4 \arccos \left (a x \right )^{3} \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{9}-\frac {8 a x \arccos \left (a x \right )^{2}}{3}+\frac {160 a x}{27}+\frac {16 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{3}-\frac {4 a^{3} x^{3} \arccos \left (a x \right )^{2}}{9}+\frac {8 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{27}+\frac {8 a^{3} x^{3}}{81}}{a^{3}} \]

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 = 1.07, size = 146, normalized size = 0.88 \[ \frac {1}{3} \, x^{3} \arccos \left (a x\right )^{4} - \frac {4}{9} \, a {\left (\frac {\sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{2}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{a^{4}}\right )} \arccos \left (a x\right )^{3} + \frac {4}{81} \, {\left (2 \, a {\left (\frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {20 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{3}} + \frac {a^{2} x^{3} + 60 \, x}{a^{4}}\right )} - \frac {9 \, {\left (a^{2} x^{3} + 6 \, x\right )} \arccos \left (a x\right )^{2}}{a^{3}}\right )} a \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*acos(a*x)^4, x)

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sympy [A]  time = 3.04, size = 165, normalized size = 0.99 \[ \begin {cases} \frac {x^{3} \operatorname {acos}^{4}{\left (a x \right )}}{3} - \frac {4 x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{9} + \frac {8 x^{3}}{81} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{9 a} + \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{27 a} - \frac {8 x \operatorname {acos}^{2}{\left (a x \right )}}{3 a^{2}} + \frac {160 x}{27 a^{2}} - \frac {8 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{9 a^{3}} + \frac {160 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{27 a^{3}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{3}}{48} & \text {otherwise} \end {cases} \]

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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