3.23 \(\int x^3 \cos ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=167 \[ \frac{3 x^3 \sqrt{1-a^2 x^2}}{128 a}+\frac{45 x \sqrt{1-a^2 x^2}}{256 a^3}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}-\frac{9 x^2 \cos ^{-1}(a x)}{32 a^2}-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}-\frac{45 \sin ^{-1}(a x)}{256 a^4}-\frac{3 \cos ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{3}{32} x^4 \cos ^{-1}(a x) \]

[Out]

(45*x*Sqrt[1 - a^2*x^2])/(256*a^3) + (3*x^3*Sqrt[1 - a^2*x^2])/(128*a) - (9*x^2*ArcCos[a*x])/(32*a^2) - (3*x^4
*ArcCos[a*x])/32 - (9*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(1
6*a) - (3*ArcCos[a*x]^3)/(32*a^4) + (x^4*ArcCos[a*x]^3)/4 - (45*ArcSin[a*x])/(256*a^4)

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Rubi [A]  time = 0.31498, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 4708, 4642, 321, 216} \[ \frac{3 x^3 \sqrt{1-a^2 x^2}}{128 a}+\frac{45 x \sqrt{1-a^2 x^2}}{256 a^3}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}-\frac{9 x^2 \cos ^{-1}(a x)}{32 a^2}-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}-\frac{45 \sin ^{-1}(a x)}{256 a^4}-\frac{3 \cos ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{3}{32} x^4 \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCos[a*x]^3,x]

[Out]

(45*x*Sqrt[1 - a^2*x^2])/(256*a^3) + (3*x^3*Sqrt[1 - a^2*x^2])/(128*a) - (9*x^2*ArcCos[a*x])/(32*a^2) - (3*x^4
*ArcCos[a*x])/32 - (9*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(32*a^3) - (3*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(1
6*a) - (3*ArcCos[a*x]^3)/(32*a^4) + (x^4*ArcCos[a*x]^3)/4 - (45*ArcSin[a*x])/(256*a^4)

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 4642

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int x^3 \cos ^{-1}(a x)^3 \, dx &=\frac{1}{4} x^4 \cos ^{-1}(a x)^3+\frac{1}{4} (3 a) \int \frac{x^4 \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{3}{8} \int x^3 \cos ^{-1}(a x) \, dx+\frac{9 \int \frac{x^2 \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{16 a}\\ &=-\frac{3}{32} x^4 \cos ^{-1}(a x)-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3+\frac{9 \int \frac{\cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{32 a^3}-\frac{9 \int x \cos ^{-1}(a x) \, dx}{16 a^2}-\frac{1}{32} (3 a) \int \frac{x^4}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 x^3 \sqrt{1-a^2 x^2}}{128 a}-\frac{9 x^2 \cos ^{-1}(a x)}{32 a^2}-\frac{3}{32} x^4 \cos ^{-1}(a x)-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}-\frac{3 \cos ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{9 \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{128 a}-\frac{9 \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx}{32 a}\\ &=\frac{45 x \sqrt{1-a^2 x^2}}{256 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2}}{128 a}-\frac{9 x^2 \cos ^{-1}(a x)}{32 a^2}-\frac{3}{32} x^4 \cos ^{-1}(a x)-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}-\frac{3 \cos ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{9 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{256 a^3}-\frac{9 \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{64 a^3}\\ &=\frac{45 x \sqrt{1-a^2 x^2}}{256 a^3}+\frac{3 x^3 \sqrt{1-a^2 x^2}}{128 a}-\frac{9 x^2 \cos ^{-1}(a x)}{32 a^2}-\frac{3}{32} x^4 \cos ^{-1}(a x)-\frac{9 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{32 a^3}-\frac{3 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{16 a}-\frac{3 \cos ^{-1}(a x)^3}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^3-\frac{45 \sin ^{-1}(a x)}{256 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0756178, size = 115, normalized size = 0.69 \[ \frac{3 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+15\right )+8 \left (8 a^4 x^4-3\right ) \cos ^{-1}(a x)^3-24 a x \sqrt{1-a^2 x^2} \left (2 a^2 x^2+3\right ) \cos ^{-1}(a x)^2-24 a^2 x^2 \left (a^2 x^2+3\right ) \cos ^{-1}(a x)-45 \sin ^{-1}(a x)}{256 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCos[a*x]^3,x]

[Out]

(3*a*x*Sqrt[1 - a^2*x^2]*(15 + 2*a^2*x^2) - 24*a^2*x^2*(3 + a^2*x^2)*ArcCos[a*x] - 24*a*x*Sqrt[1 - a^2*x^2]*(3
+ 2*a^2*x^2)*ArcCos[a*x]^2 + 8*(-3 + 8*a^4*x^4)*ArcCos[a*x]^3 - 45*ArcSin[a*x])/(256*a^4)

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Maple [A]  time = 0.056, size = 151, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{4}} \left ({\frac{{a}^{4}{x}^{4} \left ( \arccos \left ( ax \right ) \right ) ^{3}}{4}}-{\frac{3\, \left ( \arccos \left ( ax \right ) \right ) ^{2}}{32} \left ( 2\,{a}^{3}{x}^{3}\sqrt{-{a}^{2}{x}^{2}+1}+3\,ax\sqrt{-{a}^{2}{x}^{2}+1}+3\,\arccos \left ( ax \right ) \right ) }-{\frac{3\,{a}^{4}{x}^{4}\arccos \left ( ax \right ) }{32}}+{\frac{3\,ax \left ( 2\,{a}^{2}{x}^{2}+3 \right ) }{256}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{45\,\arccos \left ( ax \right ) }{256}}-{\frac{9\,{a}^{2}{x}^{2}\arccos \left ( ax \right ) }{32}}+{\frac{9\,ax}{64}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{3\, \left ( \arccos \left ( ax \right ) \right ) ^{3}}{16}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccos(a*x)^3,x)

[Out]

1/a^4*(1/4*a^4*x^4*arccos(a*x)^3-3/32*arccos(a*x)^2*(2*a^3*x^3*(-a^2*x^2+1)^(1/2)+3*a*x*(-a^2*x^2+1)^(1/2)+3*a
rccos(a*x))-3/32*a^4*x^4*arccos(a*x)+3/256*a*x*(2*a^2*x^2+3)*(-a^2*x^2+1)^(1/2)+45/256*arccos(a*x)-9/32*a^2*x^
2*arccos(a*x)+9/64*a*x*(-a^2*x^2+1)^(1/2)+3/16*arccos(a*x)^3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{3} - 3 \, a \int \frac{\sqrt{a x + 1} \sqrt{-a x + 1} x^{4} \arctan \left (\sqrt{a x + 1} \sqrt{-a x + 1}, a x\right )^{2}}{4 \,{\left (a^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^3,x, algorithm="maxima")

[Out]

1/4*x^4*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3 - 3*a*integrate(1/4*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^4*arct
an2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^2/(a^2*x^2 - 1), x)

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Fricas [A]  time = 2.43175, size = 234, normalized size = 1.4 \begin{align*} \frac{8 \,{\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right )^{3} - 3 \,{\left (8 \, a^{4} x^{4} + 24 \, a^{2} x^{2} - 15\right )} \arccos \left (a x\right ) + 3 \,{\left (2 \, a^{3} x^{3} - 8 \,{\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arccos \left (a x\right )^{2} + 15 \, a x\right )} \sqrt{-a^{2} x^{2} + 1}}{256 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^3,x, algorithm="fricas")

[Out]

1/256*(8*(8*a^4*x^4 - 3)*arccos(a*x)^3 - 3*(8*a^4*x^4 + 24*a^2*x^2 - 15)*arccos(a*x) + 3*(2*a^3*x^3 - 8*(2*a^3
*x^3 + 3*a*x)*arccos(a*x)^2 + 15*a*x)*sqrt(-a^2*x^2 + 1))/a^4

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Sympy [A]  time = 4.58681, size = 167, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{4} \operatorname{acos}^{3}{\left (a x \right )}}{4} - \frac{3 x^{4} \operatorname{acos}{\left (a x \right )}}{32} - \frac{3 x^{3} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{16 a} + \frac{3 x^{3} \sqrt{- a^{2} x^{2} + 1}}{128 a} - \frac{9 x^{2} \operatorname{acos}{\left (a x \right )}}{32 a^{2}} - \frac{9 x \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{32 a^{3}} + \frac{45 x \sqrt{- a^{2} x^{2} + 1}}{256 a^{3}} - \frac{3 \operatorname{acos}^{3}{\left (a x \right )}}{32 a^{4}} + \frac{45 \operatorname{acos}{\left (a x \right )}}{256 a^{4}} & \text{for}\: a \neq 0 \\\frac{\pi ^{3} x^{4}}{32} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acos(a*x)**3,x)

[Out]

Piecewise((x**4*acos(a*x)**3/4 - 3*x**4*acos(a*x)/32 - 3*x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(16*a) + 3*x**
3*sqrt(-a**2*x**2 + 1)/(128*a) - 9*x**2*acos(a*x)/(32*a**2) - 9*x*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(32*a**3)
+ 45*x*sqrt(-a**2*x**2 + 1)/(256*a**3) - 3*acos(a*x)**3/(32*a**4) + 45*acos(a*x)/(256*a**4), Ne(a, 0)), (pi**3
*x**4/32, True))

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Giac [A]  time = 1.21932, size = 190, normalized size = 1.14 \begin{align*} \frac{1}{4} \, x^{4} \arccos \left (a x\right )^{3} - \frac{3}{32} \, x^{4} \arccos \left (a x\right ) - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{2}}{16 \, a} + \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{3}}{128 \, a} - \frac{9 \, x^{2} \arccos \left (a x\right )}{32 \, a^{2}} - \frac{9 \, \sqrt{-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{2}}{32 \, a^{3}} - \frac{3 \, \arccos \left (a x\right )^{3}}{32 \, a^{4}} + \frac{45 \, \sqrt{-a^{2} x^{2} + 1} x}{256 \, a^{3}} + \frac{45 \, \arccos \left (a x\right )}{256 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*arccos(a*x)^3,x, algorithm="giac")

[Out]

1/4*x^4*arccos(a*x)^3 - 3/32*x^4*arccos(a*x) - 3/16*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^2/a + 3/128*sqrt(-a^2*x
^2 + 1)*x^3/a - 9/32*x^2*arccos(a*x)/a^2 - 9/32*sqrt(-a^2*x^2 + 1)*x*arccos(a*x)^2/a^3 - 3/32*arccos(a*x)^3/a^
4 + 45/256*sqrt(-a^2*x^2 + 1)*x/a^3 + 45/256*arccos(a*x)/a^4