3.22 \(\int x^4 \cos ^{-1}(a x)^3 \, dx\)

Optimal. Leaf size=201 \[ \frac{6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac{76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac{298 \sqrt{1-a^2 x^2}}{375 a^5}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{16 x \cos ^{-1}(a x)}{25 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{6}{125} x^5 \cos ^{-1}(a x) \]

[Out]

(298*Sqrt[1 - a^2*x^2])/(375*a^5) - (76*(1 - a^2*x^2)^(3/2))/(1125*a^5) + (6*(1 - a^2*x^2)^(5/2))/(625*a^5) -
(16*x*ArcCos[a*x])/(25*a^4) - (8*x^3*ArcCos[a*x])/(75*a^2) - (6*x^5*ArcCos[a*x])/125 - (8*Sqrt[1 - a^2*x^2]*Ar
cCos[a*x]^2)/(25*a^5) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x
]^2)/(25*a) + (x^5*ArcCos[a*x]^3)/5

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Rubi [A]  time = 0.402615, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4628, 4708, 4678, 4620, 261, 266, 43} \[ \frac{6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac{76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac{298 \sqrt{1-a^2 x^2}}{375 a^5}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{16 x \cos ^{-1}(a x)}{25 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{6}{125} x^5 \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCos[a*x]^3,x]

[Out]

(298*Sqrt[1 - a^2*x^2])/(375*a^5) - (76*(1 - a^2*x^2)^(3/2))/(1125*a^5) + (6*(1 - a^2*x^2)^(5/2))/(625*a^5) -
(16*x*ArcCos[a*x])/(25*a^4) - (8*x^3*ArcCos[a*x])/(75*a^2) - (6*x^5*ArcCos[a*x])/125 - (8*Sqrt[1 - a^2*x^2]*Ar
cCos[a*x]^2)/(25*a^5) - (4*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^2)/(25*a^3) - (3*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x
]^2)/(25*a) + (x^5*ArcCos[a*x]^3)/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 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 261

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

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^4 \cos ^{-1}(a x)^3 \, dx &=\frac{1}{5} x^5 \cos ^{-1}(a x)^3+\frac{1}{5} (3 a) \int \frac{x^5 \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{6}{25} \int x^4 \cos ^{-1}(a x) \, dx+\frac{12 \int \frac{x^3 \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac{6}{125} x^5 \cos ^{-1}(a x)-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3+\frac{8 \int \frac{x \cos ^{-1}(a x)^2}{\sqrt{1-a^2 x^2}} \, dx}{25 a^3}-\frac{8 \int x^2 \cos ^{-1}(a x) \, dx}{25 a^2}-\frac{1}{125} (6 a) \int \frac{x^5}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{6}{125} x^5 \cos ^{-1}(a x)-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{16 \int \cos ^{-1}(a x) \, dx}{25 a^4}-\frac{8 \int \frac{x^3}{\sqrt{1-a^2 x^2}} \, dx}{75 a}-\frac{1}{125} (3 a) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{16 x \cos ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{6}{125} x^5 \cos ^{-1}(a x)-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{16 \int \frac{x}{\sqrt{1-a^2 x^2}} \, dx}{25 a^3}-\frac{4 \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-a^2 x}} \, dx,x,x^2\right )}{75 a}-\frac{1}{125} (3 a) \operatorname{Subst}\left (\int \left (\frac{1}{a^4 \sqrt{1-a^2 x}}-\frac{2 \sqrt{1-a^2 x}}{a^4}+\frac{\left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=\frac{86 \sqrt{1-a^2 x^2}}{125 a^5}-\frac{4 \left (1-a^2 x^2\right )^{3/2}}{125 a^5}+\frac{6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac{16 x \cos ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{6}{125} x^5 \cos ^{-1}(a x)-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3-\frac{4 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 \sqrt{1-a^2 x}}-\frac{\sqrt{1-a^2 x}}{a^2}\right ) \, dx,x,x^2\right )}{75 a}\\ &=\frac{298 \sqrt{1-a^2 x^2}}{375 a^5}-\frac{76 \left (1-a^2 x^2\right )^{3/2}}{1125 a^5}+\frac{6 \left (1-a^2 x^2\right )^{5/2}}{625 a^5}-\frac{16 x \cos ^{-1}(a x)}{25 a^4}-\frac{8 x^3 \cos ^{-1}(a x)}{75 a^2}-\frac{6}{125} x^5 \cos ^{-1}(a x)-\frac{8 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^5}-\frac{4 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^2}{25 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^3\\ \end{align*}

Mathematica [A]  time = 0.0682192, size = 122, normalized size = 0.61 \[ \frac{2 \sqrt{1-a^2 x^2} \left (27 a^4 x^4+136 a^2 x^2+2072\right )+1125 a^5 x^5 \cos ^{-1}(a x)^3-30 a x \left (9 a^4 x^4+20 a^2 x^2+120\right ) \cos ^{-1}(a x)-225 \sqrt{1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cos ^{-1}(a x)^2}{5625 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCos[a*x]^3,x]

[Out]

(2*Sqrt[1 - a^2*x^2]*(2072 + 136*a^2*x^2 + 27*a^4*x^4) - 30*a*x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*ArcCos[a*x] - 2
25*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a*x]^2 + 1125*a^5*x^5*ArcCos[a*x]^3)/(5625*a^5)

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Maple [A]  time = 0.056, size = 159, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{3}{a}^{5}{x}^{5}}{5}}-{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{2} \left ( 3\,{a}^{4}{x}^{4}+4\,{a}^{2}{x}^{2}+8 \right ) }{25}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{16}{25}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{16\,ax\arccos \left ( ax \right ) }{25}}-{\frac{6\,{a}^{5}{x}^{5}\arccos \left ( ax \right ) }{125}}+{\frac{6\,{a}^{4}{x}^{4}+8\,{a}^{2}{x}^{2}+16}{625}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{8\,{a}^{3}{x}^{3}\arccos \left ( ax \right ) }{75}}+{\frac{8\,{a}^{2}{x}^{2}+16}{225}\sqrt{-{a}^{2}{x}^{2}+1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccos(a*x)^3,x)

[Out]

1/a^5*(1/5*arccos(a*x)^3*a^5*x^5-1/25*arccos(a*x)^2*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)+16/25*(-a^2*x^2
+1)^(1/2)-16/25*a*x*arccos(a*x)-6/125*a^5*x^5*arccos(a*x)+2/625*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)-8/7
5*a^3*x^3*arccos(a*x)+8/225*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.50085, size = 231, normalized size = 1.15 \begin{align*} \frac{1}{5} \, x^{5} \arccos \left (a x\right )^{3} - \frac{1}{25} \,{\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 )^{2} + \frac{2}{5625} \, a{\left (\frac{27 \, \sqrt{-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt{-a^{2} x^{2} + 1} x^{2} + \frac{2072 \, \sqrt{-a^{2} x^{2} + 1}}{a^{2}}}{a^{4}} - \frac{15 \,{\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arccos \left (a x\right )}{a^{5}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^3,x, algorithm="maxima")

[Out]

1/5*x^5*arccos(a*x)^3 - 1/25*(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 + 2/5625*a*((27*sqrt(-a^2*x^2 + 1)*a^2*x^4 + 136*sqrt(-a^2*x^2 + 1)*x^2 + 2072*sqrt(-a
^2*x^2 + 1)/a^2)/a^4 - 15*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)*arccos(a*x)/a^5)

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Fricas [A]  time = 2.3728, size = 265, normalized size = 1.32 \begin{align*} \frac{1125 \, a^{5} x^{5} \arccos \left (a x\right )^{3} - 30 \,{\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arccos \left (a x\right ) +{\left (54 \, a^{4} x^{4} + 272 \, a^{2} x^{2} - 225 \,{\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arccos \left (a x\right )^{2} + 4144\right )} \sqrt{-a^{2} x^{2} + 1}}{5625 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^3,x, algorithm="fricas")

[Out]

1/5625*(1125*a^5*x^5*arccos(a*x)^3 - 30*(9*a^5*x^5 + 20*a^3*x^3 + 120*a*x)*arccos(a*x) + (54*a^4*x^4 + 272*a^2
*x^2 - 225*(3*a^4*x^4 + 4*a^2*x^2 + 8)*arccos(a*x)^2 + 4144)*sqrt(-a^2*x^2 + 1))/a^5

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Sympy [A]  time = 7.92169, size = 202, normalized size = 1. \begin{align*} \begin{cases} \frac{x^{5} \operatorname{acos}^{3}{\left (a x \right )}}{5} - \frac{6 x^{5} \operatorname{acos}{\left (a x \right )}}{125} - \frac{3 x^{4} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{25 a} + \frac{6 x^{4} \sqrt{- a^{2} x^{2} + 1}}{625 a} - \frac{8 x^{3} \operatorname{acos}{\left (a x \right )}}{75 a^{2}} - \frac{4 x^{2} \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{25 a^{3}} + \frac{272 x^{2} \sqrt{- a^{2} x^{2} + 1}}{5625 a^{3}} - \frac{16 x \operatorname{acos}{\left (a x \right )}}{25 a^{4}} - \frac{8 \sqrt{- a^{2} x^{2} + 1} \operatorname{acos}^{2}{\left (a x \right )}}{25 a^{5}} + \frac{4144 \sqrt{- a^{2} x^{2} + 1}}{5625 a^{5}} & \text{for}\: a \neq 0 \\\frac{\pi ^{3} x^{5}}{40} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acos(a*x)**3,x)

[Out]

Piecewise((x**5*acos(a*x)**3/5 - 6*x**5*acos(a*x)/125 - 3*x**4*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(25*a) + 6*x*
*4*sqrt(-a**2*x**2 + 1)/(625*a) - 8*x**3*acos(a*x)/(75*a**2) - 4*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)**2/(25*a*
*3) + 272*x**2*sqrt(-a**2*x**2 + 1)/(5625*a**3) - 16*x*acos(a*x)/(25*a**4) - 8*sqrt(-a**2*x**2 + 1)*acos(a*x)*
*2/(25*a**5) + 4144*sqrt(-a**2*x**2 + 1)/(5625*a**5), Ne(a, 0)), (pi**3*x**5/40, True))

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Giac [A]  time = 1.14352, size = 236, normalized size = 1.17 \begin{align*} \frac{1}{5} \, x^{5} \arccos \left (a x\right )^{3} - \frac{6}{125} \, x^{5} \arccos \left (a x\right ) - \frac{3 \, \sqrt{-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )^{2}}{25 \, a} + \frac{6 \, \sqrt{-a^{2} x^{2} + 1} x^{4}}{625 \, a} - \frac{8 \, x^{3} \arccos \left (a x\right )}{75 \, a^{2}} - \frac{4 \, \sqrt{-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{2}}{25 \, a^{3}} + \frac{272 \, \sqrt{-a^{2} x^{2} + 1} x^{2}}{5625 \, a^{3}} - \frac{16 \, x \arccos \left (a x\right )}{25 \, a^{4}} - \frac{8 \, \sqrt{-a^{2} x^{2} + 1} \arccos \left (a x\right )^{2}}{25 \, a^{5}} + \frac{4144 \, \sqrt{-a^{2} x^{2} + 1}}{5625 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^3,x, algorithm="giac")

[Out]

1/5*x^5*arccos(a*x)^3 - 6/125*x^5*arccos(a*x) - 3/25*sqrt(-a^2*x^2 + 1)*x^4*arccos(a*x)^2/a + 6/625*sqrt(-a^2*
x^2 + 1)*x^4/a - 8/75*x^3*arccos(a*x)/a^2 - 4/25*sqrt(-a^2*x^2 + 1)*x^2*arccos(a*x)^2/a^3 + 272/5625*sqrt(-a^2
*x^2 + 1)*x^2/a^3 - 16/25*x*arccos(a*x)/a^4 - 8/25*sqrt(-a^2*x^2 + 1)*arccos(a*x)^2/a^5 + 4144/5625*sqrt(-a^2*
x^2 + 1)/a^5