3.1.22 \(\int x^4 \text {ArcCos}(a x)^3 \, dx\) [22]

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

[Out]

-76/1125*(-a^2*x^2+1)^(3/2)/a^5+6/625*(-a^2*x^2+1)^(5/2)/a^5-16/25*x*arccos(a*x)/a^4-8/75*x^3*arccos(a*x)/a^2-
6/125*x^5*arccos(a*x)+1/5*x^5*arccos(a*x)^3+298/375*(-a^2*x^2+1)^(1/2)/a^5-8/25*arccos(a*x)^2*(-a^2*x^2+1)^(1/
2)/a^5-4/25*x^2*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)/a^3-3/25*x^4*arccos(a*x)^2*(-a^2*x^2+1)^(1/2)/a

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Rubi [A]
time = 0.26, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4724, 4796, 4768, 4716, 267, 272, 45} \begin {gather*} -\frac {16 x \text {ArcCos}(a x)}{25 a^4}-\frac {8 x^3 \text {ArcCos}(a x)}{75 a^2}-\frac {3 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{25 a}-\frac {8 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{25 a^5}+\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 {4 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^2}{25 a^3}+\frac {1}{5} x^5 \text {ArcCos}(a x)^3-\frac {6}{125} x^5 \text {ArcCos}(a x) \end {gather*}

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 45

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])

Rule 267

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 272

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 4716

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 4724

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 4768

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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 4796

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[(f*x)^(m - 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 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) \text {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 \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )}{75 a}-\frac {1}{125} (3 a) \text {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 \text {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*}

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Mathematica [A]
time = 0.05, size = 122, normalized size = 0.61 \begin {gather*} \frac {2 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right )-30 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \text {ArcCos}(a x)-225 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {ArcCos}(a x)^2+1125 a^5 x^5 \text {ArcCos}(a x)^3}{5625 a^5} \end {gather*}

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 = 0.05, size = 159, normalized size = 0.79

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{3}}{5}-\frac {\arccos \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arccos \left (a x \right )}{125}+\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arccos \left (a x \right )}{75}+\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}+\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arccos \left (a x \right )}{25}}{a^{5}}\)	\(159\)
default	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{3}}{5}-\frac {\arccos \left (a x \right )^{2} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{25}-\frac {6 a^{5} x^{5} \arccos \left (a x \right )}{125}+\frac {2 \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}-\frac {8 a^{3} x^{3} \arccos \left (a x \right )}{75}+\frac {8 \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}+\frac {16 \sqrt {-a^{2} x^{2}+1}}{25}-\frac {16 a x \arccos \left (a x \right )}{25}}{a^{5}}\)	\(159\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccos(a*x)^3,x,method=_RETURNVERBOSE)

[Out]

1/a^5*(1/5*a^5*x^5*arccos(a*x)^3-1/25*arccos(a*x)^2*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)-6/125*a^5*x^5*a
rccos(a*x)+2/625*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)-8/75*a^3*x^3*arccos(a*x)+8/225*(a^2*x^2+2)*(-a^2*x
^2+1)^(1/2)+16/25*(-a^2*x^2+1)^(1/2)-16/25*a*x*arccos(a*x))

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Maxima [A]
time = 0.49, size = 171, normalized size = 0.85 \begin {gather*} \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 {gather*}

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 = 3.36, size = 104, normalized size = 0.52 \begin {gather*} \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 {gather*}

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 = 0.73, size = 202, normalized size = 1.00 \begin {gather*} \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 {gather*}

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 = 0.43, size = 175, normalized size = 0.87 \begin {gather*} \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 {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,{\mathrm {acos}\left (a\,x\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*acos(a*x)^3, x)

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