\(\int x^4 \arccos (a x)^4 \, dx\) [33]

Optimal result

Integrand size = 10, antiderivative size = 250 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {16576 x}{5625 a^4}+\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}-\frac {32 x \arccos (a x)^2}{25 a^4}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arccos (a x)^2-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4 \]

[Out]

16576/5625*x/a^4+1088/16875*x^3/a^2+24/3125*x^5-32/25*x*arccos(a*x)^2/a^4-16/75*x^3*arccos(a*x)^2/a^2-12/125*x
^5*arccos(a*x)^2+1/5*x^5*arccos(a*x)^4+16576/5625*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a^5+1088/5625*x^2*arccos(a*x)
*(-a^2*x^2+1)^(1/2)/a^3+24/625*x^4*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a-32/75*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/a^5
-16/75*x^2*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/a^3-4/25*x^4*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4724, 4796, 4768, 4716, 8, 30} \[ \int x^4 \arccos (a x)^4 \, dx=-\frac {32 x \arccos (a x)^2}{25 a^4}+\frac {16576 x}{5625 a^4}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}+\frac {1088 x^3}{16875 a^2}-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}+\frac {16576 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^3}+\frac {1}{5} x^5 \arccos (a x)^4-\frac {12}{125} x^5 \arccos (a x)^2+\frac {24 x^5}{3125} \]

[In]

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

[Out]

(16576*x)/(5625*a^4) + (1088*x^3)/(16875*a^2) + (24*x^5)/3125 + (16576*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(5625*a^
5) + (1088*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(5625*a^3) + (24*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(625*a) - (3
2*x*ArcCos[a*x]^2)/(25*a^4) - (16*x^3*ArcCos[a*x]^2)/(75*a^2) - (12*x^5*ArcCos[a*x]^2)/125 - (32*Sqrt[1 - a^2*
x^2]*ArcCos[a*x]^3)/(75*a^5) - (16*x^2*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(75*a^3) - (4*x^4*Sqrt[1 - a^2*x^2]*Ar
cCos[a*x]^3)/(25*a) + (x^5*ArcCos[a*x]^4)/5

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 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*} \text {integral}& = \frac {1}{5} x^5 \arccos (a x)^4+\frac {1}{5} (4 a) \int \frac {x^5 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4-\frac {12}{25} \int x^4 \arccos (a x)^2 \, dx+\frac {16 \int \frac {x^3 \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{25 a} \\ & = -\frac {12}{125} x^5 \arccos (a x)^2-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4+\frac {32 \int \frac {x \arccos (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{75 a^3}-\frac {16 \int x^2 \arccos (a x)^2 \, dx}{25 a^2}-\frac {1}{125} (24 a) \int \frac {x^5 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx \\ & = \frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arccos (a x)^2-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4+\frac {24 \int x^4 \, dx}{625}-\frac {32 \int \arccos (a x)^2 \, dx}{25 a^4}-\frac {96 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{625 a}-\frac {32 \int \frac {x^3 \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{75 a} \\ & = \frac {24 x^5}{3125}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}-\frac {32 x \arccos (a x)^2}{25 a^4}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arccos (a x)^2-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4-\frac {64 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{625 a^3}-\frac {64 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{225 a^3}-\frac {64 \int \frac {x \arccos (a x)}{\sqrt {1-a^2 x^2}} \, dx}{25 a^3}+\frac {32 \int x^2 \, dx}{625 a^2}+\frac {32 \int x^2 \, dx}{225 a^2} \\ & = \frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}-\frac {32 x \arccos (a x)^2}{25 a^4}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arccos (a x)^2-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4+\frac {64 \int 1 \, dx}{625 a^4}+\frac {64 \int 1 \, dx}{225 a^4}+\frac {64 \int 1 \, dx}{25 a^4} \\ & = \frac {16576 x}{5625 a^4}+\frac {1088 x^3}{16875 a^2}+\frac {24 x^5}{3125}+\frac {16576 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^5}+\frac {1088 x^2 \sqrt {1-a^2 x^2} \arccos (a x)}{5625 a^3}+\frac {24 x^4 \sqrt {1-a^2 x^2} \arccos (a x)}{625 a}-\frac {32 x \arccos (a x)^2}{25 a^4}-\frac {16 x^3 \arccos (a x)^2}{75 a^2}-\frac {12}{125} x^5 \arccos (a x)^2-\frac {32 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^5}-\frac {16 x^2 \sqrt {1-a^2 x^2} \arccos (a x)^3}{75 a^3}-\frac {4 x^4 \sqrt {1-a^2 x^2} \arccos (a x)^3}{25 a}+\frac {1}{5} x^5 \arccos (a x)^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.60 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {8 a x \left (31080+680 a^2 x^2+81 a^4 x^4\right )+120 \sqrt {1-a^2 x^2} \left (2072+136 a^2 x^2+27 a^4 x^4\right ) \arccos (a x)-900 a x \left (120+20 a^2 x^2+9 a^4 x^4\right ) \arccos (a x)^2-4500 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \arccos (a x)^3+16875 a^5 x^5 \arccos (a x)^4}{84375 a^5} \]

[In]

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

[Out]

(8*a*x*(31080 + 680*a^2*x^2 + 81*a^4*x^4) + 120*Sqrt[1 - a^2*x^2]*(2072 + 136*a^2*x^2 + 27*a^4*x^4)*ArcCos[a*x
] - 900*a*x*(120 + 20*a^2*x^2 + 9*a^4*x^4)*ArcCos[a*x]^2 - 4500*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*
ArcCos[a*x]^3 + 16875*a^5*x^5*ArcCos[a*x]^4)/(84375*a^5)

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(\frac {\frac {\arccos \left (a x \right )^{4} a^{5} x^{5}}{5}-\frac {4 \arccos \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 \arccos \left (a x \right )^{2} a^{5} x^{5}}{125}+\frac {8 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 \arccos \left (a x \right )^{2} a^{3} x^{3}}{75}+\frac {32 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 \arccos \left (a x \right )^{2} a x}{25}+\frac {64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\)	\(197\)
default	\(\frac {\frac {\arccos \left (a x \right )^{4} a^{5} x^{5}}{5}-\frac {4 \arccos \left (a x \right )^{3} \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {12 \arccos \left (a x \right )^{2} a^{5} x^{5}}{125}+\frac {8 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{625}+\frac {24 a^{5} x^{5}}{3125}+\frac {1088 a^{3} x^{3}}{16875}+\frac {16576 a x}{5625}-\frac {16 \arccos \left (a x \right )^{2} a^{3} x^{3}}{75}+\frac {32 \arccos \left (a x \right ) \left (a^{2} x^{2}+2\right ) \sqrt {-a^{2} x^{2}+1}}{225}-\frac {32 \arccos \left (a x \right )^{2} a x}{25}+\frac {64 \arccos \left (a x \right ) \sqrt {-a^{2} x^{2}+1}}{25}}{a^{5}}\)	\(197\)

[In]

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

[Out]

1/a^5*(1/5*arccos(a*x)^4*a^5*x^5-4/75*arccos(a*x)^3*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)-12/125*arccos(a
*x)^2*a^5*x^5+8/625*arccos(a*x)*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)+24/3125*a^5*x^5+1088/16875*a^3*x^3+
16576/5625*a*x-16/75*arccos(a*x)^2*a^3*x^3+32/225*arccos(a*x)*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2)-32/25*arccos(a*x)
^2*a*x+64/25*arccos(a*x)*(-a^2*x^2+1)^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.54 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {16875 \, a^{5} x^{5} \arccos \left (a x\right )^{4} + 648 \, a^{5} x^{5} + 5440 \, a^{3} x^{3} - 900 \, {\left (9 \, a^{5} x^{5} + 20 \, a^{3} x^{3} + 120 \, a x\right )} \arccos \left (a x\right )^{2} + 248640 \, a x - 60 \, \sqrt {-a^{2} x^{2} + 1} {\left (75 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \arccos \left (a x\right )^{3} - 2 \, {\left (27 \, a^{4} x^{4} + 136 \, a^{2} x^{2} + 2072\right )} \arccos \left (a x\right )\right )}}{84375 \, a^{5}} \]

[In]

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

[Out]

1/84375*(16875*a^5*x^5*arccos(a*x)^4 + 648*a^5*x^5 + 5440*a^3*x^3 - 900*(9*a^5*x^5 + 20*a^3*x^3 + 120*a*x)*arc
cos(a*x)^2 + 248640*a*x - 60*sqrt(-a^2*x^2 + 1)*(75*(3*a^4*x^4 + 4*a^2*x^2 + 8)*arccos(a*x)^3 - 2*(27*a^4*x^4
+ 136*a^2*x^2 + 2072)*arccos(a*x)))/a^5

Sympy [A] (verification not implemented)

Time = 0.92 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.99 \[ \int x^4 \arccos (a x)^4 \, dx=\begin {cases} \frac {x^{5} \operatorname {acos}^{4}{\left (a x \right )}}{5} - \frac {12 x^{5} \operatorname {acos}^{2}{\left (a x \right )}}{125} + \frac {24 x^{5}}{3125} - \frac {4 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{25 a} + \frac {24 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{625 a} - \frac {16 x^{3} \operatorname {acos}^{2}{\left (a x \right )}}{75 a^{2}} + \frac {1088 x^{3}}{16875 a^{2}} - \frac {16 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{75 a^{3}} + \frac {1088 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{5625 a^{3}} - \frac {32 x \operatorname {acos}^{2}{\left (a x \right )}}{25 a^{4}} + \frac {16576 x}{5625 a^{4}} - \frac {32 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{75 a^{5}} + \frac {16576 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{5625 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{5}}{80} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**5*acos(a*x)**4/5 - 12*x**5*acos(a*x)**2/125 + 24*x**5/3125 - 4*x**4*sqrt(-a**2*x**2 + 1)*acos(a*
x)**3/(25*a) + 24*x**4*sqrt(-a**2*x**2 + 1)*acos(a*x)/(625*a) - 16*x**3*acos(a*x)**2/(75*a**2) + 1088*x**3/(16
875*a**2) - 16*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(75*a**3) + 1088*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)/(56
25*a**3) - 32*x*acos(a*x)**2/(25*a**4) + 16576*x/(5625*a**4) - 32*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(75*a**5)
+ 16576*sqrt(-a**2*x**2 + 1)*acos(a*x)/(5625*a**5), Ne(a, 0)), (pi**4*x**5/80, True))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.82 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{4} - \frac {4}{75} \, {\left (\frac {3 \, \sqrt {-a^{2} x^{2} + 1} x^{4}}{a^{2}} + \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{a^{4}} + \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{a^{6}}\right )} a \arccos \left (a x\right )^{3} + \frac {4}{84375} \, {\left (2 \, a {\left (\frac {15 \, {\left (27 \, \sqrt {-a^{2} x^{2} + 1} a^{2} x^{4} + 136 \, \sqrt {-a^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {-a^{2} x^{2} + 1}}{a^{2}}\right )} \arccos \left (a x\right )}{a^{5}} + \frac {81 \, a^{4} x^{5} + 680 \, a^{2} x^{3} + 31080 \, x}{a^{6}}\right )} - \frac {225 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )} \arccos \left (a x\right )^{2}}{a^{5}}\right )} a \]

[In]

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

[Out]

1/5*x^5*arccos(a*x)^4 - 4/75*(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)^3 + 4/84375*(2*a*(15*(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)*arccos(a*x)/a^5 + (81*a^4*x^5 + 680*a^2*x^3 + 31080*x)/a^6) - 225*(9*a^4*x^5 + 20*a^2*
x^3 + 120*x)*arccos(a*x)^2/a^5)*a

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.85 \[ \int x^4 \arccos (a x)^4 \, dx=\frac {1}{5} \, x^{5} \arccos \left (a x\right )^{4} - \frac {12}{125} \, x^{5} \arccos \left (a x\right )^{2} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )^{3}}{25 \, a} + \frac {24}{3125} \, x^{5} + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{625 \, a} - \frac {16 \, x^{3} \arccos \left (a x\right )^{2}}{75 \, a^{2}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )^{3}}{75 \, a^{3}} + \frac {1088 \, x^{3}}{16875 \, a^{2}} + \frac {1088 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{5625 \, a^{3}} - \frac {32 \, x \arccos \left (a x\right )^{2}}{25 \, a^{4}} - \frac {32 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )^{3}}{75 \, a^{5}} + \frac {16576 \, x}{5625 \, a^{4}} + \frac {16576 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{5625 \, a^{5}} \]

[In]

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

[Out]

1/5*x^5*arccos(a*x)^4 - 12/125*x^5*arccos(a*x)^2 - 4/25*sqrt(-a^2*x^2 + 1)*x^4*arccos(a*x)^3/a + 24/3125*x^5 +
24/625*sqrt(-a^2*x^2 + 1)*x^4*arccos(a*x)/a - 16/75*x^3*arccos(a*x)^2/a^2 - 16/75*sqrt(-a^2*x^2 + 1)*x^2*arcc
os(a*x)^3/a^3 + 1088/16875*x^3/a^2 + 1088/5625*sqrt(-a^2*x^2 + 1)*x^2*arccos(a*x)/a^3 - 32/25*x*arccos(a*x)^2/
a^4 - 32/75*sqrt(-a^2*x^2 + 1)*arccos(a*x)^3/a^5 + 16576/5625*x/a^4 + 16576/5625*sqrt(-a^2*x^2 + 1)*arccos(a*x
)/a^5

Mupad [F(-1)]

Timed out. \[ \int x^4 \arccos (a x)^4 \, dx=\int x^4\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \]

[In]

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

[Out]

int(x^4*acos(a*x)^4, x)