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

Optimal. Leaf size=250 \[ \frac{1088 x^3}{16875 a^2}-\frac{4 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac{24 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac{16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac{16 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}+\frac{1088 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^3}-\frac{32 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}+\frac{16576 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^5}+\frac{16576 x}{5625 a^4}-\frac{32 x \cos ^{-1}(a x)^2}{25 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^4-\frac{12}{125} x^5 \cos ^{-1}(a x)^2+\frac{24 x^5}{3125} \]

[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

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Rubi [A]  time = 0.669239, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4628, 4708, 4678, 4620, 8, 30} \[ \frac{1088 x^3}{16875 a^2}-\frac{4 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{25 a}+\frac{24 x^4 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{625 a}-\frac{16 x^3 \cos ^{-1}(a x)^2}{75 a^2}-\frac{16 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^3}+\frac{1088 x^2 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^3}-\frac{32 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^3}{75 a^5}+\frac{16576 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{5625 a^5}+\frac{16576 x}{5625 a^4}-\frac{32 x \cos ^{-1}(a x)^2}{25 a^4}+\frac{1}{5} x^5 \cos ^{-1}(a x)^4-\frac{12}{125} x^5 \cos ^{-1}(a x)^2+\frac{24 x^5}{3125} \]

Antiderivative was successfully verified.

[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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.0840269, size = 150, normalized size = 0.6 \[ \frac{8 a x \left (81 a^4 x^4+680 a^2 x^2+31080\right )+16875 a^5 x^5 \cos ^{-1}(a x)^4-900 a x \left (9 a^4 x^4+20 a^2 x^2+120\right ) \cos ^{-1}(a x)^2-4500 \sqrt{1-a^2 x^2} \left (3 a^4 x^4+4 a^2 x^2+8\right ) \cos ^{-1}(a x)^3+120 \sqrt{1-a^2 x^2} \left (27 a^4 x^4+136 a^2 x^2+2072\right ) \cos ^{-1}(a x)}{84375 a^5} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.058, size = 197, normalized size = 0.8 \begin{align*}{\frac{1}{{a}^{5}} \left ({\frac{{a}^{5}{x}^{5} \left ( \arccos \left ( ax \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \arccos \left ( ax \right ) \right ) ^{3} \left ( 3\,{a}^{4}{x}^{4}+4\,{a}^{2}{x}^{2}+8 \right ) }{75}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{32\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}}{25}}+{\frac{16576\,ax}{5625}}+{\frac{64\,\arccos \left ( ax \right ) }{25}\sqrt{-{a}^{2}{x}^{2}+1}}-{\frac{12\, \left ( \arccos \left ( ax \right ) \right ) ^{2}{a}^{5}{x}^{5}}{125}}+{\frac{8\,\arccos \left ( ax \right ) \left ( 3\,{a}^{4}{x}^{4}+4\,{a}^{2}{x}^{2}+8 \right ) }{625}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{24\,{a}^{5}{x}^{5}}{3125}}+{\frac{1088\,{a}^{3}{x}^{3}}{16875}}-{\frac{16\,{a}^{3}{x}^{3} \left ( \arccos \left ( ax \right ) \right ) ^{2}}{75}}+{\frac{32\,\arccos \left ( ax \right ) \left ({a}^{2}{x}^{2}+2 \right ) }{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)^4,x)

[Out]

1/a^5*(1/5*a^5*x^5*arccos(a*x)^4-4/75*arccos(a*x)^3*(3*a^4*x^4+4*a^2*x^2+8)*(-a^2*x^2+1)^(1/2)-32/25*a*x*arcco
s(a*x)^2+16576/5625*a*x+64/25*arccos(a*x)*(-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^3-16/75*a^3*x^3*arccos(a*x)^2+32/225
*arccos(a*x)*(a^2*x^2+2)*(-a^2*x^2+1)^(1/2))

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Maxima [A]  time = 1.52494, size = 278, normalized size = 1.11 \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.33717, size = 352, normalized size = 1.41 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 13.673, size = 248, normalized size = 0.99 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.16928, size = 286, normalized size = 1.14 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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