3.32 \(\int x^5 \cos ^{-1}(a x)^4 \, dx\)

Optimal. Leaf size=282 \[ -\frac {5 \cos ^{-1}(a x)^4}{96 a^6}+\frac {245 \cos ^{-1}(a x)^2}{1152 a^6}+\frac {245 x^2}{1152 a^4}-\frac {5 x^2 \cos ^{-1}(a x)^2}{16 a^4}+\frac {65 x^4}{3456 a^2}-\frac {5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac {5 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}+\frac {245 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}+\frac {65 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4-\frac {1}{18} x^6 \cos ^{-1}(a x)^2+\frac {x^6}{324} \]

[Out]

245/1152*x^2/a^4+65/3456*x^4/a^2+1/324*x^6+245/1152*arccos(a*x)^2/a^6-5/16*x^2*arccos(a*x)^2/a^4-5/48*x^4*arcc
os(a*x)^2/a^2-1/18*x^6*arccos(a*x)^2-5/96*arccos(a*x)^4/a^6+1/6*x^6*arccos(a*x)^4+245/576*x*arccos(a*x)*(-a^2*
x^2+1)^(1/2)/a^5+65/864*x^3*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a^3+1/54*x^5*arccos(a*x)*(-a^2*x^2+1)^(1/2)/a-5/24*
x*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/a^5-5/36*x^3*arccos(a*x)^3*(-a^2*x^2+1)^(1/2)/a^3-1/9*x^5*arccos(a*x)^3*(-a
^2*x^2+1)^(1/2)/a

________________________________________________________________________________________

Rubi [A]  time = 0.87, antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4628, 4708, 4642, 30} \[ \frac {65 x^4}{3456 a^2}+\frac {245 x^2}{1152 a^4}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac {5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}+\frac {65 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}-\frac {5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac {5 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}+\frac {245 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}-\frac {5 \cos ^{-1}(a x)^4}{96 a^6}+\frac {245 \cos ^{-1}(a x)^2}{1152 a^6}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4-\frac {1}{18} x^6 \cos ^{-1}(a x)^2+\frac {x^6}{324} \]

Antiderivative was successfully verified.

[In]

Int[x^5*ArcCos[a*x]^4,x]

[Out]

(245*x^2)/(1152*a^4) + (65*x^4)/(3456*a^2) + x^6/324 + (245*x*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(576*a^5) + (65*x
^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(864*a^3) + (x^5*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(54*a) + (245*ArcCos[a*x]^2)
/(1152*a^6) - (5*x^2*ArcCos[a*x]^2)/(16*a^4) - (5*x^4*ArcCos[a*x]^2)/(48*a^2) - (x^6*ArcCos[a*x]^2)/18 - (5*x*
Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(24*a^5) - (5*x^3*Sqrt[1 - a^2*x^2]*ArcCos[a*x]^3)/(36*a^3) - (x^5*Sqrt[1 - a
^2*x^2]*ArcCos[a*x]^3)/(9*a) - (5*ArcCos[a*x]^4)/(96*a^6) + (x^6*ArcCos[a*x]^4)/6

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 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 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]

Rubi steps

\begin {align*} \int x^5 \cos ^{-1}(a x)^4 \, dx &=\frac {1}{6} x^6 \cos ^{-1}(a x)^4+\frac {1}{3} (2 a) \int \frac {x^6 \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4-\frac {1}{3} \int x^5 \cos ^{-1}(a x)^2 \, dx+\frac {5 \int \frac {x^4 \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{9 a}\\ &=-\frac {1}{18} x^6 \cos ^{-1}(a x)^2-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4+\frac {5 \int \frac {x^2 \cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{12 a^3}-\frac {5 \int x^3 \cos ^{-1}(a x)^2 \, dx}{12 a^2}-\frac {1}{9} a \int \frac {x^6 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac {5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac {1}{18} x^6 \cos ^{-1}(a x)^2-\frac {5 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4+\frac {\int x^5 \, dx}{54}+\frac {5 \int \frac {\cos ^{-1}(a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{24 a^5}-\frac {5 \int x \cos ^{-1}(a x)^2 \, dx}{8 a^4}-\frac {5 \int \frac {x^4 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{54 a}-\frac {5 \int \frac {x^4 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{24 a}\\ &=\frac {x^6}{324}+\frac {65 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac {5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac {5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac {1}{18} x^6 \cos ^{-1}(a x)^2-\frac {5 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac {5 \cos ^{-1}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4-\frac {5 \int \frac {x^2 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{72 a^3}-\frac {5 \int \frac {x^2 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3}-\frac {5 \int \frac {x^2 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 a^3}+\frac {5 \int x^3 \, dx}{216 a^2}+\frac {5 \int x^3 \, dx}{96 a^2}\\ &=\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}+\frac {245 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}+\frac {65 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{54 a}-\frac {5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac {5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac {1}{18} x^6 \cos ^{-1}(a x)^2-\frac {5 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac {5 \cos ^{-1}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4-\frac {5 \int \frac {\cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{144 a^5}-\frac {5 \int \frac {\cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 a^5}-\frac {5 \int \frac {\cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 a^5}+\frac {5 \int x \, dx}{144 a^4}+\frac {5 \int x \, dx}{64 a^4}+\frac {5 \int x \, dx}{16 a^4}\\ &=\frac {245 x^2}{1152 a^4}+\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}+\frac {245 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{576 a^5}+\frac {65 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{864 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{54 a}+\frac {245 \cos ^{-1}(a x)^2}{1152 a^6}-\frac {5 x^2 \cos ^{-1}(a x)^2}{16 a^4}-\frac {5 x^4 \cos ^{-1}(a x)^2}{48 a^2}-\frac {1}{18} x^6 \cos ^{-1}(a x)^2-\frac {5 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{24 a^5}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{36 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^3}{9 a}-\frac {5 \cos ^{-1}(a x)^4}{96 a^6}+\frac {1}{6} x^6 \cos ^{-1}(a x)^4\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 167, normalized size = 0.59 \[ \frac {108 \left (16 a^6 x^6-5\right ) \cos ^{-1}(a x)^4+a^2 x^2 \left (32 a^4 x^4+195 a^2 x^2+2205\right )-144 a x \sqrt {1-a^2 x^2} \left (8 a^4 x^4+10 a^2 x^2+15\right ) \cos ^{-1}(a x)^3+6 a x \sqrt {1-a^2 x^2} \left (32 a^4 x^4+130 a^2 x^2+735\right ) \cos ^{-1}(a x)-9 \left (64 a^6 x^6+120 a^4 x^4+360 a^2 x^2-245\right ) \cos ^{-1}(a x)^2}{10368 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*ArcCos[a*x]^4,x]

[Out]

(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) + 6*a*x*Sqrt[1 - a^2*x^2]*(735 + 130*a^2*x^2 + 32*a^4*x^4)*ArcCos[a
*x] - 9*(-245 + 360*a^2*x^2 + 120*a^4*x^4 + 64*a^6*x^6)*ArcCos[a*x]^2 - 144*a*x*Sqrt[1 - a^2*x^2]*(15 + 10*a^2
*x^2 + 8*a^4*x^4)*ArcCos[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcCos[a*x]^4)/(10368*a^6)

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 153, normalized size = 0.54 \[ \frac {32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \, {\left (16 \, a^{6} x^{6} - 5\right )} \arccos \left (a x\right )^{4} + 2205 \, a^{2} x^{2} - 9 \, {\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \arccos \left (a x\right )^{2} - 6 \, \sqrt {-a^{2} x^{2} + 1} {\left (24 \, {\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \arccos \left (a x\right )^{3} - {\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \arccos \left (a x\right )\right )}}{10368 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccos(a*x)^4,x, algorithm="fricas")

[Out]

1/10368*(32*a^6*x^6 + 195*a^4*x^4 + 108*(16*a^6*x^6 - 5)*arccos(a*x)^4 + 2205*a^2*x^2 - 9*(64*a^6*x^6 + 120*a^
4*x^4 + 360*a^2*x^2 - 245)*arccos(a*x)^2 - 6*sqrt(-a^2*x^2 + 1)*(24*(8*a^5*x^5 + 10*a^3*x^3 + 15*a*x)*arccos(a
*x)^3 - (32*a^5*x^5 + 130*a^3*x^3 + 735*a*x)*arccos(a*x)))/a^6

________________________________________________________________________________________

giac [A]  time = 0.20, size = 245, normalized size = 0.87 \[ \frac {1}{6} \, x^{6} \arccos \left (a x\right )^{4} - \frac {1}{18} \, x^{6} \arccos \left (a x\right )^{2} - \frac {\sqrt {-a^{2} x^{2} + 1} x^{5} \arccos \left (a x\right )^{3}}{9 \, a} + \frac {1}{324} \, x^{6} + \frac {\sqrt {-a^{2} x^{2} + 1} x^{5} \arccos \left (a x\right )}{54 \, a} - \frac {5 \, x^{4} \arccos \left (a x\right )^{2}}{48 \, a^{2}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )^{3}}{36 \, a^{3}} + \frac {65 \, x^{4}}{3456 \, a^{2}} + \frac {65 \, \sqrt {-a^{2} x^{2} + 1} x^{3} \arccos \left (a x\right )}{864 \, a^{3}} - \frac {5 \, x^{2} \arccos \left (a x\right )^{2}}{16 \, a^{4}} - \frac {5 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )^{3}}{24 \, a^{5}} + \frac {245 \, x^{2}}{1152 \, a^{4}} - \frac {5 \, \arccos \left (a x\right )^{4}}{96 \, a^{6}} + \frac {245 \, \sqrt {-a^{2} x^{2} + 1} x \arccos \left (a x\right )}{576 \, a^{5}} + \frac {245 \, \arccos \left (a x\right )^{2}}{1152 \, a^{6}} - \frac {9485}{82944 \, a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccos(a*x)^4,x, algorithm="giac")

[Out]

1/6*x^6*arccos(a*x)^4 - 1/18*x^6*arccos(a*x)^2 - 1/9*sqrt(-a^2*x^2 + 1)*x^5*arccos(a*x)^3/a + 1/324*x^6 + 1/54
*sqrt(-a^2*x^2 + 1)*x^5*arccos(a*x)/a - 5/48*x^4*arccos(a*x)^2/a^2 - 5/36*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)^3
/a^3 + 65/3456*x^4/a^2 + 65/864*sqrt(-a^2*x^2 + 1)*x^3*arccos(a*x)/a^3 - 5/16*x^2*arccos(a*x)^2/a^4 - 5/24*sqr
t(-a^2*x^2 + 1)*x*arccos(a*x)^3/a^5 + 245/1152*x^2/a^4 - 5/96*arccos(a*x)^4/a^6 + 245/576*sqrt(-a^2*x^2 + 1)*x
*arccos(a*x)/a^5 + 245/1152*arccos(a*x)^2/a^6 - 9485/82944/a^6

________________________________________________________________________________________

maple [A]  time = 0.16, size = 318, normalized size = 1.13 \[ \frac {\frac {a^{6} x^{6} \arccos \left (a x \right )^{4}}{6}-\frac {\arccos \left (a x \right )^{3} \left (8 a^{5} x^{5} \sqrt {-a^{2} x^{2}+1}+10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+15 a x \sqrt {-a^{2} x^{2}+1}+15 \arccos \left (a x \right )\right )}{72}-\frac {\arccos \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arccos \left (a x \right ) \left (8 a^{5} x^{5} \sqrt {-a^{2} x^{2}+1}+10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+15 a x \sqrt {-a^{2} x^{2}+1}+15 \arccos \left (a x \right )\right )}{432}-\frac {245 \arccos \left (a x \right )^{2}}{1152}+\frac {a^{6} x^{6}}{324}+\frac {65 a^{4} x^{4}}{3456}+\frac {245 a^{2} x^{2}}{1152}-\frac {5 a^{4} x^{4} \arccos \left (a x \right )^{2}}{48}+\frac {5 \arccos \left (a x \right ) \left (2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}+3 a x \sqrt {-a^{2} x^{2}+1}+3 \arccos \left (a x \right )\right )}{192}-\frac {5 a^{2} x^{2} \arccos \left (a x \right )^{2}}{16}+\frac {5 \arccos \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arccos \left (a x \right )\right )}{16}-\frac {5}{32}+\frac {5 \arccos \left (a x \right )^{4}}{32}}{a^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*arccos(a*x)^4,x)

[Out]

1/a^6*(1/6*a^6*x^6*arccos(a*x)^4-1/72*arccos(a*x)^3*(8*a^5*x^5*(-a^2*x^2+1)^(1/2)+10*a^3*x^3*(-a^2*x^2+1)^(1/2
)+15*a*x*(-a^2*x^2+1)^(1/2)+15*arccos(a*x))-1/18*arccos(a*x)^2*a^6*x^6+1/432*arccos(a*x)*(8*a^5*x^5*(-a^2*x^2+
1)^(1/2)+10*a^3*x^3*(-a^2*x^2+1)^(1/2)+15*a*x*(-a^2*x^2+1)^(1/2)+15*arccos(a*x))-245/1152*arccos(a*x)^2+1/324*
a^6*x^6+65/3456*a^4*x^4+245/1152*a^2*x^2-5/48*a^4*x^4*arccos(a*x)^2+5/192*arccos(a*x)*(2*a^3*x^3*(-a^2*x^2+1)^
(1/2)+3*a*x*(-a^2*x^2+1)^(1/2)+3*arccos(a*x))-5/16*a^2*x^2*arccos(a*x)^2+5/16*arccos(a*x)*(a*x*(-a^2*x^2+1)^(1
/2)+arccos(a*x))-5/32+5/32*arccos(a*x)^4)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, x^{6} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{4} - 2 \, a \int \frac {\sqrt {a x + 1} \sqrt {-a x + 1} x^{6} \arctan \left (\sqrt {a x + 1} \sqrt {-a x + 1}, a x\right )^{3}}{3 \, {\left (a^{2} x^{2} - 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*arccos(a*x)^4,x, algorithm="maxima")

[Out]

1/6*x^6*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^4 - 2*a*integrate(1/3*sqrt(a*x + 1)*sqrt(-a*x + 1)*x^6*arct
an2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)^3/(a^2*x^2 - 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^5\,{\mathrm {acos}\left (a\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*acos(a*x)^4,x)

[Out]

int(x^5*acos(a*x)^4, x)

________________________________________________________________________________________

sympy [A]  time = 14.82, size = 275, normalized size = 0.98 \[ \begin {cases} \frac {x^{6} \operatorname {acos}^{4}{\left (a x \right )}}{6} - \frac {x^{6} \operatorname {acos}^{2}{\left (a x \right )}}{18} + \frac {x^{6}}{324} - \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{9 a} + \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{54 a} - \frac {5 x^{4} \operatorname {acos}^{2}{\left (a x \right )}}{48 a^{2}} + \frac {65 x^{4}}{3456 a^{2}} - \frac {5 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{36 a^{3}} + \frac {65 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{864 a^{3}} - \frac {5 x^{2} \operatorname {acos}^{2}{\left (a x \right )}}{16 a^{4}} + \frac {245 x^{2}}{1152 a^{4}} - \frac {5 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}^{3}{\left (a x \right )}}{24 a^{5}} + \frac {245 x \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{576 a^{5}} - \frac {5 \operatorname {acos}^{4}{\left (a x \right )}}{96 a^{6}} + \frac {245 \operatorname {acos}^{2}{\left (a x \right )}}{1152 a^{6}} & \text {for}\: a \neq 0 \\\frac {\pi ^{4} x^{6}}{96} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*acos(a*x)**4,x)

[Out]

Piecewise((x**6*acos(a*x)**4/6 - x**6*acos(a*x)**2/18 + x**6/324 - x**5*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(9*a
) + x**5*sqrt(-a**2*x**2 + 1)*acos(a*x)/(54*a) - 5*x**4*acos(a*x)**2/(48*a**2) + 65*x**4/(3456*a**2) - 5*x**3*
sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(36*a**3) + 65*x**3*sqrt(-a**2*x**2 + 1)*acos(a*x)/(864*a**3) - 5*x**2*acos(
a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) - 5*x*sqrt(-a**2*x**2 + 1)*acos(a*x)**3/(24*a**5) + 245*x*sqrt(-a**2*
x**2 + 1)*acos(a*x)/(576*a**5) - 5*acos(a*x)**4/(96*a**6) + 245*acos(a*x)**2/(1152*a**6), Ne(a, 0)), (pi**4*x*
*6/96, True))

________________________________________________________________________________________