3.1.12 \(\int x^4 \text {ArcCos}(a x)^2 \, dx\) [12]

Optimal. Leaf size=120 \[ -\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{25 a}+\frac {1}{5} x^5 \text {ArcCos}(a x)^2 \]

[Out]

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

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Rubi [A]
time = 0.13, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4724, 4796, 4768, 8, 30} \begin {gather*} -\frac {16 x}{75 a^4}-\frac {2 x^4 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{25 a}-\frac {8 x^3}{225 a^2}-\frac {16 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)}{75 a^3}+\frac {1}{5} x^5 \text {ArcCos}(a x)^2-\frac {2 x^5}{125} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCos[a*x]^2,x]

[Out]

(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (16*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(75*a^5) - (8*x^2*Sqrt
[1 - a^2*x^2]*ArcCos[a*x])/(75*a^3) - (2*x^4*Sqrt[1 - a^2*x^2]*ArcCos[a*x])/(25*a) + (x^5*ArcCos[a*x]^2)/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 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)^2 \, dx &=\frac {1}{5} x^5 \cos ^{-1}(a x)^2+\frac {1}{5} (2 a) \int \frac {x^5 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^2-\frac {2 \int x^4 \, dx}{25}+\frac {8 \int \frac {x^3 \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac {2 x^5}{125}-\frac {8 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^2+\frac {16 \int \frac {x \cos ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx}{75 a^3}-\frac {8 \int x^2 \, dx}{75 a^2}\\ &=-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^2-\frac {16 \int 1 \, dx}{75 a^4}\\ &=-\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {16 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{75 a^5}-\frac {8 x^2 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{75 a^3}-\frac {2 x^4 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)}{25 a}+\frac {1}{5} x^5 \cos ^{-1}(a x)^2\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 82, normalized size = 0.68 \begin {gather*} -\frac {16 x}{75 a^4}-\frac {8 x^3}{225 a^2}-\frac {2 x^5}{125}-\frac {2 \sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right ) \text {ArcCos}(a x)}{75 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCos[a*x]^2,x]

[Out]

(-16*x)/(75*a^4) - (8*x^3)/(225*a^2) - (2*x^5)/125 - (2*Sqrt[1 - a^2*x^2]*(8 + 4*a^2*x^2 + 3*a^4*x^4)*ArcCos[a
*x])/(75*a^5) + (x^5*ArcCos[a*x]^2)/5

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Maple [A]
time = 0.05, size = 76, normalized size = 0.63

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{2}}{5}-\frac {2 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}}\)	\(76\)
default	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )^{2}}{5}-\frac {2 \arccos \left (a x \right ) \left (3 a^{4} x^{4}+4 a^{2} x^{2}+8\right ) \sqrt {-a^{2} x^{2}+1}}{75}-\frac {2 a^{5} x^{5}}{125}-\frac {8 a^{3} x^{3}}{225}-\frac {16 a x}{75}}{a^{5}}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccos(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.47, size = 102, normalized size = 0.85 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac {2}{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 ) - \frac {2 \, {\left (9 \, a^{4} x^{5} + 20 \, a^{2} x^{3} + 120 \, x\right )}}{1125 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^2,x, algorithm="maxima")

[Out]

1/5*x^5*arccos(a*x)^2 - 2/75*(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) - 2/1125*(9*a^4*x^5 + 20*a^2*x^3 + 120*x)/a^4

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Fricas [A]
time = 2.98, size = 76, normalized size = 0.63 \begin {gather*} \frac {225 \, a^{5} x^{5} \arccos \left (a x\right )^{2} - 18 \, a^{5} x^{5} - 40 \, a^{3} x^{3} - 30 \, {\left (3 \, a^{4} x^{4} + 4 \, a^{2} x^{2} + 8\right )} \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right ) - 240 \, a x}{1125 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^2,x, algorithm="fricas")

[Out]

1/1125*(225*a^5*x^5*arccos(a*x)^2 - 18*a^5*x^5 - 40*a^3*x^3 - 30*(3*a^4*x^4 + 4*a^2*x^2 + 8)*sqrt(-a^2*x^2 + 1
)*arccos(a*x) - 240*a*x)/a^5

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Sympy [A]
time = 0.51, size = 121, normalized size = 1.01 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acos}^{2}{\left (a x \right )}}{5} - \frac {2 x^{5}}{125} - \frac {2 x^{4} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{25 a} - \frac {8 x^{3}}{225 a^{2}} - \frac {8 x^{2} \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{75 a^{3}} - \frac {16 x}{75 a^{4}} - \frac {16 \sqrt {- a^{2} x^{2} + 1} \operatorname {acos}{\left (a x \right )}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{5}}{20} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acos(a*x)**2,x)

[Out]

Piecewise((x**5*acos(a*x)**2/5 - 2*x**5/125 - 2*x**4*sqrt(-a**2*x**2 + 1)*acos(a*x)/(25*a) - 8*x**3/(225*a**2)
- 8*x**2*sqrt(-a**2*x**2 + 1)*acos(a*x)/(75*a**3) - 16*x/(75*a**4) - 16*sqrt(-a**2*x**2 + 1)*acos(a*x)/(75*a*
*5), Ne(a, 0)), (pi**2*x**5/20, True))

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Giac [A]
time = 0.47, size = 100, normalized size = 0.83 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right )^{2} - \frac {2}{125} \, x^{5} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{4} \arccos \left (a x\right )}{25 \, a} - \frac {8 \, x^{3}}{225 \, a^{2}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} x^{2} \arccos \left (a x\right )}{75 \, a^{3}} - \frac {16 \, x}{75 \, a^{4}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} \arccos \left (a x\right )}{75 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccos(a*x)^2,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^4\,{\mathrm {acos}\left (a\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*acos(a*x)^2,x)

[Out]

int(x^4*acos(a*x)^2, x)

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