3.1.1 \(\int x^4 \text {ArcCos}(a x) \, dx\) [1]

Optimal. Leaf size=75 \[ -\frac {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x) \]

[Out]

2/15*(-a^2*x^2+1)^(3/2)/a^5-1/25*(-a^2*x^2+1)^(5/2)/a^5+1/5*x^5*arccos(a*x)-1/5*(-a^2*x^2+1)^(1/2)/a^5

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 272, 45} \begin {gather*} -\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5*Sqrt[1 - a^2*x^2]/a^5 + (2*(1 - a^2*x^2)^(3/2))/(15*a^5) - (1 - a^2*x^2)^(5/2)/(25*a^5) + (x^5*ArcCos[a*x
])/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 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 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]

Rubi steps

\begin {align*} \int x^4 \cos ^{-1}(a x) \, dx &=\frac {1}{5} x^5 \cos ^{-1}(a x)+\frac {1}{5} a \int \frac {x^5}{\sqrt {1-a^2 x^2}} \, dx\\ &=\frac {1}{5} x^5 \cos ^{-1}(a x)+\frac {1}{10} a \text {Subst}\left (\int \frac {x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} x^5 \cos ^{-1}(a x)+\frac {1}{10} 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 {\sqrt {1-a^2 x^2}}{5 a^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{15 a^5}-\frac {\left (1-a^2 x^2\right )^{5/2}}{25 a^5}+\frac {1}{5} x^5 \cos ^{-1}(a x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 51, normalized size = 0.68 \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \left (8+4 a^2 x^2+3 a^4 x^4\right )}{75 a^5}+\frac {1}{5} x^5 \text {ArcCos}(a x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 72, normalized size = 0.96

method	result	size
derivativedivides	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )}{5}-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{25}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{75}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{75}}{a^{5}}\)	\(72\)
default	\(\frac {\frac {a^{5} x^{5} \arccos \left (a x \right )}{5}-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{25}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{75}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{75}}{a^{5}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 71, normalized size = 0.95 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right ) - \frac {1}{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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*arccos(a*x) - 1/75*(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

________________________________________________________________________________________

Fricas [A]
time = 2.48, size = 50, normalized size = 0.67 \begin {gather*} \frac {15 \, a^{5} x^{5} \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 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/75*(15*a^5*x^5*arccos(a*x) - (3*a^4*x^4 + 4*a^2*x^2 + 8)*sqrt(-a^2*x^2 + 1))/a^5

________________________________________________________________________________________

Sympy [A]
time = 0.32, size = 75, normalized size = 1.00 \begin {gather*} \begin {cases} \frac {x^{5} \operatorname {acos}{\left (a x \right )}}{5} - \frac {x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} - \frac {4 x^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{3}} - \frac {8 \sqrt {- a^{2} x^{2} + 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi x^{5}}{10} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x**5*acos(a*x)/5 - x**4*sqrt(-a**2*x**2 + 1)/(25*a) - 4*x**2*sqrt(-a**2*x**2 + 1)/(75*a**3) - 8*sqr
t(-a**2*x**2 + 1)/(75*a**5), Ne(a, 0)), (pi*x**5/10, True))

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 67, normalized size = 0.89 \begin {gather*} \frac {1}{5} \, x^{5} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{25 \, a} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} x^{2}}{75 \, a^{3}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1}}{75 \, a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*acos(a*x), x)

________________________________________________________________________________________