\(\int x^3 \arccos (a x) \, dx\) [2]

Optimal result

Integrand size = 8, antiderivative size = 69 \[ \int x^3 \arccos (a x) \, dx=-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \arccos (a x)+\frac {3 \arcsin (a x)}{32 a^4} \]

[Out]

1/4*x^4*arccos(a*x)+3/32*arcsin(a*x)/a^4-3/32*x*(-a^2*x^2+1)^(1/2)/a^3-1/16*x^3*(-a^2*x^2+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4724, 327, 222} \[ \int x^3 \arccos (a x) \, dx=\frac {3 \arcsin (a x)}{32 a^4}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}-\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}+\frac {1}{4} x^4 \arccos (a x) \]

[In]

Int[x^3*ArcCos[a*x],x]

[Out]

(-3*x*Sqrt[1 - a^2*x^2])/(32*a^3) - (x^3*Sqrt[1 - a^2*x^2])/(16*a) + (x^4*ArcCos[a*x])/4 + (3*ArcSin[a*x])/(32
*a^4)

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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*} \text {integral}& = \frac {1}{4} x^4 \arccos (a x)+\frac {1}{4} a \int \frac {x^4}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \arccos (a x)+\frac {3 \int \frac {x^2}{\sqrt {1-a^2 x^2}} \, dx}{16 a} \\ & = -\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \arccos (a x)+\frac {3 \int \frac {1}{\sqrt {1-a^2 x^2}} \, dx}{32 a^3} \\ & = -\frac {3 x \sqrt {1-a^2 x^2}}{32 a^3}-\frac {x^3 \sqrt {1-a^2 x^2}}{16 a}+\frac {1}{4} x^4 \arccos (a x)+\frac {3 \arcsin (a x)}{32 a^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.78 \[ \int x^3 \arccos (a x) \, dx=\frac {-a x \sqrt {1-a^2 x^2} \left (3+2 a^2 x^2\right )+8 a^4 x^4 \arccos (a x)+3 \arcsin (a x)}{32 a^4} \]

[In]

Integrate[x^3*ArcCos[a*x],x]

[Out]

(-(a*x*Sqrt[1 - a^2*x^2]*(3 + 2*a^2*x^2)) + 8*a^4*x^4*ArcCos[a*x] + 3*ArcSin[a*x])/(32*a^4)

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{16}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (a x \right )}{32}}{a^{4}}\)	\(60\)
default	\(\frac {\frac {a^{4} x^{4} \arccos \left (a x \right )}{4}-\frac {a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}}{16}-\frac {3 a x \sqrt {-a^{2} x^{2}+1}}{32}+\frac {3 \arcsin \left (a x \right )}{32}}{a^{4}}\)	\(60\)
parts	\(\frac {x^{4} \arccos \left (a x \right )}{4}+\frac {a \left (-\frac {x^{3} \sqrt {-a^{2} x^{2}+1}}{4 a^{2}}+\frac {-\frac {3 x \sqrt {-a^{2} x^{2}+1}}{8 a^{2}}+\frac {3 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{8 a^{2} \sqrt {a^{2}}}}{a^{2}}\right )}{4}\)	\(89\)

[In]

int(x^3*arccos(a*x),x,method=_RETURNVERBOSE)

[Out]

1/a^4*(1/4*a^4*x^4*arccos(a*x)-1/16*a^3*x^3*(-a^2*x^2+1)^(1/2)-3/32*a*x*(-a^2*x^2+1)^(1/2)+3/32*arcsin(a*x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.70 \[ \int x^3 \arccos (a x) \, dx=\frac {{\left (8 \, a^{4} x^{4} - 3\right )} \arccos \left (a x\right ) - {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \sqrt {-a^{2} x^{2} + 1}}{32 \, a^{4}} \]

[In]

integrate(x^3*arccos(a*x),x, algorithm="fricas")

[Out]

1/32*((8*a^4*x^4 - 3)*arccos(a*x) - (2*a^3*x^3 + 3*a*x)*sqrt(-a^2*x^2 + 1))/a^4

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int x^3 \arccos (a x) \, dx=\begin {cases} \frac {x^{4} \operatorname {acos}{\left (a x \right )}}{4} - \frac {x^{3} \sqrt {- a^{2} x^{2} + 1}}{16 a} - \frac {3 x \sqrt {- a^{2} x^{2} + 1}}{32 a^{3}} - \frac {3 \operatorname {acos}{\left (a x \right )}}{32 a^{4}} & \text {for}\: a \neq 0 \\\frac {\pi x^{4}}{8} & \text {otherwise} \end {cases} \]

[In]

integrate(x**3*acos(a*x),x)

[Out]

Piecewise((x**4*acos(a*x)/4 - x**3*sqrt(-a**2*x**2 + 1)/(16*a) - 3*x*sqrt(-a**2*x**2 + 1)/(32*a**3) - 3*acos(a
*x)/(32*a**4), Ne(a, 0)), (pi*x**4/8, True))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int x^3 \arccos (a x) \, dx=\frac {1}{4} \, x^{4} \arccos \left (a x\right ) - \frac {1}{32} \, {\left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} x^{3}}{a^{2}} + \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{a^{4}} - \frac {3 \, \arcsin \left (a x\right )}{a^{5}}\right )} a \]

[In]

integrate(x^3*arccos(a*x),x, algorithm="maxima")

[Out]

1/4*x^4*arccos(a*x) - 1/32*(2*sqrt(-a^2*x^2 + 1)*x^3/a^2 + 3*sqrt(-a^2*x^2 + 1)*x/a^4 - 3*arcsin(a*x)/a^5)*a

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.83 \[ \int x^3 \arccos (a x) \, dx=\frac {1}{4} \, x^{4} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} x^{3}}{16 \, a} - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} x}{32 \, a^{3}} - \frac {3 \, \arccos \left (a x\right )}{32 \, a^{4}} \]

[In]

integrate(x^3*arccos(a*x),x, algorithm="giac")

[Out]

1/4*x^4*arccos(a*x) - 1/16*sqrt(-a^2*x^2 + 1)*x^3/a - 3/32*sqrt(-a^2*x^2 + 1)*x/a^3 - 3/32*arccos(a*x)/a^4

Mupad [F(-1)]

Timed out. \[ \int x^3 \arccos (a x) \, dx=\int x^3\,\mathrm {acos}\left (a\,x\right ) \,d x \]

[In]

int(x^3*acos(a*x),x)

[Out]

int(x^3*acos(a*x), x)