\(\int x \arccos (a x^2) \, dx\) [49]

Optimal result

Integrand size = 8, antiderivative size = 35 \[ \int x \arccos \left (a x^2\right ) \, dx=-\frac {\sqrt {1-a^2 x^4}}{2 a}+\frac {1}{2} x^2 \arccos \left (a x^2\right ) \]

[Out]

1/2*x^2*arccos(a*x^2)-1/2*(-a^2*x^4+1)^(1/2)/a

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6847, 4716, 267} \[ \int x \arccos \left (a x^2\right ) \, dx=\frac {1}{2} x^2 \arccos \left (a x^2\right )-\frac {\sqrt {1-a^2 x^4}}{2 a} \]

[In]

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

[Out]

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

Rule 267

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

Rule 4716

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 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \arccos (a x) \, dx,x,x^2\right ) \\ & = \frac {1}{2} x^2 \arccos \left (a x^2\right )+\frac {1}{2} a \text {Subst}\left (\int \frac {x}{\sqrt {1-a^2 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {\sqrt {1-a^2 x^4}}{2 a}+\frac {1}{2} x^2 \arccos \left (a x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int x \arccos \left (a x^2\right ) \, dx=-\frac {\sqrt {1-a^2 x^4}}{2 a}+\frac {1}{2} x^2 \arccos \left (a x^2\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {x^{2} \arccos \left (a \,x^{2}\right )}{2}-\frac {\sqrt {-a^{2} x^{4}+1}}{2 a}\)	\(30\)
derivativedivides	\(\frac {a \,x^{2} \arccos \left (a \,x^{2}\right )-\sqrt {-a^{2} x^{4}+1}}{2 a}\)	\(32\)
default	\(\frac {a \,x^{2} \arccos \left (a \,x^{2}\right )-\sqrt {-a^{2} x^{4}+1}}{2 a}\)	\(32\)

[In]

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

[Out]

1/2*x^2*arccos(a*x^2)-1/2*(-a^2*x^4+1)^(1/2)/a

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int x \arccos \left (a x^2\right ) \, dx=\frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int x \arccos \left (a x^2\right ) \, dx=\begin {cases} \frac {x^{2} \operatorname {acos}{\left (a x^{2} \right )}}{2} - \frac {\sqrt {- a^{2} x^{4} + 1}}{2 a} & \text {for}\: a \neq 0 \\\frac {\pi x^{2}}{4} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int x \arccos \left (a x^2\right ) \, dx=\frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int x \arccos \left (a x^2\right ) \, dx=\frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int x \arccos \left (a x^2\right ) \, dx=\frac {x^2\,\mathrm {acos}\left (a\,x^2\right )}{2}-\frac {\sqrt {1-a^2\,x^4}}{2\,a} \]

[In]

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

[Out]

(x^2*acos(a*x^2))/2 - (1 - a^2*x^4)^(1/2)/(2*a)