3.1.49 \(\int x \text {ArcCos}(a x^2) \, dx\) [49]

Optimal. Leaf size=35 \[ -\frac {\sqrt {1-a^2 x^4}}{2 a}+\frac {1}{2} x^2 \text {ArcCos}\left (a x^2\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6847, 4716, 267} \begin {gather*} \frac {1}{2} x^2 \text {ArcCos}\left (a x^2\right )-\frac {\sqrt {1-a^2 x^4}}{2 a} \end {gather*}

Antiderivative was successfully verified.

[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*} \int x \cos ^{-1}\left (a x^2\right ) \, dx &=\frac {1}{2} \text {Subst}\left (\int \cos ^{-1}(a x) \, dx,x,x^2\right )\\ &=\frac {1}{2} x^2 \cos ^{-1}\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 \cos ^{-1}\left (a x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-a^2 x^4}}{2 a}+\frac {1}{2} x^2 \text {ArcCos}\left (a x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[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]
time = 0.02, size = 32, normalized size = 0.91

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 31, normalized size = 0.89 \begin {gather*} \frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]
time = 1.54, size = 31, normalized size = 0.89 \begin {gather*} \frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.08, size = 32, normalized size = 0.91 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

Giac [A]
time = 0.45, size = 31, normalized size = 0.89 \begin {gather*} \frac {a x^{2} \arccos \left (a x^{2}\right ) - \sqrt {-a^{2} x^{4} + 1}}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]
time = 0.33, size = 29, normalized size = 0.83 \begin {gather*} \frac {x^2\,\mathrm {acos}\left (a\,x^2\right )}{2}-\frac {\sqrt {1-a^2\,x^4}}{2\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________