3.4 \(\int x \cos ^{-1}(a x) \, dx\)

Optimal. Leaf size=45 \[ -\frac{x \sqrt{1-a^2 x^2}}{4 a}+\frac{\sin ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0160866, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 321, 216} \[ -\frac{x \sqrt{1-a^2 x^2}}{4 a}+\frac{\sin ^{-1}(a x)}{4 a^2}+\frac{1}{2} x^2 \cos ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4628

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 321

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 216

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]

Rubi steps

\begin{align*} \int x \cos ^{-1}(a x) \, dx &=\frac{1}{2} x^2 \cos ^{-1}(a x)+\frac{1}{2} a \int \frac{x^2}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{x \sqrt{1-a^2 x^2}}{4 a}+\frac{1}{2} x^2 \cos ^{-1}(a x)+\frac{\int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{4 a}\\ &=-\frac{x \sqrt{1-a^2 x^2}}{4 a}+\frac{1}{2} x^2 \cos ^{-1}(a x)+\frac{\sin ^{-1}(a x)}{4 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0161218, size = 42, normalized size = 0.93 \[ \frac{-a x \sqrt{1-a^2 x^2}+2 a^2 x^2 \cos ^{-1}(a x)+\sin ^{-1}(a x)}{4 a^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 40, normalized size = 0.9 \begin{align*}{\frac{1}{{a}^{2}} \left ({\frac{{a}^{2}{x}^{2}\arccos \left ( ax \right ) }{2}}-{\frac{ax}{4}\sqrt{-{a}^{2}{x}^{2}+1}}+{\frac{\arcsin \left ( ax \right ) }{4}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*arccos(a*x),x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.45635, size = 70, normalized size = 1.56 \begin{align*} \frac{1}{2} \, x^{2} \arccos \left (a x\right ) - \frac{1}{4} \, a{\left (\frac{\sqrt{-a^{2} x^{2} + 1} x}{a^{2}} - \frac{\arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}} a^{2}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.96631, size = 88, normalized size = 1.96 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1} a x -{\left (2 \, a^{2} x^{2} - 1\right )} \arccos \left (a x\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.241072, size = 42, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x^{2} \operatorname{acos}{\left (a x \right )}}{2} - \frac{x \sqrt{- a^{2} x^{2} + 1}}{4 a} - \frac{\operatorname{acos}{\left (a x \right )}}{4 a^{2}} & \text{for}\: a \neq 0 \\\frac{\pi x^{2}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.1384, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{2} \, x^{2} \arccos \left (a x\right ) - \frac{\sqrt{-a^{2} x^{2} + 1} x}{4 \, a} - \frac{\arccos \left (a x\right )}{4 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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