3.19 \(\int \frac{\cos ^{-1}(a x)^2}{x^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\cos ^{-1}(a x)^2}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0783459, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4628, 4682, 29} \[ \frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\cos ^{-1}(a x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcCos[a*x]^2/x^3,x]

[Out]

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

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 4682

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n)/(d*f*(m + 1)), x] + Dist[(b*c*n*d^IntPart[p]*(d + e*x
^2)^FracPart[p])/(f*(m + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCo
s[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p
+ 3, 0] && NeQ[m, -1]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{\cos ^{-1}(a x)^2}{x^3} \, dx &=-\frac{\cos ^{-1}(a x)^2}{2 x^2}-a \int \frac{\cos ^{-1}(a x)}{x^2 \sqrt{1-a^2 x^2}} \, dx\\ &=\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}-\frac{\cos ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac{1}{x} \, dx\\ &=\frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}-\frac{\cos ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.027813, size = 43, normalized size = 1. \[ \frac{a \sqrt{1-a^2 x^2} \cos ^{-1}(a x)}{x}+a^2 \log (x)-\frac{\cos ^{-1}(a x)^2}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCos[a*x]^2/x^3,x]

[Out]

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

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Maple [A]  time = 0.051, size = 42, normalized size = 1. \begin{align*} -{\frac{ \left ( \arccos \left ( ax \right ) \right ) ^{2}}{2\,{x}^{2}}}+{\frac{a\arccos \left ( ax \right ) }{x}\sqrt{-{a}^{2}{x}^{2}+1}}+{a}^{2}\ln \left ( ax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccos(a*x)^2/x^3,x)

[Out]

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

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Maxima [A]  time = 1.52371, size = 53, normalized size = 1.23 \begin{align*} a^{2} \log \left (x\right ) + \frac{\sqrt{-a^{2} x^{2} + 1} a \arccos \left (a x\right )}{x} - \frac{\arccos \left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^2/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.39595, size = 112, normalized size = 2.6 \begin{align*} \frac{2 \, a^{2} x^{2} \log \left (x\right ) + 2 \, \sqrt{-a^{2} x^{2} + 1} a x \arccos \left (a x\right ) - \arccos \left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^2/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acos}^{2}{\left (a x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acos(a*x)**2/x**3,x)

[Out]

Integral(acos(a*x)**2/x**3, x)

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Giac [B]  time = 1.19005, size = 117, normalized size = 2.72 \begin{align*} -\frac{1}{2} \,{\left ({\left (\frac{a^{4} x}{{\left (\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a\right )}{\left | a \right |}} - \frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{x{\left | a \right |}}\right )} \arccos \left (a x\right ) - a \log \left (a^{2} x^{2}\right )\right )} a - \frac{\arccos \left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccos(a*x)^2/x^3,x, algorithm="giac")

[Out]

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