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

Optimal. Leaf size=27 \[ a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0224197, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4628, 266, 63, 208} \[ a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )-\frac{\cos ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(ArcCos[a*x]/x) + a*ArcTanh[Sqrt[1 - a^2*x^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 266

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\cos ^{-1}(a x)}{x^2} \, dx &=-\frac{\cos ^{-1}(a x)}{x}-a \int \frac{1}{x \sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{\cos ^{-1}(a x)}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-a^2 x}} \, dx,x,x^2\right )\\ &=-\frac{\cos ^{-1}(a x)}{x}+\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{1}{a^2}-\frac{x^2}{a^2}} \, dx,x,\sqrt{1-a^2 x^2}\right )}{a}\\ &=-\frac{\cos ^{-1}(a x)}{x}+a \tanh ^{-1}\left (\sqrt{1-a^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0111537, size = 34, normalized size = 1.26 \[ a \log \left (\sqrt{1-a^2 x^2}+1\right )-a \log (x)-\frac{\cos ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 29, normalized size = 1.1 \begin{align*} a \left ( -{\frac{\arccos \left ( ax \right ) }{ax}}+{\it Artanh} \left ({\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}} \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.45474, size = 51, normalized size = 1.89 \begin{align*} a \log \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) - \frac{\arccos \left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.47838, size = 203, normalized size = 7.52 \begin{align*} \frac{a x \log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - a x \log \left (\sqrt{-a^{2} x^{2} + 1} - 1\right ) + 2 \,{\left (x - 1\right )} \arccos \left (a x\right ) - 2 \, x \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1} a x}{a^{2} x^{2} - 1}\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.61538, size = 34, normalized size = 1.26 \begin{align*} - a \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{a x} \right )} & \text{for}\: \frac{1}{\left |{a^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{a x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{\operatorname{acos}{\left (a x \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*Piecewise((-acosh(1/(a*x)), 1/Abs(a**2*x**2) > 1), (I*asin(1/(a*x)), True)) - acos(a*x)/x

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Giac [A]  time = 1.13227, size = 65, normalized size = 2.41 \begin{align*} \frac{1}{2} \, a{\left (\log \left (\sqrt{-a^{2} x^{2} + 1} + 1\right ) - \log \left (-\sqrt{-a^{2} x^{2} + 1} + 1\right )\right )} - \frac{\arccos \left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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