3.8 \(\int \frac{\cot ^{-1}(a x)}{x^2} \, dx\)

Optimal. Leaf size=30 \[ \frac{1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac{\cot ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.0178592, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {4853, 266, 36, 29, 31} \[ \frac{1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac{\cot ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4853

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
t[c*x])^p)/(d*(m + 1)), x] + Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCot[c*x])^(p - 1))/(1 + c^
2*x^2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] && (EqQ[p, 1] || IntegerQ[m]) && 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 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(a x)}{x^2} \, dx &=-\frac{\cot ^{-1}(a x)}{x}-a \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(a x)}{x}-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{\cot ^{-1}(a x)}{x}-a \log (x)+\frac{1}{2} a \log \left (1+a^2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0022753, size = 30, normalized size = 1. \[ \frac{1}{2} a \log \left (a^2 x^2+1\right )-a \log (x)-\frac{\cot ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 31, normalized size = 1. \begin{align*} -{\frac{{\rm arccot} \left (ax\right )}{x}}+{\frac{a\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2}}-a\ln \left ( ax \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.966614, size = 41, normalized size = 1.37 \begin{align*} \frac{1}{2} \, a{\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac{\operatorname{arccot}\left (a x\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.91349, size = 82, normalized size = 2.73 \begin{align*} \frac{a x \log \left (a^{2} x^{2} + 1\right ) - 2 \, a x \log \left (x\right ) - 2 \, \operatorname{arccot}\left (a x\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.34515, size = 24, normalized size = 0.8 \begin{align*} - a \log{\left (x \right )} + \frac{a \log{\left (a^{2} x^{2} + 1 \right )}}{2} - \frac{\operatorname{acot}{\left (a x \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*log(x) + a*log(a**2*x**2 + 1)/2 - acot(a*x)/x

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Giac [A]  time = 1.09297, size = 46, normalized size = 1.53 \begin{align*} \frac{1}{2} \, a{\left (\log \left (a^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} - \frac{\arctan \left (\frac{1}{a x}\right )}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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