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

Optimal. Leaf size=59 \[ -\frac{1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.087978, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4853, 4919, 266, 36, 29, 31, 4885} \[ -\frac{1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{a \cot ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*ArcCot[a*x])/x - (a^2*ArcCot[a*x]^2)/2 - ArcCot[a*x]^2/(2*x^2) + a^2*Log[x] - (a^2*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 4919

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcCot[c*x])^p)/(d + e*
x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[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]

Rule 4885

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(a + b*ArcCot[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(a x)^2}{x^3} \, dx &=-\frac{\cot ^{-1}(a x)^2}{2 x^2}-a \int \frac{\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)^2}{2 x^2}-a \int \frac{\cot ^{-1}(a x)}{x^2} \, dx+a^3 \int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+a^2 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{2} a^2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} a^4 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{a \cot ^{-1}(a x)}{x}-\frac{1}{2} a^2 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac{1}{2} a^2 \log \left (1+a^2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0167775, size = 56, normalized size = 0.95 \[ -\frac{1}{2} a^2 \log \left (a^2 x^2+1\right )+\frac{\left (-a^2 x^2-1\right ) \cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)+\frac{a \cot ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.052, size = 68, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{2\,{x}^{2}}}+{a}^{2}{\rm arccot} \left (ax\right )\arctan \left ( ax \right ) +{\frac{a{\rm arccot} \left (ax\right )}{x}}-{\frac{{a}^{2}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{2}}+{a}^{2}\ln \left ( ax \right ) +{\frac{{a}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47291, size = 76, normalized size = 1.29 \begin{align*} \frac{1}{2} \,{\left (\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right ) + 2 \, \log \left (x\right )\right )} a^{2} +{\left (a \arctan \left (a x\right ) + \frac{1}{x}\right )} a \operatorname{arccot}\left (a x\right ) - \frac{\operatorname{arccot}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.02672, size = 143, normalized size = 2.42 \begin{align*} -\frac{a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \operatorname{arccot}\left (a x\right ) +{\left (a^{2} x^{2} + 1\right )} \operatorname{arccot}\left (a x\right )^{2}}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.695179, size = 53, normalized size = 0.9 \begin{align*} a^{2} \log{\left (x \right )} - \frac{a^{2} \log{\left (a^{2} x^{2} + 1 \right )}}{2} - \frac{a^{2} \operatorname{acot}^{2}{\left (a x \right )}}{2} + \frac{a \operatorname{acot}{\left (a x \right )}}{x} - \frac{\operatorname{acot}^{2}{\left (a x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x)^2/x^3, x)