3.22 \(\int \frac{\cot ^{-1}(a x)^2}{x^5} \, dx\)

Optimal. Leaf size=89 \[ -\frac{a^2}{12 x^2}+\frac{1}{3} a^4 \log \left (a^2 x^2+1\right )-\frac{2}{3} a^4 \log (x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{\cot ^{-1}(a x)^2}{4 x^4} \]

[Out]

-a^2/(12*x^2) + (a*ArcCot[a*x])/(6*x^3) - (a^3*ArcCot[a*x])/(2*x) + (a^4*ArcCot[a*x]^2)/4 - ArcCot[a*x]^2/(4*x
^4) - (2*a^4*Log[x])/3 + (a^4*Log[1 + a^2*x^2])/3

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Rubi [A]  time = 0.15512, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4853, 4919, 266, 44, 36, 29, 31, 4885} \[ -\frac{a^2}{12 x^2}+\frac{1}{3} a^4 \log \left (a^2 x^2+1\right )-\frac{2}{3} a^4 \log (x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{\cot ^{-1}(a x)^2}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^2/x^5,x]

[Out]

-a^2/(12*x^2) + (a*ArcCot[a*x])/(6*x^3) - (a^3*ArcCot[a*x])/(2*x) + (a^4*ArcCot[a*x]^2)/4 - ArcCot[a*x]^2/(4*x
^4) - (2*a^4*Log[x])/3 + (a^4*Log[1 + a^2*x^2])/3

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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^5} \, dx &=-\frac{\cot ^{-1}(a x)^2}{4 x^4}-\frac{1}{2} a \int \frac{\cot ^{-1}(a x)}{x^4 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)^2}{4 x^4}-\frac{1}{2} a \int \frac{\cot ^{-1}(a x)}{x^4} \, dx+\frac{1}{2} a^3 \int \frac{\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{\cot ^{-1}(a x)^2}{4 x^4}+\frac{1}{6} a^2 \int \frac{1}{x^3 \left (1+a^2 x^2\right )} \, dx+\frac{1}{2} a^3 \int \frac{\cot ^{-1}(a x)}{x^2} \, dx-\frac{1}{2} a^5 \int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{4 x^4}+\frac{1}{12} a^2 \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+a^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} a^4 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{4 x^4}+\frac{1}{12} a^2 \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{a^2}{x}+\frac{a^4}{1+a^2 x}\right ) \, dx,x,x^2\right )-\frac{1}{4} a^4 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a^2}{12 x^2}+\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{4 x^4}-\frac{1}{6} a^4 \log (x)+\frac{1}{12} a^4 \log \left (1+a^2 x^2\right )-\frac{1}{4} a^4 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{4} a^6 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2}{12 x^2}+\frac{a \cot ^{-1}(a x)}{6 x^3}-\frac{a^3 \cot ^{-1}(a x)}{2 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{4 x^4}-\frac{2}{3} a^4 \log (x)+\frac{1}{3} a^4 \log \left (1+a^2 x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0197266, size = 81, normalized size = 0.91 \[ -\frac{a^2}{12 x^2}+\frac{1}{3} a^4 \log \left (a^2 x^2+1\right )-\frac{a \left (3 a^2 x^2-1\right ) \cot ^{-1}(a x)}{6 x^3}+\frac{\left (a^4 x^4-1\right ) \cot ^{-1}(a x)^2}{4 x^4}-\frac{2}{3} a^4 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]^2/x^5,x]

[Out]

-a^2/(12*x^2) - (a*(-1 + 3*a^2*x^2)*ArcCot[a*x])/(6*x^3) + ((-1 + a^4*x^4)*ArcCot[a*x]^2)/(4*x^4) - (2*a^4*Log
[x])/3 + (a^4*Log[1 + a^2*x^2])/3

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Maple [A]  time = 0.058, size = 91, normalized size = 1. \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{4\,{x}^{4}}}-{\frac{{a}^{4}{\rm arccot} \left (ax\right )\arctan \left ( ax \right ) }{2}}+{\frac{a{\rm arccot} \left (ax\right )}{6\,{x}^{3}}}-{\frac{{a}^{3}{\rm arccot} \left (ax\right )}{2\,x}}+{\frac{{a}^{4}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{{a}^{2}}{12\,{x}^{2}}}-{\frac{2\,{a}^{4}\ln \left ( ax \right ) }{3}}-{\frac{{a}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)^2/x^5,x)

[Out]

-1/4*arccot(a*x)^2/x^4-1/2*a^4*arccot(a*x)*arctan(a*x)+1/6*a*arccot(a*x)/x^3-1/2*a^3*arccot(a*x)/x+1/3*a^4*ln(
a^2*x^2+1)-1/12*a^2/x^2-2/3*a^4*ln(a*x)-1/4*a^4*arctan(a*x)^2

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Maxima [A]  time = 1.52573, size = 128, normalized size = 1.44 \begin{align*} -\frac{1}{6} \,{\left (3 \, a^{3} \arctan \left (a x\right ) + \frac{3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a \operatorname{arccot}\left (a x\right ) - \frac{{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \left (x\right ) + 1\right )} a^{2}}{12 \, x^{2}} - \frac{\operatorname{arccot}\left (a x\right )^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^5,x, algorithm="maxima")

[Out]

-1/6*(3*a^3*arctan(a*x) + (3*a^2*x^2 - 1)/x^3)*a*arccot(a*x) - 1/12*(3*a^2*x^2*arctan(a*x)^2 - 4*a^2*x^2*log(a
^2*x^2 + 1) + 8*a^2*x^2*log(x) + 1)*a^2/x^2 - 1/4*arccot(a*x)^2/x^4

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Fricas [A]  time = 1.87986, size = 181, normalized size = 2.03 \begin{align*} \frac{4 \, a^{4} x^{4} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} + 3 \,{\left (a^{4} x^{4} - 1\right )} \operatorname{arccot}\left (a x\right )^{2} - 2 \,{\left (3 \, a^{3} x^{3} - a x\right )} \operatorname{arccot}\left (a x\right )}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^5,x, algorithm="fricas")

[Out]

1/12*(4*a^4*x^4*log(a^2*x^2 + 1) - 8*a^4*x^4*log(x) - a^2*x^2 + 3*(a^4*x^4 - 1)*arccot(a*x)^2 - 2*(3*a^3*x^3 -
a*x)*arccot(a*x))/x^4

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Sympy [A]  time = 1.19707, size = 80, normalized size = 0.9 \begin{align*} - \frac{2 a^{4} \log{\left (x \right )}}{3} + \frac{a^{4} \log{\left (a^{2} x^{2} + 1 \right )}}{3} + \frac{a^{4} \operatorname{acot}^{2}{\left (a x \right )}}{4} - \frac{a^{3} \operatorname{acot}{\left (a x \right )}}{2 x} - \frac{a^{2}}{12 x^{2}} + \frac{a \operatorname{acot}{\left (a x \right )}}{6 x^{3}} - \frac{\operatorname{acot}^{2}{\left (a x \right )}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**2/x**5,x)

[Out]

-2*a**4*log(x)/3 + a**4*log(a**2*x**2 + 1)/3 + a**4*acot(a*x)**2/4 - a**3*acot(a*x)/(2*x) - a**2/(12*x**2) + a
*acot(a*x)/(6*x**3) - acot(a*x)**2/(4*x**4)

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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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^5,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^2/x^5, x)