3.45 \(\int \frac{\cot ^{-1}(x)}{x^4 (1+x^2)} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{6 x^2}-\frac{2}{3} \log \left (x^2+1\right )-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.109391, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {4919, 4853, 266, 44, 36, 29, 31, 4885} \[ \frac{1}{6 x^2}-\frac{2}{3} \log \left (x^2+1\right )-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[x]/(x^4*(1 + x^2)),x]

[Out]

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

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

Mathematica [A]  time = 0.0203647, size = 47, normalized size = 1. \[ \frac{1}{6 x^2}-\frac{2}{3} \log \left (x^2+1\right )-\frac{\cot ^{-1}(x)}{3 x^3}+\frac{4 \log (x)}{3}-\frac{1}{2} \cot ^{-1}(x)^2+\frac{\cot ^{-1}(x)}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[x]/(x^4*(1 + x^2)),x]

[Out]

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

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Maple [A]  time = 0.039, size = 43, normalized size = 0.9 \begin{align*}{\rm arccot} \left (x\right )\arctan \left ( x \right ) -{\frac{{\rm arccot} \left (x\right )}{3\,{x}^{3}}}+{\frac{{\rm arccot} \left (x\right )}{x}}-{\frac{2\,\ln \left ({x}^{2}+1 \right ) }{3}}+{\frac{1}{6\,{x}^{2}}}+{\frac{4\,\ln \left ( x \right ) }{3}}+{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(x)/x^4/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 1.47906, size = 74, normalized size = 1.57 \begin{align*} \frac{1}{3} \,{\left (\frac{3 \, x^{2} - 1}{x^{3}} + 3 \, \arctan \left (x\right )\right )} \operatorname{arccot}\left (x\right ) + \frac{3 \, x^{2} \arctan \left (x\right )^{2} - 4 \, x^{2} \log \left (x^{2} + 1\right ) + 8 \, x^{2} \log \left (x\right ) + 1}{6 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/x^4/(x^2+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.95685, size = 130, normalized size = 2.77 \begin{align*} -\frac{3 \, x^{3} \operatorname{arccot}\left (x\right )^{2} + 4 \, x^{3} \log \left (x^{2} + 1\right ) - 8 \, x^{3} \log \left (x\right ) - 2 \,{\left (3 \, x^{2} - 1\right )} \operatorname{arccot}\left (x\right ) - x}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/x^4/(x^2+1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.87486, size = 42, normalized size = 0.89 \begin{align*} \frac{4 \log{\left (x \right )}}{3} - \frac{2 \log{\left (x^{2} + 1 \right )}}{3} - \frac{\operatorname{acot}^{2}{\left (x \right )}}{2} + \frac{\operatorname{acot}{\left (x \right )}}{x} + \frac{1}{6 x^{2}} - \frac{\operatorname{acot}{\left (x \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(x)/x**4/(x**2+1),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (x\right )}{{\left (x^{2} + 1\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(x)/x^4/(x^2+1),x, algorithm="giac")

[Out]

integrate(arccot(x)/((x^2 + 1)*x^4), x)