3.10 \(\int \frac{\cot ^{-1}(a x)}{x^4} \, dx\)

Optimal. Leaf size=46 \[ -\frac{1}{6} a^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^3 \log (x)+\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0271508, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4853, 266, 44} \[ -\frac{1}{6} a^3 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^3 \log (x)+\frac{a}{6 x^2}-\frac{\cot ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]/x^4,x]

[Out]

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

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])

Rubi steps

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

Mathematica [A]  time = 0.0113659, size = 44, normalized size = 0.96 \[ -\frac{1}{6} a \left (a^2 \log \left (a^2 x^2+1\right )-2 a^2 \log (x)-\frac{1}{x^2}\right )-\frac{\cot ^{-1}(a x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x]/x^4,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 41, normalized size = 0.9 \begin{align*} -{\frac{{\rm arccot} \left (ax\right )}{3\,{x}^{3}}}-{\frac{{a}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) }{6}}+{\frac{a}{6\,{x}^{2}}}+{\frac{{a}^{3}\ln \left ( ax \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)/x^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.980529, size = 57, normalized size = 1.24 \begin{align*} -\frac{1}{6} \,{\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} a - \frac{\operatorname{arccot}\left (a x\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.99549, size = 105, normalized size = 2.28 \begin{align*} -\frac{a^{3} x^{3} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{3} x^{3} \log \left (x\right ) - a x + 2 \, \operatorname{arccot}\left (a x\right )}{6 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.869719, size = 39, normalized size = 0.85 \begin{align*} \frac{a^{3} \log{\left (x \right )}}{3} - \frac{a^{3} \log{\left (a^{2} x^{2} + 1 \right )}}{6} + \frac{a}{6 x^{2}} - \frac{\operatorname{acot}{\left (a x \right )}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)/x**4,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.12538, size = 73, normalized size = 1.59 \begin{align*} -\frac{1}{6} \,{\left (a^{2} \log \left (a^{2} x^{2} + 1\right ) - a^{2} \log \left (x^{2}\right ) + \frac{a^{2} x^{2} - 1}{x^{2}}\right )} a - \frac{\arctan \left (\frac{1}{a x}\right )}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)/x^4,x, algorithm="giac")

[Out]

-1/6*(a^2*log(a^2*x^2 + 1) - a^2*log(x^2) + (a^2*x^2 - 1)/x^2)*a - 1/3*arctan(1/(a*x))/x^3