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

Optimal. Leaf size=41 \[ -\frac{a^3}{4 x}-\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{a}{12 x^3}-\frac{\cot ^{-1}(a x)}{4 x^4} \]

[Out]

a/(12*x^3) - a^3/(4*x) - ArcCot[a*x]/(4*x^4) - (a^4*ArcTan[a*x])/4

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Rubi [A]  time = 0.018415, antiderivative size = 41, 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, 325, 203} \[ -\frac{a^3}{4 x}-\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{a}{12 x^3}-\frac{\cot ^{-1}(a x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a/(12*x^3) - a^3/(4*x) - ArcCot[a*x]/(4*x^4) - (a^4*ArcTan[a*x])/4

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 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

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

Mathematica [C]  time = 0.0027656, size = 36, normalized size = 0.88 \[ \frac{a \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-a^2 x^2\right )}{12 x^3}-\frac{\cot ^{-1}(a x)}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcCot[a*x]/(4*x^4) + (a*Hypergeometric2F1[-3/2, 1, -1/2, -(a^2*x^2)])/(12*x^3)

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Maple [A]  time = 0.046, size = 34, normalized size = 0.8 \begin{align*}{\frac{a}{12\,{x}^{3}}}-{\frac{{a}^{3}}{4\,x}}-{\frac{{\rm arccot} \left (ax\right )}{4\,{x}^{4}}}-{\frac{{a}^{4}\arctan \left ( ax \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*a/x^3-1/4*a^3/x-1/4*arccot(a*x)/x^4-1/4*a^4*arctan(a*x)

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Maxima [A]  time = 1.48393, size = 50, normalized size = 1.22 \begin{align*} -\frac{1}{12} \,{\left (3 \, a^{3} \arctan \left (a x\right ) + \frac{3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a - \frac{\operatorname{arccot}\left (a x\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.86439, size = 80, normalized size = 1.95 \begin{align*} -\frac{3 \, a^{3} x^{3} - a x - 3 \,{\left (a^{4} x^{4} - 1\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)/x^5,x, algorithm="fricas")

[Out]

-1/12*(3*a^3*x^3 - a*x - 3*(a^4*x^4 - 1)*arccot(a*x))/x^4

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Sympy [A]  time = 1.01842, size = 32, normalized size = 0.78 \begin{align*} \frac{a^{4} \operatorname{acot}{\left (a x \right )}}{4} - \frac{a^{3}}{4 x} + \frac{a}{12 x^{3}} - \frac{\operatorname{acot}{\left (a x \right )}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**4*acot(a*x)/4 - a**3/(4*x) + a/(12*x**3) - acot(a*x)/(4*x**4)

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Giac [A]  time = 1.12323, size = 55, normalized size = 1.34 \begin{align*} -\frac{1}{12} \,{\left (3 \, a^{3} \arctan \left (a x\right ) + \frac{3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a - \frac{\arctan \left (\frac{1}{a x}\right )}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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