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

Optimal. Leaf size=31 \[ \frac{1}{2} a^2 \tan ^{-1}(a x)-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{a}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.0148004, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4853, 325, 203} \[ \frac{1}{2} a^2 \tan ^{-1}(a x)-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{a}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a/(2*x) - ArcCot[a*x]/(2*x^2) + (a^2*ArcTan[a*x])/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 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^3} \, dx &=-\frac{\cot ^{-1}(a x)}{2 x^2}-\frac{1}{2} a \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a}{2 x}-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{1}{2} a^3 \int \frac{1}{1+a^2 x^2} \, dx\\ &=\frac{a}{2 x}-\frac{\cot ^{-1}(a x)}{2 x^2}+\frac{1}{2} a^2 \tan ^{-1}(a x)\\ \end{align*}

Mathematica [C]  time = 0.0023036, size = 36, normalized size = 1.16 \[ \frac{a \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-a^2 x^2\right )}{2 x}-\frac{\cot ^{-1}(a x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.043, size = 26, normalized size = 0.8 \begin{align*}{\frac{a}{2\,x}}-{\frac{{\rm arccot} \left (ax\right )}{2\,{x}^{2}}}+{\frac{{a}^{2}\arctan \left ( ax \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.50052, size = 31, normalized size = 1. \begin{align*} \frac{1}{2} \,{\left (a \arctan \left (a x\right ) + \frac{1}{x}\right )} a - \frac{\operatorname{arccot}\left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.87474, size = 58, normalized size = 1.87 \begin{align*} \frac{a x -{\left (a^{2} x^{2} + 1\right )} \operatorname{arccot}\left (a x\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.631373, size = 24, normalized size = 0.77 \begin{align*} - \frac{a^{2} \operatorname{acot}{\left (a x \right )}}{2} + \frac{a}{2 x} - \frac{\operatorname{acot}{\left (a x \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.1101, size = 36, normalized size = 1.16 \begin{align*} \frac{1}{2} \,{\left (a \arctan \left (a x\right ) + \frac{1}{x}\right )} a - \frac{\arctan \left (\frac{1}{a x}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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