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

Optimal. Leaf size=35 \[ \frac{1}{4} a^2 \tan ^{-1}\left (a x^2\right )+\frac{a}{4 x^2}-\frac{\cot ^{-1}\left (a x^2\right )}{4 x^4} \]

[Out]

a/(4*x^2) - ArcCot[a*x^2]/(4*x^4) + (a^2*ArcTan[a*x^2])/4

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Rubi [A]  time = 0.0179996, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5034, 275, 325, 203} \[ \frac{1}{4} a^2 \tan ^{-1}\left (a x^2\right )+\frac{a}{4 x^2}-\frac{\cot ^{-1}\left (a x^2\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a/(4*x^2) - ArcCot[a*x^2]/(4*x^4) + (a^2*ArcTan[a*x^2])/4

Rule 5034

Int[((a_.) + ArcCot[(c_.)*(x_)^(n_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCot
[c*x^n]))/(d*(m + 1)), x] + Dist[(b*c*n)/(d*(m + 1)), Int[(x^(n - 1)*(d*x)^(m + 1))/(1 + c^2*x^(2*n)), x], x]
/; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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

Mathematica [C]  time = 0.0064748, size = 38, normalized size = 1.09 \[ \frac{a \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-a^2 x^4\right )}{4 x^2}-\frac{\cot ^{-1}\left (a x^2\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.044, size = 30, normalized size = 0.9 \begin{align*}{\frac{a}{4\,{x}^{2}}}-{\frac{{\rm arccot} \left (a{x}^{2}\right )}{4\,{x}^{4}}}+{\frac{{a}^{2}\arctan \left ( a{x}^{2} \right ) }{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*a/x^2-1/4*arccot(a*x^2)/x^4+1/4*a^2*arctan(a*x^2)

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Maxima [A]  time = 1.46295, size = 36, normalized size = 1.03 \begin{align*} \frac{1}{4} \,{\left (a \arctan \left (a x^{2}\right ) + \frac{1}{x^{2}}\right )} a - \frac{\operatorname{arccot}\left (a x^{2}\right )}{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]

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

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Fricas [A]  time = 2.12745, size = 63, normalized size = 1.8 \begin{align*} \frac{a x^{2} -{\left (a^{2} x^{4} + 1\right )} \operatorname{arccot}\left (a x^{2}\right )}{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="fricas")

[Out]

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

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Sympy [A]  time = 1.73273, size = 29, normalized size = 0.83 \begin{align*} - \frac{a^{2} \operatorname{acot}{\left (a x^{2} \right )}}{4} + \frac{a}{4 x^{2}} - \frac{\operatorname{acot}{\left (a x^{2} \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]

-a**2*acot(a*x**2)/4 + a/(4*x**2) - acot(a*x**2)/(4*x**4)

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Giac [A]  time = 1.12162, size = 39, normalized size = 1.11 \begin{align*} \frac{1}{4} \,{\left (a \arctan \left (a x^{2}\right ) + \frac{1}{x^{2}}\right )} a - \frac{\arctan \left (\frac{1}{a x^{2}}\right )}{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="giac")

[Out]

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