3.78 \(\int \frac{\cot ^{-1}(a x^2)}{x^3} \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{4} a \log \left (a^2 x^4+1\right )-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0173108, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5034, 266, 36, 29, 31} \[ \frac{1}{4} a \log \left (a^2 x^4+1\right )-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x^2]/x^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0058354, size = 34, normalized size = 1. \[ \frac{1}{4} a \log \left (a^2 x^4+1\right )-\frac{\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCot[a*x^2]/x^3,x]

[Out]

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

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Maple [A]  time = 0.048, size = 31, normalized size = 0.9 \begin{align*} -{\frac{{\rm arccot} \left (a{x}^{2}\right )}{2\,{x}^{2}}}-a\ln \left ( x \right ) +{\frac{a\ln \left ({a}^{2}{x}^{4}+1 \right ) }{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x^2)/x^3,x)

[Out]

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

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Maxima [A]  time = 0.965668, size = 43, normalized size = 1.26 \begin{align*} \frac{1}{4} \, a{\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac{\operatorname{arccot}\left (a x^{2}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^2)/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.29115, size = 93, normalized size = 2.74 \begin{align*} \frac{a x^{2} \log \left (a^{2} x^{4} + 1\right ) - 4 \, a x^{2} \log \left (x\right ) - 2 \, \operatorname{arccot}\left (a x^{2}\right )}{4 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^2)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.05433, size = 29, normalized size = 0.85 \begin{align*} - a \log{\left (x \right )} + \frac{a \log{\left (a^{2} x^{4} + 1 \right )}}{4} - \frac{\operatorname{acot}{\left (a x^{2} \right )}}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x**2)/x**3,x)

[Out]

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

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Giac [A]  time = 1.14325, size = 46, normalized size = 1.35 \begin{align*} \frac{1}{4} \, a{\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac{\arctan \left (\frac{1}{a x^{2}}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x^2)/x^3,x, algorithm="giac")

[Out]

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