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]

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

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Rubi [A]  time = 0.02, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \frac {1}{4} a \log \left (a^2 x^4+1\right )-\frac {\cot ^{-1}\left (a x^2\right )}{2 x^2}-a \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.68, size = 37, normalized size = 1.09 \[ \frac {a x^{2} \log \left (a^{2} x^{4} + 1\right ) - 4 \, a x^{2} \log \relax (x) - 2 \, \operatorname {arccot}\left (a x^{2}\right )}{4 \, x^{2}} \]

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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giac [A]  time = 0.12, size = 34, normalized size = 1.00 \[ \frac {1}{4} \, a {\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac {\arctan \left (\frac {1}{a x^{2}}\right )}{2 \, x^{2}} \]

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

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maple [A]  time = 0.04, size = 31, normalized size = 0.91 \[ -\frac {\mathrm {arccot}\left (a \,x^{2}\right )}{2 x^{2}}-a \ln \relax (x )+\frac {a \ln \left (a^{2} x^{4}+1\right )}{4} \]

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 = 0.32, size = 32, normalized size = 0.94 \[ \frac {1}{4} \, a {\left (\log \left (a^{2} x^{4} + 1\right ) - \log \left (x^{4}\right )\right )} - \frac {\operatorname {arccot}\left (a x^{2}\right )}{2 \, x^{2}} \]

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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mupad [B]  time = 0.66, size = 31, normalized size = 0.91 \[ \frac {a\,\ln \left (-a^2\,x^4-1\right )}{4}-\frac {\mathrm {acot}\left (a\,x^2\right )}{2\,x^2}-a\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.70, size = 29, normalized size = 0.85 \[ - a \log {\relax (x )} + \frac {a \log {\left (a^{2} x^{4} + 1 \right )}}{4} - \frac {\operatorname {acot}{\left (a x^{2} \right )}}{2 x^{2}} \]

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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