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

Optimal. Leaf size=59 \[ -\frac {1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+\frac {a \cot ^{-1}(a x)}{x} \]

[Out]

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

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Rubi [A]  time = 0.09, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4853, 4919, 266, 36, 29, 31, 4885} \[ -\frac {1}{2} a^2 \log \left (a^2 x^2+1\right )+a^2 \log (x)-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+\frac {a \cot ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 4885

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(a + b*ArcCot[c*x])^(p
+ 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 4919

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/d,
Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Dist[e/(d*f^2), Int[((f*x)^(m + 2)*(a + b*ArcCot[c*x])^p)/(d + e*
x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 56, normalized size = 0.95 \[ -\frac {1}{2} a^2 \log \left (a^2 x^2+1\right )+\frac {\left (-a^2 x^2-1\right ) \cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)+\frac {a \cot ^{-1}(a x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.71, size = 57, normalized size = 0.97 \[ -\frac {a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} x^{2} \log \relax (x) - 2 \, a x \operatorname {arccot}\left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.12, size = 60, normalized size = 1.02 \[ -\frac {1}{2} \, {\left ({\left (\arctan \left (\frac {1}{a x}\right )^{2} - \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x} + \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} a + \frac {\arctan \left (\frac {1}{a x}\right )^{2}}{a x^{2}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 68, normalized size = 1.15 \[ -\frac {\mathrm {arccot}\left (a x \right )^{2}}{2 x^{2}}+\frac {a \,\mathrm {arccot}\left (a x \right )}{x}+a^{2} \mathrm {arccot}\left (a x \right ) \arctan \left (a x \right )+a^{2} \ln \left (a x \right )-\frac {a^{2} \ln \left (a^{2} x^{2}+1\right )}{2}+\frac {a^{2} \arctan \left (a x \right )^{2}}{2} \]

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

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maxima [A]  time = 0.42, size = 56, normalized size = 0.95 \[ \frac {1}{2} \, {\left (\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right ) + 2 \, \log \relax (x)\right )} a^{2} + {\left (a \arctan \left (a x\right ) + \frac {1}{x}\right )} a \operatorname {arccot}\left (a x\right ) - \frac {\operatorname {arccot}\left (a x\right )^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.66, size = 50, normalized size = 0.85 \[ a^2\,\ln \relax (x)-{\mathrm {acot}\left (a\,x\right )}^2\,\left (\frac {a^2}{2}+\frac {1}{2\,x^2}\right )-\frac {a^2\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {a\,\mathrm {acot}\left (a\,x\right )}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.50, size = 53, normalized size = 0.90 \[ a^{2} \log {\relax (x )} - \frac {a^{2} \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {a^{2} \operatorname {acot}^{2}{\left (a x \right )}}{2} + \frac {a \operatorname {acot}{\left (a x \right )}}{x} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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