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

Optimal. Leaf size=66 \[ -i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x) \]

[Out]

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

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Rubi [A]  time = 0.105241, antiderivative size = 66, 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 = {4853, 4925, 4869, 2447} \[ -i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4925

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

Rule 4869

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcCot[c*x]
)^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] + Dist[(b*c*p)/d, Int[((a + b*ArcCot[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)/d)
])/(1 + c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 + e^2, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(a x)^2}{x^2} \, dx &=-\frac{\cot ^{-1}(a x)^2}{x}-(2 a) \int \frac{\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-(2 i a) \int \frac{\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\left (2 a^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0439344, size = 64, normalized size = 0.97 \[ a \left (i \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac{\cot ^{-1}(a x)^2}{a x}+i \cot ^{-1}(a x)^2-2 \cot ^{-1}(a x) \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*(I*ArcCot[a*x]^2 - ArcCot[a*x]^2/(a*x) - 2*ArcCot[a*x]*Log[1 + E^((2*I)*ArcCot[a*x])] + I*PolyLog[2, -E^((2*
I)*ArcCot[a*x])])

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Maple [B]  time = 0.138, size = 234, normalized size = 3.6 \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{x}}+a{\rm arccot} \left (ax\right )\ln \left ({a}^{2}{x}^{2}+1 \right ) -2\,a\ln \left ( ax \right ){\rm arccot} \left (ax\right )+ia\ln \left ( ax \right ) \ln \left ( 1+iax \right ) -ia\ln \left ( ax \right ) \ln \left ( 1-iax \right ) +ia{\it dilog} \left ( 1+iax \right ) -ia{\it dilog} \left ( 1-iax \right ) +{\frac{i}{4}}a \left ( \ln \left ( ax-i \right ) \right ) ^{2}+{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) +{\frac{i}{2}}a{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{i}{4}}a \left ( \ln \left ( ax+i \right ) \right ) ^{2}-{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{i}{2}}a{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (a x\right )^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(a*x)^2/x^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}^{2}{\left (a x \right )}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(a*x)**2/x**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x)^2/x^2, x)