3.21 \(\int \frac{\cot ^{-1}(a x)^2}{x^4} \, dx\)

Optimal. Leaf size=113 \[ \frac{1}{3} i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{a^2}{3 x}-\frac{1}{3} a^3 \tan ^{-1}(a x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2+\frac{2}{3} a^3 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)}{3 x^2}-\frac{\cot ^{-1}(a x)^2}{3 x^3} \]

[Out]

-a^2/(3*x) + (a*ArcCot[a*x])/(3*x^2) + (I/3)*a^3*ArcCot[a*x]^2 - ArcCot[a*x]^2/(3*x^3) - (a^3*ArcTan[a*x])/3 +
(2*a^3*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)])/3 + (I/3)*a^3*PolyLog[2, -1 + 2/(1 - I*a*x)]

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Rubi [A]  time = 0.161392, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {4853, 4919, 325, 203, 4925, 4869, 2447} \[ \frac{1}{3} i a^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{a^2}{3 x}-\frac{1}{3} a^3 \tan ^{-1}(a x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2+\frac{2}{3} a^3 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)}{3 x^2}-\frac{\cot ^{-1}(a x)^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^2/x^4,x]

[Out]

-a^2/(3*x) + (a*ArcCot[a*x])/(3*x^2) + (I/3)*a^3*ArcCot[a*x]^2 - ArcCot[a*x]^2/(3*x^3) - (a^3*ArcTan[a*x])/3 +
(2*a^3*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)])/3 + (I/3)*a^3*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 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]

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])

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^4} \, dx &=-\frac{\cot ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} (2 a) \int \frac{\cot ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} (2 a) \int \frac{\cot ^{-1}(a x)}{x^3} \, dx+\frac{1}{3} \left (2 a^3\right ) \int \frac{\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)}{3 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{3 x^3}+\frac{1}{3} a^2 \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx+\frac{1}{3} \left (2 i a^3\right ) \int \frac{\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=-\frac{a^2}{3 x}+\frac{a \cot ^{-1}(a x)}{3 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{3 x^3}+\frac{2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{1}{3} a^4 \int \frac{1}{1+a^2 x^2} \, dx+\frac{1}{3} \left (2 a^4\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2}{3 x}+\frac{a \cot ^{-1}(a x)}{3 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{3 x^3}-\frac{1}{3} a^3 \tan ^{-1}(a x)+\frac{2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )+\frac{1}{3} i a^3 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.23248, size = 96, normalized size = 0.85 \[ \frac{-i a^3 x^3 \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-a^2 x^2+\left (-1-i a^3 x^3\right ) \cot ^{-1}(a x)^2+a x \cot ^{-1}(a x) \left (a^2 x^2+2 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+1\right )}{3 x^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[a*x]^2/x^4,x]

[Out]

(-(a^2*x^2) + (-1 - I*a^3*x^3)*ArcCot[a*x]^2 + a*x*ArcCot[a*x]*(1 + a^2*x^2 + 2*a^2*x^2*Log[1 + E^((2*I)*ArcCo
t[a*x])]) - I*a^3*x^3*PolyLog[2, -E^((2*I)*ArcCot[a*x])])/(3*x^3)

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Maple [B]  time = 0.133, size = 290, normalized size = 2.6 \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{3\,{x}^{3}}}-{\frac{{a}^{3}{\rm arccot} \left (ax\right )\ln \left ({a}^{2}{x}^{2}+1 \right ) }{3}}+{\frac{a{\rm arccot} \left (ax\right )}{3\,{x}^{2}}}+{\frac{2\,{a}^{3}\ln \left ( ax \right ){\rm arccot} \left (ax\right )}{3}}+{\frac{i}{12}}{a}^{3} \left ( \ln \left ( ax+i \right ) \right ) ^{2}-{\frac{i}{3}}{a}^{3}{\it dilog} \left ( 1+iax \right ) +{\frac{i}{6}}{a}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{i}{6}}{a}^{3}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) +{\frac{i}{3}}{a}^{3}{\it dilog} \left ( 1-iax \right ) -{\frac{i}{3}}{a}^{3}\ln \left ( ax \right ) \ln \left ( 1+iax \right ) +{\frac{i}{6}}{a}^{3}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) -{\frac{i}{6}}{a}^{3}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{{a}^{3}\arctan \left ( ax \right ) }{3}}-{\frac{{a}^{2}}{3\,x}}-{\frac{i}{6}}{a}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) +{\frac{i}{3}}{a}^{3}\ln \left ( ax \right ) \ln \left ( 1-iax \right ) -{\frac{i}{6}}{a}^{3}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{i}{12}}{a}^{3} \left ( \ln \left ( ax-i \right ) \right ) ^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)^2/x^4,x)

[Out]

-1/3*arccot(a*x)^2/x^3-1/3*a^3*arccot(a*x)*ln(a^2*x^2+1)+1/3*a*arccot(a*x)/x^2+2/3*a^3*ln(a*x)*arccot(a*x)+1/1
2*I*a^3*ln(a*x+I)^2-1/3*I*a^3*dilog(1+I*a*x)+1/6*I*a^3*dilog(1/2*I*(a*x-I))+1/6*I*a^3*ln(a*x-I)*ln(a^2*x^2+1)+
1/3*I*a^3*dilog(1-I*a*x)-1/3*I*a^3*ln(a*x)*ln(1+I*a*x)+1/6*I*a^3*ln(a*x+I)*ln(1/2*I*(a*x-I))-1/6*I*a^3*ln(a*x+
I)*ln(a^2*x^2+1)-1/3*a^3*arctan(a*x)-1/3*a^2/x-1/6*I*a^3*dilog(-1/2*I*(a*x+I))+1/3*I*a^3*ln(a*x)*ln(1-I*a*x)-1
/6*I*a^3*ln(a*x-I)*ln(-1/2*I*(a*x+I))-1/12*I*a^3*ln(a*x-I)^2

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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^4,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^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^2/x^4, 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^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**2/x**4,x)

[Out]

Integral(acot(a*x)**2/x**4, 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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^2/x^4,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^2/x^4, x)