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

Optimal. Leaf size=105 \[ \frac{3}{2} i a^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{1}{2} a^2 \cot ^{-1}(a x)^3+\frac{3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac{\cot ^{-1}(a x)^3}{2 x^2}+\frac{3 a \cot ^{-1}(a x)^2}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.199437, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4853, 4919, 4925, 4869, 2447, 4885} \[ \frac{3}{2} i a^2 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{1}{2} a^2 \cot ^{-1}(a x)^3+\frac{3}{2} i a^2 \cot ^{-1}(a x)^2+3 a^2 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)-\frac{\cot ^{-1}(a x)^3}{2 x^2}+\frac{3 a \cot ^{-1}(a x)^2}{2 x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

Rubi steps

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

Mathematica [A]  time = 0.160054, size = 90, normalized size = 0.86 \[ -\frac{3}{2} i a^2 \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac{\cot ^{-1}(a x) \left (\left (a^2 x^2+1\right ) \cot ^{-1}(a x)^2-6 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+3 i a x (a x+i) \cot ^{-1}(a x)\right )}{2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(ArcCot[a*x]*((3*I)*a*x*(I + a*x)*ArcCot[a*x] + (1 + a^2*x^2)*ArcCot[a*x]^2 - 6*a^2*x^2*Log[1 + E^((2*I)*ArcC
ot[a*x])]))/(2*x^2) - ((3*I)/2)*a^2*PolyLog[2, -E^((2*I)*ArcCot[a*x])]

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Maple [A]  time = 0.322, size = 109, normalized size = 1. \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{2\,{x}^{2}}}-{\frac{{a}^{2} \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{2}}-{\frac{3\,i}{2}}{a}^{2} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}+{\frac{3\,a \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{2\,x}}+3\,{a}^{2}{\rm arccot} \left (ax\right )\ln \left ({\frac{ \left ( ax+i \right ) ^{2}}{{a}^{2}{x}^{2}+1}}+1 \right ) -{\frac{3\,i}{2}}{a}^{2}{\it polylog} \left ( 2,-{\frac{ \left ( ax+i \right ) ^{2}}{{a}^{2}{x}^{2}+1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arccot(a*x)^3/x^2-1/2*a^2*arccot(a*x)^3-3/2*I*a^2*arccot(a*x)^2+3/2*a*arccot(a*x)^2/x+3*a^2*arccot(a*x)*l
n((a*x+I)^2/(a^2*x^2+1)+1)-3/2*I*a^2*polylog(2,-(a*x+I)^2/(a^2*x^2+1))

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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)^3/x^3,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 )^{3}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(arccot(a*x)^3/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}^{3}{\left (a x \right )}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acot(a*x)**3/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x\right )^{3}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccot(a*x)^3/x^3, x)