3.33 \(\int \frac{\cot ^{-1}(a x)^3}{x^5} \, dx\)

Optimal. Leaf size=152 \[ -i a^4 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}+\frac{a^3}{4 x}+\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}-2 a^4 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{\cot ^{-1}(a x)^3}{4 x^4} \]

[Out]

a^3/(4*x) - (a^2*ArcCot[a*x])/(4*x^2) - I*a^4*ArcCot[a*x]^2 + (a*ArcCot[a*x]^2)/(4*x^3) - (3*a^3*ArcCot[a*x]^2
)/(4*x) + (a^4*ArcCot[a*x]^3)/4 - ArcCot[a*x]^3/(4*x^4) + (a^4*ArcTan[a*x])/4 - 2*a^4*ArcCot[a*x]*Log[2 - 2/(1
- I*a*x)] - I*a^4*PolyLog[2, -1 + 2/(1 - I*a*x)]

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Rubi [A]  time = 0.419914, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4853, 4919, 325, 203, 4925, 4869, 2447, 4885} \[ -i a^4 \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}+\frac{a^3}{4 x}+\frac{1}{4} a^4 \tan ^{-1}(a x)+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-i a^4 \cot ^{-1}(a x)^2-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}-2 a^4 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{\cot ^{-1}(a x)^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^3/x^5,x]

[Out]

a^3/(4*x) - (a^2*ArcCot[a*x])/(4*x^2) - I*a^4*ArcCot[a*x]^2 + (a*ArcCot[a*x]^2)/(4*x^3) - (3*a^3*ArcCot[a*x]^2
)/(4*x) + (a^4*ArcCot[a*x]^3)/4 - ArcCot[a*x]^3/(4*x^4) + (a^4*ArcTan[a*x])/4 - 2*a^4*ArcCot[a*x]*Log[2 - 2/(1
- I*a*x)] - I*a^4*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]]

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^5} \, dx &=-\frac{\cot ^{-1}(a x)^3}{4 x^4}-\frac{1}{4} (3 a) \int \frac{\cot ^{-1}(a x)^2}{x^4 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)^3}{4 x^4}-\frac{1}{4} (3 a) \int \frac{\cot ^{-1}(a x)^2}{x^4} \, dx+\frac{1}{4} \left (3 a^3\right ) \int \frac{\cot ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{\cot ^{-1}(a x)^3}{4 x^4}+\frac{1}{2} a^2 \int \frac{\cot ^{-1}(a x)}{x^3 \left (1+a^2 x^2\right )} \, dx+\frac{1}{4} \left (3 a^3\right ) \int \frac{\cot ^{-1}(a x)^2}{x^2} \, dx-\frac{1}{4} \left (3 a^5\right ) \int \frac{\cot ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{4 x^4}+\frac{1}{2} a^2 \int \frac{\cot ^{-1}(a x)}{x^3} \, dx-\frac{1}{2} a^4 \int \frac{\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx-\frac{1}{2} \left (3 a^4\right ) \int \frac{\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{4 x^4}-\frac{1}{4} a^3 \int \frac{1}{x^2 \left (1+a^2 x^2\right )} \, dx-\frac{1}{2} \left (i a^4\right ) \int \frac{\cot ^{-1}(a x)}{x (i+a x)} \, dx-\frac{1}{2} \left (3 i a^4\right ) \int \frac{\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=\frac{a^3}{4 x}-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{4 x^4}-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )+\frac{1}{4} a^5 \int \frac{1}{1+a^2 x^2} \, dx-\frac{1}{2} a^5 \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^5\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac{a^3}{4 x}-\frac{a^2 \cot ^{-1}(a x)}{4 x^2}-i a^4 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{4 x^3}-\frac{3 a^3 \cot ^{-1}(a x)^2}{4 x}+\frac{1}{4} a^4 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{4 x^4}+\frac{1}{4} a^4 \tan ^{-1}(a x)-2 a^4 \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a^4 \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.258419, size = 126, normalized size = 0.83 \[ \frac{4 i a^4 x^4 \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+a^3 x^3+\left (a^4 x^4-1\right ) \cot ^{-1}(a x)^3+\left (4 i a^4 x^4-3 a^3 x^3+a x\right ) \cot ^{-1}(a x)^2-a^2 x^2 \cot ^{-1}(a x) \left (a^2 x^2+8 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+1\right )}{4 x^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[a*x]^3/x^5,x]

[Out]

(a^3*x^3 + (a*x - 3*a^3*x^3 + (4*I)*a^4*x^4)*ArcCot[a*x]^2 + (-1 + a^4*x^4)*ArcCot[a*x]^3 - a^2*x^2*ArcCot[a*x
]*(1 + a^2*x^2 + 8*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]) + (4*I)*a^4*x^4*PolyLog[2, -E^((2*I)*ArcCot[a*x])])
/(4*x^4)

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Maple [A]  time = 0.485, size = 158, normalized size = 1. \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{4\,{x}^{4}}}+{\frac{{a}^{4} \left ({\rm arccot} \left (ax\right ) \right ) ^{3}}{4}}+i{a}^{4} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}-{\frac{{a}^{4}{\rm arccot} \left (ax\right )}{4}}-{\frac{3\,{a}^{3} \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{4\,x}}+{\frac{i}{4}}{a}^{4}+{\frac{{a}^{3}}{4\,x}}-{\frac{{a}^{2}{\rm arccot} \left (ax\right )}{4\,{x}^{2}}}+{\frac{a \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{4\,{x}^{3}}}-2\,{a}^{4}{\rm arccot} \left (ax\right )\ln \left ({\frac{ \left ( ax+i \right ) ^{2}}{{a}^{2}{x}^{2}+1}}+1 \right ) +i{a}^{4}{\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^5,x)

[Out]

-1/4*arccot(a*x)^3/x^4+1/4*a^4*arccot(a*x)^3+I*a^4*arccot(a*x)^2-1/4*a^4*arccot(a*x)-3/4*a^3*arccot(a*x)^2/x+1
/4*I*a^4+1/4*a^3/x-1/4*a^2*arccot(a*x)/x^2+1/4*a*arccot(a*x)^2/x^3-2*a^4*arccot(a*x)*ln((a*x+I)^2/(a^2*x^2+1)+
1)+I*a^4*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^5,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^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x^5,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^3/x^5, 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^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**3/x**5,x)

[Out]

Integral(acot(a*x)**3/x**5, 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x^5,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^3/x^5, x)