3.32 \(\int \frac{\cot ^{-1}(a x)^3}{x^4} \, dx\)

Optimal. Leaf size=167 \[ \frac{1}{2} a^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{1}{2} a^3 \log \left (a^2 x^2+1\right )-a^3 \log (x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3+\frac{1}{2} a^3 \cot ^{-1}(a x)^2-\frac{a^2 \cot ^{-1}(a x)}{x}+a^3 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}-\frac{\cot ^{-1}(a x)^3}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.336865, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.1, Rules used = {4853, 4919, 266, 36, 29, 31, 4885, 4925, 4869, 4993, 6610} \[ \frac{1}{2} a^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{1}{2} a^3 \log \left (a^2 x^2+1\right )-a^3 \log (x)+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3+\frac{1}{2} a^3 \cot ^{-1}(a x)^2-\frac{a^2 \cot ^{-1}(a x)}{x}+a^3 \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}-\frac{\cot ^{-1}(a x)^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[ArcCot[a*x]^3/x^4,x]

[Out]

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

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 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 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 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 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 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 4993

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

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int \frac{\cot ^{-1}(a x)^3}{x^4} \, dx &=-\frac{\cot ^{-1}(a x)^3}{3 x^3}-a \int \frac{\cot ^{-1}(a x)^2}{x^3 \left (1+a^2 x^2\right )} \, dx\\ &=-\frac{\cot ^{-1}(a x)^3}{3 x^3}-a \int \frac{\cot ^{-1}(a x)^2}{x^3} \, dx+a^3 \int \frac{\cot ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx\\ &=\frac{a \cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{3 x^3}+a^2 \int \frac{\cot ^{-1}(a x)}{x^2 \left (1+a^2 x^2\right )} \, dx+\left (i a^3\right ) \int \frac{\cot ^{-1}(a x)^2}{x (i+a x)} \, dx\\ &=\frac{a \cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )+a^2 \int \frac{\cot ^{-1}(a x)}{x^2} \, dx-a^4 \int \frac{\cot ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^4\right ) \int \frac{\cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 \cot ^{-1}(a x)}{x}+\frac{1}{2} a^3 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-a^3 \int \frac{1}{x \left (1+a^2 x^2\right )} \, dx+\left (i a^4\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{a^2 \cot ^{-1}(a x)}{x}+\frac{1}{2} a^3 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{1}{2} a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )-\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{x \left (1+a^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a^2 \cot ^{-1}(a x)}{x}+\frac{1}{2} a^3 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{3 x^3}+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{1}{2} a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )-\frac{1}{2} a^3 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\frac{1}{2} a^5 \operatorname{Subst}\left (\int \frac{1}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac{a^2 \cot ^{-1}(a x)}{x}+\frac{1}{2} a^3 \cot ^{-1}(a x)^2+\frac{a \cot ^{-1}(a x)^2}{2 x^2}+\frac{1}{3} i a^3 \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{3 x^3}-a^3 \log (x)+\frac{1}{2} a^3 \log \left (1+a^2 x^2\right )+a^3 \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )+i a^3 \cot ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{1}{2} a^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}

Mathematica [A]  time = 0.243506, size = 151, normalized size = 0.9 \[ \frac{1}{6} \left (-6 i a^3 \cot ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )+3 a^3 \text{PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )-6 a^3 \log \left (\frac{1}{\sqrt{\frac{1}{a^2 x^2}+1}}\right )-2 i a^3 \cot ^{-1}(a x)^3+3 a^3 \cot ^{-1}(a x)^2-\frac{6 a^2 \cot ^{-1}(a x)}{x}+6 a^3 \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+\frac{3 a \cot ^{-1}(a x)^2}{x^2}-\frac{2 \cot ^{-1}(a x)^3}{x^3}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCot[a*x]^3/x^4,x]

[Out]

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

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Maple [C]  time = 1.627, size = 5029, normalized size = 30.1 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccot(a*x)^3/x^4,x)

[Out]

result too large to display

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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^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 )^{3}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x^4,x, algorithm="fricas")

[Out]

integral(arccot(a*x)^3/x^4, 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^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acot(a*x)**3/x**4,x)

[Out]

Integral(acot(a*x)**3/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 )^{3}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccot(a*x)^3/x^4,x, algorithm="giac")

[Out]

integrate(arccot(a*x)^3/x^4, x)