3.30 \(\int \frac{\cot ^{-1}(a x)^3}{x^2} \, dx\)
Optimal. Leaf size=93 \[ -\frac{3}{2} a \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)^2 \]
[Out]
(-I)*a*ArcCot[a*x]^3 - ArcCot[a*x]^3/x - 3*a*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*ArcCot[a*x]*PolyLo
g[2, -1 + 2/(1 - I*a*x)] - (3*a*PolyLog[3, -1 + 2/(1 - I*a*x)])/2
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Rubi [A] time = 0.188242, antiderivative size = 93, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used =
{4853, 4925, 4869, 4885, 4993, 6610} \[ -\frac{3}{2} a \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
[In]
Int[ArcCot[a*x]^3/x^2,x]
[Out]
(-I)*a*ArcCot[a*x]^3 - ArcCot[a*x]^3/x - 3*a*ArcCot[a*x]^2*Log[2 - 2/(1 - I*a*x)] - (3*I)*a*ArcCot[a*x]*PolyLo
g[2, -1 + 2/(1 - I*a*x)] - (3*a*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 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 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 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^2} \, dx &=-\frac{\cot ^{-1}(a x)^3}{x}-(3 a) \int \frac{\cot ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx\\ &=-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-(3 i a) \int \frac{\cot ^{-1}(a x)^2}{x (i+a x)} \, dx\\ &=-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-\left (6 a^2\right ) \int \frac{\cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\left (3 i a^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-i a \cot ^{-1}(a x)^3-\frac{\cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a \cot ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{3}{2} a \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.0720911, size = 83, normalized size = 0.89 \[ 3 i a \cot ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac{3}{2} a \text{PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right )+\frac{(-1+i a x) \cot ^{-1}(a x)^3}{x}-3 a \cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
Integrate[ArcCot[a*x]^3/x^2,x]
[Out]
((-1 + I*a*x)*ArcCot[a*x]^3)/x - 3*a*ArcCot[a*x]^2*Log[1 + E^((2*I)*ArcCot[a*x])] + (3*I)*a*ArcCot[a*x]*PolyLo
g[2, -E^((2*I)*ArcCot[a*x])] - (3*a*PolyLog[3, -E^((2*I)*ArcCot[a*x])])/2
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Maple [C] time = 0.574, size = 1576, normalized size = 17. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(arccot(a*x)^3/x^2,x)
[Out]
-3/2*a*polylog(3,-(a*x+I)^2/(a^2*x^2+1))+3/2*I*a*arccot(a*x)^2*Pi*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/
(a^2*x^2+1)+1))^3-3/2*I*a*arccot(a*x)^2*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^3+3/4*I
*a*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1))^3-3/4*I*a*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*
x+I)^2/(a^2*x^2+1)-1)^2)^3-3/4*I*a*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^3-3/2*I*a*arccot(a*x)^
2*Pi*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^2-3/4*I*a*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/
(a^2*x^2+1)-1)^2)*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1))^2-3/2*I*a*arccot(a*x)^2*Pi*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1
)*((a*x+I)^2/(a^2*x^2+1)+1))*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))+3/2*I*a*arccot(a*x)^2
*Pi*csgn(1/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^2*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/
(a^2*x^2+1)+1))+3/2*I*a*arccot(a*x)^2*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1))^2*csgn(I/
((a*x+I)^2/(a^2*x^2+1)-1))+3/2*I*a*arccot(a*x)^2*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(a^2*x^2+1)+1)
)^2*csgn(I*((a*x+I)^2/(a^2*x^2+1)+1))+3/4*I*a*arccot(a*x)^2*Pi*csgn(I*(a*x+I)/(a^2*x^2+1)^(1/2))^2*csgn(I*(a*x
+I)^2/(a^2*x^2+1))-3/2*I*a*arccot(a*x)^2*Pi*csgn(I*(a*x+I)/(a^2*x^2+1)^(1/2))*csgn(I*(a*x+I)^2/(a^2*x^2+1))^2-
3/4*I*a*arccot(a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1))*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2*x^2+1)-1)^
2)^2-3/4*I*a*arccot(a*x)^2*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2
*x^2+1)-1)^2)^2+3/2*I*a*arccot(a*x)^2*Pi*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1)^2)^2*csgn(I*((a*x+I)^2/(a^2*x^2+1)-1
))+I*a*arccot(a*x)^3+3/2*a*arccot(a*x)^2*ln(a^2*x^2+1)-3*a*ln(a*x)*arccot(a*x)^2-3*a*arccot(a*x)^2*ln((a*x+I)/
(a^2*x^2+1)^(1/2))-3*a*arccot(a*x)^2*ln(2)+3*I*a*arccot(a*x)*polylog(2,-(a*x+I)^2/(a^2*x^2+1))+3/4*I*a*arccot(
a*x)^2*Pi*csgn(I*(a*x+I)^2/(a^2*x^2+1))*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)^2)*csgn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x
+I)^2/(a^2*x^2+1)-1)^2)-arccot(a*x)^3/x-3/2*I*a*arccot(a*x)^2*Pi*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1)*((a*x+I)^2/(
a^2*x^2+1)+1))*csgn(I/((a*x+I)^2/(a^2*x^2+1)-1))*csgn(I*((a*x+I)^2/(a^2*x^2+1)+1))+3/2*I*a*arccot(a*x)^2*Pi*cs
gn(I*(a*x+I)^2/(a^2*x^2+1)/((a*x+I)^2/(a^2*x^2+1)-1)^2)^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)^3/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 )^{3}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccot(a*x)^3/x^2,x, algorithm="fricas")
[Out]
integral(arccot(a*x)^3/x^2, 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^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(acot(a*x)**3/x**2,x)
[Out]
Integral(acot(a*x)**3/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 )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(arccot(a*x)^3/x^2,x, algorithm="giac")
[Out]
integrate(arccot(a*x)^3/x^2, x)