\(\int \frac {\cot ^{-1}(a x)^2}{x} \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 116 \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4943, 5109, 5005, 5113, 6745} \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right )-i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)+i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]

[In]

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

[Out]

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

Rule 4943

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)/(x_), x_Symbol] :> Simp[2*(a + b*ArcCot[c*x])^p*ArcCoth[1 - 2/(1 +
I*c*x)], x] + Dist[2*b*c*p, Int[(a + b*ArcCot[c*x])^(p - 1)*(ArcCoth[1 - 2/(1 + I*c*x)]/(1 + c^2*x^2)), x], x
] /; FreeQ[{a, b, c}, x] && IGtQ[p, 1]

Rule 5005

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 5109

Int[(ArcCoth[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[1/2, Int[L
og[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] - Dist[1/2, Int[Log[SimplifyInte
grand[1 - 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && E
qQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c*x)))^2, 0]

Rule 5113

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*(a + b*ArcCo
t[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 6745

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=-\frac {2}{3} i \cot ^{-1}(a x)^3-\cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right ) \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 5.67 (sec) , antiderivative size = 891, normalized size of antiderivative = 7.68

method	result	size
derivativedivides	\(\ln \left (a x \right ) \operatorname {arccot}\left (a x \right )^{2}+\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )-\operatorname {csgn}\left (\frac {i}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}-\operatorname {csgn}\left (i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}+{\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{3}-\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) {\operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}+\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) \operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )-{\operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{3}+{\operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}-1\right ) \operatorname {arccot}\left (a x \right )^{2}}{2}+\operatorname {arccot}\left (a x \right )^{2} \ln \left (\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1\right )-\operatorname {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\)	\(891\)
default	\(\ln \left (a x \right ) \operatorname {arccot}\left (a x \right )^{2}+\frac {i \pi \left (\operatorname {csgn}\left (\frac {i}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) \operatorname {csgn}\left (i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )\right ) \operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )-\operatorname {csgn}\left (\frac {i}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) {\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}-\operatorname {csgn}\left (i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )\right ) {\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}+{\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{3}-\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) {\operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}+\operatorname {csgn}\left (\frac {i \left (1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right ) \operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )-{\operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{3}+{\operatorname {csgn}\left (\frac {1+\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}}{\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1}\right )}^{2}-1\right ) \operatorname {arccot}\left (a x \right )^{2}}{2}+\operatorname {arccot}\left (a x \right )^{2} \ln \left (\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}-1\right )-\operatorname {arccot}\left (a x \right )^{2} \ln \left (1-\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {arccot}\left (a x \right )^{2} \ln \left (1+\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, -\frac {a x +i}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {arccot}\left (a x \right ) \operatorname {polylog}\left (2, -\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )+\frac {\operatorname {polylog}\left (3, -\frac {\left (a x +i\right )^{2}}{a^{2} x^{2}+1}\right )}{2}\)	\(891\)
parts	\(\text {Expression too large to display}\)	\(1317\)

[In]

int(arccot(a*x)^2/x,x,method=_RETURNVERBOSE)

[Out]

ln(a*x)*arccot(a*x)^2+1/2*I*Pi*(csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I*(1+(I+a*x)^2/(a^2*x^2+1)))*csgn(I/((I
+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))-csgn(I/((I+a*x)^2/(a^2*x^2+1)-1))*csgn(I/((I+a*x)^2/(a^2*x^2
+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^2-csgn(I*(1+(I+a*x)^2/(a^2*x^2+1)))*csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a
*x)^2/(a^2*x^2+1)))^2+csgn(I/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^3-csgn(I/((I+a*x)^2/(a^2*x^2
+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))*csgn(1/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^2+csgn(I/((I+a*x
)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))*csgn(1/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))-csgn
(1/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x^2+1)))^3+csgn(1/((I+a*x)^2/(a^2*x^2+1)-1)*(1+(I+a*x)^2/(a^2*x
^2+1)))^2-1)*arccot(a*x)^2+arccot(a*x)^2*ln((I+a*x)^2/(a^2*x^2+1)-1)-arccot(a*x)^2*ln(1-(I+a*x)/(a^2*x^2+1)^(1
/2))+2*I*arccot(a*x)*polylog(2,(I+a*x)/(a^2*x^2+1)^(1/2))-2*polylog(3,(I+a*x)/(a^2*x^2+1)^(1/2))-arccot(a*x)^2
*ln(1+(I+a*x)/(a^2*x^2+1)^(1/2))+2*I*arccot(a*x)*polylog(2,-(I+a*x)/(a^2*x^2+1)^(1/2))-2*polylog(3,-(I+a*x)/(a
^2*x^2+1)^(1/2))-I*arccot(a*x)*polylog(2,-(I+a*x)^2/(a^2*x^2+1))+1/2*polylog(3,-(I+a*x)^2/(a^2*x^2+1))

Fricas [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integral(arccot(a*x)^2/x, x)

Sympy [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(acot(a*x)**2/x, x)

Maxima [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integrate(arccot(a*x)^2/x, x)

Giac [F]

\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x} \,d x } \]

[In]

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

[Out]

integrate(arccot(a*x)^2/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x} \,d x \]

[In]

int(acot(a*x)^2/x,x)

[Out]

int(acot(a*x)^2/x, x)