Integrand size = 10, antiderivative size = 116 \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 i}{i+a x}\right )+i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,1-\frac {2 a x}{i+a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 i}{i+a x}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,1-\frac {2 a x}{i+a x}\right ) \]
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))
Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=-\frac {2}{3} i \cot ^{-1}(a x)^3-\cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )-i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(a x)}\right )-i \cot ^{-1}(a x) \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (3,-e^{2 i \cot ^{-1}(a x)}\right ) \]
((-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)*ArcCot[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
Time = 0.65 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5358, 5524, 5528, 7164}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)^2}{x} \, dx\) |
| \(\Big \downarrow \) 5358 |
| \(\displaystyle 4 a \int \frac {\cot ^{-1}(a x) \coth ^{-1}\left (1-\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\) |
| \(\Big \downarrow \) 5524 |
| \(\displaystyle 4 a \left (\frac {1}{2} \int \frac {\cot ^{-1}(a x) \log \left (\frac {2 a x}{a x+i}\right )}{a^2 x^2+1}dx-\frac {1}{2} \int \frac {\cot ^{-1}(a x) \log \left (\frac {2 i}{a x+i}\right )}{a^2 x^2+1}dx\right )+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\) |
| \(\Big \downarrow \) 5528 |
| \(\displaystyle 4 a \left (\frac {1}{2} \left (-\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right )}{a^2 x^2+1}dx-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )+\frac {1}{2} \left (\frac {1}{2} i \int \frac {\operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right )}{a^2 x^2+1}dx+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )\right )+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\) |
| \(\Big \downarrow \) 7164 |
| \(\displaystyle 4 a \left (\frac {1}{2} \left (-\frac {\operatorname {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right )}{4 a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )+\frac {1}{2} \left (\frac {\operatorname {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right )}{4 a}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)}{2 a}\right )\right )+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )\) |
2*ArcCot[a*x]^2*ArcCoth[1 - 2/(1 + I*a*x)] + 4*a*((((-1/2*I)*ArcCot[a*x]*P olyLog[2, 1 - (2*I)/(I + a*x)])/a - PolyLog[3, 1 - (2*I)/(I + a*x)]/(4*a)) /2 + (((I/2)*ArcCot[a*x]*PolyLog[2, 1 - (2*a*x)/(I + a*x)])/a + PolyLog[3, 1 - (2*a*x)/(I + a*x)]/(4*a))/2)
3.1.18.3.1 Defintions of rubi rules used
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] + Simp[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]
Int[(ArcCoth[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x _)^2), x_Symbol] :> Simp[1/2 Int[Log[SimplifyIntegrand[1 + 1/u, x]]*((a + b*ArcCot[c*x])^p/(d + e*x^2)), x], x] - Simp[1/2 Int[Log[SimplifyIntegra nd[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] && EqQ[e, c^2*d] && EqQ[u^2 - (1 - 2*(I/(I - c* x)))^2, 0]
Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2 ), x_Symbol] :> Simp[I*(a + b*ArcCot[c*x])^p*(PolyLog[2, 1 - u]/(2*c*d)), x ] + Simp[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]
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]
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 | \(\text {Expression too large to display}\) | \(1317\) |
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))
\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x} \,d x } \]
\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x}\, dx \]
\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x} \,d x } \]
\[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a x)^2}{x} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x} \,d x \]