Integrand size = 10, antiderivative size = 104 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=-\frac {4 x^2}{45 a^4}+\frac {x^4}{60 a^2}+\frac {x \cot ^{-1}(a x)}{3 a^5}-\frac {x^3 \cot ^{-1}(a x)}{9 a^3}+\frac {x^5 \cot ^{-1}(a x)}{15 a}+\frac {\cot ^{-1}(a x)^2}{6 a^6}+\frac {1}{6} x^6 \cot ^{-1}(a x)^2+\frac {23 \log \left (1+a^2 x^2\right )}{90 a^6} \]
-4/45*x^2/a^4+1/60*x^4/a^2+1/3*x*arccot(a*x)/a^5-1/9*x^3*arccot(a*x)/a^3+1 /15*x^5*arccot(a*x)/a+1/6*arccot(a*x)^2/a^6+1/6*x^6*arccot(a*x)^2+23/90*ln (a^2*x^2+1)/a^6
Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.76 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {-16 a^2 x^2+3 a^4 x^4+4 a x \left (15-5 a^2 x^2+3 a^4 x^4\right ) \cot ^{-1}(a x)+30 \left (1+a^6 x^6\right ) \cot ^{-1}(a x)^2+46 \log \left (1+a^2 x^2\right )}{180 a^6} \]
(-16*a^2*x^2 + 3*a^4*x^4 + 4*a*x*(15 - 5*a^2*x^2 + 3*a^4*x^4)*ArcCot[a*x] + 30*(1 + a^6*x^6)*ArcCot[a*x]^2 + 46*Log[1 + a^2*x^2])/(180*a^6)
Time = 1.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.64, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {5362, 5452, 5362, 243, 49, 2009, 5452, 5362, 243, 49, 2009, 5452, 5346, 240, 5420}
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.
| \(\displaystyle \int x^5 \cot ^{-1}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{3} a \int \frac {x^6 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\int x^4 \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{5} a \int \frac {x^5}{a^2 x^2+1}dx+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \int \frac {x^4}{a^2 x^2+1}dx^2+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \int \left (\frac {x^2}{a^2}+\frac {1}{a^4 \left (a^2 x^2+1\right )}-\frac {1}{a^4}\right )dx^2+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^4 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\int x^2 \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{3} a \int \frac {x^3}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \int \frac {x^2}{a^2 x^2+1}dx^2+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (a^2 x^2+1\right )}\right )dx^2+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x^2 \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\int \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5346 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {a \int \frac {x}{a^2 x^2+1}dx+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5420 |
| \(\displaystyle \frac {1}{3} a \left (\frac {\frac {1}{10} a \left (-\frac {x^2}{a^4}+\frac {x^4}{2 a^2}+\frac {\log \left (a^2 x^2+1\right )}{a^6}\right )+\frac {1}{5} x^5 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\frac {1}{6} a \left (\frac {x^2}{a^2}-\frac {\log \left (a^2 x^2+1\right )}{a^4}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)}{a^2}-\frac {\frac {\cot ^{-1}(a x)^2}{2 a^3}+\frac {\frac {\log \left (a^2 x^2+1\right )}{2 a}+x \cot ^{-1}(a x)}{a^2}}{a^2}}{a^2}\right )+\frac {1}{6} x^6 \cot ^{-1}(a x)^2\) |
(x^6*ArcCot[a*x]^2)/6 + (a*(((x^5*ArcCot[a*x])/5 + (a*(-(x^2/a^4) + x^4/(2 *a^2) + Log[1 + a^2*x^2]/a^6))/10)/a^2 - (((x^3*ArcCot[a*x])/3 + (a*(x^2/a ^2 - Log[1 + a^2*x^2]/a^4))/6)/a^2 - (ArcCot[a*x]^2/(2*a^3) + (x*ArcCot[a* x] + Log[1 + a^2*x^2]/(2*a))/a^2)/a^2)/a^2))/3
3.1.12.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCot[c*x^n])^p, x] + Simp[b*c*n*p Int[x^n*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0] && (EqQ[n, 1] || EqQ[p, 1])
Int[((a_.) + ArcCot[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCot[c*x^n])^p/(m + 1)), x] + Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcCot[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> 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]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[f^2/e Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x] )^p, x], x] - Simp[d*(f^2/e) 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] && GtQ[m, 1]
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
| method | result | size |
| parallelrisch | \(\frac {30 a^{6} x^{6} \operatorname {arccot}\left (a x \right )^{2}+12 a^{5} x^{5} \operatorname {arccot}\left (a x \right )+3 a^{4} x^{4}-20 a^{3} x^{3} \operatorname {arccot}\left (a x \right )+16-16 a^{2} x^{2}+60 \,\operatorname {arccot}\left (a x \right ) a x +30 \operatorname {arccot}\left (a x \right )^{2}+46 \ln \left (a^{2} x^{2}+1\right )}{180 a^{6}}\) | \(90\) |
| parts | \(\frac {x^{6} \operatorname {arccot}\left (a x \right )^{2}}{6}+\frac {\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{5}-\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{3}+\operatorname {arccot}\left (a x \right ) a x -\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\frac {a^{4} x^{4}}{20}-\frac {4 a^{2} x^{2}}{15}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{30}-\frac {\arctan \left (a x \right )^{2}}{2}}{3 a^{6}}\) | \(96\) |
| derivativedivides | \(\frac {\frac {a^{6} x^{6} \operatorname {arccot}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{15}-\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{9}+\frac {\operatorname {arccot}\left (a x \right ) a x}{3}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{3}+\frac {a^{4} x^{4}}{60}-\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90}-\frac {\arctan \left (a x \right )^{2}}{6}}{a^{6}}\) | \(98\) |
| default | \(\frac {\frac {a^{6} x^{6} \operatorname {arccot}\left (a x \right )^{2}}{6}+\frac {a^{5} x^{5} \operatorname {arccot}\left (a x \right )}{15}-\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )}{9}+\frac {\operatorname {arccot}\left (a x \right ) a x}{3}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{3}+\frac {a^{4} x^{4}}{60}-\frac {4 a^{2} x^{2}}{45}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90}-\frac {\arctan \left (a x \right )^{2}}{6}}{a^{6}}\) | \(98\) |
| risch | \(-\frac {\left (a^{6} x^{6}+1\right ) \ln \left (i a x +1\right )^{2}}{24 a^{6}}+\frac {\left (15 i \pi \,a^{6} x^{6}+15 x^{6} \ln \left (-i a x +1\right ) a^{6}+6 i a^{5} x^{5}-10 i a^{3} x^{3}+30 i a x +15 \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{180 a^{6}}+\frac {i x^{3} \ln \left (-i a x +1\right )}{18 a^{3}}-\frac {x^{6} \ln \left (-i a x +1\right )^{2}}{24}+\frac {x^{6} \pi ^{2}}{24}-\frac {i x \ln \left (-i a x +1\right )}{6 a^{5}}+\frac {\pi \,x^{5}}{30 a}-\frac {i \pi \,x^{6} \ln \left (-i a x +1\right )}{12}+\frac {x^{4}}{60 a^{2}}-\frac {\pi \,x^{3}}{18 a^{3}}-\frac {i x^{5} \ln \left (-i a x +1\right )}{30 a}-\frac {4 x^{2}}{45 a^{4}}+\frac {\pi x}{6 a^{5}}-\frac {\ln \left (-i a x +1\right )^{2}}{24 a^{6}}-\frac {\pi \arctan \left (a x \right )}{6 a^{6}}+\frac {23 \ln \left (a^{2} x^{2}+1\right )}{90 a^{6}}\) | \(267\) |
1/180*(30*a^6*x^6*arccot(a*x)^2+12*a^5*x^5*arccot(a*x)+3*a^4*x^4-20*a^3*x^ 3*arccot(a*x)+16-16*a^2*x^2+60*arccot(a*x)*a*x+30*arccot(a*x)^2+46*ln(a^2* x^2+1))/a^6
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.75 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} + 30 \, {\left (a^{6} x^{6} + 1\right )} \operatorname {arccot}\left (a x\right )^{2} + 4 \, {\left (3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 15 \, a x\right )} \operatorname {arccot}\left (a x\right ) + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \]
1/180*(3*a^4*x^4 - 16*a^2*x^2 + 30*(a^6*x^6 + 1)*arccot(a*x)^2 + 4*(3*a^5* x^5 - 5*a^3*x^3 + 15*a*x)*arccot(a*x) + 46*log(a^2*x^2 + 1))/a^6
Time = 0.36 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\begin {cases} \frac {x^{6} \operatorname {acot}^{2}{\left (a x \right )}}{6} + \frac {x^{5} \operatorname {acot}{\left (a x \right )}}{15 a} + \frac {x^{4}}{60 a^{2}} - \frac {x^{3} \operatorname {acot}{\left (a x \right )}}{9 a^{3}} - \frac {4 x^{2}}{45 a^{4}} + \frac {x \operatorname {acot}{\left (a x \right )}}{3 a^{5}} + \frac {23 \log {\left (a^{2} x^{2} + 1 \right )}}{90 a^{6}} + \frac {\operatorname {acot}^{2}{\left (a x \right )}}{6 a^{6}} & \text {for}\: a \neq 0 \\\frac {\pi ^{2} x^{6}}{24} & \text {otherwise} \end {cases} \]
Piecewise((x**6*acot(a*x)**2/6 + x**5*acot(a*x)/(15*a) + x**4/(60*a**2) - x**3*acot(a*x)/(9*a**3) - 4*x**2/(45*a**4) + x*acot(a*x)/(3*a**5) + 23*log (a**2*x**2 + 1)/(90*a**6) + acot(a*x)**2/(6*a**6), Ne(a, 0)), (pi**2*x**6/ 24, True))
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.91 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {1}{6} \, x^{6} \operatorname {arccot}\left (a x\right )^{2} + \frac {1}{45} \, a {\left (\frac {3 \, a^{4} x^{5} - 5 \, a^{2} x^{3} + 15 \, x}{a^{6}} - \frac {15 \, \arctan \left (a x\right )}{a^{7}}\right )} \operatorname {arccot}\left (a x\right ) + \frac {3 \, a^{4} x^{4} - 16 \, a^{2} x^{2} - 30 \, \arctan \left (a x\right )^{2} + 46 \, \log \left (a^{2} x^{2} + 1\right )}{180 \, a^{6}} \]
1/6*x^6*arccot(a*x)^2 + 1/45*a*((3*a^4*x^5 - 5*a^2*x^3 + 15*x)/a^6 - 15*ar ctan(a*x)/a^7)*arccot(a*x) + 1/180*(3*a^4*x^4 - 16*a^2*x^2 - 30*arctan(a*x )^2 + 46*log(a^2*x^2 + 1))/a^6
\[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\int { x^{5} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
Time = 0.95 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.82 \[ \int x^5 \cot ^{-1}(a x)^2 \, dx=\frac {x^6\,{\mathrm {acot}\left (a\,x\right )}^2}{6}+\frac {\frac {23\,\ln \left (a^2\,x^2+1\right )}{90}-\frac {4\,a^2\,x^2}{45}+\frac {a^4\,x^4}{60}+\frac {{\mathrm {acot}\left (a\,x\right )}^2}{6}-\frac {a^3\,x^3\,\mathrm {acot}\left (a\,x\right )}{9}+\frac {a^5\,x^5\,\mathrm {acot}\left (a\,x\right )}{15}+\frac {a\,x\,\mathrm {acot}\left (a\,x\right )}{3}}{a^6} \]