Integrand size = 10, antiderivative size = 111 \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\frac {x}{3 a^2}+\frac {x^2 \cot ^{-1}(a x)}{3 a}-\frac {i \cot ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} x^3 \cot ^{-1}(a x)^2-\frac {\arctan (a x)}{3 a^3}+\frac {2 \cot ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a^3} \]
1/3*x/a^2+1/3*x^2*arccot(a*x)/a-1/3*I*arccot(a*x)^2/a^3+1/3*x^3*arccot(a*x )^2-1/3*arctan(a*x)/a^3+2/3*arccot(a*x)*ln(2/(1+I*a*x))/a^3-1/3*I*polylog( 2,1-2/(1+I*a*x))/a^3
Time = 0.21 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.68 \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\frac {a x+\left (-i+a^3 x^3\right ) \cot ^{-1}(a x)^2+\cot ^{-1}(a x) \left (1+a^2 x^2+2 \log \left (1-e^{2 i \cot ^{-1}(a x)}\right )\right )-i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(a x)}\right )}{3 a^3} \]
(a*x + (-I + a^3*x^3)*ArcCot[a*x]^2 + ArcCot[a*x]*(1 + a^2*x^2 + 2*Log[1 - E^((2*I)*ArcCot[a*x])]) - I*PolyLog[2, E^((2*I)*ArcCot[a*x])])/(3*a^3)
Time = 0.68 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5362, 5452, 5362, 262, 216, 5456, 5380, 2849, 2752}
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.
| \(\displaystyle \int x^2 \cot ^{-1}(a x)^2 \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {2}{3} a \int \frac {x^3 \cot ^{-1}(a x)}{a^2 x^2+1}dx+\frac {1}{3} x^3 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5452 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\int x \cot ^{-1}(a x)dx}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {1}{2} a \int \frac {x^2}{a^2 x^2+1}dx+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\int \frac {1}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\int \frac {x \cot ^{-1}(a x)}{a^2 x^2+1}dx}{a^2}\right )+\frac {1}{3} x^3 \cot ^{-1}(a x)^2\) |
| \(\Big \downarrow \) 5456 |
| \(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\cot ^{-1}(a x)}{i-a x}dx}{a}}{a^2}\right )\) |
| \(\Big \downarrow \) 5380 |
| \(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\int \frac {\log \left (\frac {2}{i a x+1}\right )}{a^2 x^2+1}dx+\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}}{a}}{a^2}\right )\) |
| \(\Big \downarrow \) 2849 |
| \(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \int \frac {\log \left (\frac {2}{i a x+1}\right )}{1-\frac {2}{i a x+1}}d\frac {1}{i a x+1}}{a}}{a}}{a^2}\right )\) |
| \(\Big \downarrow \) 2752 |
| \(\displaystyle \frac {1}{3} x^3 \cot ^{-1}(a x)^2+\frac {2}{3} a \left (\frac {\frac {1}{2} a \left (\frac {x}{a^2}-\frac {\arctan (a x)}{a^3}\right )+\frac {1}{2} x^2 \cot ^{-1}(a x)}{a^2}-\frac {\frac {i \cot ^{-1}(a x)^2}{2 a^2}-\frac {\frac {\log \left (\frac {2}{1+i a x}\right ) \cot ^{-1}(a x)}{a}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{2 a}}{a}}{a^2}\right )\) |
(x^3*ArcCot[a*x]^2)/3 + (2*a*(((x^2*ArcCot[a*x])/2 + (a*(x/a^2 - ArcTan[a* x]/a^3))/2)/a^2 - (((I/2)*ArcCot[a*x]^2)/a^2 - ((ArcCot[a*x]*Log[2/(1 + I* a*x)])/a - ((I/2)*PolyLog[2, 1 - 2/(1 + I*a*x)])/a)/a)/a^2))/3
3.1.15.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp [-e/g Subst[Int[Log[2*d*x]/(1 - 2*d*x), x], x, 1/(d + e*x)], x] /; FreeQ[ {c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]
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_)), x_Symbol] :> Simp[(-(a + b*ArcCot[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] - Simp[b*c*( p/e) Int[(a + b*ArcCot[c*x])^(p - 1)*(Log[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 ]
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]
Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[I*((a + b*ArcCot[c*x])^(p + 1)/(b*e*(p + 1))), x] - Simp[ 1/(c*d) Int[(a + b*ArcCot[c*x])^p/(I - c*x), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Time = 0.46 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.67
| method | result | size |
| parts | \(\frac {x^{3} \operatorname {arccot}\left (a x \right )^{2}}{3}+\frac {\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}}{a^{3}}\) | \(185\) |
| derivativedivides | \(\frac {\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )^{2}}{3}+\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}}{a^{3}}\) | \(186\) |
| default | \(\frac {\frac {a^{3} x^{3} \operatorname {arccot}\left (a x \right )^{2}}{3}+\frac {\operatorname {arccot}\left (a x \right ) a^{2} x^{2}}{3}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {a x}{3}-\frac {\arctan \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}}{a^{3}}\) | \(186\) |
| risch | \(\frac {\pi ^{2} x^{3}}{12}+\frac {\pi \,x^{2}}{6 a}+\frac {\ln \left (i a x +1\right ) \ln \left (-i a x +1\right ) x^{3}}{6}-\frac {i \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a^{3}}-\frac {2 i \ln \left (a^{2} x^{2}+1\right )}{9 a^{3}}+\frac {5 i \ln \left (-i a x +1\right )}{36 a^{3}}+\frac {11 \pi }{18 a^{3}}-\frac {\ln \left (-i a x +1\right )^{2} x^{3}}{12}-\frac {\ln \left (i a x +1\right )^{2} x^{3}}{12}-\frac {17 i}{54 a^{3}}+\frac {x}{3 a^{2}}-\frac {i \ln \left (-i a x +1\right ) x^{2}}{6 a}+\frac {i \ln \left (i a x +1\right ) x^{2}}{6 a}-\frac {\pi \ln \left (a^{2} x^{2}+1\right )}{6 a^{3}}+\frac {11 i \ln \left (i a x +1\right )}{36 a^{3}}-\frac {i \pi \ln \left (-i a x +1\right ) x^{3}}{6}+\frac {i \ln \left (-i a x +1\right )^{2}}{12 a^{3}}-\frac {i \pi ^{2}}{12 a^{3}}-\frac {i \ln \left (i a x +1\right )^{2}}{12 a^{3}}-\frac {i \ln \left (i a x +1\right ) \ln \left (-i a x +1\right )}{6 a^{3}}-\frac {\arctan \left (a x \right )}{6 a^{3}}+\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{3 a^{3}}-\frac {i \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{3 a^{3}}+\frac {i \pi \ln \left (i a x +1\right ) x^{3}}{6}\) | \(342\) |
1/3*x^3*arccot(a*x)^2+2/3/a^3*(1/2*arccot(a*x)*a^2*x^2-1/2*arccot(a*x)*ln( a^2*x^2+1)+1/2*a*x-1/2*arctan(a*x)+1/4*I*(ln(a*x-I)*ln(a^2*x^2+1)-1/2*ln(a *x-I)^2-dilog(-1/2*I*(I+a*x))-ln(a*x-I)*ln(-1/2*I*(I+a*x)))-1/4*I*(ln(I+a* x)*ln(a^2*x^2+1)-1/2*ln(I+a*x)^2-dilog(1/2*I*(a*x-I))-ln(I+a*x)*ln(1/2*I*( a*x-I))))
\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int x^{2} \operatorname {acot}^{2}{\left (a x \right )}\, dx \]
\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
1/12*x^3*arctan2(1, a*x)^2 - 1/48*x^3*log(a^2*x^2 + 1)^2 + integrate(1/48* (36*a^2*x^4*arctan2(1, a*x)^2 + 4*a^2*x^4*log(a^2*x^2 + 1) + 8*a*x^3*arcta n2(1, a*x) + 36*x^2*arctan2(1, a*x)^2 + 3*(a^2*x^4 + x^2)*log(a^2*x^2 + 1) ^2)/(a^2*x^2 + 1), x)
\[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int { x^{2} \operatorname {arccot}\left (a x\right )^{2} \,d x } \]
Timed out. \[ \int x^2 \cot ^{-1}(a x)^2 \, dx=\int x^2\,{\mathrm {acot}\left (a\,x\right )}^2 \,d x \]