Integrand size = 10, antiderivative size = 66 \[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=-i a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )-i a \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \] Output:
-I*a*arccot(a*x)^2-arccot(a*x)^2/x-2*a*arccot(a*x)*ln(2-2/(1-I*a*x))-I*a*p olylog(2,-1+2/(1-I*a*x))
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=a \left (i \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{a x}-2 \cot ^{-1}(a x) \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )\right ) \] Input:
Integrate[ArcCot[a*x]^2/x^2,x]
Output:
a*(I*ArcCot[a*x]^2 - ArcCot[a*x]^2/(a*x) - 2*ArcCot[a*x]*Log[1 + E^((2*I)* ArcCot[a*x])] + I*PolyLog[2, -E^((2*I)*ArcCot[a*x])])
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5362, 5460, 5404, 2897}
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.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -2 a \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{x}\) |
| \(\Big \downarrow \) 5460 |
| \(\displaystyle -\frac {\cot ^{-1}(a x)^2}{x}-2 a \left (i \int \frac {\cot ^{-1}(a x)}{x (a x+i)}dx+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\) |
| \(\Big \downarrow \) 5404 |
| \(\displaystyle -\frac {\cot ^{-1}(a x)^2}{x}-2 a \left (i \left (-i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {\cot ^{-1}(a x)^2}{x}-2 a \left (i \left (\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\) |
Input:
Int[ArcCot[a*x]^2/x^2,x]
Output:
-(ArcCot[a*x]^2/x) - 2*a*((I/2)*ArcCot[a*x]^2 + I*((-I)*ArcCot[a*x]*Log[2 - 2/(1 - I*a*x)] + PolyLog[2, -1 + 2/(1 - I*a*x)]/2))
Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
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_.)/((x_)*((d_) + (e_.)*(x_))), x_ Symbol] :> Simp[(a + b*ArcCot[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] + Si mp[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]
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] + Simp[ 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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 221 vs. \(2 (62 ) = 124\).
Time = 0.52 (sec) , antiderivative size = 222, normalized size of antiderivative = 3.36
| method | result | size |
| parts | \(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{x}-2 a \left (\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{4}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4}\right )\) | \(222\) |
| derivativedivides | \(a \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{a x}-2 \,\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )+\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \operatorname {dilog}\left (i a x +1\right )-i \operatorname {dilog}\left (-i a x +1\right )-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}\right )\) | \(223\) |
| default | \(a \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{a x}-2 \,\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )+\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \operatorname {dilog}\left (i a x +1\right )-i \operatorname {dilog}\left (-i a x +1\right )-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}\right )\) | \(223\) |
Input:
int(arccot(a*x)^2/x^2,x,method=_RETURNVERBOSE)
Output:
-arccot(a*x)^2/x-2*a*(arccot(a*x)*ln(a*x)-1/2*arccot(a*x)*ln(a^2*x^2+1)-1/ 2*I*ln(a*x)*ln(1+I*a*x)+1/2*I*ln(a*x)*ln(1-I*a*x)-1/2*I*dilog(1+I*a*x)+1/2 *I*dilog(1-I*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))))
\[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}} \,d x } \] Input:
integrate(arccot(a*x)^2/x^2,x, algorithm="fricas")
Output:
integral(arccot(a*x)^2/x^2, x)
\[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=\int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{2}}\, dx \] Input:
integrate(acot(a*x)**2/x**2,x)
Output:
Integral(acot(a*x)**2/x**2, x)
\[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}} \,d x } \] Input:
integrate(arccot(a*x)^2/x^2,x, algorithm="maxima")
Output:
1/16*(4*(3*a*arctan(a*x)*arctan(1/(a*x))^2 + (arctan(a*x)^3/a + 3*arctan(a *x)^2*arctan(1/(a*x))/a)*a^2 + 4*a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)^2 /(a^2*x^4 + x^2), x) - 16*a^2*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^2*x^4 + x^2), x) - 32*a*integrate(1/16*x*arctan(1/(a*x))/(a^2*x^4 + x^2), x) + 48*integrate(1/16*arctan(1/(a*x))^2/(a^2*x^4 + x^2), x) + 4*integrate(1/16 *log(a^2*x^2 + 1)^2/(a^2*x^4 + x^2), x))*x - 4*arctan2(1, a*x)^2 + log(a^2 *x^2 + 1)^2)/x
\[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{2}} \,d x } \] Input:
integrate(arccot(a*x)^2/x^2,x, algorithm="giac")
Output:
integrate(arccot(a*x)^2/x^2, x)
Timed out. \[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^2} \,d x \] Input:
int(acot(a*x)^2/x^2,x)
Output:
int(acot(a*x)^2/x^2, x)
\[ \int \frac {\cot ^{-1}(a x)^2}{x^2} \, dx=\frac {-\mathit {acot} \left (a x \right )^{2}-2 \left (\int \frac {\mathit {acot} \left (a x \right )}{a^{2} x^{3}+x}d x \right ) a x}{x} \] Input:
int(acot(a*x)^2/x^2,x)
Output:
( - acot(a*x)**2 - 2*int(acot(a*x)/(a**2*x**3 + x),x)*a*x)/x