Integrand size = 10, antiderivative size = 113 \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=-\frac {a^2}{3 x}+\frac {a \cot ^{-1}(a x)}{3 x^2}+\frac {1}{3} i a^3 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} a^3 \arctan (a x)+\frac {2}{3} a^3 \cot ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )+\frac {1}{3} i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right ) \]
-1/3*a^2/x+1/3*a*arccot(a*x)/x^2+1/3*I*a^3*arccot(a*x)^2-1/3*arccot(a*x)^2 /x^3-1/3*a^3*arctan(a*x)+2/3*a^3*arccot(a*x)*ln(2-2/(1-I*a*x))+1/3*I*a^3*p olylog(2,-1+2/(1-I*a*x))
Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\frac {-a^2 x^2+\left (-1-i a^3 x^3\right ) \cot ^{-1}(a x)^2+a x \cot ^{-1}(a x) \left (1+a^2 x^2+2 a^2 x^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right )-i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )}{3 x^3} \]
(-(a^2*x^2) + (-1 - I*a^3*x^3)*ArcCot[a*x]^2 + a*x*ArcCot[a*x]*(1 + a^2*x^ 2 + 2*a^2*x^2*Log[1 + E^((2*I)*ArcCot[a*x])]) - I*a^3*x^3*PolyLog[2, -E^(( 2*I)*ArcCot[a*x])])/(3*x^3)
Time = 0.64 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5362, 5454, 5362, 264, 216, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {2}{3} a \int \frac {\cot ^{-1}(a x)}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{3 x^3}\) |
| \(\Big \downarrow \) 5454 |
| \(\displaystyle -\frac {2}{3} a \left (\int \frac {\cot ^{-1}(a x)}{x^3}dx-a^2 \int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {2}{3} a \left (-\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle -\frac {2}{3} a \left (-\frac {1}{2} a \left (a^2 \left (-\int \frac {1}{a^2 x^2+1}dx\right )-\frac {1}{x}\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {2}{3} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x \left (a^2 x^2+1\right )}dx\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )-\frac {\cot ^{-1}(a x)^2}{3 x^3}\) |
| \(\Big \downarrow \) 5460 |
| \(\displaystyle -\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a \left (-\left (a^2 \left (i \int \frac {\cot ^{-1}(a x)}{x (a x+i)}dx+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )\) |
| \(\Big \downarrow \) 5404 |
| \(\displaystyle -\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a \left (-\left (a^2 \left (i \left (-i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle -\frac {\cot ^{-1}(a x)^2}{3 x^3}-\frac {2}{3} a \left (-\left (a^2 \left (i \left (\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )-i \log \left (2-\frac {2}{1-i a x}\right ) \cot ^{-1}(a x)\right )+\frac {1}{2} i \cot ^{-1}(a x)^2\right )\right )-\frac {1}{2} a \left (-a \arctan (a x)-\frac {1}{x}\right )-\frac {\cot ^{-1}(a x)}{2 x^2}\right )\) |
-1/3*ArcCot[a*x]^2/x^3 - (2*a*(-1/2*ArcCot[a*x]/x^2 - (a*(-x^(-1) - a*ArcT an[a*x]))/2 - a^2*((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))))/3
3.1.21.3.1 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*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcCot[c*x])^p, x], x] - Simp[e/(d*f^2) 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] && LtQ[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*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. 250 vs. \(2 (95 ) = 190\).
Time = 0.72 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.22
| method | result | size |
| parts | \(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{3 x^{3}}-\frac {2 a^{3} \left (\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\operatorname {arccot}\left (a x \right )}{2 a^{2} x^{2}}-\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{4}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{4}+\frac {\arctan \left (a x \right )}{2}+\frac {1}{2 a x}+\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{2}-\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{2}+\frac {i \operatorname {dilog}\left (i a x +1\right )}{2}-\frac {i \operatorname {dilog}\left (-i a x +1\right )}{2}\right )}{3}\) | \(251\) |
| derivativedivides | \(a^{3} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\operatorname {arccot}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \,\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}-\frac {\arctan \left (a x \right )}{3}-\frac {1}{3 a x}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{3}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{3}\right )\) | \(252\) |
| default | \(a^{3} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{3 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\operatorname {arccot}\left (a x \right )}{3 a^{2} x^{2}}+\frac {2 \,\operatorname {arccot}\left (a x \right ) \ln \left (a x \right )}{3}+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{6}-\frac {\arctan \left (a x \right )}{3}-\frac {1}{3 a x}-\frac {i \ln \left (a x \right ) \ln \left (i a x +1\right )}{3}+\frac {i \ln \left (a x \right ) \ln \left (-i a x +1\right )}{3}-\frac {i \operatorname {dilog}\left (i a x +1\right )}{3}+\frac {i \operatorname {dilog}\left (-i a x +1\right )}{3}\right )\) | \(252\) |
-1/3*arccot(a*x)^2/x^3-2/3*a^3*(1/2*arccot(a*x)*ln(a^2*x^2+1)-1/2*arccot(a *x)/a^2/x^2-arccot(a*x)*ln(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)))+1/2*arctan(a*x)+1/2/a/x+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))
\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}} \,d x } \]
\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}} \,d x } \]
1/48*(48*x^3*integrate(1/48*(36*a^2*x^2*arctan2(1, a*x)^2 - 4*a^2*x^2*log( a^2*x^2 + 1) - 8*a*x*arctan2(1, a*x) + 3*(a^2*x^2 + 1)*log(a^2*x^2 + 1)^2 + 36*arctan2(1, a*x)^2)/(a^2*x^6 + x^4), x) - 4*arctan2(1, a*x)^2 + log(a^ 2*x^2 + 1)^2)/x^3
\[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int { \frac {\operatorname {arccot}\left (a x\right )^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {\cot ^{-1}(a x)^2}{x^4} \, dx=\int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x^4} \,d x \]