Integrand size = 10, antiderivative size = 89 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}+\frac {a \cot ^{-1}(a x)}{6 x^3}-\frac {a^3 \cot ^{-1}(a x)}{2 x}+\frac {1}{4} a^4 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)+\frac {1}{3} a^4 \log \left (1+a^2 x^2\right ) \]
-1/12*a^2/x^2+1/6*a*arccot(a*x)/x^3-1/2*a^3*arccot(a*x)/x+1/4*a^4*arccot(a *x)^2-1/4*arccot(a*x)^2/x^4-2/3*a^4*ln(x)+1/3*a^4*ln(a^2*x^2+1)
Time = 0.01 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {a^2}{12 x^2}-\frac {a \left (-1+3 a^2 x^2\right ) \cot ^{-1}(a x)}{6 x^3}+\frac {\left (-1+a^4 x^4\right ) \cot ^{-1}(a x)^2}{4 x^4}-\frac {2}{3} a^4 \log (x)+\frac {1}{3} a^4 \log \left (1+a^2 x^2\right ) \]
-1/12*a^2/x^2 - (a*(-1 + 3*a^2*x^2)*ArcCot[a*x])/(6*x^3) + ((-1 + a^4*x^4) *ArcCot[a*x]^2)/(4*x^4) - (2*a^4*Log[x])/3 + (a^4*Log[1 + a^2*x^2])/3
Time = 0.76 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.27, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {5362, 5454, 5362, 243, 54, 2009, 5454, 5362, 243, 47, 14, 16, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {1}{2} a \int \frac {\cot ^{-1}(a x)}{x^4 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 5454 |
| \(\displaystyle -\frac {1}{2} a \left (\int \frac {\cot ^{-1}(a x)}{x^4}dx-a^2 \int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {1}{2} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {1}{3} a \int \frac {1}{x^3 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{2} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {1}{6} a \int \frac {1}{x^4 \left (a^2 x^2+1\right )}dx^2-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {1}{2} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {1}{6} a \int \left (\frac {a^4}{a^2 x^2+1}-\frac {a^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {1}{2} a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 5454 |
| \(\displaystyle -\frac {1}{2} a \left (-\left (a^2 \left (\int \frac {\cot ^{-1}(a x)}{x^2}dx-a^2 \int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 5362 |
| \(\displaystyle -\frac {1}{2} a \left (-\left (a^2 \left (-a \int \frac {1}{x \left (a^2 x^2+1\right )}dx+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{2} a \left (-\left (a^2 \left (-\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle -\frac {1}{2} a \left (-\left (a^2 \left (-\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle -\frac {1}{2} a \left (-\left (a^2 \left (-\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {1}{2} a \left (-\left (a^2 \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\cot ^{-1}(a x)}{x}\right )\right )-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
| \(\Big \downarrow \) 5420 |
| \(\displaystyle -\frac {1}{2} a \left (-\frac {1}{6} a \left (a^2 \left (-\log \left (x^2\right )\right )+a^2 \log \left (a^2 x^2+1\right )-\frac {1}{x^2}\right )-\left (a^2 \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )\right )-\frac {\cot ^{-1}(a x)}{3 x^3}\right )-\frac {\cot ^{-1}(a x)^2}{4 x^4}\) |
-1/4*ArcCot[a*x]^2/x^4 - (a*(-1/3*ArcCot[a*x]/x^3 - a^2*(-(ArcCot[a*x]/x) + (a*ArcCot[a*x]^2)/2 - (a*(Log[x^2] - Log[1 + a^2*x^2]))/2) - (a*(-x^(-2) - a^2*Log[x^2] + a^2*Log[1 + a^2*x^2]))/6))/2
3.1.22.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
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_)^(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[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]
Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.96
| method | result | size |
| parts | \(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{4 x^{4}}-\frac {a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )}{3 a^{3} x^{3}}+\frac {\operatorname {arccot}\left (a x \right )}{a x}+\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\frac {1}{6 a^{2} x^{2}}+\frac {4 \ln \left (a x \right )}{3}-\frac {2 \ln \left (a^{2} x^{2}+1\right )}{3}+\frac {\arctan \left (a x \right )^{2}}{2}\right )}{2}\) | \(85\) |
| derivativedivides | \(a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\operatorname {arccot}\left (a x \right )}{6 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right )}{2 a x}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}-\frac {1}{12 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}+\frac {\ln \left (a^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (a x \right )^{2}}{4}\right )\) | \(88\) |
| default | \(a^{4} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{4 a^{4} x^{4}}+\frac {\operatorname {arccot}\left (a x \right )}{6 a^{3} x^{3}}-\frac {\operatorname {arccot}\left (a x \right )}{2 a x}-\frac {\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )}{2}-\frac {1}{12 a^{2} x^{2}}-\frac {2 \ln \left (a x \right )}{3}+\frac {\ln \left (a^{2} x^{2}+1\right )}{3}-\frac {\arctan \left (a x \right )^{2}}{4}\right )\) | \(88\) |
| parallelrisch | \(-\frac {-3 a^{4} x^{4} \operatorname {arccot}\left (a x \right )^{2}+8 a^{4} \ln \left (x \right ) x^{4}-4 a^{4} \ln \left (a^{2} x^{2}+1\right ) x^{4}-a^{4} x^{4}+6 a^{3} x^{3} \operatorname {arccot}\left (a x \right )+a^{2} x^{2}-2 \,\operatorname {arccot}\left (a x \right ) a x +3 \operatorname {arccot}\left (a x \right )^{2}}{12 x^{4}}\) | \(92\) |
| risch | \(-\frac {\left (a^{4} x^{4}-1\right ) \ln \left (i a x +1\right )^{2}}{16 x^{4}}-\frac {i \left (3 i a^{4} \ln \left (-i a x +1\right ) x^{4}+6 a^{3} x^{3}-3 i \ln \left (-i a x +1\right )-2 a x +3 \pi \right ) \ln \left (i a x +1\right )}{24 x^{4}}-\frac {-6 i a^{4} \ln \left (\left (-\pi a +8 i a \right ) x +8+i \pi \right ) \pi \,x^{4}+6 i a^{4} \ln \left (\left (-\pi a -8 i a \right ) x +8-i \pi \right ) \pi \,x^{4}+3 a^{4} x^{4} \ln \left (-i a x +1\right )^{2}-16 a^{4} \ln \left (\left (-\pi a +8 i a \right ) x +8+i \pi \right ) x^{4}-16 a^{4} \ln \left (\left (-\pi a -8 i a \right ) x +8-i \pi \right ) x^{4}+32 a^{4} \ln \left (-x \right ) x^{4}-12 i a^{3} x^{3} \ln \left (-i a x +1\right )+12 \pi \,a^{3} x^{3}+4 i a x \ln \left (-i a x +1\right )+4 a^{2} x^{2}-6 i \pi \ln \left (-i a x +1\right )-4 \pi a x +3 \pi ^{2}-3 \ln \left (-i a x +1\right )^{2}}{48 x^{4}}\) | \(309\) |
-1/4*arccot(a*x)^2/x^4-1/2*a^4*(-1/3/a^3/x^3*arccot(a*x)+1/a/x*arccot(a*x) +arccot(a*x)*arctan(a*x)+1/6/a^2/x^2+4/3*ln(a*x)-2/3*ln(a^2*x^2+1)+1/2*arc tan(a*x)^2)
Time = 0.28 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.88 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=\frac {4 \, a^{4} x^{4} \log \left (a^{2} x^{2} + 1\right ) - 8 \, a^{4} x^{4} \log \left (x\right ) - a^{2} x^{2} + 3 \, {\left (a^{4} x^{4} - 1\right )} \operatorname {arccot}\left (a x\right )^{2} - 2 \, {\left (3 \, a^{3} x^{3} - a x\right )} \operatorname {arccot}\left (a x\right )}{12 \, x^{4}} \]
1/12*(4*a^4*x^4*log(a^2*x^2 + 1) - 8*a^4*x^4*log(x) - a^2*x^2 + 3*(a^4*x^4 - 1)*arccot(a*x)^2 - 2*(3*a^3*x^3 - a*x)*arccot(a*x))/x^4
Time = 0.23 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=- \frac {2 a^{4} \log {\left (x \right )}}{3} + \frac {a^{4} \log {\left (a^{2} x^{2} + 1 \right )}}{3} + \frac {a^{4} \operatorname {acot}^{2}{\left (a x \right )}}{4} - \frac {a^{3} \operatorname {acot}{\left (a x \right )}}{2 x} - \frac {a^{2}}{12 x^{2}} + \frac {a \operatorname {acot}{\left (a x \right )}}{6 x^{3}} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{4 x^{4}} \]
-2*a**4*log(x)/3 + a**4*log(a**2*x**2 + 1)/3 + a**4*acot(a*x)**2/4 - a**3* acot(a*x)/(2*x) - a**2/(12*x**2) + a*acot(a*x)/(6*x**3) - acot(a*x)**2/(4* x**4)
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=-\frac {1}{6} \, {\left (3 \, a^{3} \arctan \left (a x\right ) + \frac {3 \, a^{2} x^{2} - 1}{x^{3}}\right )} a \operatorname {arccot}\left (a x\right ) - \frac {{\left (3 \, a^{2} x^{2} \arctan \left (a x\right )^{2} - 4 \, a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) + 8 \, a^{2} x^{2} \log \left (x\right ) + 1\right )} a^{2}}{12 \, x^{2}} - \frac {\operatorname {arccot}\left (a x\right )^{2}}{4 \, x^{4}} \]
-1/6*(3*a^3*arctan(a*x) + (3*a^2*x^2 - 1)/x^3)*a*arccot(a*x) - 1/12*(3*a^2 *x^2*arctan(a*x)^2 - 4*a^2*x^2*log(a^2*x^2 + 1) + 8*a^2*x^2*log(x) + 1)*a^ 2/x^2 - 1/4*arccot(a*x)^2/x^4
Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx=\frac {1}{12} \, {\left ({\left (3 \, \arctan \left (\frac {1}{a x}\right )^{2} - \frac {6 \, \arctan \left (\frac {1}{a x}\right )}{a x} - \frac {1}{a^{2} x^{2}} + \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a^{3} x^{3}} + 4 \, \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} a^{3} - \frac {3 \, \arctan \left (\frac {1}{a x}\right )^{2}}{a x^{4}}\right )} a \]
1/12*((3*arctan(1/(a*x))^2 - 6*arctan(1/(a*x))/(a*x) - 1/(a^2*x^2) + 2*arc tan(1/(a*x))/(a^3*x^3) + 4*log(1/(a^2*x^2) + 1))*a^3 - 3*arctan(1/(a*x))^2 /(a*x^4))*a
Time = 0.81 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {\cot ^{-1}(a x)^2}{x^5} \, dx={\mathrm {acot}\left (a\,x\right )}^2\,\left (\frac {a^4}{4}-\frac {1}{4\,x^4}\right )-\frac {2\,a^4\,\ln \left (x\right )}{3}+\frac {a^4\,\ln \left (a^2\,x^2+1\right )}{3}-\frac {a^2}{12\,x^2}-\frac {a^2\,\mathrm {acot}\left (a\,x\right )\,\left (\frac {a\,x^2}{2}-\frac {1}{6\,a}\right )}{x^3} \]