Integrand size = 10, antiderivative size = 59 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=\frac {a \cot ^{-1}(a x)}{x}-\frac {1}{2} a^2 \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1+a^2 x^2\right ) \]
Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=\frac {a \cot ^{-1}(a x)}{x}+\frac {\left (-1-a^2 x^2\right ) \cot ^{-1}(a x)^2}{2 x^2}+a^2 \log (x)-\frac {1}{2} a^2 \log \left (1+a^2 x^2\right ) \]
(a*ArcCot[a*x])/x + ((-1 - a^2*x^2)*ArcCot[a*x]^2)/(2*x^2) + a^2*Log[x] - (a^2*Log[1 + a^2*x^2])/2
Time = 0.44 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5362, 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.
\(\displaystyle \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle -a \int \frac {\cot ^{-1}(a x)}{x^2 \left (a^2 x^2+1\right )}dx-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 5454 |
\(\displaystyle -a \left (\int \frac {\cot ^{-1}(a x)}{x^2}dx-a^2 \int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle -a \left (-a \int \frac {1}{x \left (a^2 x^2+1\right )}dx+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -a \left (-\frac {1}{2} a \int \frac {1}{x^2 \left (a^2 x^2+1\right )}dx^2+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle -a \left (-\frac {1}{2} a \left (\int \frac {1}{x^2}dx^2-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle -a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-a^2 \int \frac {1}{a^2 x^2+1}dx^2\right )+a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle -a \left (a^2 \left (-\int \frac {\cot ^{-1}(a x)}{a^2 x^2+1}dx\right )-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
\(\Big \downarrow \) 5420 |
\(\displaystyle -a \left (-\frac {1}{2} a \left (\log \left (x^2\right )-\log \left (a^2 x^2+1\right )\right )+\frac {1}{2} a \cot ^{-1}(a x)^2-\frac {\cot ^{-1}(a x)}{x}\right )-\frac {\cot ^{-1}(a x)^2}{2 x^2}\) |
-1/2*ArcCot[a*x]^2/x^2 - a*(-(ArcCot[a*x]/x) + (a*ArcCot[a*x]^2)/2 - (a*(L og[x^2] - Log[1 + a^2*x^2]))/2)
3.1.20.3.1 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[(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 = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arccot}\left (a x \right )}{a x}+\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\ln \left (a x \right )-\frac {\ln \left (a^{2} x^{2}+1\right )}{2}+\frac {\arctan \left (a x \right )^{2}}{2}\right )\) | \(64\) |
default | \(a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )^{2}}{2 a^{2} x^{2}}+\frac {\operatorname {arccot}\left (a x \right )}{a x}+\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )+\ln \left (a x \right )-\frac {\ln \left (a^{2} x^{2}+1\right )}{2}+\frac {\arctan \left (a x \right )^{2}}{2}\right )\) | \(64\) |
parallelrisch | \(\frac {-a^{2} x^{2} \operatorname {arccot}\left (a x \right )^{2}+2 a^{2} \ln \left (x \right ) x^{2}-a^{2} \ln \left (a^{2} x^{2}+1\right ) x^{2}+2 \,\operatorname {arccot}\left (a x \right ) a x -\operatorname {arccot}\left (a x \right )^{2}}{2 x^{2}}\) | \(65\) |
parts | \(-\frac {\operatorname {arccot}\left (a x \right )^{2}}{2 x^{2}}-a^{2} \left (-\frac {\operatorname {arccot}\left (a x \right )}{a x}-\operatorname {arccot}\left (a x \right ) \arctan \left (a x \right )-\ln \left (a x \right )+\frac {\ln \left (a^{2} x^{2}+1\right )}{2}-\frac {\arctan \left (a x \right )^{2}}{2}\right )\) | \(67\) |
risch | \(\frac {\left (a^{2} x^{2}+1\right ) \ln \left (i a x +1\right )^{2}}{8 x^{2}}-\frac {i \left (-i x^{2} \ln \left (-i a x +1\right ) a^{2}-2 a x +\pi -i \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{4 x^{2}}-\frac {2 i a^{2} \ln \left (\left (-\pi a +6 i a \right ) x +6+i \pi \right ) \pi \,x^{2}-2 i a^{2} \ln \left (\left (-\pi a -6 i a \right ) x +6-i \pi \right ) \pi \,x^{2}-a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+4 a^{2} \ln \left (\left (-\pi a +6 i a \right ) x +6+i \pi \right ) x^{2}+4 a^{2} \ln \left (\left (-\pi a -6 i a \right ) x +6-i \pi \right ) x^{2}-8 a^{2} \ln \left (-x \right ) x^{2}+4 i a x \ln \left (-i a x +1\right )-2 i \pi \ln \left (-i a x +1\right )-4 \pi a x +\pi ^{2}-\ln \left (-i a x +1\right )^{2}}{8 x^{2}}\) | \(263\) |
a^2*(-1/2/a^2/x^2*arccot(a*x)^2+1/a/x*arccot(a*x)+arccot(a*x)*arctan(a*x)+ ln(a*x)-1/2*ln(a^2*x^2+1)+1/2*arctan(a*x)^2)
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.97 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=-\frac {a^{2} x^{2} \log \left (a^{2} x^{2} + 1\right ) - 2 \, a^{2} x^{2} \log \left (x\right ) - 2 \, a x \operatorname {arccot}\left (a x\right ) + {\left (a^{2} x^{2} + 1\right )} \operatorname {arccot}\left (a x\right )^{2}}{2 \, x^{2}} \]
-1/2*(a^2*x^2*log(a^2*x^2 + 1) - 2*a^2*x^2*log(x) - 2*a*x*arccot(a*x) + (a ^2*x^2 + 1)*arccot(a*x)^2)/x^2
Time = 0.18 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.90 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=a^{2} \log {\left (x \right )} - \frac {a^{2} \log {\left (a^{2} x^{2} + 1 \right )}}{2} - \frac {a^{2} \operatorname {acot}^{2}{\left (a x \right )}}{2} + \frac {a \operatorname {acot}{\left (a x \right )}}{x} - \frac {\operatorname {acot}^{2}{\left (a x \right )}}{2 x^{2}} \]
a**2*log(x) - a**2*log(a**2*x**2 + 1)/2 - a**2*acot(a*x)**2/2 + a*acot(a*x )/x - acot(a*x)**2/(2*x**2)
Time = 0.28 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=\frac {1}{2} \, {\left (\arctan \left (a x\right )^{2} - \log \left (a^{2} x^{2} + 1\right ) + 2 \, \log \left (x\right )\right )} a^{2} + {\left (a \arctan \left (a x\right ) + \frac {1}{x}\right )} a \operatorname {arccot}\left (a x\right ) - \frac {\operatorname {arccot}\left (a x\right )^{2}}{2 \, x^{2}} \]
1/2*(arctan(a*x)^2 - log(a^2*x^2 + 1) + 2*log(x))*a^2 + (a*arctan(a*x) + 1 /x)*a*arccot(a*x) - 1/2*arccot(a*x)^2/x^2
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.02 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=-\frac {1}{2} \, {\left ({\left (\arctan \left (\frac {1}{a x}\right )^{2} - \frac {2 \, \arctan \left (\frac {1}{a x}\right )}{a x} + \log \left (\frac {1}{a^{2} x^{2}} + 1\right )\right )} a + \frac {\arctan \left (\frac {1}{a x}\right )^{2}}{a x^{2}}\right )} a \]
-1/2*((arctan(1/(a*x))^2 - 2*arctan(1/(a*x))/(a*x) + log(1/(a^2*x^2) + 1)) *a + arctan(1/(a*x))^2/(a*x^2))*a
Time = 0.75 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.85 \[ \int \frac {\cot ^{-1}(a x)^2}{x^3} \, dx=a^2\,\ln \left (x\right )-{\mathrm {acot}\left (a\,x\right )}^2\,\left (\frac {a^2}{2}+\frac {1}{2\,x^2}\right )-\frac {a^2\,\ln \left (a^2\,x^2+1\right )}{2}+\frac {a\,\mathrm {acot}\left (a\,x\right )}{x} \]