Integrand size = 13, antiderivative size = 47 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=\frac {1}{6 x^2}-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {\cot ^{-1}(x)}{x}-\frac {1}{2} \cot ^{-1}(x)^2+\frac {4 \log (x)}{3}-\frac {2}{3} \log \left (1+x^2\right ) \]
Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=\frac {1}{6 x^2}-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {\cot ^{-1}(x)}{x}-\frac {1}{2} \cot ^{-1}(x)^2+\frac {4 \log (x)}{3}-\frac {2}{3} \log \left (1+x^2\right ) \]
1/(6*x^2) - ArcCot[x]/(3*x^3) + ArcCot[x]/x - ArcCot[x]^2/2 + (4*Log[x])/3 - (2*Log[1 + x^2])/3
Time = 0.54 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {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.
\(\displaystyle \int \frac {\cot ^{-1}(x)}{x^4 \left (x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 5454 |
\(\displaystyle \int \frac {\cot ^{-1}(x)}{x^4}dx-\int \frac {\cot ^{-1}(x)}{x^2 \left (x^2+1\right )}dx\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle -\int \frac {\cot ^{-1}(x)}{x^2 \left (x^2+1\right )}dx-\frac {1}{3} \int \frac {1}{x^3 \left (x^2+1\right )}dx-\frac {\cot ^{-1}(x)}{3 x^3}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle -\int \frac {\cot ^{-1}(x)}{x^2 \left (x^2+1\right )}dx-\frac {1}{6} \int \frac {1}{x^4 \left (x^2+1\right )}dx^2-\frac {\cot ^{-1}(x)}{3 x^3}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -\int \frac {\cot ^{-1}(x)}{x^2 \left (x^2+1\right )}dx-\frac {1}{6} \int \left (-\frac {1}{x^2}+\frac {1}{x^4}+\frac {1}{x^2+1}\right )dx^2-\frac {\cot ^{-1}(x)}{3 x^3}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\int \frac {\cot ^{-1}(x)}{x^2 \left (x^2+1\right )}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )\) |
\(\Big \downarrow \) 5454 |
\(\displaystyle -\int \frac {\cot ^{-1}(x)}{x^2}dx+\int \frac {\cot ^{-1}(x)}{x^2+1}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )\) |
\(\Big \downarrow \) 5362 |
\(\displaystyle \int \frac {1}{x \left (x^2+1\right )}dx+\int \frac {\cot ^{-1}(x)}{x^2+1}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {\cot ^{-1}(x)}{x}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2 \left (x^2+1\right )}dx^2+\int \frac {\cot ^{-1}(x)}{x^2+1}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {\cot ^{-1}(x)}{x}\) |
\(\Big \downarrow \) 47 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{x^2}dx^2-\int \frac {1}{x^2+1}dx^2\right )+\int \frac {\cot ^{-1}(x)}{x^2+1}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {\cot ^{-1}(x)}{x}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle \frac {1}{2} \left (\log \left (x^2\right )-\int \frac {1}{x^2+1}dx^2\right )+\int \frac {\cot ^{-1}(x)}{x^2+1}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {\cot ^{-1}(x)}{x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \int \frac {\cot ^{-1}(x)}{x^2+1}dx-\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{2} \left (\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {\cot ^{-1}(x)}{x}\) |
\(\Big \downarrow \) 5420 |
\(\displaystyle -\frac {\cot ^{-1}(x)}{3 x^3}+\frac {1}{2} \left (\log \left (x^2\right )-\log \left (x^2+1\right )\right )+\frac {1}{6} \left (\frac {1}{x^2}+\log \left (x^2\right )-\log \left (x^2+1\right )\right )-\frac {1}{2} \cot ^{-1}(x)^2+\frac {\cot ^{-1}(x)}{x}\) |
-1/3*ArcCot[x]/x^3 + ArcCot[x]/x - ArcCot[x]^2/2 + (Log[x^2] - Log[1 + x^2 ])/2 + (x^(-2) + Log[x^2] - Log[1 + x^2])/6
3.1.45.3.1 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 = 0.42 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91
method | result | size |
default | \(-\frac {\operatorname {arccot}\left (x \right )}{3 x^{3}}+\frac {\operatorname {arccot}\left (x \right )}{x}+\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+\frac {1}{6 x^{2}}+\frac {4 \ln \left (x \right )}{3}-\frac {2 \ln \left (x^{2}+1\right )}{3}+\frac {\arctan \left (x \right )^{2}}{2}\) | \(43\) |
parts | \(-\frac {\operatorname {arccot}\left (x \right )}{3 x^{3}}+\frac {\operatorname {arccot}\left (x \right )}{x}+\operatorname {arccot}\left (x \right ) \arctan \left (x \right )+\frac {1}{6 x^{2}}+\frac {4 \ln \left (x \right )}{3}-\frac {2 \ln \left (x^{2}+1\right )}{3}+\frac {\arctan \left (x \right )^{2}}{2}\) | \(43\) |
parallelrisch | \(\frac {-3 \operatorname {arccot}\left (x \right )^{2} x^{3}+8 \ln \left (x \right ) x^{3}-4 \ln \left (x^{2}+1\right ) x^{3}+6 x^{2} \operatorname {arccot}\left (x \right )+x -2 \,\operatorname {arccot}\left (x \right )}{6 x^{3}}\) | \(46\) |
risch | \(\frac {\ln \left (i x +1\right )^{2}}{8}-\frac {\left (3 \ln \left (-i x +1\right ) x^{3}-6 i x^{2}+2 i\right ) \ln \left (i x +1\right )}{12 x^{3}}+\frac {-6 i \ln \left (\left (-\pi +8 i\right ) x +8+i \pi \right ) \pi \,x^{3}+6 i \ln \left (\left (-\pi -8 i\right ) x +8-i \pi \right ) \pi \,x^{3}+3 \ln \left (-i x +1\right )^{2} x^{3}-12 i x^{2} \ln \left (-i x +1\right )-16 \ln \left (\left (-\pi +8 i\right ) x +8+i \pi \right ) x^{3}-16 \ln \left (\left (-\pi -8 i\right ) x +8-i \pi \right ) x^{3}+32 \ln \left (-x \right ) x^{3}+12 \pi \,x^{2}+4 i \ln \left (-i x +1\right )-4 \pi +4 x}{24 x^{3}}\) | \(194\) |
-1/3*arccot(x)/x^3+arccot(x)/x+arccot(x)*arctan(x)+1/6/x^2+4/3*ln(x)-2/3*l n(x^2+1)+1/2*arctan(x)^2
Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=-\frac {3 \, x^{3} \operatorname {arccot}\left (x\right )^{2} + 4 \, x^{3} \log \left (x^{2} + 1\right ) - 8 \, x^{3} \log \left (x\right ) - 2 \, {\left (3 \, x^{2} - 1\right )} \operatorname {arccot}\left (x\right ) - x}{6 \, x^{3}} \]
-1/6*(3*x^3*arccot(x)^2 + 4*x^3*log(x^2 + 1) - 8*x^3*log(x) - 2*(3*x^2 - 1 )*arccot(x) - x)/x^3
Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.89 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=\frac {4 \log {\left (x \right )}}{3} - \frac {2 \log {\left (x^{2} + 1 \right )}}{3} - \frac {\operatorname {acot}^{2}{\left (x \right )}}{2} + \frac {\operatorname {acot}{\left (x \right )}}{x} + \frac {1}{6 x^{2}} - \frac {\operatorname {acot}{\left (x \right )}}{3 x^{3}} \]
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.17 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=\frac {1}{3} \, {\left (\frac {3 \, x^{2} - 1}{x^{3}} + 3 \, \arctan \left (x\right )\right )} \operatorname {arccot}\left (x\right ) + \frac {3 \, x^{2} \arctan \left (x\right )^{2} - 4 \, x^{2} \log \left (x^{2} + 1\right ) + 8 \, x^{2} \log \left (x\right ) + 1}{6 \, x^{2}} \]
1/3*((3*x^2 - 1)/x^3 + 3*arctan(x))*arccot(x) + 1/6*(3*x^2*arctan(x)^2 - 4 *x^2*log(x^2 + 1) + 8*x^2*log(x) + 1)/x^2
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.83 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=-\frac {1}{2} \, \arctan \left (\frac {1}{x}\right )^{2} + \frac {\arctan \left (\frac {1}{x}\right )}{x} + \frac {1}{6 \, x^{2}} - \frac {\arctan \left (\frac {1}{x}\right )}{3 \, x^{3}} - \frac {2}{3} \, \log \left (\frac {1}{x^{2}} + 1\right ) \]
Time = 0.10 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\cot ^{-1}(x)}{x^4 \left (1+x^2\right )} \, dx=\frac {4\,\ln \left (x\right )}{3}-\frac {2\,\ln \left (x^2+1\right )}{3}-\frac {{\mathrm {acot}\left (x\right )}^2}{2}+\frac {1}{6\,x^2}+\frac {\mathrm {acot}\left (x\right )\,\left (x^2-\frac {1}{3}\right )}{x^3} \]