Integrand size = 12, antiderivative size = 67 \[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=-\frac {1}{16 \left (1-x^2\right )^2}-\frac {3}{16 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {3}{16} \coth ^{-1}(x)^2 \]
[Out]
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6108, 6104, 267} \[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=-\frac {3}{16 \left (1-x^2\right )}-\frac {1}{16 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3}{16} \coth ^{-1}(x)^2 \]
[In]
[Out]
Rule 267
Rule 6104
Rule 6108
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{16 \left (1-x^2\right )^2}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3}{4} \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^2} \, dx \\ & = -\frac {1}{16 \left (1-x^2\right )^2}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {3}{16} \coth ^{-1}(x)^2-\frac {3}{8} \int \frac {x}{\left (1-x^2\right )^2} \, dx \\ & = -\frac {1}{16 \left (1-x^2\right )^2}-\frac {3}{16 \left (1-x^2\right )}+\frac {x \coth ^{-1}(x)}{4 \left (1-x^2\right )^2}+\frac {3 x \coth ^{-1}(x)}{8 \left (1-x^2\right )}+\frac {3}{16} \coth ^{-1}(x)^2 \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=-\frac {4-3 x^2+2 x \left (-5+3 x^2\right ) \coth ^{-1}(x)-3 \left (-1+x^2\right )^2 \coth ^{-1}(x)^2}{16 \left (-1+x^2\right )^2} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(127\) vs. \(2(57)=114\).
Time = 0.36 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.91
method | result | size |
risch | \(\frac {3 \ln \left (x -1\right )^{2}}{64}-\frac {\left (3 x^{4} \ln \left (1+x \right )-6 x^{3}-6 x^{2} \ln \left (1+x \right )+10 x +3 \ln \left (1+x \right )\right ) \ln \left (x -1\right )}{32 \left (x^{2}-1\right )^{2}}+\frac {3 x^{4} \ln \left (1+x \right )^{2}-12 x^{3} \ln \left (1+x \right )-6 x^{2} \ln \left (1+x \right )^{2}+12 x^{2}+20 \ln \left (1+x \right ) x +3 \ln \left (1+x \right )^{2}-16}{64 \left (x -1\right ) \left (1+x \right ) \left (x^{2}-1\right )}\) | \(128\) |
default | \(-\frac {\operatorname {arccoth}\left (x \right )}{16 \left (1+x \right )^{2}}-\frac {3 \,\operatorname {arccoth}\left (x \right )}{16 \left (1+x \right )}+\frac {3 \,\operatorname {arccoth}\left (x \right ) \ln \left (1+x \right )}{16}+\frac {\operatorname {arccoth}\left (x \right )}{16 \left (x -1\right )^{2}}-\frac {3 \,\operatorname {arccoth}\left (x \right )}{16 \left (x -1\right )}-\frac {3 \,\operatorname {arccoth}\left (x \right ) \ln \left (x -1\right )}{16}-\frac {3 \ln \left (1+x \right )^{2}}{64}+\frac {3 \left (\ln \left (1+x \right )-\ln \left (\frac {1}{2}+\frac {x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{2}\right )}{32}+\frac {3 \ln \left (x -1\right ) \ln \left (\frac {1}{2}+\frac {x}{2}\right )}{32}-\frac {3 \ln \left (x -1\right )^{2}}{64}-\frac {1}{64 \left (x -1\right )^{2}}+\frac {7}{64 \left (x -1\right )}-\frac {1}{64 \left (1+x \right )^{2}}-\frac {7}{64 \left (1+x \right )}\) | \(131\) |
parts | \(-\frac {\operatorname {arccoth}\left (x \right )}{16 \left (1+x \right )^{2}}-\frac {3 \,\operatorname {arccoth}\left (x \right )}{16 \left (1+x \right )}+\frac {3 \,\operatorname {arccoth}\left (x \right ) \ln \left (1+x \right )}{16}+\frac {\operatorname {arccoth}\left (x \right )}{16 \left (x -1\right )^{2}}-\frac {3 \,\operatorname {arccoth}\left (x \right )}{16 \left (x -1\right )}-\frac {3 \,\operatorname {arccoth}\left (x \right ) \ln \left (x -1\right )}{16}-\frac {3 \ln \left (1+x \right )^{2}}{64}+\frac {3 \left (\ln \left (1+x \right )-\ln \left (\frac {1}{2}+\frac {x}{2}\right )\right ) \ln \left (\frac {1}{2}-\frac {x}{2}\right )}{32}+\frac {3 \ln \left (x -1\right ) \ln \left (\frac {1}{2}+\frac {x}{2}\right )}{32}-\frac {3 \ln \left (x -1\right )^{2}}{64}-\frac {1}{64 \left (x -1\right )^{2}}+\frac {7}{64 \left (x -1\right )}-\frac {1}{64 \left (1+x \right )^{2}}-\frac {7}{64 \left (1+x \right )}\) | \(131\) |
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.99 \[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=\frac {3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (\frac {x + 1}{x - 1}\right )^{2} + 12 \, x^{2} - 4 \, {\left (3 \, x^{3} - 5 \, x\right )} \log \left (\frac {x + 1}{x - 1}\right ) - 16}{64 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
[In]
[Out]
\[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=- \int \frac {\operatorname {acoth}{\left (x \right )}}{x^{6} - 3 x^{4} + 3 x^{2} - 1}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (49) = 98\).
Time = 0.20 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.76 \[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=-\frac {1}{16} \, {\left (\frac {2 \, {\left (3 \, x^{3} - 5 \, x\right )}}{x^{4} - 2 \, x^{2} + 1} - 3 \, \log \left (x + 1\right ) + 3 \, \log \left (x - 1\right )\right )} \operatorname {arcoth}\left (x\right ) - \frac {3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right )^{2} - 6 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right ) \log \left (x - 1\right ) + 3 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x - 1\right )^{2} - 12 \, x^{2} + 16}{64 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
[In]
[Out]
\[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=\int { -\frac {\operatorname {arcoth}\left (x\right )}{{\left (x^{2} - 1\right )}^{3}} \,d x } \]
[In]
[Out]
Time = 4.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.67 \[ \int \frac {\coth ^{-1}(x)}{\left (1-x^2\right )^3} \, dx=\frac {3\,{\ln \left (\frac {1}{x}+1\right )}^2}{64}-\ln \left (1-\frac {1}{x}\right )\,\left (\frac {3\,\ln \left (\frac {1}{x}+1\right )}{32}+\frac {\frac {5\,x}{16}-\frac {3\,x^3}{16}}{x^4-2\,x^2+1}\right )+\frac {3\,{\ln \left (1-\frac {1}{x}\right )}^2}{64}+\frac {\frac {3\,x^2}{16}-\frac {1}{4}}{x^4-2\,x^2+1}+\frac {\ln \left (\frac {1}{x}+1\right )\,\left (\frac {5\,x}{16}-\frac {3\,x^3}{16}\right )}{x^4-2\,x^2+1} \]
[In]
[Out]