Integrand size = 19, antiderivative size = 89 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=\frac {1}{18 (1-2 x)}+\frac {1}{36 (1-x)}-\frac {1}{36 (1+x)}-\frac {1}{18 (1+2 x)}+\frac {2 x}{3 \left (3-4 x^2\right )}+\frac {67 \text {arctanh}(x)}{54}-\frac {7}{27} \text {arctanh}(2 x)-\frac {5 \text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2082, 213, 205} \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=\frac {67 \text {arctanh}(x)}{54}-\frac {7}{27} \text {arctanh}(2 x)-\frac {5 \text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2 x}{3 \left (3-4 x^2\right )}+\frac {1}{18 (1-2 x)}+\frac {1}{36 (1-x)}-\frac {1}{36 (x+1)}-\frac {1}{18 (2 x+1)} \]
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Rule 205
Rule 213
Rule 2082
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{36 (-1+x)^2}+\frac {1}{36 (1+x)^2}+\frac {1}{9 (-1+2 x)^2}+\frac {1}{9 (1+2 x)^2}-\frac {67}{54 \left (-1+x^2\right )}+\frac {4}{\left (-3+4 x^2\right )^2}+\frac {4}{-3+4 x^2}+\frac {14}{27 \left (-1+4 x^2\right )}\right ) \, dx \\ & = \frac {1}{18 (1-2 x)}+\frac {1}{36 (1-x)}-\frac {1}{36 (1+x)}-\frac {1}{18 (1+2 x)}+\frac {14}{27} \int \frac {1}{-1+4 x^2} \, dx-\frac {67}{54} \int \frac {1}{-1+x^2} \, dx+4 \int \frac {1}{\left (-3+4 x^2\right )^2} \, dx+4 \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{18 (1-2 x)}+\frac {1}{36 (1-x)}-\frac {1}{36 (1+x)}-\frac {1}{18 (1+2 x)}+\frac {2 x}{3 \left (3-4 x^2\right )}+\frac {67}{54} \tanh ^{-1}(x)-\frac {7}{27} \tanh ^{-1}(2 x)-\frac {2 \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2}{3} \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{18 (1-2 x)}+\frac {1}{36 (1-x)}-\frac {1}{36 (1+x)}-\frac {1}{18 (1+2 x)}+\frac {2 x}{3 \left (3-4 x^2\right )}+\frac {67}{54} \tanh ^{-1}(x)-\frac {7}{27} \tanh ^{-1}(2 x)-\frac {5 \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{3 \sqrt {3}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=\frac {1}{108} \left (-\frac {6 x \left (27-104 x^2+80 x^4\right )}{-3+19 x^2-32 x^4+16 x^6}+14 \log (1-2 x)+30 \sqrt {3} \log \left (\sqrt {3}-2 x\right )-67 \log (1-x)+67 \log (1+x)-14 \log (1+2 x)-30 \sqrt {3} \log \left (\sqrt {3}+2 x\right )\right ) \]
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Time = 0.12 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {1}{18 \left (1+2 x \right )}-\frac {7 \ln \left (1+2 x \right )}{54}-\frac {1}{18 \left (2 x -1\right )}+\frac {7 \ln \left (2 x -1\right )}{54}-\frac {1}{36 \left (x +1\right )}+\frac {67 \ln \left (x +1\right )}{108}-\frac {1}{36 \left (x -1\right )}-\frac {67 \ln \left (x -1\right )}{108}-\frac {x}{6 \left (x^{2}-\frac {3}{4}\right )}-\frac {5 \,\operatorname {arctanh}\left (\frac {2 x \sqrt {3}}{3}\right ) \sqrt {3}}{9}\) | \(84\) |
risch | \(\frac {-\frac {5}{18} x^{5}+\frac {13}{36} x^{3}-\frac {3}{32} x}{x^{6}-2 x^{4}+\frac {19}{16} x^{2}-\frac {3}{16}}+\frac {67 \ln \left (x +1\right )}{108}-\frac {7 \ln \left (1+2 x \right )}{54}+\frac {7 \ln \left (2 x -1\right )}{54}+\frac {5 \sqrt {3}\, \ln \left (2 x -\sqrt {3}\right )}{18}-\frac {5 \sqrt {3}\, \ln \left (2 x +\sqrt {3}\right )}{18}-\frac {67 \ln \left (x -1\right )}{108}\) | \(90\) |
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Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (67) = 134\).
Time = 0.26 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.99 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=-\frac {480 \, x^{5} - 624 \, x^{3} - 30 \, \sqrt {3} {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (\frac {4 \, x^{2} - 4 \, \sqrt {3} x + 3}{4 \, x^{2} - 3}\right ) + 14 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x + 1\right ) - 14 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (2 \, x - 1\right ) - 67 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x + 1\right ) + 67 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )} \log \left (x - 1\right ) + 162 \, x}{108 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} \]
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Time = 0.82 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=\frac {- 80 x^{5} + 104 x^{3} - 27 x}{288 x^{6} - 576 x^{4} + 342 x^{2} - 54} - \frac {67 \log {\left (x - 1 \right )}}{108} + \frac {7 \log {\left (x - \frac {1}{2} \right )}}{54} - \frac {7 \log {\left (x + \frac {1}{2} \right )}}{54} + \frac {67 \log {\left (x + 1 \right )}}{108} + \frac {5 \sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} \right )}}{18} - \frac {5 \sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} \right )}}{18} \]
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Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=\frac {5}{18} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3}}{2 \, x + \sqrt {3}}\right ) - \frac {80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac {7}{54} \, \log \left (2 \, x + 1\right ) + \frac {7}{54} \, \log \left (2 \, x - 1\right ) + \frac {67}{108} \, \log \left (x + 1\right ) - \frac {67}{108} \, \log \left (x - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=\frac {5}{18} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} \right |}}\right ) - \frac {80 \, x^{5} - 104 \, x^{3} + 27 \, x}{18 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}} - \frac {7}{54} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) + \frac {7}{54} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {67}{108} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {67}{108} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^2} \, dx=-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,67{}\mathrm {i}}{54}+\frac {\mathrm {atan}\left (x\,2{}\mathrm {i}\right )\,7{}\mathrm {i}}{27}-\frac {\frac {5\,x^5}{18}-\frac {13\,x^3}{36}+\frac {3\,x}{32}}{x^6-2\,x^4+\frac {19\,x^2}{16}-\frac {3}{16}}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x\,2{}\mathrm {i}}{3}\right )\,5{}\mathrm {i}}{9} \]
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