Integrand size = 19, antiderivative size = 161 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=\frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {5 x}{3 \left (3-4 x^2\right )}+\frac {3913 \text {arctanh}(x)}{648}+\frac {67}{162} \text {arctanh}(2 x)+\frac {5 \text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{6 \sqrt {3}}-4 \sqrt {3} \text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right ) \]
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Time = 0.09 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2082, 213, 205} \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=\frac {3913 \text {arctanh}(x)}{648}+\frac {67}{162} \text {arctanh}(2 x)-4 \sqrt {3} \text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )+\frac {5 \text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{6 \sqrt {3}}+\frac {5 x}{3 \left (3-4 x^2\right )}-\frac {2 x}{3 \left (3-4 x^2\right )^2}-\frac {7}{108 (1-2 x)}+\frac {67}{432 (1-x)}-\frac {67}{432 (x+1)}+\frac {7}{108 (2 x+1)}+\frac {1}{108 (1-2 x)^2}+\frac {1}{432 (1-x)^2}-\frac {1}{432 (x+1)^2}-\frac {1}{108 (2 x+1)^2} \]
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Rule 205
Rule 213
Rule 2082
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{216 (-1+x)^3}+\frac {67}{432 (-1+x)^2}+\frac {1}{216 (1+x)^3}+\frac {67}{432 (1+x)^2}-\frac {1}{27 (-1+2 x)^3}-\frac {7}{54 (-1+2 x)^2}+\frac {1}{27 (1+2 x)^3}-\frac {7}{54 (1+2 x)^2}-\frac {3913}{648 \left (-1+x^2\right )}+\frac {8}{\left (-3+4 x^2\right )^3}+\frac {12}{\left (-3+4 x^2\right )^2}+\frac {24}{-3+4 x^2}-\frac {67}{81 \left (-1+4 x^2\right )}\right ) \, dx \\ & = \frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {67}{81} \int \frac {1}{-1+4 x^2} \, dx-\frac {3913}{648} \int \frac {1}{-1+x^2} \, dx+8 \int \frac {1}{\left (-3+4 x^2\right )^3} \, dx+12 \int \frac {1}{\left (-3+4 x^2\right )^2} \, dx+24 \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {2 x}{3-4 x^2}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )-2 \int \frac {1}{\left (-3+4 x^2\right )^2} \, dx-2 \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {5 x}{3 \left (3-4 x^2\right )}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)+\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}}-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )+\frac {1}{3} \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{108 (1-2 x)^2}-\frac {7}{108 (1-2 x)}+\frac {1}{432 (1-x)^2}+\frac {67}{432 (1-x)}-\frac {1}{432 (1+x)^2}-\frac {67}{432 (1+x)}-\frac {1}{108 (1+2 x)^2}+\frac {7}{108 (1+2 x)}-\frac {2 x}{3 \left (3-4 x^2\right )^2}+\frac {5 x}{3 \left (3-4 x^2\right )}+\frac {3913}{648} \tanh ^{-1}(x)+\frac {67}{162} \tanh ^{-1}(2 x)+\frac {5 \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{6 \sqrt {3}}-4 \sqrt {3} \tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=\frac {\frac {36 x \left (27-104 x^2+80 x^4\right )}{\left (3-19 x^2+32 x^4-16 x^6\right )^2}-\frac {6 x \left (345-2384 x^2+2288 x^4\right )}{-3+19 x^2-32 x^4+16 x^6}-268 \log (1-2 x)+2412 \sqrt {3} \log \left (\sqrt {3}-2 x\right )-3913 \log (1-x)+3913 \log (1+x)+268 \log (1+2 x)-2412 \sqrt {3} \log \left (\sqrt {3}+2 x\right )}{1296} \]
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Time = 0.14 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.67
method | result | size |
risch | \(\frac {-\frac {4576}{27} x^{11}+\frac {4640}{9} x^{9}-580 x^{7}+\frac {7960}{27} x^{5}-\frac {4777}{72} x^{3}+\frac {133}{24} x}{\left (16 x^{6}-32 x^{4}+19 x^{2}-3\right )^{2}}+\frac {67 \ln \left (1+2 x \right )}{324}+\frac {3913 \ln \left (x +1\right )}{1296}+\frac {67 \sqrt {3}\, \ln \left (2 x -\sqrt {3}\right )}{36}-\frac {67 \sqrt {3}\, \ln \left (2 x +\sqrt {3}\right )}{36}-\frac {67 \ln \left (2 x -1\right )}{324}-\frac {3913 \ln \left (x -1\right )}{1296}\) | \(108\) |
default | \(-\frac {1}{108 \left (1+2 x \right )^{2}}+\frac {7}{108 \left (1+2 x \right )}+\frac {67 \ln \left (1+2 x \right )}{324}+\frac {1}{108 \left (2 x -1\right )^{2}}+\frac {7}{108 \left (2 x -1\right )}-\frac {67 \ln \left (2 x -1\right )}{324}-\frac {1}{432 \left (x +1\right )^{2}}-\frac {67}{432 \left (x +1\right )}+\frac {3913 \ln \left (x +1\right )}{1296}+\frac {1}{432 \left (x -1\right )^{2}}-\frac {67}{432 \left (x -1\right )}-\frac {3913 \ln \left (x -1\right )}{1296}+\frac {-\frac {20}{3} x^{3}+\frac {13}{3} x}{\left (4 x^{2}-3\right )^{2}}-\frac {67 \,\operatorname {arctanh}\left (\frac {2 x \sqrt {3}}{3}\right ) \sqrt {3}}{18}\) | \(126\) |
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (111) = 222\).
Time = 0.30 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.75 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=-\frac {219648 \, x^{11} - 668160 \, x^{9} + 751680 \, x^{7} - 382080 \, x^{5} + 85986 \, x^{3} - 2412 \, \sqrt {3} {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (\frac {4 \, x^{2} - 4 \, \sqrt {3} x + 3}{4 \, x^{2} - 3}\right ) - 268 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (2 \, x + 1\right ) + 268 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (2 \, x - 1\right ) - 3913 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (x + 1\right ) + 3913 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )} \log \left (x - 1\right ) - 7182 \, x}{1296 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )}} \]
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Time = 0.88 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=- \frac {36608 x^{11} - 111360 x^{9} + 125280 x^{7} - 63680 x^{5} + 14331 x^{3} - 1197 x}{55296 x^{12} - 221184 x^{10} + 352512 x^{8} - 283392 x^{6} + 119448 x^{4} - 24624 x^{2} + 1944} - \frac {3913 \log {\left (x - 1 \right )}}{1296} - \frac {67 \log {\left (x - \frac {1}{2} \right )}}{324} + \frac {67 \log {\left (x + \frac {1}{2} \right )}}{324} + \frac {3913 \log {\left (x + 1 \right )}}{1296} + \frac {67 \sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} \right )}}{36} - \frac {67 \sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} \right )}}{36} \]
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Time = 0.28 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=\frac {67}{36} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3}}{2 \, x + \sqrt {3}}\right ) - \frac {36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x}{216 \, {\left (256 \, x^{12} - 1024 \, x^{10} + 1632 \, x^{8} - 1312 \, x^{6} + 553 \, x^{4} - 114 \, x^{2} + 9\right )}} + \frac {67}{324} \, \log \left (2 \, x + 1\right ) - \frac {67}{324} \, \log \left (2 \, x - 1\right ) + \frac {3913}{1296} \, \log \left (x + 1\right ) - \frac {3913}{1296} \, \log \left (x - 1\right ) \]
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Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=\frac {67}{36} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} \right |}}\right ) - \frac {36608 \, x^{11} - 111360 \, x^{9} + 125280 \, x^{7} - 63680 \, x^{5} + 14331 \, x^{3} - 1197 \, x}{216 \, {\left (16 \, x^{6} - 32 \, x^{4} + 19 \, x^{2} - 3\right )}^{2}} + \frac {67}{324} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {67}{324} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {3913}{1296} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {3913}{1296} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (3-19 x^2+32 x^4-16 x^6\right )^3} \, dx=\frac {-\frac {143\,x^{11}}{216}+\frac {145\,x^9}{72}-\frac {145\,x^7}{64}+\frac {995\,x^5}{864}-\frac {4777\,x^3}{18432}+\frac {133\,x}{6144}}{x^{12}-4\,x^{10}+\frac {51\,x^8}{8}-\frac {41\,x^6}{8}+\frac {553\,x^4}{256}-\frac {57\,x^2}{128}+\frac {9}{256}}-\frac {\mathrm {atan}\left (x\,2{}\mathrm {i}\right )\,67{}\mathrm {i}}{162}-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,3913{}\mathrm {i}}{648}+\frac {\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,x\,2{}\mathrm {i}}{3}\right )\,67{}\mathrm {i}}{18} \]
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