Integrand size = 19, antiderivative size = 31 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\text {arctanh}(x)}{3}+\frac {1}{3} \text {arctanh}(2 x)-\frac {\text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2082, 213} \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\text {arctanh}(x)}{3}+\frac {1}{3} \text {arctanh}(2 x)-\frac {\text {arctanh}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rule 213
Rule 2082
Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{3 \left (-1+x^2\right )}+\frac {2}{-3+4 x^2}-\frac {2}{3 \left (-1+4 x^2\right )}\right ) \, dx \\ & = -\left (\frac {1}{3} \int \frac {1}{-1+x^2} \, dx\right )-\frac {2}{3} \int \frac {1}{-1+4 x^2} \, dx+2 \int \frac {1}{-3+4 x^2} \, dx \\ & = \frac {1}{3} \tanh ^{-1}(x)+\frac {1}{3} \tanh ^{-1}(2 x)-\frac {\tanh ^{-1}\left (\frac {2 x}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \left (\sqrt {3} \log \left (\sqrt {3}-2 x\right )-\sqrt {3} \log \left (\sqrt {3}+2 x\right )-\log \left (1-3 x+2 x^2\right )+\log \left (1+3 x+2 x^2\right )\right ) \]
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Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.35
method | result | size |
default | \(\frac {\ln \left (1+2 x \right )}{6}-\frac {\ln \left (2 x -1\right )}{6}+\frac {\ln \left (x +1\right )}{6}-\frac {\ln \left (x -1\right )}{6}-\frac {\operatorname {arctanh}\left (\frac {2 x \sqrt {3}}{3}\right ) \sqrt {3}}{3}\) | \(42\) |
risch | \(\frac {\sqrt {3}\, \ln \left (2 x -\sqrt {3}\right )}{6}-\frac {\sqrt {3}\, \ln \left (2 x +\sqrt {3}\right )}{6}+\frac {\ln \left (2 x^{2}+3 x +1\right )}{6}-\frac {\ln \left (2 x^{2}-3 x +1\right )}{6}\) | \(56\) |
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Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.81 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {4 \, x^{2} - 4 \, \sqrt {3} x + 3}{4 \, x^{2} - 3}\right ) + \frac {1}{6} \, \log \left (2 \, x^{2} + 3 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x^{2} - 3 \, x + 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (29) = 58\).
Time = 0.07 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.03 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\sqrt {3} \log {\left (x - \frac {\sqrt {3}}{2} \right )}}{6} - \frac {\sqrt {3} \log {\left (x + \frac {\sqrt {3}}{2} \right )}}{6} - \frac {\log {\left (x^{2} - \frac {3 x}{2} + \frac {1}{2} \right )}}{6} + \frac {\log {\left (x^{2} + \frac {3 x}{2} + \frac {1}{2} \right )}}{6} \]
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Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.74 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {2 \, x - \sqrt {3}}{2 \, x + \sqrt {3}}\right ) + \frac {1}{6} \, \log \left (2 \, x + 1\right ) - \frac {1}{6} \, \log \left (2 \, x - 1\right ) + \frac {1}{6} \, \log \left (x + 1\right ) - \frac {1}{6} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | 8 \, x - 4 \, \sqrt {3} \right |}}{{\left | 8 \, x + 4 \, \sqrt {3} \right |}}\right ) + \frac {1}{6} \, \log \left ({\left | 2 \, x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{6} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 10.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {1}{3-19 x^2+32 x^4-16 x^6} \, dx=\frac {\mathrm {atanh}\left (\frac {x}{4608\,\left (\frac {x^2}{6912}+\frac {1}{13824}\right )}\right )}{3}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x}{3}\right )}{3} \]
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