Integrand size = 20, antiderivative size = 13 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {\arctan (x)}{2}+\frac {\text {arctanh}(x)}{2} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {28, 21, 218, 212, 209} \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {\arctan (x)}{2}+\frac {\text {arctanh}(x)}{2} \]
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Rule 21
Rule 28
Rule 209
Rule 212
Rule 218
Rubi steps \begin{align*} \text {integral}& = \int \frac {1-x^4}{\left (-1+x^4\right )^2} \, dx \\ & = -\int \frac {1}{-1+x^4} \, dx \\ & = \frac {1}{2} \int \frac {1}{1-x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx \\ & = \frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.92 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {\arctan (x)}{2}-\frac {1}{4} \log (1-x)+\frac {1}{4} \log (1+x) \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77
method | result | size |
default | \(\frac {\arctan \left (x \right )}{2}+\frac {\operatorname {arctanh}\left (x \right )}{2}\) | \(10\) |
risch | \(-\frac {\ln \left (x -1\right )}{4}+\frac {\arctan \left (x \right )}{2}+\frac {\ln \left (x +1\right )}{4}\) | \(18\) |
parallelrisch | \(-\frac {\ln \left (x -1\right )}{4}+\frac {i \ln \left (x +i\right )}{4}-\frac {i \ln \left (x -i\right )}{4}+\frac {\ln \left (x +1\right )}{4}\) | \(30\) |
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none
Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (8) = 16\).
Time = 0.06 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=- \frac {\log {\left (x - 1 \right )}}{4} + \frac {\log {\left (x + 1 \right )}}{4} + \frac {\operatorname {atan}{\left (x \right )}}{2} \]
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none
Time = 0.30 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x - 1\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (9) = 18\).
Time = 0.29 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {1}{2} \, \arctan \left (x\right ) + \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int \frac {1-x^4}{1-2 x^4+x^8} \, dx=\frac {\mathrm {atan}\left (x\right )}{2}+\frac {\mathrm {atanh}\left (x\right )}{2} \]
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