Integrand size = 24, antiderivative size = 45 \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\frac {\arctan (x)}{2}+\frac {\sqrt {x^6} \arctan (x)}{2 x^3}+\frac {\text {arctanh}(x)}{2}-\frac {\sqrt {x^6} \text {arctanh}(x)}{2 x^3} \]
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Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6857, 218, 212, 209, 15, 304} \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\frac {\sqrt {x^6} \arctan (x)}{2 x^3}+\frac {\arctan (x)}{2}-\frac {\sqrt {x^6} \text {arctanh}(x)}{2 x^3}+\frac {\text {arctanh}(x)}{2} \]
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Rule 15
Rule 209
Rule 212
Rule 218
Rule 304
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{1-x^4}+\frac {\sqrt {x^6}}{x \left (-1+x^4\right )}\right ) \, dx \\ & = \int \frac {1}{1-x^4} \, dx+\int \frac {\sqrt {x^6}}{x \left (-1+x^4\right )} \, dx \\ & = \frac {1}{2} \int \frac {1}{1-x^2} \, dx+\frac {1}{2} \int \frac {1}{1+x^2} \, dx+\frac {\sqrt {x^6} \int \frac {x^2}{-1+x^4} \, dx}{x^3} \\ & = \frac {1}{2} \tan ^{-1}(x)+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {x^6} \int \frac {1}{1-x^2} \, dx}{2 x^3}+\frac {\sqrt {x^6} \int \frac {1}{1+x^2} \, dx}{2 x^3} \\ & = \frac {1}{2} \tan ^{-1}(x)+\frac {\sqrt {x^6} \tan ^{-1}(x)}{2 x^3}+\frac {1}{2} \tanh ^{-1}(x)-\frac {\sqrt {x^6} \tanh ^{-1}(x)}{2 x^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.27 \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\frac {\arctan (x)}{2}-\frac {1}{2} \arctan \left (\frac {\sqrt {x^6}}{x^4}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt {x^6}}{x^4}\right )-\frac {1}{4} \log (1-x)+\frac {1}{4} \log (1+x) \]
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Time = 1.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78
method | result | size |
default | \(\frac {\sqrt {x^{6}}\, \left (\ln \left (x -1\right )-\ln \left (x +1\right )+2 \arctan \left (x \right )\right )}{4 x^{3}}+\frac {\arctan \left (x \right )}{2}+\frac {\operatorname {arctanh}\left (x \right )}{2}\) | \(35\) |
meijerg | \(-\frac {x \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )-2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {\sqrt {x^{6}}\, \left (\ln \left (1-\left (x^{4}\right )^{\frac {1}{4}}\right )-\ln \left (1+\left (x^{4}\right )^{\frac {1}{4}}\right )+2 \arctan \left (\left (x^{4}\right )^{\frac {1}{4}}\right )\right )}{4 \left (x^{4}\right )^{\frac {3}{4}}}\) | \(80\) |
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Time = 0.32 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.04 \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\arctan \left (x\right ) \]
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\[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=- \int \frac {x}{x^{5} - x}\, dx - \int \left (- \frac {\sqrt {x^{6}}}{x^{5} - x}\right )\, dx \]
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Time = 0.27 (sec) , antiderivative size = 2, normalized size of antiderivative = 0.04 \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\arctan \left (x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.69 \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\frac {1}{2} \, {\left (\mathrm {sgn}\left (x\right ) + 1\right )} \arctan \left (x\right ) - \frac {1}{4} \, {\left (\mathrm {sgn}\left (x\right ) - 1\right )} \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, {\left (\mathrm {sgn}\left (x\right ) - 1\right )} \log \left ({\left | x - 1 \right |}\right ) \]
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Timed out. \[ \int \frac {1-\frac {\sqrt {x^6}}{x}}{1-x^4} \, dx=\int \frac {\frac {\sqrt {x^6}}{x}-1}{x^4-1} \,d x \]
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