Integrand size = 17, antiderivative size = 81 \[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}} \]
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Time = 0.03 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {399, 246, 218, 212, 209, 385} \[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4-1}}\right )}{\sqrt [4]{2}} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 399
Rubi steps \begin{align*} \text {integral}& = -\left (2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (1+x^4\right )} \, dx\right )+\int \frac {1}{\sqrt [4]{-1+x^4}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\right )+\text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}}+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt [4]{2}} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.95 \[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\frac {1}{2} \left (\arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-2^{3/4} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )\right ) \]
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Time = 2.78 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.32
method | result | size |
pseudoelliptic | \(-\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}-x}{x}\right )}{4}-\frac {\arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{2}+\frac {\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}}+x}{x}\right )}{4}+\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}}}{2}-\frac {\ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{4}-1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{4}\) | \(107\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.81 \[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=-\frac {1}{4} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} i \cdot 2^{\frac {3}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} i \cdot 2^{\frac {3}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]
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\[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\int \frac {\left (\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{4}}}{x^{4} + 1}\, dx \]
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\[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )}^{\frac {3}{4}}}{x^{4} + 1} \,d x } \]
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\[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\int { \frac {{\left (x^{4} - 1\right )}^{\frac {3}{4}}}{x^{4} + 1} \,d x } \]
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Timed out. \[ \int \frac {\left (-1+x^4\right )^{3/4}}{1+x^4} \, dx=\int \frac {{\left (x^4-1\right )}^{3/4}}{x^4+1} \,d x \]
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