Integrand size = 24, antiderivative size = 91 \[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{1+x^4}}+\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \]
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Time = 0.12 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6857, 246, 218, 212, 209, 1418, 390, 385} \[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt [4]{2}}+\text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+1}}\right )}{4 \sqrt [4]{2}}-\frac {x}{2 \sqrt [4]{x^4+1}} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 385
Rule 390
Rule 1418
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt [4]{1+x^4}}+\frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt [4]{1+x^4}} \, dx+\int \frac {1}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx \\ & = 2 \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )+\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{5/4}} \, dx \\ & = -\frac {x}{2 \sqrt [4]{1+x^4}}+\frac {1}{2} \int \frac {1}{\left (-1+x^4\right ) \sqrt [4]{1+x^4}} \, dx+\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+x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = -\frac {x}{2 \sqrt [4]{1+x^4}}+\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+2 x^4} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = -\frac {x}{2 \sqrt [4]{1+x^4}}+\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{1+x^4}}\right ) \\ & = -\frac {x}{2 \sqrt [4]{1+x^4}}+\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{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 )}{4 \sqrt [4]{2}} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00 \[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{1+x^4}}+\arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}}+\text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{1+x^4}}\right )}{4 \sqrt [4]{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(67)=134\).
Time = 4.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.66
method | result | size |
pseudoelliptic | \(-\frac {\frac {\ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{4}+1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{8}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{4}+1\right )^{\frac {1}{4}}}{4}+2 \arctan \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}}{x}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}+\ln \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}-x}{x}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}-\ln \left (\frac {\left (x^{4}+1\right )^{\frac {1}{4}}+x}{x}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}+x}{2 \left (x^{4}+1\right )^{\frac {1}{4}}}\) | \(151\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.24 \[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=-\frac {2^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (\frac {2^{\frac {1}{4}} x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - 2^{\frac {3}{4}} {\left (x^{4} + 1\right )} \log \left (-\frac {2^{\frac {1}{4}} x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - 2^{\frac {3}{4}} {\left (i \, x^{4} + i\right )} \log \left (\frac {i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - 2^{\frac {3}{4}} {\left (-i \, x^{4} - i\right )} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 16 \, {\left (x^{4} + 1\right )} \arctan \left (\frac {{\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) - 8 \, {\left (x^{4} + 1\right )} \log \left (\frac {x + {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 8 \, {\left (x^{4} + 1\right )} \log \left (-\frac {x - {\left (x^{4} + 1\right )}^{\frac {1}{4}}}{x}\right ) + 8 \, {\left (x^{4} + 1\right )}^{\frac {3}{4}} x}{16 \, {\left (x^{4} + 1\right )}} \]
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\[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int \frac {2 x^{8} - 1}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )^{\frac {5}{4}}}\, dx \]
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\[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 1}{{\left (x^{8} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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\[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int { \frac {2 \, x^{8} - 1}{{\left (x^{8} - 1\right )} {\left (x^{4} + 1\right )}^{\frac {1}{4}}} \,d x } \]
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Timed out. \[ \int \frac {-1+2 x^8}{\sqrt [4]{1+x^4} \left (-1+x^8\right )} \, dx=\int \frac {2\,x^8-1}{{\left (x^4+1\right )}^{1/4}\,\left (x^8-1\right )} \,d x \]
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