Integrand size = 14, antiderivative size = 54 \[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\frac {\left (\left (1+x^4\right )^2\right )^{7/8} \left (\frac {1}{2} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )\right )}{\left (1+x^4\right )^{7/4}} \]
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Time = 0.02 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1357, 246, 218, 212, 209} \[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\frac {\sqrt [4]{x^4+1} \arctan \left (\frac {x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [8]{x^8+2 x^4+1}}+\frac {\sqrt [4]{x^4+1} \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt [8]{x^8+2 x^4+1}} \]
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Rule 209
Rule 212
Rule 218
Rule 246
Rule 1357
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{2+2 x^4} \int \frac {1}{\sqrt [4]{2+2 x^4}} \, dx}{\sqrt [8]{1+2 x^4+x^8}} \\ & = \frac {\sqrt [4]{2+2 x^4} \text {Subst}\left (\int \frac {1}{1-2 x^4} \, dx,x,\frac {x}{\sqrt [4]{2+2 x^4}}\right )}{\sqrt [8]{1+2 x^4+x^8}} \\ & = \frac {\sqrt [4]{2+2 x^4} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+2 x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}}+\frac {\sqrt [4]{2+2 x^4} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{2+2 x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}} \\ & = \frac {\sqrt [4]{1+x^4} \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}}+\frac {\sqrt [4]{1+x^4} \text {arctanh}\left (\frac {x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt [8]{1+2 x^4+x^8}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.30 \[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\frac {\sqrt [4]{1+x^4} \left (2 \arctan \left (\frac {x}{\sqrt [4]{1+x^4}}\right )-\log \left (1-\frac {x}{\sqrt [4]{1+x^4}}\right )+\log \left (1+\frac {x}{\sqrt [4]{1+x^4}}\right )\right )}{4 \sqrt [8]{\left (1+x^4\right )^2}} \]
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\[\int \frac {1}{\left (x^{8}+2 x^{4}+1\right )^{\frac {1}{8}}}d x\]
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none
Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=-\frac {1}{2} \, \arctan \left (\frac {{\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}{x}\right ) + \frac {1}{4} \, \log \left (\frac {x + {\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}{x}\right ) - \frac {1}{4} \, \log \left (-\frac {x - {\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}}{x}\right ) \]
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\[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\int \frac {1}{\sqrt [8]{x^{8} + 2 x^{4} + 1}}\, dx \]
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\[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}} \,d x } \]
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\[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\int { \frac {1}{{\left (x^{8} + 2 \, x^{4} + 1\right )}^{\frac {1}{8}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt [8]{1+2 x^4+x^8}} \, dx=\int \frac {1}{{\left (x^8+2\,x^4+1\right )}^{1/8}} \,d x \]
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