Integrand size = 42, antiderivative size = 44 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=-\frac {\left (\left (1+x^4\right )^5\right )^{9/10} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2} \left (1+x^4\right )^{9/2}} \]
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Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.55, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6820, 1972, 1713, 209} \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}} \]
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Rule 209
Rule 1713
Rule 1972
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{\left (1+x^4\right )^5}} \, dx \\ & = \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx \\ & = -\text {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=-\frac {\sqrt {1+x^4} \arctan \left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2} \sqrt [10]{\left (1+x^4\right )^5}} \]
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\[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \left (x^{20}+5 x^{16}+10 x^{12}+10 x^{8}+5 x^{4}+1\right )^{\frac {1}{10}}}d x\]
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none
Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.02 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=-\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} x}{x^{4} + 1}\right ) \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt [10]{\left (x^{4} + 1\right )^{5}}}\, dx \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=\int { \frac {x^{2} - 1}{{\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,{\left (x^{20}+5\,x^{16}+10\,x^{12}+10\,x^8+5\,x^4+1\right )}^{1/10}} \,d x \]
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