Integrand size = 32, antiderivative size = 20 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{x \left (1+x^2\right )} \]
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\[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2}{\sqrt {1+x^6}}-\frac {1}{x^2 \sqrt {1+x^6}}-\frac {3}{\left (1+x^2\right ) \sqrt {1+x^6}}\right ) \, dx \\ & = 2 \int \frac {1}{\sqrt {1+x^6}} \, dx-3 \int \frac {1}{\left (1+x^2\right ) \sqrt {1+x^6}} \, dx-\int \frac {1}{x^2 \sqrt {1+x^6}} \, dx \\ & = \frac {\sqrt {1+x^6}}{x}+\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-2 \int \frac {x^4}{\sqrt {1+x^6}} \, dx-3 \int \left (\frac {i}{2 (i-x) \sqrt {1+x^6}}+\frac {i}{2 (i+x) \sqrt {1+x^6}}\right ) \, dx \\ & = \frac {\sqrt {1+x^6}}{x}+\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-\frac {3}{2} i \int \frac {1}{(i-x) \sqrt {1+x^6}} \, dx-\frac {3}{2} i \int \frac {1}{(i+x) \sqrt {1+x^6}} \, dx-\left (-1+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^6}} \, dx+\int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {1+x^6}} \, dx \\ & = \frac {\sqrt {1+x^6}}{x}-\frac {\left (1+\sqrt {3}\right ) x \sqrt {1+x^6}}{1+\left (1+\sqrt {3}\right ) x^2}+\frac {\sqrt [4]{3} x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} E\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}+\frac {x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}+\frac {\left (1-\sqrt {3}\right ) x \left (1+x^2\right ) \sqrt {\frac {1-x^2+x^4}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {1+\left (1-\sqrt {3}\right ) x^2}{1+\left (1+\sqrt {3}\right ) x^2}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{2 \sqrt [4]{3} \sqrt {\frac {x^2 \left (1+x^2\right )}{\left (1+\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {1+x^6}}-\frac {3}{2} i \int \frac {1}{(i-x) \sqrt {1+x^6}} \, dx-\frac {3}{2} i \int \frac {1}{(i+x) \sqrt {1+x^6}} \, dx \\ \end{align*}
Time = 8.36 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {1+x^6}}{x+x^3} \]
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Time = 1.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
trager | \(\frac {\sqrt {x^{6}+1}}{x \left (x^{2}+1\right )}\) | \(19\) |
gosper | \(\frac {x^{4}-x^{2}+1}{x \sqrt {x^{6}+1}}\) | \(22\) |
risch | \(\frac {x^{4}-x^{2}+1}{x \sqrt {x^{6}+1}}\) | \(22\) |
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Time = 0.25 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{6} + 1}}{x^{3} + x} \]
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\[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\int \frac {2 x^{4} - 2 x^{2} - 1}{x^{2} \sqrt {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^{4} - x^{2} + 1}}{\sqrt {x^{2} + 1} x} \]
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\[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\int { \frac {2 \, x^{4} - 2 \, x^{2} - 1}{\sqrt {x^{6} + 1} {\left (x^{2} + 1\right )} x^{2}} \,d x } \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-1-2 x^2+2 x^4}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx=\frac {\sqrt {x^6+1}}{x\,\left (x^2+1\right )} \]
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