Integrand size = 22, antiderivative size = 53 \[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt [4]{2}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.40 (sec) , antiderivative size = 314, normalized size of antiderivative = 5.92, number of steps used = 21, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1600, 6857, 415, 226, 418, 1231, 1721} \[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {x^4+1}}\right )}{2 \sqrt [4]{2}}+\frac {i \left (\sqrt {2}+(1+i)\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \sqrt {x^4+1}}+\frac {i \left (\sqrt {2}+(-1+i)\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \sqrt {x^4+1}}+\frac {\left ((-1-i)-i \sqrt {2}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \sqrt {x^4+1}}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (1+(-1)^{3/4}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {x^4+1}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {x^4+1}}\right )}{2 \sqrt [4]{2}} \]
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Rule 226
Rule 415
Rule 418
Rule 1231
Rule 1600
Rule 1721
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (-1+x^4\right ) \sqrt {1+x^4}}{1+x^8} \, dx \\ & = \int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {1+x^4}}{i-x^4}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {1+x^4}}{i+x^4}\right ) \, dx \\ & = \left (-\frac {1}{2}-\frac {i}{2}\right ) \int \frac {\sqrt {1+x^4}}{i-x^4} \, dx+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {\sqrt {1+x^4}}{i+x^4} \, dx \\ & = -\left (i \int \frac {1}{\left (i-x^4\right ) \sqrt {1+x^4}} \, dx\right )-i \int \frac {1}{\left (i+x^4\right ) \sqrt {1+x^4}} \, dx+\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{2} \int \frac {1}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {1}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx \\ & = \frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\left (\left (\frac {1}{4}+\frac {i}{4}\right ) \left (1+\sqrt [4]{-1}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\left (\left (-\frac {1}{4}-\frac {i}{4}\right ) \sqrt [4]{-1} \left (1+\sqrt [4]{-1}\right )\right ) \int \frac {1+x^2}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx-\left (\left (\frac {1}{4}-\frac {i}{4}\right ) (-1)^{3/4} \left (1-(-1)^{3/4}\right )\right ) \int \frac {1+x^2}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx-\left (\left (\frac {1}{4}-\frac {i}{4}\right ) \left (1+(-1)^{3/4}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\left (\left (-\frac {1}{4}+\frac {i}{4}\right ) (-1)^{3/4} \left (1+(-1)^{3/4}\right )\right ) \int \frac {1+x^2}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{4} \left ((1+i)-i \sqrt {2}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{4} \left (i \left ((-1-i)+\sqrt {2}\right )\right ) \int \frac {1+x^2}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{4} \left (i \left ((1+i)+\sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx \\ & = -\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt [4]{2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (1+\sqrt [4]{-1}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {1+x^4}}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \left (1+(-1)^{3/4}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{\sqrt {1+x^4}}-\frac {\left ((1+i)-i \sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \sqrt {1+x^4}}+\frac {i \left ((1+i)+\sqrt {2}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{8 \sqrt {1+x^4}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt {1+x^4}}\right )}{2 \sqrt [4]{2}} \]
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Time = 3.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}+1}}{2 x}\right )+\ln \left (\frac {2^{\frac {1}{4}} x +\sqrt {x^{4}+1}}{-2^{\frac {1}{4}} x +\sqrt {x^{4}+1}}\right )\right )}{8}\) | \(56\) |
pseudoelliptic | \(-\frac {2^{\frac {3}{4}} \left (-2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}+1}}{2 x}\right )+\ln \left (\frac {2^{\frac {1}{4}} x +\sqrt {x^{4}+1}}{-2^{\frac {1}{4}} x +\sqrt {x^{4}+1}}\right )\right )}{8}\) | \(56\) |
elliptic | \(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {x^{4}+1}}{2 x}\right )-\ln \left (\frac {\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}+\frac {2^{\frac {3}{4}}}{2}}{\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}-\frac {2^{\frac {3}{4}}}{2}}\right )\right )}{8}\) | \(73\) |
trager | \(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}-8 \sqrt {x^{4}+1}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{2 x^{4}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+2}\right )}{8}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )-8 \sqrt {x^{4}+1}\, x +2 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x^{4}-2}\right )}{8}\) | \(182\) |
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Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 264, normalized size of antiderivative = 4.98 \[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=-\frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (-\frac {2^{\frac {3}{4}} {\left (x^{8} + 4 \, x^{4} + 1\right )} + 4 \, {\left (x^{5} + \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} + 1}\right ) + \frac {1}{16} \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (x^{8} + 4 \, x^{4} + 1\right )} - 4 \, {\left (x^{5} + \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}}{x^{8} + 1}\right ) + \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (i \, x^{8} + 4 i \, x^{4} + i\right )} - 4 \, {\left (x^{5} - \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} - 4 \cdot 2^{\frac {1}{4}} {\left (i \, x^{6} + i \, x^{2}\right )}}{x^{8} + 1}\right ) - \frac {1}{16} i \cdot 2^{\frac {3}{4}} \log \left (\frac {2^{\frac {3}{4}} {\left (-i \, x^{8} - 4 i \, x^{4} - i\right )} - 4 \, {\left (x^{5} - \sqrt {2} x^{3} + x\right )} \sqrt {x^{4} + 1} - 4 \cdot 2^{\frac {1}{4}} {\left (-i \, x^{6} - i \, x^{2}\right )}}{x^{8} + 1}\right ) \]
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\[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \sqrt {x^{4} + 1}}{x^{8} + 1}\, dx \]
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\[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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\[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + 1\right )} \sqrt {x^{4} + 1}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^8}{\sqrt {1+x^4} \left (1+x^8\right )} \, dx=\int \frac {x^8-1}{\sqrt {x^4+1}\,\left (x^8+1\right )} \,d x \]
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