Integrand size = 15, antiderivative size = 58 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {52, 65, 218, 212, 209} \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )+4 \sqrt [4]{1-x} \]
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 218
Rubi steps \begin{align*} \text {integral}& = 4 \sqrt [4]{1-x}+2 \int \frac {1}{(1-x)^{3/4} (1+x)} \, dx \\ & = 4 \sqrt [4]{1-x}-8 \text {Subst}\left (\int \frac {1}{2-x^4} \, dx,x,\sqrt [4]{1-x}\right ) \\ & = 4 \sqrt [4]{1-x}-\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{1-x}\right )-\left (2 \sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}+x^2} \, dx,x,\sqrt [4]{1-x}\right ) \\ & = 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]
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Time = 1.34 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.03
method | result | size |
derivativedivides | \(4 \left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}} \left (\ln \left (\frac {\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right )\) | \(60\) |
default | \(4 \left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}} \left (\ln \left (\frac {\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right )\) | \(60\) |
pseudoelliptic | \(4 \left (1-x \right )^{\frac {1}{4}}-\ln \left (\frac {-\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}{-\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-2 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\) | \(66\) |
trager | \(4 \left (1-x \right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {-x \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (1-x \right )^{\frac {1}{4}}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )-4 \left (1-x \right )^{\frac {3}{4}}-4 \sqrt {1-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{1+x}\right )-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (1-x \right )^{\frac {1}{4}}-4 \sqrt {1-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )-4 \left (1-x \right )^{\frac {3}{4}}}{1+x}\right )\) | \(208\) |
risch | \(-\frac {4 \left (-1+x \right )}{\left (1-x \right )^{\frac {3}{4}}}+\frac {\left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x -x^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {3}{4}}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x -2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{3}+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {3}{4}}+7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )\right ) \left (-\left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1-x \right )^{\frac {3}{4}}}\) | \(529\) |
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) - i \cdot 2^{\frac {1}{4}} \log \left (i \cdot 2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + i \cdot 2^{\frac {1}{4}} \log \left (-i \cdot 2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left (-2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \]
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Result contains complex when optimal does not.
Time = 2.90 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.19 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=\frac {5 \sqrt [4]{-1} \sqrt [4]{x - 1} \Gamma \left (\frac {5}{4}\right )}{\Gamma \left (\frac {9}{4}\right )} + \frac {5 \sqrt [4]{-2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {5 \left (-1\right )^{\frac {3}{4}} \cdot \sqrt [4]{2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {3 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {5 \sqrt [4]{-2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {5 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {5 \left (-1\right )^{\frac {3}{4}} \cdot \sqrt [4]{2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {7 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} - {\left (-x + 1\right )}^{\frac {1}{4}}}{2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \]
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Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \]
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Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=4\,{\left (1-x\right )}^{1/4}-2\,2^{1/4}\,\mathrm {atanh}\left (\frac {2^{3/4}\,{\left (1-x\right )}^{1/4}}{2}\right )-2\,2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (1-x\right )}^{1/4}}{2}\right ) \]
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