Optimal. Leaf size=58 \[ 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {50, 63, 212, 206, 203} \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 203
Rule 206
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1-x}}{1+x} \, dx &=4 \sqrt [4]{1-x}+2 \int \frac {1}{(1-x)^{3/4} (1+x)} \, dx\\ &=4 \sqrt [4]{1-x}-8 \operatorname {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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{1-x}\right )-\left (2 \sqrt {2}\right ) \operatorname {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*}
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Mathematica [A] time = 0.02, size = 58, normalized size = 1.00 \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 58, normalized size = 1.00 \begin {gather*} 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 {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 82, normalized size = 1.41 \begin {gather*} 4 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {\sqrt {2} + \sqrt {-x + 1}} - \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 (-2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.17, size = 64, normalized size = 1.10 \begin {gather*} -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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 62, normalized size = 1.07 \begin {gather*} -2 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (-x +1\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )-2^{\frac {1}{4}} \ln \left (\frac {\left (-x +1\right )^{\frac {1}{4}}+2^{\frac {1}{4}}}{\left (-x +1\right )^{\frac {1}{4}}-2^{\frac {1}{4}}}\right )+4 \left (-x +1\right )^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.91, size = 61, normalized size = 1.05 \begin {gather*} -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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 46, normalized size = 0.79 \begin {gather*} 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.35, size = 243, normalized size = 4.19 \begin {gather*} \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}} \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}} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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