Optimal. Leaf size=203 \[ -\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}-2 \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
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Rubi [A] time = 0.202828, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ -\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}-2 \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + x)^(1/4)/((1 - x)^(1/4)*x),x]
[Out]
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Rubi in Sympy [A] time = 20.788, size = 172, normalized size = 0.85 \[ - \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{2} + \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + \frac{\sqrt{- x + 1}}{\sqrt{x + 1}} + 1 \right )}}{2} - 2 \operatorname{atan}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} - 1 \right )} - \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{- x + 1}}{\sqrt [4]{x + 1}} + 1 \right )} - 2 \operatorname{atanh}{\left (\frac{\sqrt [4]{x + 1}}{\sqrt [4]{- x + 1}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x)**(1/4)/(1-x)**(1/4)/x,x)
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Mathematica [C] time = 0.207344, size = 119, normalized size = 0.59 \[ \frac{72 (x+1)^{5/4} F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{x+1}{2},x+1\right )}{5 \sqrt [4]{1-x} x \left (18 F_1\left (\frac{5}{4};\frac{1}{4},1;\frac{9}{4};\frac{x+1}{2},x+1\right )+(x+1) \left (8 F_1\left (\frac{9}{4};\frac{1}{4},2;\frac{13}{4};\frac{x+1}{2},x+1\right )+F_1\left (\frac{9}{4};\frac{5}{4},1;\frac{13}{4};\frac{x+1}{2},x+1\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(1 + x)^(1/4)/((1 - x)^(1/4)*x),x]
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Maple [F] time = 0.057, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x)^(1/4)/(1-x)^(1/4)/x,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{1}{4}}}{x{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(1/4)/(x*(-x + 1)^(1/4)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.245397, size = 443, normalized size = 2.18 \[ \frac{1}{2} \, \sqrt{2}{\left (2 \, \sqrt{2} \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + \sqrt{2} \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - \sqrt{2} \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) + 4 \, \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - 1}\right ) + 4 \, \arctan \left (\frac{x - 1}{\sqrt{2}{\left (x - 1\right )} \sqrt{-\frac{\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1}{x - 1}} + \sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + 1}\right ) - \log \left (\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} + x - \sqrt{x + 1} \sqrt{-x + 1} - 1\right )}}{x - 1}\right ) + \log \left (-\frac{2 \,{\left (\sqrt{2}{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - x + \sqrt{x + 1} \sqrt{-x + 1} + 1\right )}}{x - 1}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(1/4)/(x*(-x + 1)^(1/4)),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [4]{x + 1}}{x \sqrt [4]{- x + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x)**(1/4)/(1-x)**(1/4)/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (x + 1\right )}^{\frac{1}{4}}}{x{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 1)^(1/4)/(x*(-x + 1)^(1/4)),x, algorithm="giac")
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