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 = 0.107982, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {105, 63, 331, 297, 1162, 617, 204, 1165, 628, 93, 212, 206, 203} \[ -\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{\sqrt{2}}-2 \tan ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )-\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
Antiderivative was successfully verified.
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Rule 105
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rule 93
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x} x} \, dx &=\int \frac{1}{\sqrt [4]{1-x} (1+x)^{3/4}} \, dx+\int \frac{1}{\sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\left (4 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-x}\right )\right )+4 \operatorname{Subst}\left (\int \frac{1}{-1+x^4} \, dx,x,\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\right )-2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-4 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )+2 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt{2}}-\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt{2}}-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )\\ &=-2 \tan ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )+\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-2 \tanh ^{-1}\left (\frac{\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-x}}{\sqrt{1+x}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0235178, size = 74, normalized size = 0.36 \[ -\frac{2 (1-x)^{3/4} \left (\sqrt [4]{2} (x+1)^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{1-x}{2}\right )+2 \, _2F_1\left (\frac{3}{4},1;\frac{7}{4};\frac{1-x}{x+1}\right )\right )}{3 (x+1)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + 1\right )}^{\frac{1}{4}}}{x{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72465, size = 981, normalized size = 4.83 \begin{align*} 2 \, \sqrt{2} \arctan \left (\frac{\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}{x - 1}\right ) + 2 \, \sqrt{2} \arctan \left (\frac{\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}{x - 1}\right ) - \frac{1}{2} \, \sqrt{2} \log \left (\frac{4 \,{\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 ) + \frac{1}{2} \, \sqrt{2} \log \left (-\frac{4 \,{\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 ) + 2 \, \arctan \left (\frac{{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}}}{x - 1}\right ) + \log \left (\frac{x +{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) - \log \left (-\frac{x -{\left (x + 1\right )}^{\frac{1}{4}}{\left (-x + 1\right )}^{\frac{3}{4}} - 1}{x - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt [4]{x + 1}}{x \sqrt [4]{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + 1\right )}^{\frac{1}{4}}}{x{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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