Optimal. Leaf size=125 \[ \frac {2}{3} \left (x^4-1\right )^{3/4}+\tan ^{-1}\left (\frac {\sqrt [4]{x^4-1}}{x}\right )+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt {x^4-1}}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {\sqrt [4]{x^4-1}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+1}\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.14, antiderivative size = 170, normalized size of antiderivative = 1.36, number of steps used = 18, number of rules used = 14, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {1833, 240, 212, 206, 203, 446, 80, 63, 297, 1162, 617, 204, 1165, 628} \begin {gather*} \frac {2}{3} \left (x^4-1\right )^{3/4}+\frac {\log \left (\sqrt {x^4-1}-\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\frac {\log \left (\sqrt {x^4-1}+\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{4 \sqrt {2}}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{x^4-1}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (\sqrt {2} \sqrt [4]{x^4-1}+1\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 203
Rule 204
Rule 206
Rule 212
Rule 240
Rule 297
Rule 446
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1833
Rubi steps
\begin {align*} \int \frac {1-2 x+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx &=\int \left (-\frac {2}{\sqrt [4]{-1+x^4}}+\frac {1+2 x^4}{x \sqrt [4]{-1+x^4}}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\sqrt [4]{-1+x^4}} \, dx\right )+\int \frac {1+2 x^4}{x \sqrt [4]{-1+x^4}} \, dx\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1+2 x}{\sqrt [4]{-1+x} x} \, dx,x,x^4\right )-2 \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+x} x} \, dx,x,x^4\right )-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt [4]{-1+x^4}\right )\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt [4]{-1+x^4}\right )}{4 \sqrt {2}}\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}\\ &=\frac {2}{3} \left (-1+x^4\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tan ^{-1}\left (1-\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}+\frac {\tan ^{-1}\left (1+\sqrt {2} \sqrt [4]{-1+x^4}\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {\log \left (1-\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt [4]{-1+x^4}+\sqrt {-1+x^4}\right )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 70, normalized size = 0.56 \begin {gather*} \frac {1}{3} \left (x^4-1\right )^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};1-x^4\right )+\frac {2}{3} \left (x^4-1\right )^{3/4}-\tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-1}}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 4.73, size = 120, normalized size = 0.96 \begin {gather*} \frac {2}{3} \left (x^4-1\right )^{3/4}+\tan ^{-1}\left (\frac {\sqrt [4]{x^4-1}}{x}\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^4-1}}{\sqrt {x^4-1}-1}\right )}{2 \sqrt {2}}-\tanh ^{-1}\left (\frac {\sqrt [4]{x^4-1}}{x}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{x^4-1}}{\sqrt {x^4-1}+1}\right )}{2 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 17.87, size = 517, normalized size = 4.14 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} + 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - {\left (4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{4} + 2 \, \sqrt {2} \sqrt {x^{4} - 1} {\left (x^{4} - 4\right )} + \sqrt {2} {\left (x^{8} - 10 \, x^{4} + 8\right )} + 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}}}{x^{8} - 16 \, x^{4} + 16}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {x^{8} + 4 \, \sqrt {x^{4} - 1} x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} {\left (x^{4} - 4\right )} - 2 \, \sqrt {2} {\left (3 \, x^{4} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} - {\left (4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{4} - 2 \, \sqrt {2} \sqrt {x^{4} - 1} {\left (x^{4} - 4\right )} - \sqrt {2} {\left (x^{8} - 10 \, x^{4} + 8\right )} + 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}}{x^{4}}}}{x^{8} - 16 \, x^{4} + 16}\right ) - \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}\right )}}{x^{4}}\right ) + \frac {1}{16} \, \sqrt {2} \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + 4 \, \sqrt {x^{4} - 1}\right )}}{x^{4}}\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \frac {1}{2} \, \arctan \left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x\right ) + \frac {1}{2} \, \log \left (-2 \, x^{4} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 \, \sqrt {x^{4} - 1} x^{2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{4} - 2 \, x + 1}{{\left (x^{4} - 1\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 142, normalized size = 1.14 \begin {gather*} \frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} \left (\frac {\pi \sqrt {2}\, x^{4} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], x^{4}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \relax (2)-\frac {\pi }{2}+4 \ln \relax (x )+i \pi \right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{8 \pi \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}+\frac {\left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} x^{4} \hypergeom \left (\left [\frac {1}{4}, 1\right ], \relax [2], x^{4}\right )}{2 \mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {2 \left (-\mathrm {signum}\left (x^{4}-1\right )\right )^{\frac {1}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{4}\right ], \left [\frac {5}{4}\right ], x^{4}\right )}{\mathrm {signum}\left (x^{4}-1\right )^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 155, normalized size = 1.24 \begin {gather*} \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) + \frac {1}{4} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {1}{8} \, \sqrt {2} \log \left (-\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} + \sqrt {x^{4} - 1} + 1\right ) + \frac {2}{3} \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} + \arctan \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + 1\right ) + \frac {1}{2} \, \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.30, size = 81, normalized size = 0.65 \begin {gather*} \frac {2\,{\left (x^4-1\right )}^{3/4}}{3}-\frac {2\,x\,{\left (1-x^4\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{4};\ \frac {5}{4};\ x^4\right )}{{\left (x^4-1\right )}^{1/4}}+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\sqrt {2}\,{\left (x^4-1\right )}^{1/4}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 3.18, size = 71, normalized size = 0.57 \begin {gather*} - \frac {x e^{- \frac {i \pi }{4}} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {x^{4}} \right )}}{2 \Gamma \left (\frac {5}{4}\right )} + \frac {2 \left (x^{4} - 1\right )^{\frac {3}{4}}}{3} - \frac {\Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 x \Gamma \left (\frac {5}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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