Optimal. Leaf size=257 \[ \frac {1}{2} \sqrt [4]{x^4-x^2} x-\frac {3}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-x^2}}\right )+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-x^2}}\right )+\frac {3}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^2}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-x^2}}\right )-\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-x^2}}\right ) \]
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Rubi [A] time = 0.81, antiderivative size = 435, normalized size of antiderivative = 1.69, number of steps used = 25, number of rules used = 12, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.387, Rules used = {2056, 6728, 279, 329, 331, 298, 203, 206, 1269, 1518, 1528, 494} \begin {gather*} \frac {1}{2} \sqrt [4]{x^4-x^2} x-\frac {3 \sqrt [4]{x^4-x^2} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{x^2-1} \sqrt {x}}+\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}+\frac {3 \sqrt [4]{x^4-x^2} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{4 \sqrt [4]{x^2-1} \sqrt {x}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}}-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{x^4-x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{x^2-1}}\right )}{\sqrt {5} \sqrt [4]{x^2-1} \sqrt {x}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 203
Rule 206
Rule 279
Rule 298
Rule 329
Rule 331
Rule 494
Rule 1269
Rule 1518
Rule 1528
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt [4]{-x^2+x^4}}{-1-x^2+x^4} \, dx &=\frac {\sqrt [4]{-x^2+x^4} \int \frac {\sqrt {x} \sqrt [4]{-1+x^2} \left (-1+x^4\right )}{-1-x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\sqrt [4]{-x^2+x^4} \int \left (\sqrt {x} \sqrt [4]{-1+x^2}+\frac {x^{5/2} \sqrt [4]{-1+x^2}}{-1-x^2+x^4}\right ) \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {\sqrt [4]{-x^2+x^4} \int \sqrt {x} \sqrt [4]{-1+x^2} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \int \frac {x^{5/2} \sqrt [4]{-1+x^2}}{-1-x^2+x^4} \, dx}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \int \frac {\sqrt {x}}{\left (-1+x^2\right )^{3/4}} \, dx}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{-1+x^4}}{-1-x^4+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{2 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {2 x^2}{\sqrt {5} \left (1+\sqrt {5}-2 x^4\right ) \left (-1+x^4\right )^{3/4}}-\frac {2 x^2}{\sqrt {5} \left (-1+x^4\right )^{3/4} \left (-1+\sqrt {5}+2 x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {\sqrt [4]{-x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{-x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{-x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+\sqrt {5}-2 x^4\right ) \left (-1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4} \left (-1+\sqrt {5}+2 x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3 \sqrt [4]{-x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {3 \sqrt [4]{-x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+\sqrt {5}-\left (-1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (4 \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3 \sqrt [4]{-x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {3 \sqrt [4]{-x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\left (2 \sqrt {\frac {2}{5}} \sqrt [4]{-x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\left (1-\sqrt {5}\right ) \sqrt {x} \sqrt [4]{-1+x^2}}\\ &=\frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3 \sqrt [4]{-x^2+x^4} \tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}+\frac {3 \sqrt [4]{-x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{4 \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}-\frac {\sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \sqrt [4]{-x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )}{\sqrt {5} \sqrt {x} \sqrt [4]{-1+x^2}}\\ \end {align*}
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Mathematica [C] time = 2.07, size = 231, normalized size = 0.90 \begin {gather*} \frac {\sqrt [4]{x^2 \left (x^2-1\right )} \left (\frac {4 \left (\sqrt {5}-1\right )^{3/4} \left (5+\sqrt {5}\right ) x^{3/2} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};\frac {\left (1+\sqrt {5}\right ) x^2}{2 x^2+\sqrt {5}-1}\right )}{\sqrt [4]{1-x^2} \left (2 x^2+\sqrt {5}-1\right )^{3/4}}+\frac {4 \left (\sqrt {5}-5\right ) x^{3/2} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (-1+\sqrt {5}\right ) x^2}{\left (1+\sqrt {5}\right ) \left (x^2-1\right )}\right )}{x^2-1}-\frac {45 \left (\tan ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )-\tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{x^2-1}}\right )\right )}{\sqrt [4]{x^2-1}}\right )}{60 \sqrt {x}}+\frac {1}{2} \sqrt [4]{x^2 \left (x^2-1\right )} x \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.87, size = 257, normalized size = 1.00 \begin {gather*} \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3}{4} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\frac {3}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 35.91, size = 1255, normalized size = 4.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.92, size = 248, normalized size = 0.96 \begin {gather*} -\frac {1}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{20} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} - {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{20} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {3}{4} \, \arctan \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {3}{8} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {3}{8} \, \log \left (-{\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{4}}}{x^{4}-x^{2}-1}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}{x^{4} - x^{2} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {\left (x^4-1\right )\,{\left (x^4-x^2\right )}^{1/4}}{-x^4+x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{4} - x^{2} - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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