Optimal. Leaf size=257 \[ \frac {1}{2} x \sqrt [4]{-x^2+x^4}-\frac {3}{4} \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )+\frac {3}{4} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^2+x^4}}\right ) \]
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Rubi [A]
time = 0.67, 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 = {2081, 6860,
285, 335, 338, 304, 209, 212, 1283, 1532, 1542, 508} \begin {gather*} -\frac {3 \sqrt [4]{x^4-x^2} \text {ArcTan}\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} \text {ArcTan}\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} \text {ArcTan}\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 {1}{2} \sqrt [4]{x^4-x^2} 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*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 285
Rule 304
Rule 335
Rule 338
Rule 508
Rule 1283
Rule 1532
Rule 1542
Rule 2081
Rule 6860
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 ) \text {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} \text {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 ) \text {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 ) \text {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} \text {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 ) \text {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 ) \text {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} \text {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} \text {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} \text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 [A]
time = 0.70, size = 268, normalized size = 1.04 \begin {gather*} \frac {x^{3/2} \left (-1+x^2\right )^{3/4} \left (10 x^{3/2} \sqrt [4]{-1+x^2}-15 \text {ArcTan}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \sqrt {10 \left (1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+2 \sqrt {10 \left (-1+\sqrt {5}\right )} \text {ArcTan}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )+15 \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-2 \sqrt {10 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )-2 \sqrt {10 \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \sqrt {x}}{\sqrt [4]{-1+x^2}}\right )\right )}{20 \left (x^2 \left (-1+x^2\right )\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1255 vs.
\(2 (187) = 374\).
time = 19.25, size = 1255, normalized size = 4.88 \begin {gather*} -\frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {10} \sqrt {x^{4} - x^{2}} {\left (430 \, x^{3} - \sqrt {5} {\left (448 \, x^{3} - 439 \, x\right )} - 1335 \, x\right )} - \sqrt {10} {\left (1120 \, x^{5} - 1550 \, x^{3} - \sqrt {5} {\left (215 \, x^{5} - 663 \, x^{3} + 224 \, x\right )} + 215 \, x\right )}\right )} \sqrt {40157 \, \sqrt {5} + 36899} \sqrt {\sqrt {5} + 1} + 81862 \, {\left (\sqrt {10} {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (\sqrt {5} {\left (2 \, x^{2} - 1\right )} + 5\right )} + \sqrt {10} {\left (5 \, x^{4} - 5 \, x^{2} - \sqrt {5} {\left (x^{4} - 3 \, x^{2}\right )}\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\sqrt {5} + 1}}{818620 \, {\left (x^{5} - x^{3} - x\right )}}\right ) - \frac {1}{10} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {10} \sqrt {x^{4} - x^{2}} {\left (430 \, x^{3} + \sqrt {5} {\left (448 \, x^{3} - 439 \, x\right )} - 1335 \, x\right )} + \sqrt {10} {\left (1120 \, x^{5} - 1550 \, x^{3} + \sqrt {5} {\left (215 \, x^{5} - 663 \, x^{3} + 224 \, x\right )} + 215 \, x\right )}\right )} \sqrt {40157 \, \sqrt {5} - 36899} \sqrt {\sqrt {5} - 1} + 81862 \, {\left (\sqrt {10} {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (\sqrt {5} {\left (2 \, x^{2} - 1\right )} - 5\right )} + \sqrt {10} {\left (5 \, x^{4} - 5 \, x^{2} + \sqrt {5} {\left (x^{4} - 3 \, x^{2}\right )}\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}\right )} \sqrt {\sqrt {5} - 1}}{818620 \, {\left (x^{5} - x^{3} - x\right )}}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\frac {20 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (448 \, x^{2} + \sqrt {5} {\left (86 \, x^{2} + 181\right )} - 9\right )} + {\left (2 \, \sqrt {10} \sqrt {x^{4} - x^{2}} {\left (905 \, x^{3} - \sqrt {5} {\left (9 \, x^{3} - 457 \, x\right )} - 475 \, x\right )} - \sqrt {10} {\left (45 \, x^{5} - 1855 \, x^{3} - \sqrt {5} {\left (905 \, x^{5} - 923 \, x^{3} + 9 \, x\right )} + 905 \, x\right )}\right )} \sqrt {\sqrt {5} + 1} - 20 \, {\left (9 \, x^{4} - 457 \, x^{2} - \sqrt {5} {\left (181 \, x^{4} - 95 \, x^{2}\right )}\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}{x^{5} - x^{3} - x}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} + 1} \log \left (\frac {20 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (448 \, x^{2} + \sqrt {5} {\left (86 \, x^{2} + 181\right )} - 9\right )} - {\left (2 \, \sqrt {10} \sqrt {x^{4} - x^{2}} {\left (905 \, x^{3} - \sqrt {5} {\left (9 \, x^{3} - 457 \, x\right )} - 475 \, x\right )} - \sqrt {10} {\left (45 \, x^{5} - 1855 \, x^{3} - \sqrt {5} {\left (905 \, x^{5} - 923 \, x^{3} + 9 \, x\right )} + 905 \, x\right )}\right )} \sqrt {\sqrt {5} + 1} - 20 \, {\left (9 \, x^{4} - 457 \, x^{2} - \sqrt {5} {\left (181 \, x^{4} - 95 \, x^{2}\right )}\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}{x^{5} - x^{3} - x}\right ) - \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {20 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (448 \, x^{2} - \sqrt {5} {\left (86 \, x^{2} + 181\right )} - 9\right )} + {\left (2 \, \sqrt {10} \sqrt {x^{4} - x^{2}} {\left (905 \, x^{3} + \sqrt {5} {\left (9 \, x^{3} - 457 \, x\right )} - 475 \, x\right )} + \sqrt {10} {\left (45 \, x^{5} - 1855 \, x^{3} + \sqrt {5} {\left (905 \, x^{5} - 923 \, x^{3} + 9 \, x\right )} + 905 \, x\right )}\right )} \sqrt {\sqrt {5} - 1} + 20 \, {\left (9 \, x^{4} - 457 \, x^{2} + \sqrt {5} {\left (181 \, x^{4} - 95 \, x^{2}\right )}\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}{x^{5} - x^{3} - x}\right ) + \frac {1}{40} \, \sqrt {10} \sqrt {\sqrt {5} - 1} \log \left (\frac {20 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}} {\left (448 \, x^{2} - \sqrt {5} {\left (86 \, x^{2} + 181\right )} - 9\right )} - {\left (2 \, \sqrt {10} \sqrt {x^{4} - x^{2}} {\left (905 \, x^{3} + \sqrt {5} {\left (9 \, x^{3} - 457 \, x\right )} - 475 \, x\right )} + \sqrt {10} {\left (45 \, x^{5} - 1855 \, x^{3} + \sqrt {5} {\left (905 \, x^{5} - 923 \, x^{3} + 9 \, x\right )} + 905 \, x\right )}\right )} \sqrt {\sqrt {5} - 1} + 20 \, {\left (9 \, x^{4} - 457 \, x^{2} + \sqrt {5} {\left (181 \, x^{4} - 95 \, x^{2}\right )}\right )} {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}}}{x^{5} - x^{3} - x}\right ) + \frac {1}{2} \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x + \frac {3}{8} \, \arctan \left (\frac {2 \, {\left ({\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}\right )}}{x}\right ) + \frac {3}{8} \, \log \left (\frac {2 \, x^{3} + 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x^{2}} x - x + 2 \, {\left (x^{4} - x^{2}\right )}^{\frac {3}{4}}}{x}\right ) \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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Giac [A]
time = 0.48, 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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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.
[In]
[Out]
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