Optimal. Leaf size=75 \[ \tan ^{-1}\left (\frac {x \sqrt {-4 x^4-x^2+2}}{4 x^4+x^2-2}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-4 x^4-x^2+2}}{4 x^4+x^2-2}\right ) \]
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Rubi [C] time = 1.79, antiderivative size = 475, normalized size of antiderivative = 6.33, number of steps used = 32, number of rules used = 8, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.170, Rules used = {6725, 1208, 1180, 524, 424, 419, 1212, 537} \begin {gather*} -\frac {1}{4} \sqrt {3 \left (11 \sqrt {33}-59\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {\left (1+4 \sqrt {2}-\sqrt {33}\right ) F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {\left (1-4 \sqrt {2}-\sqrt {33}\right ) F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (-\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 419
Rule 424
Rule 524
Rule 537
Rule 1180
Rule 1208
Rule 1212
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt {2-x^2-4 x^4} \left (1+2 x^4\right )}{\left (-1+2 x^4\right ) \left (-1-x^2+2 x^4\right )} \, dx &=\int \left (\frac {\sqrt {2-x^2-4 x^4}}{-1+x^2}+\frac {2 \sqrt {2-x^2-4 x^4}}{1+2 x^2}-\frac {4 x^2 \sqrt {2-x^2-4 x^4}}{-1+2 x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt {2-x^2-4 x^4}}{1+2 x^2} \, dx-4 \int \frac {x^2 \sqrt {2-x^2-4 x^4}}{-1+2 x^4} \, dx+\int \frac {\sqrt {2-x^2-4 x^4}}{-1+x^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {-2+8 x^2}{\sqrt {2-x^2-4 x^4}} \, dx\right )-3 \int \frac {1}{\left (-1+x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx+3 \int \frac {1}{\left (1+2 x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx-4 \int \left (-\frac {\sqrt {2-x^2-4 x^4}}{2 \sqrt {2} \left (1-\sqrt {2} x^2\right )}+\frac {\sqrt {2-x^2-4 x^4}}{2 \sqrt {2} \left (1+\sqrt {2} x^2\right )}\right ) \, dx-\int \frac {5+4 x^2}{\sqrt {2-x^2-4 x^4}} \, dx\\ &=-\left (2 \int \frac {-2+8 x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx\right )-4 \int \frac {5+4 x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx-12 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \left (-1+x^2\right ) \sqrt {1+\sqrt {33}+8 x^2}} \, dx+12 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \left (1+2 x^2\right ) \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\sqrt {2} \int \frac {\sqrt {2-x^2-4 x^4}}{1-\sqrt {2} x^2} \, dx-\sqrt {2} \int \frac {\sqrt {2-x^2-4 x^4}}{1+\sqrt {2} x^2} \, dx\\ &=3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-2 \left (2 \int \frac {\sqrt {1+\sqrt {33}+8 x^2}}{\sqrt {-1+\sqrt {33}-8 x^2}} \, dx\right )-\frac {\int \frac {-4-\sqrt {2}-4 \sqrt {2} x^2}{\sqrt {2-x^2-4 x^4}} \, dx}{\sqrt {2}}+\frac {\int \frac {-4+\sqrt {2}+4 \sqrt {2} x^2}{\sqrt {2-x^2-4 x^4}} \, dx}{\sqrt {2}}-\left (2 \left (9-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\left (2 \left (3+\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx-\int \frac {1}{\left (1-\sqrt {2} x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx-\int \frac {1}{\left (1+\sqrt {2} x^2\right ) \sqrt {2-x^2-4 x^4}} \, dx\\ &=-\sqrt {2 \left (1+\sqrt {33}\right )} E\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\frac {1}{4} \sqrt {3 \left (-59+11 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-4 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2} \left (1-\sqrt {2} x^2\right )} \, dx-4 \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2} \left (1+\sqrt {2} x^2\right )} \, dx-\left (2 \sqrt {2}\right ) \int \frac {-4-\sqrt {2}-4 \sqrt {2} x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\left (2 \sqrt {2}\right ) \int \frac {-4+\sqrt {2}+4 \sqrt {2} x^2}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx\\ &=-\sqrt {2 \left (1+\sqrt {33}\right )} E\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\frac {1}{4} \sqrt {3 \left (-59+11 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (-\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+2 \left (2 \int \frac {\sqrt {1+\sqrt {33}+8 x^2}}{\sqrt {-1+\sqrt {33}-8 x^2}} \, dx\right )+\left (2 \left (1-4 \sqrt {2}-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx+\left (2 \left (1+4 \sqrt {2}-\sqrt {33}\right )\right ) \int \frac {1}{\sqrt {-1+\sqrt {33}-8 x^2} \sqrt {1+\sqrt {33}+8 x^2}} \, dx\\ &=\frac {\left (1-4 \sqrt {2}-\sqrt {33}\right ) F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {\left (1+4 \sqrt {2}-\sqrt {33}\right ) F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )}{\sqrt {2 \left (1+\sqrt {33}\right )}}+\frac {1}{4} \sqrt {3 \left (13+3 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\frac {1}{4} \sqrt {3 \left (-59+11 \sqrt {33}\right )} F\left (\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{4} \left (1-\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (-\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )-\sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1-\sqrt {33}}{4 \sqrt {2}};\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )+3 \sqrt {\frac {2}{1+\sqrt {33}}} \Pi \left (\frac {1}{8} \left (-1+\sqrt {33}\right );\sin ^{-1}\left (2 \sqrt {\frac {2}{-1+\sqrt {33}}} x\right )|\frac {1}{16} \left (-17+\sqrt {33}\right )\right )\\ \end {align*}
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Mathematica [C] time = 7.04, size = 2667, normalized size = 35.56 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.43, size = 75, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x \sqrt {-4 x^4-x^2+2}}{4 x^4+x^2-2}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-4 x^4-x^2+2}}{4 x^4+x^2-2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 71, normalized size = 0.95 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt {-4 \, x^{4} - x^{2} + 2} x}{2 \, x^{4} + 2 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-4 \, x^{4} - x^{2} + 2} x}{2 \, x^{4} + x^{2} - 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 {{\left (2 \, x^{4} + 1\right )} \sqrt {-4 \, x^{4} - x^{2} + 2}}{{\left (2 \, x^{4} - x^{2} - 1\right )} {\left (2 \, x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 823, normalized size = 10.97
result too large to display
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 (2 \, x^{4} + 1\right )} \sqrt {-4 \, x^{4} - x^{2} + 2}}{{\left (2 \, x^{4} - x^{2} - 1\right )} {\left (2 \, x^{4} - 1\right )}}\,{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 = -0.01 \begin {gather*} -\int \frac {\left (2\,x^4+1\right )\,\sqrt {-4\,x^4-x^2+2}}{\left (2\,x^4-1\right )\,\left (-2\,x^4+x^2+1\right )} \,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 {\left (2 x^{4} + 1\right ) \sqrt {- 4 x^{4} - x^{2} + 2}}{\left (x - 1\right ) \left (x + 1\right ) \left (2 x^{2} + 1\right ) \left (2 x^{4} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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