Optimal. Leaf size=158 \[ \tan ^{-1}\left (\frac {x \sqrt {-2 x^4-3 x^2+1}}{2 x^4+3 x^2-1}\right )-i \sqrt {2} \tanh ^{-1}\left (\frac {2 \sqrt {2} x^3-2 i \sqrt {-2 x^4-3 x^2+1} x-2 i x}{2 \sqrt {2} x^4+3 \sqrt {2} x^2-2 i \sqrt {-2 x^4-3 x^2+1} x^2-\sqrt {2} \sqrt {-2 x^4-3 x^2+1}-\sqrt {2}}\right ) \]
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Rubi [C] time = 1.47, antiderivative size = 467, normalized size of antiderivative = 2.96, number of steps used = 32, number of rules used = 8, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6728, 1208, 1180, 524, 424, 419, 1212, 537} \begin {gather*} -\sqrt {\frac {1}{2} \left (9 \sqrt {17}-37\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {3 \sqrt {17}-5} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\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 6728
Rubi steps
\begin {align*} \int \frac {\sqrt {1-3 x^2-2 x^4} \left (1+2 x^4\right )}{\left (-1+x^2+2 x^4\right ) \left (-1+2 x^2+2 x^4\right )} \, dx &=\int \left (\frac {\sqrt {1-3 x^2-2 x^4}}{1+x^2}+\frac {2 \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2}-\frac {2 \left (1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2+2 x^4}\right ) \, dx\\ &=2 \int \frac {\sqrt {1-3 x^2-2 x^4}}{-1+2 x^2} \, dx-2 \int \frac {\left (1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}}{-1+2 x^2+2 x^4} \, dx+\int \frac {\sqrt {1-3 x^2-2 x^4}}{1+x^2} \, dx\\ &=-\left (\frac {1}{2} \int \frac {8+4 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx\right )+2 \int \frac {1}{\left (1+x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-2 \int \frac {1}{\left (-1+2 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-2 \int \left (\frac {2 \sqrt {1-3 x^2-2 x^4}}{2-2 \sqrt {3}+4 x^2}+\frac {2 \sqrt {1-3 x^2-2 x^4}}{2+2 \sqrt {3}+4 x^2}\right ) \, dx-\int \frac {1+2 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx\\ &=-\left (4 \int \frac {\sqrt {1-3 x^2-2 x^4}}{2-2 \sqrt {3}+4 x^2} \, dx\right )-4 \int \frac {\sqrt {1-3 x^2-2 x^4}}{2+2 \sqrt {3}+4 x^2} \, dx-\sqrt {2} \int \frac {8+4 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (2 \sqrt {2}\right ) \int \frac {1+2 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (1+x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (4 \sqrt {2}\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (-1+2 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{4} \int \frac {12-2 \left (2-2 \sqrt {3}\right )+8 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx+\frac {1}{4} \int \frac {12-2 \left (2+2 \sqrt {3}\right )+8 x^2}{\sqrt {1-3 x^2-2 x^4}} \, dx-2 \left (\sqrt {2} \int \frac {\sqrt {3+\sqrt {17}+4 x^2}}{\sqrt {-3+\sqrt {17}-4 x^2}} \, dx\right )-\left (2 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\left (2-2 \sqrt {3}+4 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-\left (2 \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\left (2+2 \sqrt {3}+4 x^2\right ) \sqrt {1-3 x^2-2 x^4}} \, dx-\left (\sqrt {2} \left (5-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (\sqrt {2} \left (1+\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=-\sqrt {2 \left (3+\sqrt {17}\right )} E\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {\int \frac {12-2 \left (2-2 \sqrt {3}\right )+8 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx}{\sqrt {2}}+\frac {\int \frac {12-2 \left (2+2 \sqrt {3}\right )+8 x^2}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx}{\sqrt {2}}-\left (4 \sqrt {2} \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (2-2 \sqrt {3}+4 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx-\left (4 \sqrt {2} \left (1+\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \left (2+2 \sqrt {3}+4 x^2\right ) \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=-\sqrt {2 \left (3+\sqrt {17}\right )} E\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \left (\sqrt {2} \int \frac {\sqrt {3+\sqrt {17}+4 x^2}}{\sqrt {-3+\sqrt {17}-4 x^2}} \, dx\right )+\left (\sqrt {2} \left (1-2 \sqrt {3}-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx+\left (\sqrt {2} \left (1+2 \sqrt {3}-\sqrt {17}\right )\right ) \int \frac {1}{\sqrt {-3+\sqrt {17}-4 x^2} \sqrt {3+\sqrt {17}+4 x^2}} \, dx\\ &=\frac {\left (1-2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {\left (1+2 \sqrt {3}-\sqrt {17}\right ) F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )}{\sqrt {2 \left (3+\sqrt {17}\right )}}+\frac {1}{2} \sqrt {-5+3 \sqrt {17}} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {1}{2} \left (-37+9 \sqrt {17}\right )} F\left (\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{4} \left (3-\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1-\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )-\sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {3-\sqrt {17}}{2 \left (1+\sqrt {3}\right )};\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )+2 \sqrt {\frac {2}{3+\sqrt {17}}} \Pi \left (\frac {1}{2} \left (-3+\sqrt {17}\right );\sin ^{-1}\left (\frac {2 x}{\sqrt {-3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13+3 \sqrt {17}\right )\right )\\ \end {align*}
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Mathematica [C] time = 0.94, size = 251, normalized size = 1.59 \begin {gather*} i \sqrt {\frac {2}{\sqrt {17}-3}} \left (F\left (i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )-2 \Pi \left (-\frac {3}{2}-\frac {\sqrt {17}}{2};i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )-2 \Pi \left (\frac {1}{4} \left (3+\sqrt {17}\right );i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|\frac {1}{4} \left (-13-3 \sqrt {17}\right )\right )+\Pi \left (\frac {3+\sqrt {17}}{2-2 \sqrt {3}};i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )+\Pi \left (\frac {3+\sqrt {17}}{2+2 \sqrt {3}};i \sinh ^{-1}\left (\frac {2 x}{\sqrt {3+\sqrt {17}}}\right )|-\frac {13}{4}-\frac {3 \sqrt {17}}{4}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.47, size = 79, normalized size = 0.50 \begin {gather*} \tan ^{-1}\left (\frac {x \sqrt {-2 x^4-3 x^2+1}}{2 x^4+3 x^2-1}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {-2 x^4-3 x^2+1}}{2 x^4+3 x^2-1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 75, normalized size = 0.47 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 5 \, x^{2} - 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{4} - 3 \, x^{2} + 1} x}{2 \, x^{4} + 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 {-2 \, x^{4} - 3 \, x^{2} + 1}}{{\left (2 \, x^{4} + 2 \, x^{2} - 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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maple [C] time = 0.29, size = 862, normalized size = 5.46
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 {-2 \, x^{4} - 3 \, x^{2} + 1}}{{\left (2 \, x^{4} + 2 \, x^{2} - 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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (2\,x^4+1\right )\,\sqrt {-2\,x^4-3\,x^2+1}}{\left (2\,x^4+x^2-1\right )\,\left (2\,x^4+2\,x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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