Optimal. Leaf size=96 \[ \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )-\sin ^{-1}(x) \]
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Rubi [A] time = 0.20, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1293, 216, 1692, 377, 207, 203} \[ \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )-\sin ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 207
Rule 216
Rule 377
Rule 1293
Rule 1692
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt {1-x^2}}{-1+x^2+x^4} \, dx &=-\int \frac {1}{\sqrt {1-x^2}} \, dx-\int \frac {1-2 x^2}{\sqrt {1-x^2} \left (-1+x^2+x^4\right )} \, dx\\ &=-\sin ^{-1}(x)-\int \left (\frac {-2+\frac {4}{\sqrt {5}}}{\sqrt {1-x^2} \left (1-\sqrt {5}+2 x^2\right )}+\frac {-2-\frac {4}{\sqrt {5}}}{\sqrt {1-x^2} \left (1+\sqrt {5}+2 x^2\right )}\right ) \, dx\\ &=-\sin ^{-1}(x)+\frac {1}{5} \left (2 \left (5-2 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx+\frac {1}{5} \left (2 \left (5+2 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx\\ &=-\sin ^{-1}(x)+\frac {1}{5} \left (2 \left (5-2 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {5}-\left (-3+\sqrt {5}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )+\frac {1}{5} \left (2 \left (5+2 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {5}-\left (-3-\sqrt {5}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=-\sin ^{-1}(x)+\sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.50, size = 743, normalized size = 7.74 \[ \frac {i \sqrt {5 \left (\sqrt {5}-2\right )} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}-i \sqrt {2 \left (1+\sqrt {5}\right )} x+2\right )+2 i \sqrt {\sqrt {5}-2} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}-i \sqrt {2 \left (1+\sqrt {5}\right )} x+2\right )-i \sqrt {5 \left (\sqrt {5}-2\right )} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}+i \sqrt {2 \left (1+\sqrt {5}\right )} x+2\right )-2 i \sqrt {\sqrt {5}-2} \log \left (\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}+i \sqrt {2 \left (1+\sqrt {5}\right )} x+2\right )-\sqrt {5 \left (2+\sqrt {5}\right )} \log \left (\sqrt {2} \sqrt {\left (\sqrt {5}-3\right ) \left (x^2-1\right )}-\sqrt {2 \left (\sqrt {5}-1\right )} x+2\right )+2 \sqrt {2+\sqrt {5}} \log \left (\sqrt {2} \sqrt {\left (\sqrt {5}-3\right ) \left (x^2-1\right )}-\sqrt {2 \left (\sqrt {5}-1\right )} x+2\right )+\sqrt {5 \left (2+\sqrt {5}\right )} \log \left (\sqrt {2} \sqrt {\left (\sqrt {5}-3\right ) \left (x^2-1\right )}+\sqrt {2 \left (\sqrt {5}-1\right )} x+2\right )-2 \sqrt {2+\sqrt {5}} \log \left (\sqrt {2} \sqrt {\left (\sqrt {5}-3\right ) \left (x^2-1\right )}+\sqrt {2 \left (\sqrt {5}-1\right )} x+2\right )+\left (\sqrt {5}-2\right ) \sqrt {2+\sqrt {5}} \log \left (x-\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )}\right )-\sqrt {5 \left (2+\sqrt {5}\right )} \log \left (x+\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )}\right )+2 \sqrt {2+\sqrt {5}} \log \left (x+\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )}\right )-i \sqrt {5 \left (\sqrt {5}-2\right )} \log \left (x-i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-2 i \sqrt {\sqrt {5}-2} \log \left (x-i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )+i \sqrt {5 \left (\sqrt {5}-2\right )} \log \left (x+i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )+2 i \sqrt {\sqrt {5}-2} \log \left (x+i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-2 \sqrt {5} \sin ^{-1}(x)}{2 \sqrt {5}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.09, size = 290, normalized size = 3.02 \[ \frac {2}{5} \, \sqrt {5} \sqrt {\sqrt {5} + 2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {-x^{2} + 1} {\left (\sqrt {5} - 3\right )} + \sqrt {5} - 3\right )} \sqrt {\sqrt {5} + 2} \sqrt {\frac {x^{4} - 4 \, x^{2} - \sqrt {5} {\left (x^{4} - 2 \, x^{2}\right )} - 2 \, {\left (\sqrt {5} x^{2} - x^{2} + 2\right )} \sqrt {-x^{2} + 1} + 4}{x^{4}}} + 2 \, \sqrt {-x^{2} + 1} \sqrt {\sqrt {5} + 2} {\left (\sqrt {5} - 3\right )}}{4 \, x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-\frac {2 \, x^{2} + {\left (\sqrt {-x^{2} + 1} {\left (\sqrt {5} x + x\right )} - \sqrt {5} x - x\right )} \sqrt {\sqrt {5} - 2} + 2 \, \sqrt {-x^{2} + 1} - 2}{x^{2}}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {\sqrt {5} - 2} \log \left (-\frac {2 \, x^{2} - {\left (\sqrt {-x^{2} + 1} {\left (\sqrt {5} x + x\right )} - \sqrt {5} x - x\right )} \sqrt {\sqrt {5} - 2} + 2 \, \sqrt {-x^{2} + 1} - 2}{x^{2}}\right ) + 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.72, size = 209, normalized size = 2.18 \[ -\frac {1}{2} \, \pi \mathrm {sgn}\relax (x) - \frac {1}{5} \, \sqrt {5 \, \sqrt {5} + 10} \arctan \left (-\frac {\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}}{\sqrt {2 \, \sqrt {5} + 2}}\right ) - \frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | \sqrt {2 \, \sqrt {5} - 2} - \frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} \right |}\right ) + \frac {1}{10} \, \sqrt {5 \, \sqrt {5} - 10} \log \left ({\left | -\sqrt {2 \, \sqrt {5} - 2} - \frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} \right |}\right ) - \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 160, normalized size = 1.67 \[ -\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {2+\sqrt {5}}\, x}\right )}{5 \sqrt {2+\sqrt {5}}}+\frac {\sqrt {\sqrt {5}-2}\, \sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {\sqrt {5}-2}\, x}\right )}{5}+2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )-\frac {\sqrt {2+\sqrt {5}}\, \sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {2+\sqrt {5}}\, x}\right )}{5}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {\sqrt {5}-2}\, x}\right )}{5 \sqrt {\sqrt {5}-2}} \]
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 \[ \int \frac {\sqrt {-x^{2} + 1} x^{2}}{x^{4} + x^{2} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 383, normalized size = 3.99 \[ -\mathrm {asin}\relax (x)-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {3}{2}-\frac {\sqrt {5}}{2}}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}\right )\,\left (\sqrt {5}-2\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {3}{2}-\frac {\sqrt {5}}{2}}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}-1\right )\,1{}\mathrm {i}}{\sqrt {\frac {\sqrt {5}}{2}+\frac {3}{2}}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}}\right )\,\left (\sqrt {5}+2\right )}{\left (2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {3}{2}}}+\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {3}{2}-\frac {\sqrt {5}}{2}}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}}\right )\,\left (\sqrt {5}-2\right )}{\left (2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {3}{2}-\frac {\sqrt {5}}{2}}}-\frac {\ln \left (\frac {\frac {\left (x\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+1\right )\,1{}\mathrm {i}}{\sqrt {\frac {\sqrt {5}}{2}+\frac {3}{2}}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}}\right )\,\left (\sqrt {5}+2\right )}{\left (2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}\right )\,\sqrt {\frac {\sqrt {5}}{2}+\frac {3}{2}}} \]
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 \[ \int \frac {x^{2} \sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{x^{4} + x^{2} - 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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