Optimal. Leaf size=185 \[ x \log \left (x^2+\sqrt {1-x^2}\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )-2 x-\sin ^{-1}(x)+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right ) \]
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Rubi [A] time = 0.98, antiderivative size = 349, normalized size of antiderivative = 1.89, number of steps used = 31, number of rules used = 12, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {2548, 6742, 1293, 216, 1692, 377, 207, 203, 1166, 1130, 1174, 402} \[ x \log \left (x^2+\sqrt {1-x^2}\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )-\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )-2 \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} x}{\sqrt {1-x^2}}\right )-2 x-\sin ^{-1}(x)+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right )+2 \sqrt {\frac {1}{5} \left (\sqrt {5}-2\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{\sqrt {5}-1}} x\right ) \]
Warning: Unable to verify antiderivative.
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Rule 203
Rule 207
Rule 216
Rule 377
Rule 402
Rule 1130
Rule 1166
Rule 1174
Rule 1293
Rule 1692
Rule 2548
Rule 6742
Rubi steps
\begin {align*} \int \log \left (x^2+\sqrt {1-x^2}\right ) \, dx &=x \log \left (x^2+\sqrt {1-x^2}\right )-\int \frac {x^2 \left (2-\frac {1}{\sqrt {1-x^2}}\right )}{x^2+\sqrt {1-x^2}} \, dx\\ &=x \log \left (x^2+\sqrt {1-x^2}\right )-\int \left (\frac {2 x^2}{x^2+\sqrt {1-x^2}}-\frac {x^2}{1-x^2+x^2 \sqrt {1-x^2}}\right ) \, dx\\ &=x \log \left (x^2+\sqrt {1-x^2}\right )-2 \int \frac {x^2}{x^2+\sqrt {1-x^2}} \, dx+\int \frac {x^2}{1-x^2+x^2 \sqrt {1-x^2}} \, dx\\ &=x \log \left (x^2+\sqrt {1-x^2}\right )-2 \int \left (1-\frac {x^2 \sqrt {1-x^2}}{-1+x^2+x^4}+\frac {1-x^2}{-1+x^2+x^4}\right ) \, dx+\int \left (\frac {1}{\sqrt {1-x^2}}-\frac {x^2}{-1+x^2+x^4}+\frac {\sqrt {1-x^2}}{-1+x^2+x^4}\right ) \, dx\\ &=-2 x+x \log \left (x^2+\sqrt {1-x^2}\right )+2 \int \frac {x^2 \sqrt {1-x^2}}{-1+x^2+x^4} \, dx-2 \int \frac {1-x^2}{-1+x^2+x^4} \, dx+\int \frac {1}{\sqrt {1-x^2}} \, dx-\int \frac {x^2}{-1+x^2+x^4} \, dx+\int \frac {\sqrt {1-x^2}}{-1+x^2+x^4} \, dx\\ &=-2 x+\sin ^{-1}(x)+x \log \left (x^2+\sqrt {1-x^2}\right )-2 \int \frac {1}{\sqrt {1-x^2}} \, dx-2 \int \frac {1-2 x^2}{\sqrt {1-x^2} \left (-1+x^2+x^4\right )} \, dx+\frac {2 \int \frac {\sqrt {1-x^2}}{1-\sqrt {5}+2 x^2} \, dx}{\sqrt {5}}-\frac {2 \int \frac {\sqrt {1-x^2}}{1+\sqrt {5}+2 x^2} \, dx}{\sqrt {5}}+\frac {1}{10} \left (-5+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx-\frac {1}{10} \left (5+\sqrt {5}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx-\frac {1}{5} \left (-5+3 \sqrt {5}\right ) \int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x^2} \, dx+\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x^2} \, dx\\ &=-2 x-\sin ^{-1}(x)-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+2 \sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+x \log \left (x^2+\sqrt {1-x^2}\right )-2 \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-\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1-\sqrt {5}+2 x^2\right )} \, dx-\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {1}{\sqrt {1-x^2} \left (1+\sqrt {5}+2 x^2\right )} \, dx\\ &=-2 x-\sin ^{-1}(x)-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+2 \sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+x \log \left (x^2+\sqrt {1-x^2}\right )-\frac {1}{5} \left (5-3 \sqrt {5}\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 (4 \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 (4 \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 (5+3 \sqrt {5}\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 )\\ &=-2 x-\sin ^{-1}(x)-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+2 \sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+x \log \left (x^2+\sqrt {1-x^2}\right )+\frac {1}{5} \left (4 \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 (4 \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 )\\ &=-2 x-\sin ^{-1}(x)-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+2 \sqrt {\frac {1}{5} \left (2+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+2 \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 )+2 \sqrt {\frac {1}{5} \left (-2+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-2 \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 )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+x \log \left (x^2+\sqrt {1-x^2}\right )\\ \end {align*}
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Mathematica [C] time = 0.58, size = 910, normalized size = 4.92 \[ \frac {4 \sqrt {5} \log \left (x^2+\sqrt {1-x^2}\right ) x-8 \sqrt {5} x-4 \sqrt {5} \sin ^{-1}(x)+\sqrt {10 \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+5 \sqrt {2 \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )-\left (-5+\sqrt {5}\right ) \sqrt {2 \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )+3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (x-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}\right )-5 \sqrt {2+\sqrt {5}} \log \left (x-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}\right )-3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (x+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}\right )+5 \sqrt {2+\sqrt {5}} \log \left (x+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}\right )-3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (x-i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )-5 i \sqrt {-2+\sqrt {5}} \log \left (x-i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )+3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (x+i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )+5 i \sqrt {-2+\sqrt {5}} \log \left (x+i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}\right )+3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (-i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}+2\right )+5 i \sqrt {-2+\sqrt {5}} \log \left (-i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}+2\right )-3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}+2\right )-5 i \sqrt {-2+\sqrt {5}} \log \left (i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}+2\right )-3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (-\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (x^2-1\right )}+2\right )+5 \sqrt {2+\sqrt {5}} \log \left (-\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (x^2-1\right )}+2\right )+3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (x^2-1\right )}+2\right )-5 \sqrt {2+\sqrt {5}} \log \left (\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (x^2-1\right )}+2\right )}{4 \sqrt {5}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.47, size = 452, normalized size = 2.44 \[ -\sqrt {2} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {1}{8} \, \sqrt {4 \, x^{2} + 2 \, \sqrt {5} + 2} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} - \frac {1}{4} \, {\left (\sqrt {5} \sqrt {2} x - \sqrt {2} x\right )} \sqrt {\sqrt {5} + 1}\right ) - \sqrt {2} \sqrt {\sqrt {5} + 1} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {-x^{2} + 1} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} + \sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1} \sqrt {\frac {x^{4} - 4 \, x^{2} - \sqrt {5} {\left (x^{4} - 2 \, x^{2}\right )} - 2 \, {\left (\sqrt {5} x^{2} - x^{2} + 2\right )} \sqrt {-x^{2} + 1} + 4}{x^{4}}} + 2 \, \sqrt {-x^{2} + 1} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} \sqrt {\sqrt {5} + 1}}{8 \, x}\right ) + x \log \left (x^{2} + \sqrt {-x^{2} + 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (2 \, x + \sqrt {2} \sqrt {\sqrt {5} - 1}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (2 \, x - \sqrt {2} \sqrt {\sqrt {5} - 1}\right ) + \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (-\frac {2 \, x^{2} + {\left (\sqrt {2} \sqrt {-x^{2} + 1} x - \sqrt {2} x\right )} \sqrt {\sqrt {5} - 1} + 2 \, \sqrt {-x^{2} + 1} - 2}{x^{2}}\right ) - \frac {1}{4} \, \sqrt {2} \sqrt {\sqrt {5} - 1} \log \left (-\frac {2 \, x^{2} - {\left (\sqrt {2} \sqrt {-x^{2} + 1} x - \sqrt {2} x\right )} \sqrt {\sqrt {5} - 1} + 2 \, \sqrt {-x^{2} + 1} - 2}{x^{2}}\right ) - 2 \, x + 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.39, size = 301, normalized size = 1.63 \[ x \log \left (x^{2} + \sqrt {-x^{2} + 1}\right ) - \frac {1}{2} \, \pi \mathrm {sgn}\relax (x) + \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {x}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {5} + 2} \arctan \left (-\frac {\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}}{\sqrt {2 \, \sqrt {5} + 2}}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x + \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | x - \sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} \right |}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | \sqrt {2 \, \sqrt {5} - 2} - \frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} \right |}\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left ({\left | -\sqrt {2 \, \sqrt {5} - 2} - \frac {x}{\sqrt {-x^{2} + 1} - 1} + \frac {\sqrt {-x^{2} + 1} - 1}{x} \right |}\right ) - 2 \, x - \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 392, normalized size = 2.12 \[ x \ln \left (x^{2}+\sqrt {-x^{2}+1}\right )-2 x +\frac {3 \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {-2+\sqrt {5}}\, x}\right )}{2 \sqrt {-2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {-2+\sqrt {5}}\, x}\right )}{2 \sqrt {-2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {2+\sqrt {5}}\, x}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {2+\sqrt {5}}\, x}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}-\arcsin \relax (x )-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {-2+\sqrt {5}}\, x}\right )}{2 \sqrt {-2+\sqrt {5}}}+\frac {\arctan \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {-2+\sqrt {5}}\, x}\right )}{2 \sqrt {-2+\sqrt {5}}}-\frac {3 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {2+\sqrt {5}}\, x}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{\sqrt {2+\sqrt {5}}\, x}\right )}{2 \sqrt {2+\sqrt {5}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}} \]
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 \[ x \log \left (x^{2} + \sqrt {x + 1} \sqrt {-x + 1}\right ) - x - \int \frac {x^{4} - 2 \, x^{2}}{x^{4} - x^{2} + {\left (x^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (-x + 1\right )\right )}}\,{d x} + \frac {1}{2} \, \log \left (x + 1\right ) - \frac {1}{2} \, \log \left (-x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.18, size = 608, normalized size = 3.29 \[ x\,\ln \left (x^2+\sqrt {1-x^2}\right )-\mathrm {asin}\relax (x)-2\,x+\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}-\frac {5}{2}\right )}{2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}-\frac {5}{2}\right )}{2\,\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}-\frac {\ln \left (x-\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {5}{2}\right )}{2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{3/2}}+\frac {\ln \left (x+\frac {\sqrt {2}\,\sqrt {-\sqrt {5}-1}}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {5}{2}\right )}{2\,\sqrt {-\frac {\sqrt {5}}{2}-\frac {1}{2}}+4\,{\left (-\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}^{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 (\frac {3\,\sqrt {5}}{2}-\frac {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 (\frac {3\,\sqrt {5}}{2}+\frac {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 (\frac {3\,\sqrt {5}}{2}-\frac {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 (\frac {3\,\sqrt {5}}{2}+\frac {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 \[ \int \log {\left (x^{2} + \sqrt {1 - x^{2}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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