Optimal. Leaf size=185 \[ -2 x-\sin ^{-1}(x)+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{-1+\sqrt {5}}} x\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} x}{\sqrt {1-x^2}}\right )+x \log \left (x^2+\sqrt {1-x^2}\right ) \]
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Rubi [A]
time = 0.70, 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 = {2628, 6874,
1307, 222, 1706, 385, 213, 209, 1180, 1144, 1188, 399} \begin {gather*} 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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 213
Rule 222
Rule 385
Rule 399
Rule 1144
Rule 1180
Rule 1188
Rule 1307
Rule 1706
Rule 2628
Rule 6874
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] Result contains complex when optimal does not.
time = 0.34, size = 910, normalized size = 4.92 \begin {gather*} \frac {-8 \sqrt {5} x-4 \sqrt {5} \sin ^{-1}(x)+5 \sqrt {2 \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} x\right )+\sqrt {10 \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 )-5 \sqrt {2+\sqrt {5}} \log \left (-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}+x\right )+3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}+x\right )+5 \sqrt {2+\sqrt {5}} \log \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}+x\right )-3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )}+x\right )-5 i \sqrt {-2+\sqrt {5}} \log \left (-i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )-3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (-i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )+5 i \sqrt {-2+\sqrt {5}} \log \left (i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )+3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (i \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )}+x\right )+4 \sqrt {5} x \log \left (x^2+\sqrt {1-x^2}\right )+5 i \sqrt {-2+\sqrt {5}} \log \left (2-i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}\right )+3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (2-i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}\right )-5 i \sqrt {-2+\sqrt {5}} \log \left (2+i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}\right )-3 i \sqrt {5 \left (-2+\sqrt {5}\right )} \log \left (2+i \sqrt {2 \left (1+\sqrt {5}\right )} x+\sqrt {2 \left (3+\sqrt {5}\right )} \sqrt {1-x^2}\right )+5 \sqrt {2+\sqrt {5}} \log \left (2-\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (-1+x^2\right )}\right )-3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (2-\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (-1+x^2\right )}\right )-5 \sqrt {2+\sqrt {5}} \log \left (2+\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (-1+x^2\right )}\right )+3 \sqrt {5 \left (2+\sqrt {5}\right )} \log \left (2+\sqrt {2 \left (-1+\sqrt {5}\right )} x+\sqrt {2} \sqrt {\left (-3+\sqrt {5}\right ) \left (-1+x^2\right )}\right )}{4 \sqrt {5}} \end {gather*}
Warning: Unable to verify antiderivative.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(391\) vs.
\(2(138)=276\).
time = 0.10, size = 392, normalized size = 2.12
method | result | size |
default | \(x \ln \left (x^{2}+\sqrt {-x^{2}+1}\right )+\frac {\arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctan \left (\frac {2 x}{\sqrt {2+2 \sqrt {5}}}\right )}{\sqrt {2+2 \sqrt {5}}}-\frac {\arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}+\frac {\sqrt {5}\, \arctanh \left (\frac {2 x}{\sqrt {-2+2 \sqrt {5}}}\right )}{\sqrt {-2+2 \sqrt {5}}}-2 x -\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}+\frac {\arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}-\frac {3 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {2+\sqrt {5}}}\right )}{2 \sqrt {2+\sqrt {5}}}+\frac {3 \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}-\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {-x^{2}+1}-1}{x \sqrt {-2+\sqrt {5}}}\right )}{2 \sqrt {-2+\sqrt {5}}}-\arcsin \left (x \right )\) | \(392\) |
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. 452 vs.
\(2 (138) = 276\).
time = 0.36, size = 452, normalized size = 2.44 \begin {gather*} -\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 ) \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 \log {\left (x^{2} + \sqrt {1 - x^{2}} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 301 vs.
\(2 (138) = 276\).
time = 0.03, size = 404, normalized size = 2.18 \begin {gather*} -\frac {1}{4} \sqrt {2 \left (\sqrt {5}-1\right )} \ln \left |x-\sqrt {\frac {-1+\sqrt {5}}{2}}\right |+\frac {1}{4} \sqrt {2 \left (\sqrt {5}-1\right )} \ln \left |x+\sqrt {\frac {-1+\sqrt {5}}{2}}\right |+\frac {1}{2} \sqrt {2 \left (\sqrt {5}+1\right )} \arctan \left (\frac {x}{\sqrt {-\frac {-1-\sqrt {5}}{2}}}\right )-2 x-\frac {1}{2} \pi \mathrm {sign}\left (x\right )-\frac {1}{2} \sqrt {2 \left (\sqrt {5}+1\right )} \arctan \left (\frac {\frac {2 x}{-2 \sqrt {-x^{2}+1}+2}-\frac {-2 \sqrt {-x^{2}+1}+2}{2 x}}{\sqrt {2 \left (\sqrt {5}+1\right )}}\right )-\frac {1}{4} \sqrt {2 \left (\sqrt {5}-1\right )} \ln \left |\sqrt {-2 \left (-\sqrt {5}+1\right )}+\frac {2 x}{-2 \sqrt {-x^{2}+1}+2}-\frac {-2 \sqrt {-x^{2}+1}+2}{2 x}\right |+\frac {1}{4} \sqrt {2 \left (\sqrt {5}-1\right )} \ln \left |-\sqrt {-2 \left (-\sqrt {5}+1\right )}+\frac {2 x}{-2 \sqrt {-x^{2}+1}+2}-\frac {-2 \sqrt {-x^{2}+1}+2}{2 x}\right |-\arctan \left (\frac {x \left (\left (-\frac {-2 \sqrt {-x^{2}+1}+2}{2 x}\right )^{2}-1\right )}{-2 \sqrt {-x^{2}+1}+2}\right )+x \ln \left (x^{2}+\sqrt {-x^{2}+1}\right ) \end {gather*}
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 \begin {gather*} x\,\ln \left (x^2+\sqrt {1-x^2}\right )-\mathrm {asin}\left (x\right )-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}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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