Integrand size = 33, antiderivative size = 42 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\arcsin (x)-\frac {\arctan \left (\frac {1+4 x \sqrt {1-x^2}}{\sqrt {3} \left (1-2 x^2\right )}\right )}{\sqrt {3}} \]
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Result contains complex when optimal does not.
Time = 0.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 3.55, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.303, Rules used = {6874, 1307, 222, 1188, 385, 211, 1265, 787, 632, 210} \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\arcsin (x)-\frac {\arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\arctan \left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {x^2}{2}+\frac {1}{4} (1-x)^2+\frac {1}{4} (x+1)^2 \]
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Rule 210
Rule 211
Rule 222
Rule 385
Rule 632
Rule 787
Rule 1188
Rule 1265
Rule 1307
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2} (-1+x)+\frac {1+x}{2}-\frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4}+\frac {x^3 \left (1-x^2\right )}{1-x^2+x^4}\right ) \, dx \\ & = \frac {1}{4} (1-x)^2+\frac {1}{4} (1+x)^2-\int \frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4} \, dx+\int \frac {x^3 \left (1-x^2\right )}{1-x^2+x^4} \, dx \\ & = \frac {1}{4} (1-x)^2+\frac {1}{4} (1+x)^2+\frac {1}{2} \text {Subst}\left (\int \frac {(1-x) x}{1-x+x^2} \, dx,x,x^2\right )+\int \frac {1}{\sqrt {1-x^2}} \, dx-\int \frac {1}{\sqrt {1-x^2} \left (1-x^2+x^4\right )} \, dx \\ & = \frac {1}{4} (1-x)^2-\frac {x^2}{2}+\frac {1}{4} (1+x)^2+\sin ^{-1}(x)+\frac {1}{2} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )+\frac {(2 i) \int \frac {1}{\sqrt {1-x^2} \left (-1-i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}}-\frac {(2 i) \int \frac {1}{\sqrt {1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}} \\ & = \frac {1}{4} (1-x)^2-\frac {x^2}{2}+\frac {1}{4} (1+x)^2+\sin ^{-1}(x)-\frac {(2 i) \text {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-1-i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+\frac {(2 i) \text {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-1+i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}-\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right ) \\ & = \frac {1}{4} (1-x)^2-\frac {x^2}{2}+\frac {1}{4} (1+x)^2+\sin ^{-1}(x)-\frac {\tan ^{-1}\left (\frac {x}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {-1+2 x^2}{\sqrt {3}}\right )}{\sqrt {3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.07 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=-2 \arctan \left (\frac {\sqrt {1-x^2}}{1+x}\right )+\text {RootSum}\left [1+2 \text {$\#$1}+2 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log (-1+x)+\log \left (\sqrt {1-x^2}+\text {$\#$1}-x \text {$\#$1}\right )-\log (-1+x) \text {$\#$1}^2+\log \left (\sqrt {1-x^2}+\text {$\#$1}-x \text {$\#$1}\right ) \text {$\#$1}^2}{1+2 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.07 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.12
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{2}+1}+x \right )+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}+3 x \sqrt {-x^{2}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+3\right ) x^{2}-x^{2}+2}\right )}{3}\) | \(89\) |
default | \(\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{3}+\left (\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {1}{4}-\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )-\left (\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )-\left (-\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (\frac {1}{4}+\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )+\left (-\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )-2 \arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )\) | \(456\) |
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Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.74 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x^{2} - 1\right )} \sqrt {-x^{2} + 1}}{3 \, {\left (x^{3} - x\right )}}\right ) - 2 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
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\[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=- \int \frac {x \sqrt {1 - x^{2}}}{x^{3} - x - \sqrt {1 - x^{2}}}\, dx \]
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\[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\int { -\frac {\sqrt {-x^{2} + 1} x}{x^{3} - x - \sqrt {-x^{2} + 1}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 193 vs. \(2 (37) = 74\).
Time = 0.35 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.60 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\frac {1}{2} \, \pi \mathrm {sgn}\left (x\right ) - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {1}{6} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} + \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) + \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) \]
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Time = 0.51 (sec) , antiderivative size = 549, normalized size of antiderivative = 13.07 \[ \int \frac {x \sqrt {1-x^2}}{x-x^3+\sqrt {1-x^2}} \, dx=\mathrm {asin}\left (x\right )-\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )-1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{\frac {\sqrt {3}}{2}-x+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}+\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )-1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}}-\sqrt {1-x^2}\,1{}\mathrm {i}}{x-\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}-\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )+1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}-\frac {\ln \left (x-\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}-\frac {\ln \left (x+\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}{\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}+\frac {\ln \left (\frac {\frac {\left (x\,\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )+1\right )\,1{}\mathrm {i}}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}}+\sqrt {1-x^2}\,1{}\mathrm {i}}{x+\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}}\right )}{\sqrt {1-{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^2}\,\left (\sqrt {3}-4\,{\left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}\right )}+\frac {\ln \left (x-\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}}+\frac {\ln \left (x+\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}{-\sqrt {3}+4\,{\left (\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i}\right )}^3+1{}\mathrm {i}} \]
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