Integrand size = 24, antiderivative size = 160 \[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2-b}-2 x}{\sqrt {2+b}}\right )}{2 \sqrt {2+b}}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2-b}+2 x}{\sqrt {2+b}}\right )}{2 \sqrt {2+b}}-\frac {\left (1+\sqrt {2}\right ) \log \left (1-\sqrt {2-b} x+x^2\right )}{4 \sqrt {2-b}}+\frac {\left (1+\sqrt {2}\right ) \log \left (1+\sqrt {2-b} x+x^2\right )}{4 \sqrt {2-b}} \]
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Time = 0.09 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1183, 648, 632, 210, 642} \[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2-b}-2 x}{\sqrt {b+2}}\right )}{2 \sqrt {b+2}}-\frac {\left (1-\sqrt {2}\right ) \arctan \left (\frac {\sqrt {2-b}+2 x}{\sqrt {b+2}}\right )}{2 \sqrt {b+2}}-\frac {\left (1+\sqrt {2}\right ) \log \left (-\sqrt {2-b} x+x^2+1\right )}{4 \sqrt {2-b}}+\frac {\left (1+\sqrt {2}\right ) \log \left (\sqrt {2-b} x+x^2+1\right )}{4 \sqrt {2-b}} \]
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Rule 210
Rule 632
Rule 642
Rule 648
Rule 1183
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {2} \sqrt {2-b}-\left (1+\sqrt {2}\right ) x}{1-\sqrt {2-b} x+x^2} \, dx}{2 \sqrt {2-b}}+\frac {\int \frac {\sqrt {2} \sqrt {2-b}+\left (1+\sqrt {2}\right ) x}{1+\sqrt {2-b} x+x^2} \, dx}{2 \sqrt {2-b}} \\ & = \frac {1}{4} \left (-1+\sqrt {2}\right ) \int \frac {1}{1-\sqrt {2-b} x+x^2} \, dx+\frac {1}{4} \left (-1+\sqrt {2}\right ) \int \frac {1}{1+\sqrt {2-b} x+x^2} \, dx-\frac {\left (1+\sqrt {2}\right ) \int \frac {-\sqrt {2-b}+2 x}{1-\sqrt {2-b} x+x^2} \, dx}{4 \sqrt {2-b}}+\frac {\left (1+\sqrt {2}\right ) \int \frac {\sqrt {2-b}+2 x}{1+\sqrt {2-b} x+x^2} \, dx}{4 \sqrt {2-b}} \\ & = -\frac {\left (1+\sqrt {2}\right ) \log \left (1-\sqrt {2-b} x+x^2\right )}{4 \sqrt {2-b}}+\frac {\left (1+\sqrt {2}\right ) \log \left (1+\sqrt {2-b} x+x^2\right )}{4 \sqrt {2-b}}+\frac {1}{2} \left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2-b-x^2} \, dx,x,-\sqrt {2-b}+2 x\right )+\frac {1}{2} \left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{-2-b-x^2} \, dx,x,\sqrt {2-b}+2 x\right ) \\ & = \frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt {2-b}-2 x}{\sqrt {2+b}}\right )}{2 \sqrt {2+b}}-\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (\frac {\sqrt {2-b}+2 x}{\sqrt {2+b}}\right )}{2 \sqrt {2+b}}-\frac {\left (1+\sqrt {2}\right ) \log \left (1-\sqrt {2-b} x+x^2\right )}{4 \sqrt {2-b}}+\frac {\left (1+\sqrt {2}\right ) \log \left (1+\sqrt {2-b} x+x^2\right )}{4 \sqrt {2-b}} \\ \end{align*}
Timed out. \[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=\text {\$Aborted} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.28
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z}^{2} b +1\right )}{\sum }\frac {\left (-\textit {\_R}^{2}+\sqrt {2}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3}+\textit {\_R} b}\right )}{2}\) | \(44\) |
default | \(\frac {\left (-\sqrt {\left (b -2\right ) \left (2+b \right )}-b -2 \sqrt {2}\right ) \arctan \left (\frac {2 x}{\sqrt {2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}\right )}{\sqrt {\left (b -2\right ) \left (2+b \right )}\, \sqrt {2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}+\frac {\left (-\sqrt {\left (b -2\right ) \left (2+b \right )}+b +2 \sqrt {2}\right ) \arctan \left (\frac {2 x}{\sqrt {-2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}\right )}{\sqrt {\left (b -2\right ) \left (2+b \right )}\, \sqrt {-2 \sqrt {\left (b -2\right ) \left (2+b \right )}+2 b}}\) | \(136\) |
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Leaf count of result is larger than twice the leaf count of optimal. 517 vs. \(2 (123) = 246\).
Time = 0.26 (sec) , antiderivative size = 517, normalized size of antiderivative = 3.23 \[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=-\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} + \sqrt {b^{2} - 4}}{b^{2} - 4}} \log \left (2 \, {\left (2 \, b^{2} - 9\right )} x + \sqrt {\frac {1}{2}} {\left (2 \, b^{3} - 3 \, \sqrt {2} {\left (b^{2} - 4\right )} - 8 \, b - \frac {2 \, b^{4} - 14 \, b^{2} - \sqrt {2} {\left (b^{3} - 4 \, b\right )} + 24}{\sqrt {b^{2} - 4}}\right )} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} + \sqrt {b^{2} - 4}}{b^{2} - 4}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} + \sqrt {b^{2} - 4}}{b^{2} - 4}} \log \left (2 \, {\left (2 \, b^{2} - 9\right )} x - \sqrt {\frac {1}{2}} {\left (2 \, b^{3} - 3 \, \sqrt {2} {\left (b^{2} - 4\right )} - 8 \, b - \frac {2 \, b^{4} - 14 \, b^{2} - \sqrt {2} {\left (b^{3} - 4 \, b\right )} + 24}{\sqrt {b^{2} - 4}}\right )} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} + \sqrt {b^{2} - 4}}{b^{2} - 4}}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} - \sqrt {b^{2} - 4}}{b^{2} - 4}} \log \left (2 \, {\left (2 \, b^{2} - 9\right )} x + \sqrt {\frac {1}{2}} {\left (2 \, b^{3} - 3 \, \sqrt {2} {\left (b^{2} - 4\right )} - 8 \, b + \frac {2 \, b^{4} - 14 \, b^{2} - \sqrt {2} {\left (b^{3} - 4 \, b\right )} + 24}{\sqrt {b^{2} - 4}}\right )} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} - \sqrt {b^{2} - 4}}{b^{2} - 4}}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} - \sqrt {b^{2} - 4}}{b^{2} - 4}} \log \left (2 \, {\left (2 \, b^{2} - 9\right )} x - \sqrt {\frac {1}{2}} {\left (2 \, b^{3} - 3 \, \sqrt {2} {\left (b^{2} - 4\right )} - 8 \, b + \frac {2 \, b^{4} - 14 \, b^{2} - \sqrt {2} {\left (b^{3} - 4 \, b\right )} + 24}{\sqrt {b^{2} - 4}}\right )} \sqrt {-\frac {3 \, b + 4 \, \sqrt {2} - \sqrt {b^{2} - 4}}{b^{2} - 4}}\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 1469 vs. \(2 (128) = 256\).
Time = 1.34 (sec) , antiderivative size = 1469, normalized size of antiderivative = 9.18 \[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=\int { -\frac {x^{2} - \sqrt {2}}{x^{4} + b x^{2} + 1} \,d x } \]
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\[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=\int { -\frac {x^{2} - \sqrt {2}}{x^{4} + b x^{2} + 1} \,d x } \]
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Time = 14.43 (sec) , antiderivative size = 1227, normalized size of antiderivative = 7.67 \[ \int \frac {\sqrt {2}-x^2}{1+b x^2+x^4} \, dx=-\mathrm {atan}\left (\frac {x\,\sqrt {\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,32{}\mathrm {i}-b\,x\,{\left (\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}\right )}^{3/2}\,256{}\mathrm {i}+b^2\,x\,\sqrt {\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,8{}\mathrm {i}-b^4\,x\,\sqrt {\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,4{}\mathrm {i}+b^3\,x\,{\left (\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}\right )}^{3/2}\,128{}\mathrm {i}-b^5\,x\,{\left (\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}\right )}^{3/2}\,16{}\mathrm {i}+\sqrt {2}\,b\,x\,\sqrt {\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,32{}\mathrm {i}-\sqrt {2}\,b^3\,x\,\sqrt {\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,8{}\mathrm {i}}{4\,\sqrt {2}\,b-\sqrt {2}\,b^3+\sqrt {2}\,\sqrt {b^6-12\,b^4+48\,b^2-64}-2\,b^2+8}\right )\,\sqrt {\frac {12\,b+16\,\sqrt {2}-4\,\sqrt {2}\,b^2-3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {x\,\sqrt {-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,32{}\mathrm {i}-b\,x\,{\left (-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}\right )}^{3/2}\,256{}\mathrm {i}+b^2\,x\,\sqrt {-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,8{}\mathrm {i}-b^4\,x\,\sqrt {-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,4{}\mathrm {i}+b^3\,x\,{\left (-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}\right )}^{3/2}\,128{}\mathrm {i}-b^5\,x\,{\left (-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}\right )}^{3/2}\,16{}\mathrm {i}+\sqrt {2}\,b\,x\,\sqrt {-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,32{}\mathrm {i}-\sqrt {2}\,b^3\,x\,\sqrt {-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,8{}\mathrm {i}}{\sqrt {2}\,b^3-4\,\sqrt {2}\,b+\sqrt {2}\,\sqrt {b^6-12\,b^4+48\,b^2-64}+2\,b^2-8}\right )\,\sqrt {-\frac {4\,\sqrt {2}\,b^2-16\,\sqrt {2}-12\,b+3\,b^3+\sqrt {b^6-12\,b^4+48\,b^2-64}}{8\,b^4-64\,b^2+128}}\,2{}\mathrm {i} \]
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