Optimal. Leaf size=62 \[ \frac {\log \left (\sqrt {2-b} x+x^2+1\right )}{2 \sqrt {2-b}}-\frac {\log \left (-\sqrt {2-b} x+x^2+1\right )}{2 \sqrt {2-b}} \]
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Rubi [A] time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1164, 628} \[ \frac {\log \left (\sqrt {2-b} x+x^2+1\right )}{2 \sqrt {2-b}}-\frac {\log \left (-\sqrt {2-b} x+x^2+1\right )}{2 \sqrt {2-b}} \]
Antiderivative was successfully verified.
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Rule 628
Rule 1164
Rubi steps
\begin {align*} \int \frac {1-x^2}{1+b x^2+x^4} \, dx &=-\frac {\int \frac {\sqrt {2-b}+2 x}{-1-\sqrt {2-b} x-x^2} \, dx}{2 \sqrt {2-b}}-\frac {\int \frac {\sqrt {2-b}-2 x}{-1+\sqrt {2-b} x-x^2} \, dx}{2 \sqrt {2-b}}\\ &=-\frac {\log \left (1-\sqrt {2-b} x+x^2\right )}{2 \sqrt {2-b}}+\frac {\log \left (1+\sqrt {2-b} x+x^2\right )}{2 \sqrt {2-b}}\\ \end {align*}
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Mathematica [B] time = 0.07, size = 125, normalized size = 2.02 \[ \frac {\frac {\left (-\sqrt {b^2-4}+b+2\right ) \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b-\sqrt {b^2-4}}}\right )}{\sqrt {b-\sqrt {b^2-4}}}-\frac {\left (\sqrt {b^2-4}+b+2\right ) \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {b^2-4}+b}}\right )}{\sqrt {\sqrt {b^2-4}+b}}}{\sqrt {2} \sqrt {b^2-4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 100, normalized size = 1.61 \[ \left [-\frac {\sqrt {-b + 2} \log \left (\frac {x^{4} - {\left (b - 4\right )} x^{2} + 2 \, {\left (x^{3} + x\right )} \sqrt {-b + 2} + 1}{x^{4} + b x^{2} + 1}\right )}{2 \, {\left (b - 2\right )}}, \frac {\sqrt {b - 2} \arctan \left (\frac {x^{3} + {\left (b - 1\right )} x}{\sqrt {b - 2}}\right ) - \sqrt {b - 2} \arctan \left (\frac {x}{\sqrt {b - 2}}\right )}{b - 2}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 279, normalized size = 4.50 \[ \frac {b \arctan \left (\frac {2 x}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}-\frac {b \arctan \left (\frac {2 x}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}+\frac {2 \arctan \left (\frac {2 x}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}-\frac {2 \arctan \left (\frac {2 x}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}-\frac {\arctan \left (\frac {2 x}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}} \]
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 \[ -\int \frac {x^{2} - 1}{x^{4} + b x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.34, size = 76, normalized size = 1.23 \[ -\frac {\mathrm {atan}\left (\frac {x}{\sqrt {b-2}}\right )-\mathrm {atan}\left (\left (b-2\right )\,\left (x\,\left (\frac {1}{\sqrt {b-2}}+\frac {\frac {4}{b-2}+1}{\sqrt {b-2}\,\left (b+2\right )}\right )+\frac {x^3\,\left (\frac {2\,b}{b-2}-1\right )}{\sqrt {b-2}\,\left (b+2\right )}\right )\right )}{\sqrt {b-2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.35, size = 87, normalized size = 1.40 \[ \frac {\sqrt {- \frac {1}{b - 2}} \log {\left (x^{2} + x \left (- b \sqrt {- \frac {1}{b - 2}} + 2 \sqrt {- \frac {1}{b - 2}}\right ) + 1 \right )}}{2} - \frac {\sqrt {- \frac {1}{b - 2}} \log {\left (x^{2} + x \left (b \sqrt {- \frac {1}{b - 2}} - 2 \sqrt {- \frac {1}{b - 2}}\right ) + 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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