Optimal. Leaf size=62 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2-b}+2 x}{\sqrt {b+2}}\right )}{\sqrt {b+2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-b}-2 x}{\sqrt {b+2}}\right )}{\sqrt {b+2}} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1161, 618, 204} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {2-b}+2 x}{\sqrt {b+2}}\right )}{\sqrt {b+2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {2-b}-2 x}{\sqrt {b+2}}\right )}{\sqrt {b+2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+x^2}{1+b x^2+x^4} \, dx &=\frac {1}{2} \int \frac {1}{1-\sqrt {2-b} x+x^2} \, dx+\frac {1}{2} \int \frac {1}{1+\sqrt {2-b} x+x^2} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{-2-b-x^2} \, dx,x,-\sqrt {2-b}+2 x\right )-\operatorname {Subst}\left (\int \frac {1}{-2-b-x^2} \, dx,x,\sqrt {2-b}+2 x\right )\\ &=\frac {\tan ^{-1}\left (\frac {-\sqrt {2-b}+2 x}{\sqrt {2+b}}\right )}{\sqrt {2+b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2-b}+2 x}{\sqrt {2+b}}\right )}{\sqrt {2+b}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 124, normalized size = 2.00 \[ \frac {\frac {\left (\sqrt {b^2-4}-b+2\right ) \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {b-\sqrt {b^2-4}}}\right )}{\sqrt {b-\sqrt {b^2-4}}}+\frac {\left (\sqrt {b^2-4}+b-2\right ) \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {b^2-4}+b}}\right )}{\sqrt {\sqrt {b^2-4}+b}}}{\sqrt {2} \sqrt {b^2-4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 101, normalized size = 1.63 \[ \left [-\frac {\sqrt {-b - 2} \log \left (\frac {x^{4} - {\left (b + 4\right )} x^{2} - 2 \, {\left (x^{3} - x\right )} \sqrt {-b - 2} + 1}{x^{4} + b x^{2} + 1}\right )}{2 \, {\left (b + 2\right )}}, \frac {\sqrt {b + 2} \arctan \left (\frac {x^{3} + {\left (b + 1\right )} x}{\sqrt {b + 2}}\right ) + \sqrt {b + 2} \arctan \left (\frac {x}{\sqrt {b + 2}}\right )}{b + 2}\right ] \]
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 = 0.04, size = 277, normalized size = 4.47 \[ -\frac {b \arctan \left (\frac {2 x}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}+\frac {b \arctan \left (\frac {2 x}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}+\frac {2 \arctan \left (\frac {2 x}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {2 b -2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}-\frac {2 \arctan \left (\frac {2 x}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {\left (b -2\right ) \left (b +2\right )}\, \sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}+\frac {\arctan \left (\frac {2 x}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}}\right )}{\sqrt {2 b +2 \sqrt {\left (b -2\right ) \left (b +2\right )}}} \]
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 = 0.06, size = 73, normalized size = 1.18 \[ \frac {\mathrm {atan}\left (\frac {x}{\sqrt {b+2}}\right )+\mathrm {atan}\left (\left (b+2\right )\,\left (x\,\left (\frac {1}{\sqrt {b+2}}+\frac {\frac {4}{b+2}-1}{\left (b-2\right )\,\sqrt {b+2}}\right )+\frac {x^3\,\left (\frac {2\,b}{b+2}-1\right )}{\left (b-2\right )\,\sqrt {b+2}}\right )\right )}{\sqrt {b+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 88, normalized size = 1.42 \[ - \frac {\sqrt {- \frac {1}{b + 2}} \log {\left (x^{2} + x \left (- b \sqrt {- \frac {1}{b + 2}} - 2 \sqrt {- \frac {1}{b + 2}}\right ) - 1 \right )}}{2} + \frac {\sqrt {- \frac {1}{b + 2}} \log {\left (x^{2} + x \left (b \sqrt {- \frac {1}{b + 2}} + 2 \sqrt {- \frac {1}{b + 2}}\right ) - 1 \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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