Optimal. Leaf size=62 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {4-b}+4 x}{\sqrt {b+4}}\right )}{\sqrt {b+4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {4-b}-4 x}{\sqrt {b+4}}\right )}{\sqrt {b+4}} \]
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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1161, 618, 204} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {4-b}+4 x}{\sqrt {b+4}}\right )}{\sqrt {b+4}}-\frac {\tan ^{-1}\left (\frac {\sqrt {4-b}-4 x}{\sqrt {b+4}}\right )}{\sqrt {b+4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 1161
Rubi steps
\begin {align*} \int \frac {1+2 x^2}{1+b x^2+4 x^4} \, dx &=\frac {1}{4} \int \frac {1}{\frac {1}{2}-\frac {1}{2} \sqrt {4-b} x+x^2} \, dx+\frac {1}{4} \int \frac {1}{\frac {1}{2}+\frac {1}{2} \sqrt {4-b} x+x^2} \, dx\\ &=-\left (\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{4} (-4-b)-x^2} \, dx,x,-\frac {\sqrt {4-b}}{2}+2 x\right )\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{4} (-4-b)-x^2} \, dx,x,\frac {\sqrt {4-b}}{2}+2 x\right )\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {4-b}-4 x}{\sqrt {4+b}}\right )}{\sqrt {4+b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {4-b}+4 x}{\sqrt {4+b}}\right )}{\sqrt {4+b}}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 126, normalized size = 2.03 \[ \frac {\frac {\left (\sqrt {b^2-16}-b+4\right ) \tan ^{-1}\left (\frac {2 \sqrt {2} x}{\sqrt {b-\sqrt {b^2-16}}}\right )}{\sqrt {b-\sqrt {b^2-16}}}+\frac {\left (\sqrt {b^2-16}+b-4\right ) \tan ^{-1}\left (\frac {2 \sqrt {2} x}{\sqrt {\sqrt {b^2-16}+b}}\right )}{\sqrt {\sqrt {b^2-16}+b}}}{\sqrt {2} \sqrt {b^2-16}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 110, normalized size = 1.77 \[ \left [-\frac {\sqrt {-b - 4} \log \left (\frac {4 \, x^{4} - {\left (b + 8\right )} x^{2} - 2 \, {\left (2 \, x^{3} - x\right )} \sqrt {-b - 4} + 1}{4 \, x^{4} + b x^{2} + 1}\right )}{2 \, {\left (b + 4\right )}}, \frac {\sqrt {b + 4} \arctan \left (\frac {4 \, x^{3} + {\left (b + 2\right )} x}{\sqrt {b + 4}}\right ) + \sqrt {b + 4} \arctan \left (\frac {2 \, x}{\sqrt {b + 4}}\right )}{b + 4}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 77, normalized size = 1.24 \[ \frac {\sqrt {b + 4} {\left (b - 8\right )} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} x}{\sqrt {b + \sqrt {b^{2} - 16}}}\right )}{b^{2} - 4 \, b - 32} + \frac {\sqrt {b + 4} {\left (b - 8\right )} \arctan \left (\frac {4 \, \sqrt {\frac {1}{2}} x}{\sqrt {b - \sqrt {b^{2} - 16}}}\right )}{b^{2} - 4 \, b - 32} \]
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 \[ -\frac {b \arctan \left (\frac {4 x}{\sqrt {2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {b \arctan \left (\frac {4 x}{\sqrt {2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {4 \arctan \left (\frac {4 x}{\sqrt {2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {\arctan \left (\frac {4 x}{\sqrt {2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {2 b -2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}-\frac {4 \arctan \left (\frac {4 x}{\sqrt {2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {\left (b -4\right ) \left (b +4\right )}\, \sqrt {2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}+\frac {\arctan \left (\frac {4 x}{\sqrt {2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}}\right )}{\sqrt {2 b +2 \sqrt {\left (b -4\right ) \left (b +4\right )}}} \]
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 \[ \int \frac {2 \, x^{2} + 1}{4 \, x^{4} + b x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.39, size = 66, normalized size = 1.06 \[ -\frac {\mathrm {atan}\left (\frac {-b^3\,x-4\,b^2\,x^3-2\,b^2\,x+16\,b\,x+64\,x^3+32\,x}{\left (b^2-16\right )\,\sqrt {b+4}}\right )-\mathrm {atan}\left (\frac {2\,x}{\sqrt {b+4}}\right )}{\sqrt {b+4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 95, normalized size = 1.53 \[ - \frac {\sqrt {- \frac {1}{b + 4}} \log {\left (x^{2} + x \left (- \frac {b \sqrt {- \frac {1}{b + 4}}}{2} - 2 \sqrt {- \frac {1}{b + 4}}\right ) - \frac {1}{2} \right )}}{2} + \frac {\sqrt {- \frac {1}{b + 4}} \log {\left (x^{2} + x \left (\frac {b \sqrt {- \frac {1}{b + 4}}}{2} + 2 \sqrt {- \frac {1}{b + 4}}\right ) - \frac {1}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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