Optimal. Leaf size=66 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {b+4}+4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b+4}-4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1161, 618, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {b+4}+4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}}-\frac {\tan ^{-1}\left (\frac {\sqrt {b+4}-4 x}{\sqrt {4-b}}\right )}{\sqrt {4-b}} \end {gather*}
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 = 134, normalized size = 2.03 \begin {gather*} \frac {\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}}+\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}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+2 x^2}{1-b x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.80, size = 120, normalized size = 1.82 \begin {gather*} \left [\frac {\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 \, \sqrt {b - 4}}, \frac {\sqrt {-b + 4} \arctan \left (\frac {{\left (4 \, x^{3} - {\left (b - 2\right )} x\right )} \sqrt {-b + 4}}{b - 4}\right ) + \sqrt {-b + 4} \arctan \left (\frac {2 \, \sqrt {-b + 4} x}{b - 4}\right )}{b - 4}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 80, normalized size = 1.21 \begin {gather*} \frac {{\left (b + 8\right )} \sqrt {-b + 4} \arctan \left (\frac {x}{\sqrt {-\frac {1}{8} \, b + \frac {1}{8} \, \sqrt {b^{2} - 16}}}\right )}{b^{2} + 4 \, b - 32} - \frac {{\left (b + 8\right )} \sqrt {-b + 4} \arctan \left (\frac {x}{\sqrt {-\frac {1}{8} \, b - \frac {1}{8} \, \sqrt {b^{2} - 16}}}\right )}{b^{2} + 4 \, b - 32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 277, normalized size = 4.20 \begin {gather*} \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 )}}} \end {gather*}
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 \begin {gather*} \int \frac {2 \, x^{2} + 1}{4 \, x^{4} - b x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.41, size = 24, normalized size = 0.36 \begin {gather*} -\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {b-4}}{2\,x^2-1}\right )}{\sqrt {b-4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.39, size = 83, normalized size = 1.26 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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