Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{2 x \left (a^2-b+1\right )+a x^2+a}{\sqrt{2} \sqrt{1-b} \sqrt{2 a x^3+2 a x+2 b x^2+x^4+1}}\right )}{\sqrt{2} \sqrt{1-b}} \]
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Rubi [A] time = 0.204305, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.023, Rules used = {2084} \[ \frac{\tan ^{-1}\left (\frac{2 x \left (a^2-b+1\right )+a x^2+a}{\sqrt{2} \sqrt{1-b} \sqrt{2 a x^3+2 a x+2 b x^2+x^4+1}}\right )}{\sqrt{2} \sqrt{1-b}} \]
Antiderivative was successfully verified.
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Rule 2084
Rubi steps
\begin{align*} \int \frac{1-x^2}{\left (1+2 a x+x^2\right ) \sqrt{1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx &=\frac{\tan ^{-1}\left (\frac{a+2 \left (1+a^2-b\right ) x+a x^2}{\sqrt{2} \sqrt{1-b} \sqrt{1+2 a x+2 b x^2+2 a x^3+x^4}}\right )}{\sqrt{2} \sqrt{1-b}}\\ \end{align*}
Mathematica [C] time = 6.45998, size = 17955, normalized size = 242.64 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.114, size = 247419, normalized size = 3343.5 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 1}{\sqrt{2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1}{\left (2 \, a x + x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.10011, size = 633, normalized size = 8.55 \begin{align*} \left [\frac{\sqrt{2} \log \left (\frac{4 \, a^{3} x^{3} +{\left (a^{2} + 2 \, b - 2\right )} x^{4} + 4 \, a^{3} x + 2 \,{\left (2 \, a^{4} + 5 \, a^{2} - 2 \,{\left (2 \, a^{2} + 3\right )} b + 4 \, b^{2} + 2\right )} x^{2} + a^{2} - \frac{2 \, \sqrt{2} \sqrt{2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1}{\left ({\left (a b - a\right )} x^{2} + a b - 2 \,{\left (a^{2} -{\left (a^{2} + 2\right )} b + b^{2} + 1\right )} x - a\right )}}{\sqrt{b - 1}} + 2 \, b - 2}{4 \, a x^{3} + x^{4} + 2 \,{\left (2 \, a^{2} + 1\right )} x^{2} + 4 \, a x + 1}\right )}{4 \, \sqrt{b - 1}}, \frac{1}{2} \, \sqrt{2} \sqrt{-\frac{1}{b - 1}} \arctan \left (\frac{\sqrt{2} \sqrt{2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1}{\left (b - 1\right )} \sqrt{-\frac{1}{b - 1}}}{a x^{2} + 2 \,{\left (a^{2} - b + 1\right )} x + a}\right )\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{x^{2}}{2 a x \sqrt{2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + x^{2} \sqrt{2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + \sqrt{2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1}}\, dx - \int - \frac{1}{2 a x \sqrt{2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + x^{2} \sqrt{2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + \sqrt{2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} - 1}{\sqrt{2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1}{\left (2 \, a x + x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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