Integrand size = 44, antiderivative size = 74 \[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=\frac {\arctan \left (\frac {a+2 \left (1+a^2-b\right ) x+a x^2}{\sqrt {2} \sqrt {1-b} \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}}\right )}{\sqrt {2} \sqrt {1-b}} \]
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Time = 0.14 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {2109} \[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=\frac {\arctan \left (\frac {2 x \left (a^2-b+1\right )+a x^2+a}{\sqrt {2} \sqrt {1-b} \sqrt {2 a x^3+2 a x+2 b x^2+x^4+1}}\right )}{\sqrt {2} \sqrt {1-b}} \]
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Rule 2109
Rubi steps \begin{align*} \text {integral}& = \frac {\arctan \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*}
Time = 0.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-1+b} x}{1+2 a x+x^2-\sqrt {1+2 b x^2+x^4+2 a \left (x+x^3\right )}}\right )}{\sqrt {-1+b}} \]
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Time = 2.51 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\ln \left (2\right )+\ln \left (\frac {\sqrt {2 b -2}\, \sqrt {2 a \,x^{3}+x^{4}+2 x^{2} b +2 a x +1}-a \,x^{2}+\left (-2 a^{2}+2 b -2\right ) x -a}{2 a x +x^{2}+1}\right )}{\sqrt {2 b -2}}\) | \(78\) |
pseudoelliptic | \(\frac {\ln \left (2\right )+\ln \left (\frac {\sqrt {2 b -2}\, \sqrt {2 a \,x^{3}+x^{4}+2 x^{2} b +2 a x +1}-a \,x^{2}+\left (-2 a^{2}+2 b -2\right ) x -a}{2 a x +x^{2}+1}\right )}{\sqrt {2 b -2}}\) | \(78\) |
elliptic | \(\text {Expression too large to display}\) | \(258804\) |
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Time = 0.38 (sec) , antiderivative size = 252, normalized size of antiderivative = 3.41 \[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=\left [\frac {\sqrt {2} \log \left (\frac {4 \, a^{3} x^{3} + {\left (a^{2} + 2 \, b - 2\right )} x^{4} + 4 \, a^{3} x + 2 \, {\left (2 \, a^{4} + 5 \, a^{2} - 2 \, {\left (2 \, a^{2} + 3\right )} b + 4 \, b^{2} + 2\right )} x^{2} + a^{2} - \frac {2 \, \sqrt {2} \sqrt {2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1} {\left ({\left (a b - a\right )} x^{2} + a b - 2 \, {\left (a^{2} - {\left (a^{2} + 2\right )} b + b^{2} + 1\right )} x - a\right )}}{\sqrt {b - 1}} + 2 \, b - 2}{4 \, a x^{3} + x^{4} + 2 \, {\left (2 \, a^{2} + 1\right )} x^{2} + 4 \, a x + 1}\right )}{4 \, \sqrt {b - 1}}, \frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{b - 1}} \arctan \left (\frac {\sqrt {2} \sqrt {2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1} {\left (b - 1\right )} \sqrt {-\frac {1}{b - 1}}}{a x^{2} + 2 \, {\left (a^{2} - b + 1\right )} x + a}\right )\right ] \]
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\[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=- \int \frac {x^{2}}{2 a x \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + x^{2} \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1}}\, dx - \int \left (- \frac {1}{2 a x \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + x^{2} \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1} + \sqrt {2 a x^{3} + 2 a x + 2 b x^{2} + x^{4} + 1}}\right )\, dx \]
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\[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=\int { -\frac {x^{2} - 1}{\sqrt {2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1} {\left (2 \, a x + x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=\int { -\frac {x^{2} - 1}{\sqrt {2 \, a x^{3} + x^{4} + 2 \, b x^{2} + 2 \, a x + 1} {\left (2 \, a x + x^{2} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {1-x^2}{\left (1+2 a x+x^2\right ) \sqrt {1+2 a x+2 b x^2+2 a x^3+x^4}} \, dx=-\int \frac {x^2-1}{\left (x^2+2\,a\,x+1\right )\,\sqrt {x^4+2\,a\,x^3+2\,b\,x^2+2\,a\,x+1}} \,d x \]
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