Integrand size = 35, antiderivative size = 62 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=-\frac {\log (x)}{\sqrt {a}}+\frac {\log \left (-2 a-b x-2 a x^2+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \]
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\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x \sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {x}{\sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = \int \frac {1}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx-\int \frac {x}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.90 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {-\log (x)+\log \left (-b x-2 a \left (1+x^2\right )+2 \sqrt {a} \sqrt {a+b x+c x^2+b x^3+a x^4}\right )}{\sqrt {a}} \]
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Time = 1.90 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.94
method | result | size |
pseudoelliptic | \(\frac {\ln \left (2\right )-\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )}{\sqrt {a}}\) | \(58\) |
default | \(-\frac {-\ln \left (2\right )+\ln \left (\frac {2 a \,x^{2}+2 \sqrt {a}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b x +2 a}{x \sqrt {a}}\right )}{\sqrt {a}}\) | \(59\) |
elliptic | \(\text {Expression too large to display}\) | \(3365\) |
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Time = 0.82 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.39 \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [\frac {\log \left (\frac {8 \, a^{2} x^{4} + 8 \, a b x^{3} + 8 \, a b x + {\left (8 \, a^{2} + b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (2 \, a x^{2} + b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {2 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} \sqrt {-a}}{2 \, a x^{2} + b x + 2 \, a}\right )}{a}\right ] \]
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\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=- \int \left (- \frac {1}{x \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\right )\, dx - \int \frac {x}{\sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]
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\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} x} \,d x } \]
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\[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { -\frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} x} \,d x } \]
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Timed out. \[ \int \frac {1-x^2}{x \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int -\frac {x^2-1}{x\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]
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