Integrand size = 37, antiderivative size = 70 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {2 a-c} x}{\sqrt {a}+\sqrt {a} x^2-\sqrt {a+b x+c x^2+b x^3+a x^4}}\right )}{\sqrt {2 a-c}} \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}}-\frac {2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (2 \int \left (\frac {i}{2 (i-x) \sqrt {a+b x+c x^2+b x^3+a x^4}}+\frac {i}{2 (i+x) \sqrt {a+b x+c x^2+b x^3+a x^4}}\right ) \, dx\right )+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ & = -\left (i \int \frac {1}{(i-x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx\right )-i \int \frac {1}{(i+x) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx+\int \frac {1}{\sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.96 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt {2 a-c} x}{\sqrt {a} \left (1+x^2\right )-\sqrt {x \left (b+c x+b x^2\right )+a \left (1+x^4\right )}}\right )}{\sqrt {2 a-c}} \]
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Time = 2.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {\ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x^{2}+1}\right )}{\sqrt {-2 a +c}}\) | \(66\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {2 \sqrt {-2 a +c}\, \sqrt {a \,x^{4}+b \,x^{3}+c \,x^{2}+b x +a}+b \,x^{2}+\left (-4 a +2 c \right ) x +b}{x^{2}+1}\right )}{\sqrt {-2 a +c}}\) | \(66\) |
elliptic | \(\text {Expression too large to display}\) | \(90322\) |
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Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (60) = 120\).
Time = 0.45 (sec) , antiderivative size = 291, normalized size of antiderivative = 4.16 \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\left [-\frac {\sqrt {-2 \, a + c} \log \left (-\frac {{\left (8 \, a^{2} - b^{2} - 4 \, a c\right )} x^{4} + 8 \, {\left (2 \, a b - b c\right )} x^{3} - 2 \, {\left (8 \, a^{2} + b^{2} - 12 \, a c + 4 \, c^{2}\right )} x^{2} + 8 \, a^{2} + 4 \, \sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (b x^{2} - 2 \, {\left (2 \, a - c\right )} x + b\right )} \sqrt {-2 \, a + c} - b^{2} - 4 \, a c + 8 \, {\left (2 \, a b - b c\right )} x}{x^{4} + 2 \, x^{2} + 1}\right )}{2 \, {\left (2 \, a - c\right )}}, -\frac {\arctan \left (-\frac {\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (b x^{2} - 2 \, {\left (2 \, a - c\right )} x + b\right )} \sqrt {2 \, a - c}}{2 \, {\left ({\left (2 \, a^{2} - a c\right )} x^{4} + {\left (2 \, a b - b c\right )} x^{3} + {\left (2 \, a c - c^{2}\right )} x^{2} + 2 \, a^{2} - a c + {\left (2 \, a b - b c\right )} x\right )}}\right )}{\sqrt {2 \, a - c}}\right ] \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {a x^{4} + a + b x^{3} + b x + c x^{2}}}\, dx \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{2} + 1\right )}} \,d x } \]
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\[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int { \frac {x^{2} - 1}{\sqrt {a x^{4} + b x^{3} + c x^{2} + b x + a} {\left (x^{2} + 1\right )}} \,d x } \]
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Timed out. \[ \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {a+b x+c x^2+b x^3+a x^4}} \, dx=\int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {a\,x^4+b\,x^3+c\,x^2+b\,x+a}} \,d x \]
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