Integrand size = 28, antiderivative size = 70 \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=-\frac {\log \left (d-\sqrt {2 d e+f} x+e x^2\right )}{2 \sqrt {2 d e+f}}+\frac {\log \left (d+\sqrt {2 d e+f} x+e x^2\right )}{2 \sqrt {2 d e+f}} \]
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Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1178, 642} \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=\frac {\log \left (x \sqrt {2 d e+f}+d+e x^2\right )}{2 \sqrt {2 d e+f}}-\frac {\log \left (-x \sqrt {2 d e+f}+d+e x^2\right )}{2 \sqrt {2 d e+f}} \]
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Rule 642
Rule 1178
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {\frac {\sqrt {2 d e+f}}{e}+2 x}{-\frac {d}{e}-\frac {\sqrt {2 d e+f} x}{e}-x^2} \, dx}{2 \sqrt {2 d e+f}}-\frac {\int \frac {\frac {\sqrt {2 d e+f}}{e}-2 x}{-\frac {d}{e}+\frac {\sqrt {2 d e+f} x}{e}-x^2} \, dx}{2 \sqrt {2 d e+f}} \\ & = -\frac {\log \left (d-\sqrt {2 d e+f} x+e x^2\right )}{2 \sqrt {2 d e+f}}+\frac {\log \left (d+\sqrt {2 d e+f} x+e x^2\right )}{2 \sqrt {2 d e+f}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(70)=140\).
Time = 0.08 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.71 \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=\frac {-\frac {\left (-2 d e+f+\sqrt {-4 d^2 e^2+f^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {-f-\sqrt {-4 d^2 e^2+f^2}}}\right )}{\sqrt {-f-\sqrt {-4 d^2 e^2+f^2}}}+\frac {\left (-2 d e+f-\sqrt {-4 d^2 e^2+f^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {-f+\sqrt {-4 d^2 e^2+f^2}}}\right )}{\sqrt {-f+\sqrt {-4 d^2 e^2+f^2}}}}{\sqrt {2} \sqrt {-4 d^2 e^2+f^2}} \]
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Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\ln \left (d +e \,x^{2}+x \sqrt {2 e d +f}\right )}{2 \sqrt {2 e d +f}}-\frac {\ln \left (-e \,x^{2}+x \sqrt {2 e d +f}-d \right )}{2 \sqrt {2 e d +f}}\) | \(61\) |
risch | \(\frac {\ln \left (\sqrt {2 e d +f}\, e \,x^{2}+\left (2 e d +f \right ) x +\sqrt {2 e d +f}\, d \right )}{2 \sqrt {2 e d +f}}-\frac {\ln \left (\sqrt {2 e d +f}\, e \,x^{2}+\left (-2 e d -f \right ) x +\sqrt {2 e d +f}\, d \right )}{2 \sqrt {2 e d +f}}\) | \(90\) |
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Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.40 \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=\left [\frac {\log \left (\frac {e^{2} x^{4} + {\left (4 \, d e + f\right )} x^{2} + d^{2} + 2 \, {\left (e x^{3} + d x\right )} \sqrt {2 \, d e + f}}{e^{2} x^{4} - f x^{2} + d^{2}}\right )}{2 \, \sqrt {2 \, d e + f}}, -\frac {\sqrt {-2 \, d e - f} \arctan \left (\frac {\sqrt {-2 \, d e - f} e x}{2 \, d e + f}\right ) - \sqrt {-2 \, d e - f} \arctan \left (\frac {{\left (e^{2} x^{3} - {\left (d e + f\right )} x\right )} \sqrt {-2 \, d e - f}}{2 \, d^{2} e + d f}\right )}{2 \, d e + f}\right ] \]
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Time = 0.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.60 \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=- \frac {\sqrt {\frac {1}{2 d e + f}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (- 2 d e \sqrt {\frac {1}{2 d e + f}} - f \sqrt {\frac {1}{2 d e + f}}\right )}{e} \right )}}{2} + \frac {\sqrt {\frac {1}{2 d e + f}} \log {\left (\frac {d}{e} + x^{2} + \frac {x \left (2 d e \sqrt {\frac {1}{2 d e + f}} + f \sqrt {\frac {1}{2 d e + f}}\right )}{e} \right )}}{2} \]
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\[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=\int { -\frac {e x^{2} - d}{e^{2} x^{4} - f x^{2} + d^{2}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (58) = 116\).
Time = 0.69 (sec) , antiderivative size = 200, normalized size of antiderivative = 2.86 \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=\frac {{\left (2 \, d^{2} e^{3} - d e^{4} - d e^{2} f\right )} \sqrt {-2 \, d e - f} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {f + \sqrt {-4 \, d^{2} e^{2} + f^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} - 2 \, d^{2} e^{5} - d e^{4} f - d e^{2} f^{2}} - \frac {{\left (2 \, d^{2} e^{3} - d e^{4} - d e^{2} f\right )} \sqrt {-2 \, d e - f} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {f - \sqrt {-4 \, d^{2} e^{2} + f^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} - 2 \, d^{2} e^{5} - d e^{4} f - d e^{2} f^{2}} \]
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Time = 13.20 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.41 \[ \int \frac {d-e x^2}{d^2-f x^2+e^2 x^4} \, dx=\frac {\mathrm {atanh}\left (\frac {x\,\sqrt {f+2\,d\,e}}{e\,x^2+d}\right )}{\sqrt {f+2\,d\,e}} \]
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