Integrand size = 27, antiderivative size = 86 \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2 d e+f}-2 e x}{\sqrt {2 d e-f}}\right )}{\sqrt {2 d e-f}}+\frac {\arctan \left (\frac {\sqrt {2 d e+f}+2 e x}{\sqrt {2 d e-f}}\right )}{\sqrt {2 d e-f}} \]
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Time = 0.07 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1175, 632, 210} \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=\frac {\arctan \left (\frac {\sqrt {2 d e+f}+2 e x}{\sqrt {2 d e-f}}\right )}{\sqrt {2 d e-f}}-\frac {\arctan \left (\frac {\sqrt {2 d e+f}-2 e x}{\sqrt {2 d e-f}}\right )}{\sqrt {2 d e-f}} \]
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Rule 210
Rule 632
Rule 1175
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{\frac {d}{e}-\frac {\sqrt {2 d e+f} x}{e}+x^2} \, dx}{2 e}+\frac {\int \frac {1}{\frac {d}{e}+\frac {\sqrt {2 d e+f} x}{e}+x^2} \, dx}{2 e} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{-\frac {2 d e-f}{e^2}-x^2} \, dx,x,-\frac {\sqrt {2 d e+f}}{e}+2 x\right )}{e}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {2 d e-f}{e^2}-x^2} \, dx,x,\frac {\sqrt {2 d e+f}}{e}+2 x\right )}{e} \\ & = -\frac {\tan ^{-1}\left (\frac {\sqrt {2 d e+f}-2 e x}{\sqrt {2 d e-f}}\right )}{\sqrt {2 d e-f}}+\frac {\tan ^{-1}\left (\frac {\sqrt {2 d e+f}+2 e x}{\sqrt {2 d e-f}}\right )}{\sqrt {2 d e-f}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(189\) vs. \(2(86)=172\).
Time = 0.07 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.20 \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=\frac {\frac {\left (2 d e+f+\sqrt {-4 d^2 e^2+f^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {-f-\sqrt {-4 d^2 e^2+f^2}}}\right )}{\sqrt {-f-\sqrt {-4 d^2 e^2+f^2}}}+\frac {\left (-2 d e-f+\sqrt {-4 d^2 e^2+f^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {-f+\sqrt {-4 d^2 e^2+f^2}}}\right )}{\sqrt {-f+\sqrt {-4 d^2 e^2+f^2}}}}{\sqrt {2} \sqrt {-4 d^2 e^2+f^2}} \]
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Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {\arctan \left (\frac {-2 e x +\sqrt {2 e d +f}}{\sqrt {2 e d -f}}\right )}{\sqrt {2 e d -f}}+\frac {\arctan \left (\frac {2 e x +\sqrt {2 e d +f}}{\sqrt {2 e d -f}}\right )}{\sqrt {2 e d -f}}\) | \(75\) |
risch | \(-\frac {\ln \left (e \,x^{2} \sqrt {-2 e d +f}+\left (-2 e d +f \right ) x -d \sqrt {-2 e d +f}\right )}{2 \sqrt {-2 e d +f}}+\frac {\ln \left (e \,x^{2} \sqrt {-2 e d +f}+\left (2 e d -f \right ) x -d \sqrt {-2 e d +f}\right )}{2 \sqrt {-2 e d +f}}\) | \(92\) |
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Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.08 \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=\left [-\frac {\sqrt {-2 \, d e + f} \log \left (\frac {e^{2} x^{4} - {\left (4 \, d e - f\right )} x^{2} + d^{2} - 2 \, {\left (e x^{3} - d x\right )} \sqrt {-2 \, d e + f}}{e^{2} x^{4} - f x^{2} + d^{2}}\right )}{2 \, {\left (2 \, d e - f\right )}}, -\frac {\sqrt {2 \, d e - f} \arctan \left (-\frac {e x}{\sqrt {2 \, d e - f}}\right ) + \sqrt {2 \, d e - f} \arctan \left (-\frac {{\left (e^{2} x^{3} + {\left (d e - f\right )} x\right )} \sqrt {2 \, d e - f}}{2 \, d^{2} e - d f}\right )}{2 \, d e - f}\right ] \]
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Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.41 \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=- \frac {\sqrt {- \frac {1}{2 d e - f}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (- 2 d e \sqrt {- \frac {1}{2 d e - f}} + f \sqrt {- \frac {1}{2 d e - f}}\right )}{e} \right )}}{2} + \frac {\sqrt {- \frac {1}{2 d e - f}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (2 d e \sqrt {- \frac {1}{2 d e - f}} - f \sqrt {- \frac {1}{2 d e - f}}\right )}{e} \right )}}{2} \]
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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. 197 vs. \(2 (77) = 154\).
Time = 0.71 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.29 \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=-\frac {{\left (2 \, d^{2} e^{3} + d e^{4} + d e^{2} f\right )} \sqrt {2 \, d e - f} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {f + \sqrt {-4 \, d^{2} e^{2} + f^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} - d e^{4} f - d e^{2} f^{2}} - \frac {{\left (2 \, d^{2} e^{3} + d e^{4} + d e^{2} f\right )} \sqrt {2 \, d e - f} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {-\frac {f - \sqrt {-4 \, d^{2} e^{2} + f^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} - d e^{4} f - d e^{2} f^{2}} \]
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Time = 0.11 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {d+e x^2}{d^2-f x^2+e^2 x^4} \, dx=-\frac {\mathrm {atan}\left (\frac {e^2\,x^3\,\sqrt {2\,d\,e-f}-f\,x\,\sqrt {2\,d\,e-f}+d\,e\,x\,\sqrt {2\,d\,e-f}}{d\,\left (f-2\,d\,e\right )}\right )-\mathrm {atan}\left (\frac {e\,x}{\sqrt {2\,d\,e-f}}\right )}{\sqrt {2\,d\,e-f}} \]
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