Integrand size = 26, antiderivative size = 82 \[ \int \frac {d+e x^2}{d^2+f x^2+e^2 x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt {2 d e-f}-2 e x}{\sqrt {2 d e+f}}\right )}{\sqrt {2 d e+f}}+\frac {\arctan \left (\frac {\sqrt {2 d e-f}+2 e x}{\sqrt {2 d e+f}}\right )}{\sqrt {2 d e+f}} \]
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Time = 0.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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. \(181\) vs. \(2(82)=164\).
Time = 0.07 (sec) , antiderivative size = 181, normalized size of antiderivative = 2.21 \[ \int \frac {d+e x^2}{d^2+f x^2+e^2 x^4} \, dx=\frac {\frac {\left (2 d e-f+\sqrt {-4 d^2 e^2+f^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {f-\sqrt {-4 d^2 e^2+f^2}}}\right )}{\sqrt {f-\sqrt {-4 d^2 e^2+f^2}}}+\frac {\left (-2 d e+f+\sqrt {-4 d^2 e^2+f^2}\right ) \arctan \left (\frac {\sqrt {2} e x}{\sqrt {f+\sqrt {-4 d^2 e^2+f^2}}}\right )}{\sqrt {f+\sqrt {-4 d^2 e^2+f^2}}}}{\sqrt {2} \sqrt {-4 d^2 e^2+f^2}} \]
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Time = 0.09 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {\arctan \left (\frac {-2 e x +\sqrt {2 e d -f}}{\sqrt {2 e d +f}}\right )}{\sqrt {2 e d +f}}+\frac {\arctan \left (\frac {2 e x +\sqrt {2 e d -f}}{\sqrt {2 e d +f}}\right )}{\sqrt {2 e d +f}}\) | \(71\) |
risch | \(-\frac {\ln \left (e \,x^{2} \sqrt {-2 e d -f}+\left (-2 e d -f \right ) x -d \sqrt {-2 e d -f}\right )}{2 \sqrt {-2 e d -f}}+\frac {\ln \left (e \,x^{2} \sqrt {-2 e d -f}+\left (2 e d +f \right ) x -d \sqrt {-2 e d -f}\right )}{2 \sqrt {-2 e d -f}}\) | \(104\) |
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Time = 0.26 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.98 \[ \int \frac {d+e x^2}{d^2+f x^2+e^2 x^4} \, dx=\left [-\frac {\sqrt {-2 \, d e - f} \log \left (\frac {e^{2} x^{4} - {\left (4 \, d e + f\right )} x^{2} + d^{2} - 2 \, {\left (e x^{3} - d x\right )} \sqrt {-2 \, d e - f}}{e^{2} x^{4} + f x^{2} + d^{2}}\right )}{2 \, {\left (2 \, d e + f\right )}}, \frac {\sqrt {2 \, d e + f} \arctan \left (\frac {e x}{\sqrt {2 \, d e + f}}\right ) + \sqrt {2 \, d e + f} \arctan \left (\frac {{\left (e^{2} x^{3} + {\left (d e + f\right )} x\right )} \sqrt {2 \, d e + f}}{2 \, d^{2} e + d f}\right )}{2 \, d e + f}\right ] \]
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Time = 0.29 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.49 \[ \int \frac {d+e x^2}{d^2+f x^2+e^2 x^4} \, dx=- \frac {\sqrt {- \frac {1}{2 d e + f}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (- 2 d e \sqrt {- \frac {1}{2 d e + f}} - f \sqrt {- \frac {1}{2 d e + f}}\right )}{e} \right )}}{2} + \frac {\sqrt {- \frac {1}{2 d e + f}} \log {\left (- \frac {d}{e} + x^{2} + \frac {x \left (2 d e \sqrt {- \frac {1}{2 d e + f}} + f \sqrt {- \frac {1}{2 d e + f}}\right )}{e} \right )}}{2} \]
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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. 191 vs. \(2 (73) = 146\).
Time = 0.69 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.33 \[ \int \frac {d+e x^2}{d^2+f x^2+e^2 x^4} \, dx=-\frac {{\left (2 \, d^{2} e^{3} + d e^{4} - d e^{2} f\right )} \sqrt {2 \, d e + f} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {f + \sqrt {-4 \, d^{2} e^{2} + f^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} + d e^{4} f - d e^{2} f^{2}} - \frac {{\left (2 \, d^{2} e^{3} + d e^{4} - d e^{2} f\right )} \sqrt {2 \, d e + f} \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} x}{\sqrt {\frac {f - \sqrt {-4 \, d^{2} e^{2} + f^{2}}}{e^{2}}}}\right )}{4 \, d^{3} e^{4} + 2 \, d^{2} e^{5} + d e^{4} f - d e^{2} f^{2}} \]
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Time = 0.10 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.20 \[ \int \frac {d+e x^2}{d^2+f x^2+e^2 x^4} \, dx=\frac {\mathrm {atan}\left (\frac {f^2\,x-\frac {x\,{\left (f+2\,d\,e\right )}^2}{2}+\frac {f\,x\,\left (f+2\,d\,e\right )}{2}+2\,e^2\,f\,x^3-e^2\,x^3\,\left (f+2\,d\,e\right )}{\left (2\,d\,f-d\,\left (f+2\,d\,e\right )\right )\,\sqrt {f+2\,d\,e}}\right )+\mathrm {atan}\left (\frac {e\,x}{\sqrt {f+2\,d\,e}}\right )}{\sqrt {f+2\,d\,e}} \]
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