Optimal. Leaf size=24 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]
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Rubi [A] time = 0.0122413, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1150, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 1150
Rule 208
Rubi steps
\begin{align*} \int \frac{d+e x^2}{d^2-e^2 x^4} \, dx &=\int \frac{1}{d-e x^2} \, dx\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0043265, size = 24, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{d} \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 16, normalized size = 0.7 \begin{align*}{{\it Artanh} \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.90441, size = 151, normalized size = 6.29 \begin{align*} \left [\frac{\sqrt{d e} \log \left (\frac{e x^{2} + 2 \, \sqrt{d e} x + d}{e x^{2} - d}\right )}{2 \, d e}, -\frac{\sqrt{-d e} \arctan \left (\frac{\sqrt{-d e} x}{d}\right )}{d e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.186596, size = 46, normalized size = 1.92 \begin{align*} - \frac{\sqrt{\frac{1}{d e}} \log{\left (- d \sqrt{\frac{1}{d e}} + x \right )}}{2} + \frac{\sqrt{\frac{1}{d e}} \log{\left (d \sqrt{\frac{1}{d e}} + x \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16564, size = 157, normalized size = 6.54 \begin{align*} \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{7}{2}} -{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{7}{2}}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{{\left (d^{2}\right )}^{\frac{1}{4}}}\right ) e^{\left (-4\right )}}{2 \, d^{2}} + \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{11}{2}} +{\left (d^{2}\right )}^{\frac{3}{4}} e^{\frac{11}{2}}\right )} e^{\left (-6\right )} \log \left ({\left |{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} - \frac{{\left ({\left (d^{2}\right )}^{\frac{1}{4}} d e^{\frac{7}{2}} +{\left (d^{2}\right )}^{\frac{1}{4}}{\left | d \right |} e^{\frac{7}{2}}\right )} e^{\left (-4\right )} \log \left ({\left | -{\left (d^{2}\right )}^{\frac{1}{4}} e^{\left (-\frac{1}{2}\right )} + x \right |}\right )}{4 \, d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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