Optimal. Leaf size=29 \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-x \]
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Rubi [A] time = 0.023093, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1150, 388, 208} \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-x \]
Antiderivative was successfully verified.
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Rule 1150
Rule 388
Rule 208
Rubi steps
\begin{align*} \int \frac{\left (d+e x^2\right )^2}{d^2-e^2 x^4} \, dx &=\int \frac{d+e x^2}{d-e x^2} \, dx\\ &=-x+(2 d) \int \frac{1}{d-e x^2} \, dx\\ &=-x+\frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.0089009, size = 29, normalized size = 1. \[ \frac{2 \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 22, normalized size = 0.8 \begin{align*} -x+2\,{\frac{d}{\sqrt{de}}{\it Artanh} \left ({\frac{ex}{\sqrt{de}}} \right ) } \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 = 1.87925, size = 149, normalized size = 5.14 \begin{align*} \left [\sqrt{\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{\frac{d}{e}} + d}{e x^{2} - d}\right ) - x, -2 \, \sqrt{-\frac{d}{e}} \arctan \left (\frac{e x \sqrt{-\frac{d}{e}}}{d}\right ) - x\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.359195, size = 34, normalized size = 1.17 \begin{align*} - x - \sqrt{\frac{d}{e}} \log{\left (x - \sqrt{\frac{d}{e}} \right )} + \sqrt{\frac{d}{e}} \log{\left (x + \sqrt{\frac{d}{e}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15496, size = 159, normalized size = 5.48 \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 )}}{d} + \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 )}{2 \, d} - \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 )}{2 \, d} - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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