Optimal. Leaf size=38 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]
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Rubi [A] time = 0.0256809, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {1150, 377, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 1150
Rule 377
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{d+e x^2}}{d^2-e^2 x^4} \, dx &=\int \frac{1}{\left (d-e x^2\right ) \sqrt{d+e x^2}} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{d-2 d e x^2} \, dx,x,\frac{x}{\sqrt{d+e x^2}}\right )\\ &=\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.156767, size = 38, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{e} x}{\sqrt{d+e x^2}}\right )}{\sqrt{2} d \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.027, size = 986, normalized size = 26. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{\sqrt{e x^{2} + d}}{e^{2} x^{4} - d^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15559, size = 340, normalized size = 8.95 \begin{align*} \left [\frac{\sqrt{2} \log \left (\frac{17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt{2}{\left (3 \, e x^{3} + d x\right )} \sqrt{e x^{2} + d} \sqrt{e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right )}{8 \, d \sqrt{e}}, -\frac{\sqrt{2} \sqrt{-e} \arctan \left (\frac{\sqrt{2}{\left (3 \, e x^{2} + d\right )} \sqrt{e x^{2} + d} \sqrt{-e}}{4 \,{\left (e^{2} x^{3} + d e x\right )}}\right )}{4 \, d e}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- d \sqrt{d + e x^{2}} + e x^{2} \sqrt{d + e x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.45393, size = 177, normalized size = 4.66 \begin{align*} -\frac{{\left (\sqrt{2} i \arctan \left (\frac{e^{\frac{1}{2}}}{\sqrt{-\frac{d e + \sqrt{d^{2}} e}{d}}}\right ) e^{\frac{1}{2}} - \sqrt{2} i \arctan \left (\frac{e^{\frac{1}{2}}}{\sqrt{-\frac{d e - \sqrt{d^{2}} e}{d}}}\right ) e^{\frac{1}{2}}\right )} e^{\left (-1\right )} \mathrm{sgn}\left (x\right )}{4 \,{\left | d \right |}} + \frac{\sqrt{2} i \arctan \left (\frac{\sqrt{\frac{d}{x^{2}} + e}}{\sqrt{-\frac{d e \mathrm{sgn}\left (x\right ) + \sqrt{d^{2}} e}{d \mathrm{sgn}\left (x\right )}}}\right ) e^{\left (-\frac{1}{2}\right )}}{2 \,{\left | d \right |}{\left | \mathrm{sgn}\left (x\right ) \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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