Optimal. Leaf size=50 \[ \frac{\sqrt{-\frac{e}{d}} \text{PolyLog}\left (2,\frac{2 x \left (d \sqrt{-\frac{e}{d}}-e x\right )}{d+e x^2}+1\right )}{2 e} \]
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Rubi [A] time = 0.0872988, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.025, Rules used = {2447} \[ \frac{\sqrt{-\frac{e}{d}} \text{PolyLog}\left (2,\frac{2 x \left (d \sqrt{-\frac{e}{d}}-e x\right )}{d+e x^2}+1\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin{align*} \int \frac{\log \left (-\frac{2 x \left (d \sqrt{-\frac{e}{d}}-e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=\frac{\sqrt{-\frac{e}{d}} \text{Li}_2\left (1+\frac{2 x \left (d \sqrt{-\frac{e}{d}}-e x\right )}{d+e x^2}\right )}{2 e}\\ \end{align*}
Mathematica [B] time = 0.366076, size = 642, normalized size = 12.84 \[ \frac{-2 \text{PolyLog}\left (2,\frac{\sqrt{-\frac{e}{d}} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{-\frac{e}{d}}+\sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-\frac{e}{d}} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{-\frac{e}{d}}-\sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )-2 \text{PolyLog}\left (2,\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right )+2 \log \left (\frac{2 e x \left (\frac{1}{\sqrt{-\frac{e}{d}}}+x\right )}{d+e x^2}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{2 e x \left (\frac{1}{\sqrt{-\frac{e}{d}}}+x\right )}{d+e x^2}\right )+\log ^2\left (\sqrt{-d}-\sqrt{e} x\right )-\log ^2\left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{e} \left (x \sqrt{-\frac{e}{d}}+1\right )}{\sqrt{-d} \sqrt{-\frac{e}{d}}+\sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )+2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{e} \left (x \sqrt{-\frac{e}{d}}+1\right )}{\sqrt{e}-\sqrt{-d} \sqrt{-\frac{e}{d}}}\right )}{4 \sqrt{-d} \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{e{x}^{2}+d}\ln \left ( -2\,{\frac{x}{e{x}^{2}+d} \left ( -ex+d\sqrt{-{\frac{e}{d}}} \right ) } \right ) }\, dx \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 = 1.95143, size = 93, normalized size = 1.86 \begin{align*} \frac{\sqrt{-\frac{e}{d}}{\rm Li}_2\left (-\frac{2 \,{\left (e x^{2} - d x \sqrt{-\frac{e}{d}}\right )}}{e x^{2} + d} + 1\right )}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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