Optimal. Leaf size=50 \[ -\frac{\text{PolyLog}\left (2,\frac{2 x \left (\sqrt{d} \sqrt{-e}-e x\right )}{d+e x^2}+1\right )}{2 \sqrt{d} \sqrt{-e}} \]
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Rubi [A] time = 0.0783351, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.024, Rules used = {2447} \[ -\frac{\text{PolyLog}\left (2,\frac{2 x \left (\sqrt{d} \sqrt{-e}-e x\right )}{d+e x^2}+1\right )}{2 \sqrt{d} \sqrt{-e}} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin{align*} \int \frac{\log \left (-\frac{2 x \left (\sqrt{d} \sqrt{-e}-e x\right )}{d+e x^2}\right )}{d+e x^2} \, dx &=-\frac{\text{Li}_2\left (1+\frac{2 x \left (\sqrt{d} \sqrt{-e}-e x\right )}{d+e x^2}\right )}{2 \sqrt{d} \sqrt{-e}}\\ \end{align*}
Mathematica [B] time = 0.251433, size = 645, normalized size = 12.9 \[ \frac{2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}+1\right )-2 \text{PolyLog}\left (2,\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )-2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right )-2 \text{PolyLog}\left (2,\frac{e x-\sqrt{-d} \sqrt{e}}{\sqrt{d} \sqrt{-e}-\sqrt{-d} \sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt{e}+e x}{\sqrt{d} \sqrt{-e}+\sqrt{-d} \sqrt{e}}\right )+2 \log \left (\frac{2 x \left (e x-\sqrt{d} \sqrt{-e}\right )}{d+e x^2}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{2 x \left (e x-\sqrt{d} \sqrt{-e}\right )}{d+e x^2}\right )+\log ^2\left (\sqrt{-d}-\sqrt{e} x\right )-\log ^2\left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d-\sqrt{-d} \sqrt{e} x}{2 d}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )-2 \log \left (\frac{\sqrt{d} \sqrt{-e}-e x}{\sqrt{d} \sqrt{-e}-\sqrt{-d} \sqrt{e}}\right ) \log \left (\sqrt{-d}-\sqrt{e} x\right )+2 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )-2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{-d} \sqrt{e} x+d}{2 d}\right )+2 \log \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (\frac{\sqrt{d} \sqrt{-e}-e x}{\sqrt{d} \sqrt{-e}+\sqrt{-d} \sqrt{e}}\right )}{4 \sqrt{-d} \sqrt{e}} \]
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 \left ( -ex+\sqrt{d}\sqrt{-e} \right ) }{e{x}^{2}+d}} \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 = 1.84806, size = 109, normalized size = 2.18 \begin{align*} \frac{\sqrt{-e}{\rm Li}_2\left (-\frac{2 \,{\left (e x^{2} - \sqrt{d} \sqrt{-e} x\right )}}{e x^{2} + d} + 1\right )}{2 \, \sqrt{d} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (\frac{2 \,{\left (e x - \sqrt{d} \sqrt{-e}\right )} x}{e x^{2} + d}\right )}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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