Optimal. Leaf size=29 \[ \frac{\text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a} \]
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Rubi [A] time = 0.034437, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {2518} \[ \frac{\text{PolyLog}\left (2,\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 2518
Rubi steps
\begin{align*} \int \frac{\log \left (1-\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{1-a^2 x^2} \, dx &=\frac{\text{Li}_2\left (\frac{i \sqrt{1-a x}}{\sqrt{1+a x}}\right )}{a}\\ \end{align*}
Mathematica [B] time = 0.536149, size = 134, normalized size = 4.62 \[ \frac{\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(a x)}\right )-2 \left (\text{PolyLog}\left (2,-i e^{-\tanh ^{-1}(a x)}\right )-\text{PolyLog}\left (2,i e^{-\tanh ^{-1}(a x)}\right )+\tanh ^{-1}(a x) \left (\log \left (e^{-2 \tanh ^{-1}(a x)}+1\right )+\log \left (1-i e^{-\tanh ^{-1}(a x)}\right )-\log \left (1+i e^{-\tanh ^{-1}(a x)}\right )\right )\right )+4 \log \left (1-\frac{i \sqrt{1-a x}}{\sqrt{a x+1}}\right ) \tanh ^{-1}(a x)}{4 a} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{a}^{2}{x}^{2}+1}\ln \left ( 1-{i\sqrt{-ax+1}{\frac{1}{\sqrt{ax+1}}}} \right ) }\, dx \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*} -\frac{2 \,{\left (\log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} \log \left (a x + 1\right ) - \log \left (a x + 1\right )^{2} + 2 \, \log \left (a x + 1\right ) \log \left (-a x + 1\right ) - \log \left (-a x + 1\right )^{2} - 4 \,{\left (\log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )} \log \left (\sqrt{a x + 1} - i \, \sqrt{-a x + 1}\right )}{8 \, a} + \int \frac{\sqrt{a x + 1}{\left (\log \left (a x + 1\right ) - \log \left (-a x + 1\right )\right )}}{2 \,{\left (a^{2} x^{2} - 1\right )} \sqrt{a x + 1} -{\left (2 i \, a^{2} x^{2} - 2 i\right )} \sqrt{-a x + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.18479, size = 92, normalized size = 3.17 \begin{align*} \frac{{\rm Li}_2\left (-\frac{a x - i \, \sqrt{a x + 1} \sqrt{-a x + 1} + 1}{a x + 1} + 1\right )}{a} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\log \left (-\frac{i \, \sqrt{-a x + 1}}{\sqrt{a x + 1}} + 1\right )}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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