Optimal. Leaf size=98 \[ \frac{i b \text{PolyLog}\left (2,-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right )}{2 c}-\frac{i b \text{PolyLog}\left (2,\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right )}{2 c}-\frac{a \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}{c} \]
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Rubi [A] time = 0.0690427, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {206, 6681, 4849, 2391} \[ \frac{i b \text{PolyLog}\left (2,-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right )}{2 c}-\frac{i b \text{PolyLog}\left (2,\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right )}{2 c}-\frac{a \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 206
Rule 6681
Rule 4849
Rule 2391
Rubi steps
\begin{align*} \int \frac{a+b \cot ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a+b \cot ^{-1}(x)}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{a \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}-\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i}{x}\right )}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{2 c}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i}{x}\right )}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{2 c}\\ &=-\frac{a \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}+\frac{i b \text{Li}_2\left (-\frac{i \sqrt{1+c x}}{\sqrt{1-c x}}\right )}{2 c}-\frac{i b \text{Li}_2\left (\frac{i \sqrt{1+c x}}{\sqrt{1-c x}}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0361056, size = 93, normalized size = 0.95 \[ -\frac{-\frac{1}{2} i b \text{PolyLog}\left (2,-\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right )+\frac{1}{2} i b \text{PolyLog}\left (2,\frac{i \sqrt{c x+1}}{\sqrt{1-c x}}\right )+a \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}{c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.773, size = 364, normalized size = 3.7 \begin{align*}{\frac{a\ln \left ( cx+1 \right ) }{2\,c}}-{\frac{a\ln \left ( cx-1 \right ) }{2\,c}}+{\frac{b}{c}{\rm arccot} \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}\right )\ln \left ( 1+{ \left ( i+{\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ){\frac{1}{\sqrt{{\frac{-cx+1}{cx+1}}+1}}}} \right ) }-{\frac{ib}{c}{\it polylog} \left ( 2,-{ \left ( i+{\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ){\frac{1}{\sqrt{{\frac{-cx+1}{cx+1}}+1}}}} \right ) }+{\frac{b}{c}{\rm arccot} \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}\right )\ln \left ( 1-{ \left ( i+{\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ){\frac{1}{\sqrt{{\frac{-cx+1}{cx+1}}+1}}}} \right ) }-{\frac{ib}{c}{\it polylog} \left ( 2,{ \left ( i+{\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ){\frac{1}{\sqrt{{\frac{-cx+1}{cx+1}}+1}}}} \right ) }-{\frac{b}{c}{\rm arccot} \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}}\right )\ln \left ({ \left ( i+{\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) ^{2} \left ({\frac{-cx+1}{cx+1}}+1 \right ) ^{-1}}+1 \right ) }+{\frac{{\frac{i}{2}}b}{c}{\it polylog} \left ( 2,-{ \left ( i+{\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) ^{2} \left ({\frac{-cx+1}{cx+1}}+1 \right ) ^{-1}} \right ) } \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{1}{2} \, a{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} + \frac{{\left ({\left (\log \left (c x + 1\right ) - \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{c x + 1}, \sqrt{-c x + 1}\right ) + c \int \frac{e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) - e^{\left (\frac{1}{2} \, \log \left (c x + 1\right ) + \frac{1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right )}{{\left (c^{2} x^{2} - 1\right )}{\left (c x + 1\right )} -{\left (c^{2} x^{2} - 1\right )}{\left (c x - 1\right )}}\,{d x}\right )} b}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{arccot}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a}{c^{2} x^{2} - 1}, x\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{a}{c^{2} x^{2} - 1}\, dx - \int \frac{b \operatorname{acot}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx \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{b \operatorname{arccot}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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