Optimal. Leaf size=268 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{b \text{PolyLog}\left (2,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{2 c}+\frac{b^2 \text{PolyLog}\left (3,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right )}{2 c}-\frac{2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
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Rubi [A] time = 0.31119, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {6681, 5914, 6052, 5948, 6058, 6610} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{b \text{PolyLog}\left (2,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{2 c}+\frac{b^2 \text{PolyLog}\left (3,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right )}{2 c}-\frac{2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 6681
Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (1-\frac{2}{1-x}\right ) \left (a+b \tanh ^{-1}(x)\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right ) \log \left (2-\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right ) \log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-1+\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{b^2 \text{Li}_3\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.100641, size = 324, normalized size = 1.21 \[ -\frac{b \text{PolyLog}\left (2,-\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )-b \text{PolyLog}\left (2,\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )-\frac{1}{2} b^2 \text{PolyLog}\left (3,-\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right )+\frac{1}{2} b^2 \text{PolyLog}\left (3,\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right )+2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.904, size = 676, normalized size = 2.5 \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*} \frac{1}{2} \, a^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} + \frac{{\left (b^{2} \log \left (c x + 1\right ) - b^{2} \log \left (-c x + 1\right )\right )} \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right )^{2}}{8 \, c} + \int -\frac{2 \,{\left (\sqrt{c x + 1} b^{2} - \sqrt{-c x + 1} b^{2}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{2} + 8 \,{\left (\sqrt{c x + 1} a b - \sqrt{-c x + 1} a b\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) -{\left (4 \,{\left (\sqrt{c x + 1} b^{2} - \sqrt{-c x + 1} b^{2}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) +{\left (8 \, a b -{\left (b^{2} c x - b^{2}\right )} \log \left (c x + 1\right ) +{\left (b^{2} c x - b^{2}\right )} \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} -{\left (8 \, a b -{\left (b^{2} c x + b^{2}\right )} \log \left (c x + 1\right ) +{\left (b^{2} c x + b^{2}\right )} \log \left (-c x + 1\right )\right )} \sqrt{-c x + 1}\right )} \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right )}{8 \,{\left ({\left (c^{2} x^{2} - 1\right )} \sqrt{c x + 1} -{\left (c^{2} x^{2} - 1\right )} \sqrt{-c x + 1}\right )}}\,{d x} \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^{2} \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 2 \, a b \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{2}}{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^{2}}{c^{2} x^{2} - 1}\, dx - \int \frac{b^{2} \operatorname{atanh}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac{2 a b \operatorname{atanh}{\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{{\left (b \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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