Optimal. Leaf size=283 \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{i b \text{PolyLog}\left (2,-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{2 c}-\frac{b^2 \text{PolyLog}\left (3,-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{2 c}-\frac{2 \tanh ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
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Rubi [A] time = 0.295276, antiderivative size = 283, 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, 4850, 4988, 4884, 4994, 6610} \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}-\frac{i b \text{PolyLog}\left (2,-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{c}+\frac{b^2 \text{PolyLog}\left (3,1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{2 c}-\frac{b^2 \text{PolyLog}\left (3,-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{2 c}-\frac{2 \tanh ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{c} \]
Antiderivative was successfully verified.
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Rule 6681
Rule 4850
Rule 4988
Rule 4884
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-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 \tan ^{-1}(x)\right )^2}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tan ^{-1}(x)\right ) \tanh ^{-1}\left (1-\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tan ^{-1}(x)\right ) \log \left (2-\frac{2}{1+i 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 \tan ^{-1}(x)\right ) \log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{i b \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{i b \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}+\frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-1+\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{2 \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{i b \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}-\frac{i b \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_2\left (-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{b^2 \text{Li}_3\left (1-\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{b^2 \text{Li}_3\left (-1+\frac{2}{1+\frac{i \sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.106713, size = 354, normalized size = 1.25 \[ -\frac{2 i b \text{PolyLog}\left (2,-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )-2 i b \text{PolyLog}\left (2,\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )+b^2 \text{PolyLog}\left (3,-\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right )-b^2 \text{PolyLog}\left (3,\frac{\sqrt{1-c x}+i \sqrt{c x+1}}{\sqrt{1-c x}-i \sqrt{c x+1}}\right )+4 \tanh ^{-1}\left (1-\frac{2 i}{-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}+i}\right ) \left (a+b \tan ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.218, size = 947, normalized size = 3.4 \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{b^{2} \log \left (2\right )^{2} \log \left (c x + 1\right ) - b^{2} \log \left (2\right )^{2} \log \left (-c x + 1\right ) - 4 \,{\left (b^{2} \log \left (c x + 1\right ) - b^{2} \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt{-c x + 1}, \sqrt{c x + 1}\right )^{2} -{\left (12 \, b^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} \arctan \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + b^{2}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} \log \left (2\right )^{2} - 4 \, b^{2} \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) \log \left (c x + 1\right )}{c^{2} x^{2} - 1}\,{d x} + 4 \, b^{2} \int \frac{\sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) \log \left (-c x + 1\right )}{c^{2} x^{2} - 1}\,{d x} + \frac{32 \,{\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 )} a b}{c}\right )} c}{32 \, 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^{2} \arctan \left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 2 \, a b \arctan \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(-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{{\left (b \arctan \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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