Optimal. Leaf size=409 \[ -\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b^2 \text{PolyLog}\left (3,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 b \text{PolyLog}\left (2,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{4 c}-\frac{3 b^3 \text{PolyLog}\left (4,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right )}{4 c}-\frac{2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
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Rubi [A] time = 0.506209, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {6681, 5914, 6052, 5948, 6058, 6062, 6610} \[ -\frac{3 b^2 \text{PolyLog}\left (3,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b^2 \text{PolyLog}\left (3,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b \text{PolyLog}\left (2,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 b \text{PolyLog}\left (2,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}+\frac{3 b^3 \text{PolyLog}\left (4,1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right )}{4 c}-\frac{3 b^3 \text{PolyLog}\left (4,\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}-1\right )}{4 c}-\frac{2 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
Antiderivative was successfully verified.
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Rule 6681
Rule 5914
Rule 6052
Rule 5948
Rule 6058
Rule 6062
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 )^3}{1-c^2 x^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^3}{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 )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{\tanh ^{-1}\left (1-\frac{2}{1-x}\right ) \left (a+b \tanh ^{-1}(x)\right )^2}{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 )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2 \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{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right )^2 \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 )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{3 b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{3 b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right ) \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{\left (3 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \tanh ^{-1}(x)\right ) \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 )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{3 b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{3 b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{3 b^2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b^2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (1-\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{2 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (-1+\frac{2}{1-x}\right )}{1-x^2} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{2 c}\\ &=-\frac{2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{c}+\frac{3 b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{3 b \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}-\frac{3 b^2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b^2 \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{2 c}+\frac{3 b^3 \text{Li}_4\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{4 c}-\frac{3 b^3 \text{Li}_4\left (-1+\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{1+c x}}}\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.174075, size = 482, normalized size = 1.18 \[ -\frac{-6 b^2 \text{PolyLog}\left (3,-\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )+6 b^2 \text{PolyLog}\left (3,\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )+6 b \text{PolyLog}\left (2,-\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2-6 b \text{PolyLog}\left (2,\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2+3 b^3 \text{PolyLog}\left (4,-\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right )-3 b^3 \text{PolyLog}\left (4,\frac{\sqrt{1-c x}+\sqrt{c x+1}}{\sqrt{1-c x}-\sqrt{c x+1}}\right )+8 \tanh ^{-1}\left (1-\frac{2}{1-\frac{\sqrt{1-c x}}{\sqrt{c x+1}}}\right ) \left (a+b \tanh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{4 c} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.372, size = 1449, normalized size = 3.5 \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^{3}{\left (\frac{\log \left (c x + 1\right )}{c} - \frac{\log \left (c x - 1\right )}{c}\right )} - \frac{{\left (b^{3} \log \left (c x + 1\right ) - b^{3} \log \left (-c x + 1\right )\right )} \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right )^{3}}{16 \, c} - \int \frac{4 \,{\left (\sqrt{c x + 1} b^{3} - \sqrt{-c x + 1} b^{3}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{3} + 24 \,{\left (\sqrt{c x + 1} a b^{2} - \sqrt{-c x + 1} a b^{2}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{2} + 3 \,{\left (4 \,{\left (\sqrt{c x + 1} b^{3} - \sqrt{-c x + 1} b^{3}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) +{\left (8 \, a b^{2} -{\left (b^{3} c x - b^{3}\right )} \log \left (c x + 1\right ) +{\left (b^{3} c x - b^{3}\right )} \log \left (-c x + 1\right )\right )} \sqrt{c x + 1} -{\left (8 \, a b^{2} -{\left (b^{3} c x + b^{3}\right )} \log \left (c x + 1\right ) +{\left (b^{3} c x + b^{3}\right )} \log \left (-c x + 1\right )\right )} \sqrt{-c x + 1}\right )} \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right )^{2} + 48 \,{\left (\sqrt{c x + 1} a^{2} b - \sqrt{-c x + 1} a^{2} b\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right ) - 12 \,{\left (4 \, \sqrt{c x + 1} a^{2} b - 4 \, \sqrt{-c x + 1} a^{2} b +{\left (\sqrt{c x + 1} b^{3} - \sqrt{-c x + 1} b^{3}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )^{2} + 4 \,{\left (\sqrt{c x + 1} a b^{2} - \sqrt{-c x + 1} a b^{2}\right )} \log \left (\sqrt{c x + 1} + \sqrt{-c x + 1}\right )\right )} \log \left (\sqrt{c x + 1} - \sqrt{-c x + 1}\right )}{32 \,{\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^{3} \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{3} + 3 \, a b^{2} \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 3 \, a^{2} b \operatorname{artanh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right ) + a^{3}}{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^{3}}{c^{2} x^{2} - 1}\, dx - \int \frac{b^{3} \operatorname{atanh}^{3}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac{3 a b^{2} \operatorname{atanh}^{2}{\left (\frac{\sqrt{- c x + 1}}{\sqrt{c x + 1}} \right )}}{c^{2} x^{2} - 1}\, dx - \int \frac{3 a^{2} 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 )}^{3}}{c^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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