Optimal. Leaf size=261 \[ \frac{3 b^2 \text{PolyLog}\left (3,e^{-2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}+\frac{3 b \text{PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}+\frac{3 b^3 \text{PolyLog}\left (4,e^{-2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{4 c}-\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c}-\frac{\log \left (1-e^{-2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
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Rubi [A] time = 0.225726, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6681, 5659, 3716, 2190, 2531, 6609, 2282, 6589} \[ \frac{3 b^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )}{2 c}-\frac{3 b \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2}{2 c}-\frac{3 b^3 \text{PolyLog}\left (4,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )}{4 c}+\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c}-\frac{\log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{c} \]
Warning: Unable to verify antiderivative.
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Rule 6681
Rule 5659
Rule 3716
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\left (a+b \sinh ^{-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 \sinh ^{-1}(x)\right )^3}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x)^3 \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} (a+b x)^3}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}+\frac{(3 b) \operatorname{Subst}\left (\int (a+b x)^2 \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{\left (3 b^2\right ) \operatorname{Subst}\left (\int (a+b x) \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{c}\\ &=\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{3 b^2 \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \text{Li}_3\left (e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{2 c}\\ &=\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{3 b^2 \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ &=\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^4}{4 b c}-\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^3 \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{c}-\frac{3 b \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )^2 \text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}+\frac{3 b^2 \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right ) \text{Li}_3\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{2 c}-\frac{3 b^3 \text{Li}_4\left (e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}\right )}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0572943, size = 244, normalized size = 0.93 \[ \frac{6 b^2 \text{PolyLog}\left (3,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )-6 b \text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^2-3 b^3 \text{PolyLog}\left (4,e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right )+\frac{\left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{b}-4 \log \left (1-e^{2 \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )}\right ) \left (a+b \sinh ^{-1}\left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^3}{4 c} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.72, size = 1175, normalized size = 4.5 \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{2} + \sqrt{-c x + 1}\right )^{3}}{2 \, c} + \int \frac{{\left (\sqrt{2} b^{3} + \sqrt{-c x + 1} b^{3}\right )} \log \left (c x + 1\right )^{3} - 6 \,{\left (\sqrt{2} a b^{2} + \sqrt{-c x + 1} a b^{2}\right )} \log \left (c x + 1\right )^{2} - 6 \,{\left (4 \, \sqrt{2} a b^{2} - 2 \,{\left (\sqrt{2} b^{3} + \sqrt{-c x + 1} b^{3}\right )} \log \left (c x + 1\right ) +{\left (4 \, 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{2} + \sqrt{-c x + 1}\right )^{2} + 12 \,{\left (\sqrt{2} a^{2} b + \sqrt{-c x + 1} a^{2} b\right )} \log \left (c x + 1\right ) - 6 \,{\left (4 \, \sqrt{2} a^{2} b + 4 \, \sqrt{-c x + 1} a^{2} b +{\left (\sqrt{2} b^{3} + \sqrt{-c x + 1} b^{3}\right )} \log \left (c x + 1\right )^{2} - 4 \,{\left (\sqrt{2} a b^{2} + \sqrt{-c x + 1} a b^{2}\right )} \log \left (c x + 1\right )\right )} \log \left (\sqrt{2} + \sqrt{-c x + 1}\right )}{8 \,{\left (\sqrt{2} c^{2} x^{2} +{\left (c^{2} x^{2} - 1\right )} \sqrt{-c x + 1} - \sqrt{2}\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{arsinh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{3} + 3 \, a b^{2} \operatorname{arsinh}\left (\frac{\sqrt{-c x + 1}}{\sqrt{c x + 1}}\right )^{2} + 3 \, a^{2} b \operatorname{arsinh}\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(-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 \operatorname{arsinh}\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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