Optimal. Leaf size=258 \[ \frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \]
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Rubi [A] time = 0.49265, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.45, Rules used = {5689, 5718, 199, 207, 5694, 4182, 2531, 2282, 6589} \[ \frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{a x-1} \sqrt{a x+1}}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (a x-1)^{3/2} (a x+1)^{3/2}}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3} \]
Antiderivative was successfully verified.
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Rule 5689
Rule 5718
Rule 199
Rule 207
Rule 5694
Rule 4182
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^3} \, dx &=\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}-\frac{a \int \frac{x \cosh ^{-1}(a x)}{(-1+a x)^{5/2} (1+a x)^{5/2}} \, dx}{2 c^3}+\frac{3 \int \frac{\cosh ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}-\frac{\int \frac{1}{\left (-1+a^2 x^2\right )^2} \, dx}{6 c^3}+\frac{(3 a) \int \frac{x \cosh ^{-1}(a x)}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{4 c^3}+\frac{3 \int \frac{\cosh ^{-1}(a x)^2}{c-a^2 c x^2} \, dx}{8 c^2}\\ &=-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{12 c^3}+\frac{3 \int \frac{1}{-1+a^2 x^2} \, dx}{4 c^3}-\frac{3 \operatorname{Subst}\left (\int x^2 \text{csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{8 a c^3}\\ &=-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \operatorname{Subst}\left (\int x \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}-\frac{3 \operatorname{Subst}\left (\int x \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \operatorname{Subst}\left (\int \text{Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}+\frac{3 \operatorname{Subst}\left (\int \text{Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{4 a c^3}\\ &=-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ &=-\frac{x}{12 c^3 \left (1-a^2 x^2\right )}+\frac{\cosh ^{-1}(a x)}{6 a c^3 (-1+a x)^{3/2} (1+a x)^{3/2}}-\frac{3 \cosh ^{-1}(a x)}{4 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}+\frac{x \cosh ^{-1}(a x)^2}{4 c^3 \left (1-a^2 x^2\right )^2}+\frac{3 x \cosh ^{-1}(a x)^2}{8 c^3 \left (1-a^2 x^2\right )}+\frac{3 \cosh ^{-1}(a x)^2 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{5 \tanh ^{-1}(a x)}{6 a c^3}+\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \cosh ^{-1}(a x) \text{Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}-\frac{3 \text{Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}+\frac{3 \text{Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{4 a c^3}\\ \end{align*}
Mathematica [A] time = 4.8041, size = 319, normalized size = 1.24 \[ -\frac{72 \left (2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{-\cosh ^{-1}(a x)}\right )-2 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{-\cosh ^{-1}(a x)}\right )+2 \text{PolyLog}\left (3,-e^{-\cosh ^{-1}(a x)}\right )-2 \text{PolyLog}\left (3,e^{-\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^2 \log \left (1-e^{-\cosh ^{-1}(a x)}\right )-\cosh ^{-1}(a x)^2 \log \left (e^{-\cosh ^{-1}(a x)}+1\right )\right )-\frac{32 \cosh ^{-1}(a x) \sinh ^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )}{\left (\frac{a x-1}{a x+1}\right )^{3/2} (a x+1)^3}-80 \cosh ^{-1}(a x) \tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )+80 \cosh ^{-1}(a x) \coth \left (\frac{1}{2} \cosh ^{-1}(a x)\right )-3 \cosh ^{-1}(a x)^2 \text{csch}^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-2 \sqrt{\frac{a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x) \text{csch}^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+2 \left (9 \cosh ^{-1}(a x)^2-2\right ) \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+3 \cosh ^{-1}(a x)^2 \text{sech}^4\left (\frac{1}{2} \cosh ^{-1}(a x)\right )+2 \left (9 \cosh ^{-1}(a x)^2-2\right ) \text{sech}^2\left (\frac{1}{2} \cosh ^{-1}(a x)\right )-160 \log \left (\tanh \left (\frac{1}{2} \cosh ^{-1}(a x)\right )\right )}{192 a c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.143, size = 443, normalized size = 1.7 \begin{align*} -{\frac{3\,{a}^{2} \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}{x}^{3}}{ \left ( 8\,{x}^{4}{a}^{4}-16\,{a}^{2}{x}^{2}+8 \right ){c}^{3}}}-{\frac{3\,a{\rm arccosh} \left (ax\right ){x}^{2}}{ \left ( 4\,{x}^{4}{a}^{4}-8\,{a}^{2}{x}^{2}+4 \right ){c}^{3}}\sqrt{ax-1}\sqrt{ax+1}}+{\frac{{a}^{2}{x}^{3}}{ \left ( 12\,{x}^{4}{a}^{4}-24\,{a}^{2}{x}^{2}+12 \right ){c}^{3}}}+{\frac{5\,x \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{ \left ( 8\,{x}^{4}{a}^{4}-16\,{a}^{2}{x}^{2}+8 \right ){c}^{3}}}+{\frac{11\,{\rm arccosh} \left (ax\right )}{12\,a \left ({x}^{4}{a}^{4}-2\,{a}^{2}{x}^{2}+1 \right ){c}^{3}}\sqrt{ax-1}\sqrt{ax+1}}-{\frac{x}{ \left ( 12\,{x}^{4}{a}^{4}-24\,{a}^{2}{x}^{2}+12 \right ){c}^{3}}}-{\frac{5}{3\,a{c}^{3}}{\it Artanh} \left ( ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{8\,a{c}^{3}}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{3\,{\rm arccosh} \left (ax\right )}{4\,a{c}^{3}}{\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{3}{4\,a{c}^{3}}{\it polylog} \left ( 3,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{3\, \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}}{8\,a{c}^{3}}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }+{\frac{3\,{\rm arccosh} \left (ax\right )}{4\,a{c}^{3}}{\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }-{\frac{3}{4\,a{c}^{3}}{\it polylog} \left ( 3,-ax-\sqrt{ax-1}\sqrt{ax+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{{\left (6 \, a^{3} x^{3} - 10 \, a x - 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )^{2}}{16 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} - \int -\frac{{\left (6 \, a^{5} x^{5} - 16 \, a^{3} x^{3} +{\left (6 \, a^{4} x^{4} - 10 \, a^{2} x^{2} - 3 \,{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \log \left (a x - 1\right )\right )} \sqrt{a x + 1} \sqrt{a x - 1} + 10 \, a x - 3 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) + 3 \,{\left (a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt{a x + 1} \sqrt{a x - 1}\right )}{8 \,{\left (a^{7} c^{3} x^{7} - 3 \, a^{5} c^{3} x^{5} + 3 \, a^{3} c^{3} x^{3} - a c^{3} x +{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt{a x + 1} \sqrt{a 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{\operatorname{arcosh}\left (a x\right )^{2}}{a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}, 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*} - \frac{\int \frac{\operatorname{acosh}^{2}{\left (a x \right )}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \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{\operatorname{arcosh}\left (a x\right )^{2}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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