Optimal. Leaf size=241 \[ -\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.348938, antiderivative size = 241, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5713, 5688, 5715, 3716, 2190, 2531, 2282, 6589} \[ -\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{a x-1} \sqrt{a x+1} \text{PolyLog}\left (3,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt{c-a^2 c x^2}}+\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{a x-1} \sqrt{a x+1} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 5713
Rule 5688
Rule 5715
Rule 3716
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^{3/2}} \, dx &=-\frac{\left (\sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{\cosh ^{-1}(a x)^3}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\left (3 a \sqrt{-1+a x} \sqrt{1+a x}\right ) \int \frac{x \cosh ^{-1}(a x)^2}{1-a^2 x^2} \, dx}{c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}-\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x^2 \coth (x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}+\frac{\left (6 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{e^{2 x} x^2}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{\left (6 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int x \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \text{Li}_2\left (e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{\left (3 \sqrt{-1+a x} \sqrt{1+a x}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt{c-a^2 c x^2}}\\ &=\frac{x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}}+\frac{\sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^3}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x)^2 \log \left (1-e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}-\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \cosh ^{-1}(a x) \text{Li}_2\left (e^{2 \cosh ^{-1}(a x)}\right )}{a c \sqrt{c-a^2 c x^2}}+\frac{3 \sqrt{-1+a x} \sqrt{1+a x} \text{Li}_3\left (e^{2 \cosh ^{-1}(a x)}\right )}{2 a c \sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.239766, size = 145, normalized size = 0.6 \[ \frac{\frac{\sqrt{a x-1} \sqrt{a x+1} \left (-6 \cosh ^{-1}(a x) \text{PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )-6 \cosh ^{-1}(a x) \text{PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )+6 \text{PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x)^3-3 \cosh ^{-1}(a x)^2 \log \left (1-e^{\cosh ^{-1}(a x)}\right )-3 \cosh ^{-1}(a x)^2 \log \left (e^{\cosh ^{-1}(a x)}+1\right )\right )}{a}+x \cosh ^{-1}(a x)^3}{c \sqrt{c-a^2 c x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.21, size = 548, normalized size = 2.3 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ( -\sqrt{ax-1}\sqrt{ax+1}+ax \right ) }-2\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\rm arccosh} \left (ax\right ) \right ) ^{3}}{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+3\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\ln \left ( 1-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}-6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 3,ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+3\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) } \left ({\rm arccosh} \left (ax\right ) \right ) ^{2}\ln \left ( 1+ax+\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}+6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\rm arccosh} \left (ax\right ){\it polylog} \left ( 2,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-1 \right ) }}-6\,{\frac{\sqrt{ax-1}\sqrt{ax+1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\it polylog} \left ( 3,-ax-\sqrt{ax-1}\sqrt{ax+1} \right ) }{a{c}^{2} \left ({a}^{2}{x}^{2}-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*} \int \frac{\operatorname{arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{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{\sqrt{-a^{2} c x^{2} + c} \operatorname{arcosh}\left (a x\right )^{3}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, 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{\operatorname{acosh}^{3}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac{3}{2}}}\, 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{\operatorname{arcosh}\left (a x\right )^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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