Optimal. Leaf size=260 \[ \frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
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Rubi [A] time = 0.44, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5689, 5718, 5694, 4182, 2279, 2391, 2531, 6609, 2282, 6589} \[ \frac {3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {PolyLog}\left (3,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {PolyLog}\left (4,-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {PolyLog}\left (4,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2282
Rule 2391
Rule 2531
Rule 4182
Rule 5689
Rule 5694
Rule 5718
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{\left (c-a^2 c x^2\right )^2} \, dx &=\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {(3 a) \int \frac {x \cosh ^{-1}(a x)^2}{(-1+a x)^{3/2} (1+a x)^{3/2}} \, dx}{2 c^2}+\frac {\int \frac {\cosh ^{-1}(a x)^3}{c-a^2 c x^2} \, dx}{2 c}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {3 \int \frac {\cosh ^{-1}(a x)}{-1+a^2 x^2} \, dx}{c^2}-\frac {\operatorname {Subst}\left (\int x^3 \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \operatorname {Subst}\left (\int x^2 \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}-\frac {3 \operatorname {Subst}\left (\int x^2 \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{2 a c^2}+\frac {3 \operatorname {Subst}\left (\int x \text {csch}(x) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac {3 \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac {3 \operatorname {Subst}\left (\int x \text {Li}_2\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}+\frac {3 \operatorname {Subst}\left (\int x \text {Li}_2\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \operatorname {Subst}\left (\int \text {Li}_3\left (-e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}-\frac {3 \operatorname {Subst}\left (\int \text {Li}_3\left (e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ &=-\frac {3 \cosh ^{-1}(a x)^2}{2 a c^2 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {x \cosh ^{-1}(a x)^3}{2 c^2 \left (1-a^2 x^2\right )}-\frac {6 \cosh ^{-1}(a x) \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {\cosh ^{-1}(a x)^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (-e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}+\frac {3 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )}{2 a c^2}-\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}+\frac {3 \text {Li}_4\left (-e^{\cosh ^{-1}(a x)}\right )}{a c^2}-\frac {3 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )}{a c^2}\\ \end {align*}
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Mathematica [A] time = 2.16, size = 276, normalized size = 1.06 \[ \frac {-24 \cosh ^{-1}(a x)^2 \text {Li}_2\left (e^{\cosh ^{-1}(a x)}\right )-48 \cosh ^{-1}(a x) \text {Li}_3\left (-e^{-\cosh ^{-1}(a x)}\right )+48 \cosh ^{-1}(a x) \text {Li}_3\left (e^{\cosh ^{-1}(a x)}\right )-24 \left (\cosh ^{-1}(a x)^2-2\right ) \text {Li}_2\left (-e^{-\cosh ^{-1}(a x)}\right )-48 \text {Li}_2\left (e^{-\cosh ^{-1}(a x)}\right )-48 \text {Li}_4\left (-e^{-\cosh ^{-1}(a x)}\right )-48 \text {Li}_4\left (e^{\cosh ^{-1}(a x)}\right )+2 \cosh ^{-1}(a x)^4+8 \cosh ^{-1}(a x)^3 \log \left (e^{-\cosh ^{-1}(a x)}+1\right )-8 \cosh ^{-1}(a x)^3 \log \left (1-e^{\cosh ^{-1}(a x)}\right )+48 \cosh ^{-1}(a x) \log \left (1-e^{-\cosh ^{-1}(a x)}\right )-48 \cosh ^{-1}(a x) \log \left (e^{-\cosh ^{-1}(a x)}+1\right )+12 \cosh ^{-1}(a x)^2 \tanh \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-12 \cosh ^{-1}(a x)^2 \coth \left (\frac {1}{2} \cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^3 \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-2 \cosh ^{-1}(a x)^3 \text {sech}^2\left (\frac {1}{2} \cosh ^{-1}(a x)\right )-\pi ^4}{16 a c^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{3}}{a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.27, size = 464, normalized size = 1.78 \[ -\frac {\mathrm {arccosh}\left (a x \right )^{3} x}{2 \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{2 a \left (a^{2} x^{2}-1\right ) c^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 a \,c^{2}}-\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 a \,c^{2}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}-\frac {3 \polylog \left (4, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}+\frac {\mathrm {arccosh}\left (a x \right )^{3} \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 a \,c^{2}}+\frac {3 \mathrm {arccosh}\left (a x \right )^{2} \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{2 a \,c^{2}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \polylog \left (3, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}+\frac {3 \polylog \left (4, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}+\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}+\frac {3 \polylog \left (2, a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}-\frac {3 \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}}-\frac {3 \polylog \left (2, -a x -\sqrt {a x -1}\, \sqrt {a x +1}\right )}{a \,c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (2 \, a x - {\left (a^{2} x^{2} - 1\right )} \log \left (a x + 1\right ) + {\left (a^{2} x^{2} - 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{4 \, {\left (a^{3} c^{2} x^{2} - a c^{2}\right )}} - \int -\frac {3 \, {\left (2 \, a^{3} x^{3} + {\left (2 \, a^{2} x^{2} - {\left (a^{3} x^{3} - a x\right )} \log \left (a x + 1\right ) + {\left (a^{3} x^{3} - a x\right )} \log \left (a x - 1\right )\right )} \sqrt {a x + 1} \sqrt {a x - 1} - 2 \, a x - {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x + 1\right ) + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (a x - 1\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{4 \, {\left (a^{5} c^{2} x^{5} - 2 \, a^{3} c^{2} x^{3} + a c^{2} x + {\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{a^{4} x^{4} - 2 a^{2} x^{2} + 1}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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