Optimal. Leaf size=174 \[ -\frac {1}{2} a^4 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2-a^4 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {a^3 \sqrt {a x-1} \sqrt {a x+1}}{4 x}+\frac {a^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{4 x^3} \]
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Rubi [A] time = 0.58, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {5662, 5748, 5724, 5660, 3718, 2190, 2279, 2391, 95} \[ -\frac {1}{2} a^4 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}-\frac {a^3 \sqrt {a x-1} \sqrt {a x+1}}{4 x}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x}-a^4 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{4 x^3}-\frac {\cosh ^{-1}(a x)^3}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 95
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5662
Rule 5724
Rule 5748
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x^5} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{4 x^4}+\frac {1}{4} (3 a) \int \frac {\cosh ^{-1}(a x)^2}{x^4 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-\frac {1}{2} a^2 \int \frac {\cosh ^{-1}(a x)}{x^3} \, dx+\frac {1}{2} a^3 \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-\frac {1}{4} a^3 \int \frac {1}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx-a^4 \int \frac {\cosh ^{-1}(a x)}{x} \, dx\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-\left (2 a^4\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+a^4 \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {1}{2} a^4 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=-\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x}}{4 x}+\frac {a^2 \cosh ^{-1}(a x)}{4 x^2}+\frac {1}{2} a^4 \cosh ^{-1}(a x)^2+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{4 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{4 x^4}-a^4 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac {1}{2} a^4 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.67, size = 220, normalized size = 1.26 \[ \frac {-a^5 x^5+2 a^4 x^4 \sqrt {\frac {a x-1}{a x+1}} (a x+1) \text {Li}_2\left (-e^{-2 \cosh ^{-1}(a x)}\right )+a^3 x^3-a^2 x^2 \sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x) \left (4 a^2 x^2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )-1\right )-a x (a x+1) \left (2 a^3 x^3 \left (\sqrt {\frac {a x-1}{a x+1}}-1\right )+2 a^2 x^2-a x+1\right ) \cosh ^{-1}(a x)^2-\sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^3}{4 x^4 \sqrt {a x-1} \sqrt {a x+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{5}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 180, normalized size = 1.03 \[ \frac {a^{3} \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{2 x}+\frac {a^{4} \mathrm {arccosh}\left (a x \right )^{2}}{2}-\frac {a^{3} \sqrt {a x -1}\, \sqrt {a x +1}}{4 x}+\frac {a^{4}}{4}+\frac {a \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{4 x^{3}}+\frac {a^{2} \mathrm {arccosh}\left (a x \right )}{4 x^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{3}}{4 x^{4}}-a^{4} \mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {a^{4} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2} \]
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 \[ -\frac {\log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{4 \, x^{4}} + \int \frac {3 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{4 \, {\left (a^{3} x^{7} - a x^{5} + {\left (a^{2} x^{6} - x^{4}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^5} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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