Optimal. Leaf size=98 \[ -\frac {3}{2} a^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \cosh ^{-1}(a x)^2-3 a^2 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {\cosh ^{-1}(a x)^3}{2 x^2}+\frac {3 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x} \]
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Rubi [A] time = 0.32, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {5662, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac {3}{2} a^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )+\frac {3}{2} a^2 \cosh ^{-1}(a x)^2-3 a^2 \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )-\frac {\cosh ^{-1}(a x)^3}{2 x^2}+\frac {3 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{2 x} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5662
Rule 5724
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^3}{x^3} \, dx &=-\frac {\cosh ^{-1}(a x)^3}{2 x^2}+\frac {1}{2} (3 a) \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \int \frac {\cosh ^{-1}(a x)}{x} \, dx\\ &=\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-\left (3 a^2\right ) \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-\left (6 a^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\left (3 a^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )+\frac {1}{2} \left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )\\ &=\frac {3}{2} a^2 \cosh ^{-1}(a x)^2+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 x}-\frac {\cosh ^{-1}(a x)^3}{2 x^2}-3 a^2 \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )-\frac {3}{2} a^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.87, size = 92, normalized size = 0.94 \[ \frac {1}{2} \left (3 a^2 \left (\text {Li}_2\left (-e^{-2 \cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )-\frac {\cosh ^{-1}(a x)^3}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{3}}{x^{3}}, x\right ) \]
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 = 0.26, size = 113, normalized size = 1.15 \[ \frac {3 a^{2} \mathrm {arccosh}\left (a x \right )^{2}}{2}-\frac {\mathrm {arccosh}\left (a x \right )^{3}}{2 x^{2}}-3 a^{2} \mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )-\frac {3 a^{2} \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}+\frac {3 a \mathrm {arccosh}\left (a x \right )^{2} \sqrt {a x -1}\, \sqrt {a x +1}}{2 x} \]
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 \[ -\frac {\log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{3}}{2 \, x^{2}} + \int \frac {3 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{2 \, {\left (a^{3} x^{5} - a x^{3} + {\left (a^{2} x^{4} - x^{2}\right )} \sqrt {a x + 1} \sqrt {a x - 1}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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