Optimal. Leaf size=60 \[ -2 i a \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )-\frac {\cosh ^{-1}(a x)^2}{x}+4 a \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.21, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5662, 5761, 4180, 2279, 2391} \[ -2 i a \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(a x)}\right )+2 i a \text {PolyLog}\left (2,i e^{\cosh ^{-1}(a x)}\right )-\frac {\cosh ^{-1}(a x)^2}{x}+4 a \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 5662
Rule 5761
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^2} \, dx &=-\frac {\cosh ^{-1}(a x)^2}{x}+(2 a) \int \frac {\cosh ^{-1}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {\cosh ^{-1}(a x)^2}{x}+(2 a) \operatorname {Subst}\left (\int x \text {sech}(x) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {\cosh ^{-1}(a x)^2}{x}+4 a \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-(2 i a) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )+(2 i a) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\cosh ^{-1}(a x)\right )\\ &=-\frac {\cosh ^{-1}(a x)^2}{x}+4 a \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-(2 i a) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )+(2 i a) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\cosh ^{-1}(a x)}\right )\\ &=-\frac {\cosh ^{-1}(a x)^2}{x}+4 a \cosh ^{-1}(a x) \tan ^{-1}\left (e^{\cosh ^{-1}(a x)}\right )-2 i a \text {Li}_2\left (-i e^{\cosh ^{-1}(a x)}\right )+2 i a \text {Li}_2\left (i e^{\cosh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 92, normalized size = 1.53 \[ -i a \left (2 \text {Li}_2\left (-i e^{-\cosh ^{-1}(a x)}\right )-2 \text {Li}_2\left (i e^{-\cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (-\frac {i \cosh ^{-1}(a x)}{a x}+2 \log \left (1-i e^{-\cosh ^{-1}(a x)}\right )-2 \log \left (1+i e^{-\cosh ^{-1}(a x)}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 137, normalized size = 2.28 \[ -\frac {\mathrm {arccosh}\left (a x \right )^{2}}{x}-2 i a \,\mathrm {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i a \,\mathrm {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )-2 i a \dilog \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i a \dilog \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right ) \]
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 \[ -\frac {\log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{x} + \int \frac {2 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{a^{3} x^{4} - a x^{2} + {\left (a^{2} x^{3} - x\right )} \sqrt {a x + 1} \sqrt {a x - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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