Optimal. Leaf size=50 \[ -2 a \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 a \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^2}{x}-4 a \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
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Rubi [A] time = 0.10, antiderivative size = 50, 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 = {5661, 5760, 4182, 2279, 2391} \[ -2 a \text {PolyLog}\left (2,-e^{\sinh ^{-1}(a x)}\right )+2 a \text {PolyLog}\left (2,e^{\sinh ^{-1}(a x)}\right )-\frac {\sinh ^{-1}(a x)^2}{x}-4 a \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right ) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4182
Rule 5661
Rule 5760
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^2}{x^2} \, dx &=-\frac {\sinh ^{-1}(a x)^2}{x}+(2 a) \int \frac {\sinh ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx\\ &=-\frac {\sinh ^{-1}(a x)^2}{x}+(2 a) \operatorname {Subst}\left (\int x \text {csch}(x) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^2}{x}-4 a \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-(2 a) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )+(2 a) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(a x)\right )\\ &=-\frac {\sinh ^{-1}(a x)^2}{x}-4 a \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-(2 a) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )+(2 a) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(a x)}\right )\\ &=-\frac {\sinh ^{-1}(a x)^2}{x}-4 a \sinh ^{-1}(a x) \tanh ^{-1}\left (e^{\sinh ^{-1}(a x)}\right )-2 a \text {Li}_2\left (-e^{\sinh ^{-1}(a x)}\right )+2 a \text {Li}_2\left (e^{\sinh ^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.21, size = 75, normalized size = 1.50 \[ a \left (2 \text {Li}_2\left (-e^{-\sinh ^{-1}(a x)}\right )-2 \text {Li}_2\left (e^{-\sinh ^{-1}(a x)}\right )-\sinh ^{-1}(a x) \left (\frac {\sinh ^{-1}(a x)}{a x}-2 \log \left (1-e^{-\sinh ^{-1}(a x)}\right )+2 \log \left (e^{-\sinh ^{-1}(a x)}+1\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {arsinh}\left (a x\right )^{2}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.24, size = 107, normalized size = 2.14 \[ -\frac {\arcsinh \left (a x \right )^{2}}{x}+2 a \arcsinh \left (a x \right ) \ln \left (1-a x -\sqrt {a^{2} x^{2}+1}\right )+2 a \polylog \left (2, a x +\sqrt {a^{2} x^{2}+1}\right )-2 a \arcsinh \left (a x \right ) \ln \left (a x +\sqrt {a^{2} x^{2}+1}+1\right )-2 a \polylog \left (2, -a x -\sqrt {a^{2} x^{2}+1}\right ) \]
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 \[ -\frac {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2}}{x} + \int \frac {2 \, {\left (a^{3} x^{2} + \sqrt {a^{2} x^{2} + 1} a^{2} x + a\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{a^{3} x^{4} + a x^{2} + {\left (a^{2} x^{3} + x\right )} \sqrt {a^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {{\mathrm {asinh}\left (a\,x\right )}^2}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {asinh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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