Optimal. Leaf size=64 \[ -\text {sech}^{-1}(a x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6285, 3718, 2190, 2531, 2282, 6589} \[ -\text {sech}^{-1}(a x) \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (e^{2 \text {sech}^{-1}(a x)}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 6285
Rule 6589
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a x)^2}{x} \, dx &=-\operatorname {Subst}\left (\int x^2 \tanh (x) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{3} \text {sech}^{-1}(a x)^3-2 \operatorname {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )+2 \operatorname {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a x)\right )\\ &=\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a x)}\right )\\ &=\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a x)}\right )-\text {sech}^{-1}(a x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 63, normalized size = 0.98 \[ \text {sech}^{-1}(a x) \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \text {sech}^{-1}(a x)}\right )-\frac {1}{3} \text {sech}^{-1}(a x)^3-\text {sech}^{-1}(a x)^2 \log \left (e^{-2 \text {sech}^{-1}(a x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (a x\right )^{2}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 136, normalized size = 2.12 \[ \frac {\mathrm {arcsech}\left (a x \right )^{3}}{3}-\mathrm {arcsech}\left (a x \right )^{2} \ln \left (1+\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )-\mathrm {arcsech}\left (a x \right ) \polylog \left (2, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )+\frac {\polylog \left (3, -\left (\frac {1}{a x}+\sqrt {\frac {1}{a x}-1}\, \sqrt {1+\frac {1}{a x}}\right )^{2}\right )}{2} \]
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 \[ \int \frac {\operatorname {arsech}\left (a x\right )^{2}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a\,x}\right )}^2}{x} \,d x \]
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 \[ \int \frac {\operatorname {asech}^{2}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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