Optimal. Leaf size=56 \[ -\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-\log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{2} b \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6281, 5660, 3718, 2190, 2279, 2391} \[ -\frac {1}{2} b \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-\log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 6281
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}^{-1}(c x)}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-2 \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )+b \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(c x)}\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {sech}^{-1}(c x)\right ) \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )-\frac {1}{2} b \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.05, size = 47, normalized size = 0.84 \[ a \log (x)+\frac {1}{2} b \left (\text {Li}_2\left (-e^{-2 \text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (e^{-2 \text {sech}^{-1}(c x)}+1\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsech}\left (c x\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsech}\left (c x\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 100, normalized size = 1.79 \[ a \ln \left (c x \right )+\frac {b \mathrm {arcsech}\left (c x \right )^{2}}{2}-b \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-\frac {b \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2} \]
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 \[ b \int \frac {\log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )}{x}\,{d x} + a \log \relax (x) \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )}{x} \,d x \]
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 \[ \int \frac {a + b \operatorname {asech}{\left (c x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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