Optimal. Leaf size=56 \[ -\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b}-\log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{2} b \text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right ) \]
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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 = {6282, 5659, 3716, 2190, 2279, 2391} \[ -\frac {1}{2} b \text {PolyLog}\left (2,e^{2 \text {csch}^{-1}(c x)}\right )+\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b}-\log \left (1-e^{2 \text {csch}^{-1}(c x)}\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \]
Warning: Unable to verify antiderivative.
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Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5659
Rule 6282
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-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 {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+b \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )+\frac {1}{2} b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(c x)}\right )\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right )^2}{2 b}-\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-e^{2 \text {csch}^{-1}(c x)}\right )-\frac {1}{2} b \text {Li}_2\left (e^{2 \text {csch}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 47, normalized size = 0.84 \[ a \log (x)+\frac {1}{2} b \left (\text {Li}_2\left (e^{-2 \text {csch}^{-1}(c x)}\right )-\text {csch}^{-1}(c x) \left (\text {csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text {csch}^{-1}(c x)}\right )\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.07, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcsch}\left (c x\right ) + a}{x}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{x}\, dx \]
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 \[ -\frac {1}{2} \, {\left (4 \, c^{2} \int \frac {x^{2} \log \relax (x)}{c^{2} x^{3} + x}\,{d x} - 2 \, c^{2} \int \frac {x \log \relax (x)}{c^{2} x^{2} + {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 1}\,{d x} - {\left (\log \left (c^{2} x^{2} + 1\right ) - 2 \, \log \relax (x)\right )} \log \relax (c) + \log \left (c^{2} x^{2} + 1\right ) \log \relax (c) - 2 \, \log \relax (x) \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right ) + 2 \, \int \frac {\log \relax (x)}{c^{2} x^{3} + x}\,{d x}\right )} b + a \log \relax (x) \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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