Optimal. Leaf size=162 \[ \text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\frac{1}{2} \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(a+b x)}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right ) \]
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Rubi [A] time = 0.286667, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {6322, 5596, 5569, 3716, 2190, 2279, 2391, 5561} \[ \text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{PolyLog}\left (2,\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\frac{1}{2} \text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(a+b x)}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{a^2+1}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{\sqrt{a^2+1}+1}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right ) \]
Antiderivative was successfully verified.
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Rule 6322
Rule 5596
Rule 5569
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rule 5561
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}(a+b x)}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{x \coth (x) \text{csch}(x)}{-a+\text{csch}(x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{x \coth (x)}{1-a \sinh (x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=-\left (a \operatorname{Subst}\left (\int \frac{x \cosh (x)}{1-a \sinh (x)} \, dx,x,\text{csch}^{-1}(a+b x)\right )\right )-\operatorname{Subst}\left (\int x \coth (x) \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\text{csch}^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^x x}{1-\sqrt{1+a^2}-a e^x} \, dx,x,\text{csch}^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^x x}{1+\sqrt{1+a^2}-a e^x} \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \log \left (1-\frac{a e^x}{1-\sqrt{1+a^2}}\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )-\operatorname{Subst}\left (\int \log \left (1-\frac{a e^x}{1+\sqrt{1+a^2}}\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )+\operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text{csch}^{-1}(a+b x)\right )\\ &=\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )+\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \text{csch}^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{a x}{1-\sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(a+b x)}\right )-\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{a x}{1+\sqrt{1+a^2}}\right )}{x} \, dx,x,e^{\text{csch}^{-1}(a+b x)}\right )\\ &=\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{csch}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )+\text{Li}_2\left (\frac{a e^{\text{csch}^{-1}(a+b x)}}{1-\sqrt{1+a^2}}\right )+\text{Li}_2\left (\frac{a e^{\text{csch}^{-1}(a+b x)}}{1+\sqrt{1+a^2}}\right )-\frac{1}{2} \text{Li}_2\left (e^{2 \text{csch}^{-1}(a+b x)}\right )\\ \end{align*}
Mathematica [C] time = 0.378753, size = 427, normalized size = 2.64 \[ \frac{1}{8} \left (8 \text{PolyLog}\left (2,\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+8 \text{PolyLog}\left (2,-\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+4 \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(a+b x)}\right )+8 \text{csch}^{-1}(a+b x) \log \left (1-\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+8 \text{csch}^{-1}(a+b x) \log \left (\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}+1\right )+4 i \pi \log \left (1-\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )+4 i \pi \log \left (\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}+1\right )+16 i \sin ^{-1}\left (\frac{\sqrt{\frac{a-i}{a}}}{\sqrt{2}}\right ) \log \left (1-\frac{\left (\sqrt{a^2+1}-1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}\right )-16 i \sin ^{-1}\left (\frac{\sqrt{\frac{a-i}{a}}}{\sqrt{2}}\right ) \log \left (\frac{\left (\sqrt{a^2+1}+1\right ) e^{\text{csch}^{-1}(a+b x)}}{a}+1\right )-32 \sin ^{-1}\left (\frac{\sqrt{\frac{a-i}{a}}}{\sqrt{2}}\right ) \tan ^{-1}\left (\frac{(1-i a) \cot \left (\frac{1}{4} \left (\pi +2 i \text{csch}^{-1}(a+b x)\right )\right )}{\sqrt{a^2+1}}\right )-4 i \pi \log \left (\frac{b x}{a+b x}\right )-8 \text{csch}^{-1}(a+b x)^2-4 i \pi \text{csch}^{-1}(a+b x)-8 \text{csch}^{-1}(a+b x) \log \left (1-e^{-2 \text{csch}^{-1}(a+b x)}\right )+\pi ^2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.25, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccsch} \left (bx+a\right )}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcsch}\left (b x + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcsch}\left (b x + a\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acsch}{\left (a + b x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcsch}\left (b x + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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