Integrand size = 10, antiderivative size = 162 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {1+a^2}}\right )+\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1+\sqrt {1+a^2}}\right )-\text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1+\sqrt {1+a^2}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6457, 5715, 5688, 3797, 2221, 2317, 2438, 5680} \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )+\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )+\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right )-\text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3797
Rule 5680
Rule 5688
Rule 5715
Rule 6457
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x \coth (x) \text {csch}(x)}{-a+\text {csch}(x)} \, dx,x,\text {csch}^{-1}(a+b x)\right ) \\ & = -\text {Subst}\left (\int \frac {x \coth (x)}{1-a \sinh (x)} \, dx,x,\text {csch}^{-1}(a+b x)\right ) \\ & = -\left (a \text {Subst}\left (\int \frac {x \cosh (x)}{1-a \sinh (x)} \, dx,x,\text {csch}^{-1}(a+b x)\right )\right )-\text {Subst}\left (\int x \coth (x) \, dx,x,\text {csch}^{-1}(a+b x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {csch}^{-1}(a+b x)\right )-a \text {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 \text {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 )-\text {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 )-\text {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 )+\text {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} \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {csch}^{-1}(a+b x)}\right )-\text {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 {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 )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {1+a^2}}\right )+\operatorname {PolyLog}\left (2,\frac {a e^{\text {csch}^{-1}(a+b x)}}{1+\sqrt {1+a^2}}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{2 \text {csch}^{-1}(a+b x)}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.48 (sec) , antiderivative size = 427, normalized size of antiderivative = 2.64 \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\frac {1}{8} \left (\pi ^2-4 i \pi \text {csch}^{-1}(a+b x)-8 \text {csch}^{-1}(a+b x)^2-32 \arcsin \left (\frac {\sqrt {\frac {-i+a}{a}}}{\sqrt {2}}\right ) \arctan \left (\frac {(1-i a) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(a+b x)\right )\right )}{\sqrt {1+a^2}}\right )-8 \text {csch}^{-1}(a+b x) \log \left (1-e^{-2 \text {csch}^{-1}(a+b x)}\right )+4 i \pi \log \left (1-\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {csch}^{-1}(a+b x) \log \left (1-\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+16 i \arcsin \left (\frac {\sqrt {\frac {-i+a}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+4 i \pi \log \left (1+\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {csch}^{-1}(a+b x) \log \left (1+\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )-16 i \arcsin \left (\frac {\sqrt {\frac {-i+a}{a}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )-4 i \pi \log \left (\frac {b x}{a+b x}\right )+4 \operatorname {PolyLog}\left (2,e^{-2 \text {csch}^{-1}(a+b x)}\right )+8 \operatorname {PolyLog}\left (2,\frac {\left (-1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \operatorname {PolyLog}\left (2,-\frac {\left (1+\sqrt {1+a^2}\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )\right ) \]
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\[\int \frac {\operatorname {arccsch}\left (b x +a \right )}{x}d x\]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x} \,d x } \]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x}\, dx \]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x} \,d x } \]
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\[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int { \frac {\operatorname {arcsch}\left (b x + a\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {csch}^{-1}(a+b x)}{x} \, dx=\int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \]
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