Optimal. Leaf size=162 \[ \text {Li}_2\left (\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )+\text {Li}_2\left (\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )+\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{1-\sqrt {a^2+1}}\right )+\text {csch}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {csch}^{-1}(a+b x)}}{\sqrt {a^2+1}+1}\right )-\frac {1}{2} \text {Li}_2\left (e^{2 \text {csch}^{-1}(a+b x)}\right )-\text {csch}^{-1}(a+b x) \log \left (1-e^{2 \text {csch}^{-1}(a+b x)}\right ) \]
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Rubi [A] time = 0.29, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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 2190
Rule 2279
Rule 2391
Rule 3716
Rule 5561
Rule 5569
Rule 5596
Rule 6322
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*}
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Mathematica [C] time = 0.40, size = 427, normalized size = 2.64 \[ \frac {1}{8} \left (8 \text {Li}_2\left (\frac {\left (\sqrt {a^2+1}-1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {Li}_2\left (-\frac {\left (\sqrt {a^2+1}+1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {csch}^{-1}(a+b x) \log \left (1-\frac {\left (\sqrt {a^2+1}-1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+8 \text {csch}^{-1}(a+b x) \log \left (\frac {\left (\sqrt {a^2+1}+1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}+1\right )+4 i \pi \log \left (1-\frac {\left (\sqrt {a^2+1}-1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )+4 i \pi \log \left (\frac {\left (\sqrt {a^2+1}+1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}+1\right )+16 i \sin ^{-1}\left (\frac {\sqrt {\frac {a-i}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (\sqrt {a^2+1}-1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}\right )-16 i \sin ^{-1}\left (\frac {\sqrt {\frac {a-i}{a}}}{\sqrt {2}}\right ) \log \left (\frac {\left (\sqrt {a^2+1}+1\right ) e^{\text {csch}^{-1}(a+b x)}}{a}+1\right )-32 \sin ^{-1}\left (\frac {\sqrt {\frac {a-i}{a}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {(1-i a) \cot \left (\frac {1}{4} \left (\pi +2 i \text {csch}^{-1}(a+b x)\right )\right )}{\sqrt {a^2+1}}\right )+4 \text {Li}_2\left (e^{-2 \text {csch}^{-1}(a+b x)}\right )-4 i \pi \log \left (\frac {b x}{a+b x}\right )-8 \text {csch}^{-1}(a+b x)^2-4 i \pi \text {csch}^{-1}(a+b x)-8 \text {csch}^{-1}(a+b x) \log \left (1-e^{-2 \text {csch}^{-1}(a+b x)}\right )+\pi ^2\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcsch}\left (b x + a\right )}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcsch}\left (b x + a\right )}{x}\,{d x} \]
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 \[ \int \frac {\mathrm {arccsch}\left (b x +a \right )}{x}\, dx \]
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 \[ \int \frac {\operatorname {arcsch}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {asinh}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \]
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 \[ \int \frac {\operatorname {acsch}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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