Optimal. Leaf size=56 \[ \frac{1}{2} b \text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\frac{\left (a+b \text{csch}^{-1}(c x)\right )^2}{2 b}-\log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right ) \left (a+b \text{csch}^{-1}(c x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0898566, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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.
[In]
[Out]
Rule 6282
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
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*}
Mathematica [A] time = 0.0404819, size = 47, normalized size = 0.84 \[ \frac{1}{2} b \left (\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(c x)}\right )-\text{csch}^{-1}(c x) \left (\text{csch}^{-1}(c x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(c x)}\right )\right )\right )+a \log (x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.198, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b{\rm arccsch} \left (cx\right )}{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \,{\left (4 \, c^{2} \int \frac{x^{2} \log \left (x\right )}{c^{2} x^{3} + x}\,{d x} - 2 \, c^{2} \int \frac{x \log \left (x\right )}{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 \left (x\right )\right )} \log \left (c\right ) + \log \left (c^{2} x^{2} + 1\right ) \log \left (c\right ) - 2 \, \log \left (x\right ) \log \left (\sqrt{c^{2} x^{2} + 1} + 1\right ) + 2 \, \int \frac{\log \left (x\right )}{c^{2} x^{3} + x}\,{d x}\right )} b + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcsch}\left (c x\right ) + a}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{acsch}{\left (c x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arcsch}\left (c x\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]