Optimal. Leaf size=77 \[ -\frac{\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac{\text{csch}^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac{\text{csch}^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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Rubi [A] time = 0.0879159, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {2282, 6282, 5659, 3716, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac{\text{csch}^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac{\text{csch}^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 6282
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \text{csch}^{-1}\left (c e^{a+b x}\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\text{csch}^{-1}(c x)}{x} \, dx,x,e^{a+b x}\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}\left (\frac{x}{c}\right )}{x} \, dx,x,e^{-a-b x}\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}-\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{b}+\frac{\operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )\right )}{b}\\ &=\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}-\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{b}+\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{2 b}\\ &=\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )^2}{2 b}-\frac{\sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{b}-\frac{\text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{e^{-a-b x}}{c}\right )}\right )}{2 b}\\ \end{align*}
Mathematica [B] time = 0.568507, size = 236, normalized size = 3.06 \[ \frac{e^{-a-b x} \sqrt{c^2 e^{2 (a+b x)}+1} \left (-4 \text{PolyLog}\left (2,\frac{1}{2} \left (1-\sqrt{c^2 e^{2 (a+b x)}+1}\right )\right )+\log ^2\left (-c^2 e^{2 (a+b x)}\right )+2 \log ^2\left (\frac{1}{2} \left (\sqrt{c^2 e^{2 (a+b x)}+1}+1\right )\right )-4 \log \left (\frac{1}{2} \left (\sqrt{c^2 e^{2 (a+b x)}+1}+1\right )\right ) \log \left (-c^2 e^{2 (a+b x)}\right )+\left (4 \log \left (-c^2 e^{2 (a+b x)}\right )-8 b x\right ) \tanh ^{-1}\left (\sqrt{c^2 e^{2 (a+b x)}+1}\right )\right )}{8 b c \sqrt{\frac{e^{-2 (a+b x)}}{c^2}+1}}+x \text{csch}^{-1}\left (c e^{a+b x}\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.434, size = 0, normalized size = 0. \begin{align*} \int{\rm arccsch} \left (c{{\rm e}^{bx+a}}\right )\, 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*} b c^{2} \int \frac{x e^{\left (2 \, b x + 2 \, a\right )}}{c^{2} e^{\left (2 \, b x + 2 \, a\right )} +{\left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{\frac{3}{2}} + 1}\,{d x} - \frac{1}{2} \, b x^{2} -{\left (a + \log \left (c\right )\right )} x + x \log \left (\sqrt{c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1} + 1\right ) - \frac{2 \, b x \log \left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) +{\rm Li}_2\left (-c^{2} e^{\left (2 \, b x + 2 \, a\right )}\right )}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 \operatorname{acsch}{\left (c e^{a + b x} \right )}\, 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 \operatorname{arcsch}\left (c e^{\left (b x + a\right )}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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