Optimal. Leaf size=77 \[ -\frac {\text {Li}_2\left (e^{2 \text {csch}^{-1}\left (c e^{a+b x}\right )}\right )}{2 b}+\frac {\text {csch}^{-1}\left (c e^{a+b x}\right )^2}{2 b}-\frac {\text {csch}^{-1}\left (c e^{a+b x}\right ) \log \left (1-e^{2 \text {csch}^{-1}\left (c e^{a+b x}\right )}\right )}{b} \]
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Rubi [A] time = 0.09, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, 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 2190
Rule 2279
Rule 2282
Rule 2391
Rule 3716
Rule 5659
Rule 6282
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*}
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Mathematica [B] time = 0.60, size = 236, normalized size = 3.06 \[ \frac {e^{-a-b x} \sqrt {c^2 e^{2 (a+b x)}+1} \left (-4 \text {Li}_2\left (\frac {1}{2} \left (1-\sqrt {e^{2 (a+b x)} c^2+1}\right )\right )+\log ^2\left (-c^2 e^{2 (a+b x)}\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {c^2 e^{2 (a+b x)}+1}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {c^2 e^{2 (a+b x)}+1}+1\right )\right ) \log \left (-c^2 e^{2 (a+b x)}\right )+\left (4 \log \left (-c^2 e^{2 (a+b x)}\right )-8 b x\right ) \tanh ^{-1}\left (\sqrt {c^2 e^{2 (a+b x)}+1}\right )\right )}{8 b c \sqrt {\frac {e^{-2 (a+b x)}}{c^2}+1}}+x \text {csch}^{-1}\left (c e^{a+b x}\right ) \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \operatorname {arcsch}\left (c e^{\left (b x + a\right )}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int \mathrm {arccsch}\left (c \,{\mathrm e}^{b x +a}\right )\, dx \]
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 \[ b c^{2} \int \frac {x e^{\left (2 \, b x + 2 \, a\right )}}{c^{2} e^{\left (2 \, b x + 2 \, a\right )} + {\left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{\frac {3}{2}} + 1}\,{d x} - \frac {1}{2} \, b x^{2} - {\left (a + \log \relax (c)\right )} x + x \log \left (\sqrt {c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1} + 1\right ) - \frac {2 \, b x \log \left (c^{2} e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + {\rm Li}_2\left (-c^{2} e^{\left (2 \, b x + 2 \, a\right )}\right )}{4 \, b} \]
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 \[ \int \mathrm {asinh}\left (\frac {{\mathrm {e}}^{-a-b\,x}}{c}\right ) \,d x \]
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 \[ \int \operatorname {acsch}{\left (c e^{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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