Optimal. Leaf size=61 \[ -\frac{\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(a+b x)}\right )}{2 d}+\frac{\text{csch}^{-1}(a+b x)^2}{2 d}-\frac{\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )}{d} \]
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Rubi [A] time = 0.0977974, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {6320, 12, 6282, 5659, 3716, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}(a+b x)}\right )}{2 d}+\frac{\text{csch}^{-1}(a+b x)^2}{2 d}-\frac{\text{csch}^{-1}(a+b x) \log \left (1-e^{2 \text{csch}^{-1}(a+b x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 6320
Rule 12
Rule 6282
Rule 5659
Rule 3716
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{csch}^{-1}(a+b x)}{\frac{a d}{b}+d x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b \text{csch}^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\text{csch}^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sinh ^{-1}(x)}{x} \, dx,x,\frac{1}{a+b x}\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (\frac{1}{a+b x}\right )\right )}{d}\\ &=\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right )^2}{2 d}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1-e^{2 x}} \, dx,x,\sinh ^{-1}\left (\frac{1}{a+b x}\right )\right )}{d}\\ &=\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right )^2}{2 d}-\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{1}{a+b x}\right )}\right )}{d}+\frac{\operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (\frac{1}{a+b x}\right )\right )}{d}\\ &=\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right )^2}{2 d}-\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{1}{a+b x}\right )}\right )}{d}+\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (\frac{1}{a+b x}\right )}\right )}{2 d}\\ &=\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right )^2}{2 d}-\frac{\sinh ^{-1}\left (\frac{1}{a+b x}\right ) \log \left (1-e^{2 \sinh ^{-1}\left (\frac{1}{a+b x}\right )}\right )}{d}-\frac{\text{Li}_2\left (e^{2 \sinh ^{-1}\left (\frac{1}{a+b x}\right )}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.058268, size = 52, normalized size = 0.85 \[ \frac{\text{PolyLog}\left (2,e^{-2 \text{csch}^{-1}(a+b x)}\right )-\text{csch}^{-1}(a+b x) \left (\text{csch}^{-1}(a+b x)+2 \log \left (1-e^{-2 \text{csch}^{-1}(a+b x)}\right )\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.509, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccsch} \left (bx+a\right ) \left ({\frac{ad}{b}}+dx \right ) ^{-1}}\, 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*} -\frac{2 \, \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) \log \left (b x + a\right ) +{\rm Li}_2\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}{4 \, d} - \frac{\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + 1\right )}{2 \, d} + \int \frac{{\left (b^{2} x + a b\right )} \log \left (b x + a\right )}{b^{2} d x^{2} + 2 \, a b d x + a^{2} d +{\left (b^{2} d x^{2} + 2 \, a b d x + a^{2} d + d\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{arcsch}\left (b x + a\right )}{b d x + a d}, x\right ) \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*} \frac{b \int \frac{\operatorname{acsch}{\left (a + b x \right )}}{a + b x}\, dx}{d} \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 \frac{\operatorname{arcsch}\left (b x + a\right )}{d x + \frac{a d}{b}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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