Optimal. Leaf size=85 \[ \frac{2 b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d}-\frac{2 b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d}+\frac{(c+d x) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{d}+\frac{4 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d} \]
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Rubi [A] time = 0.0831275, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6316, 6280, 5452, 4182, 2279, 2391} \[ \frac{2 b^2 \text{PolyLog}\left (2,-e^{\text{csch}^{-1}(c+d x)}\right )}{d}-\frac{2 b^2 \text{PolyLog}\left (2,e^{\text{csch}^{-1}(c+d x)}\right )}{d}+\frac{(c+d x) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{d}+\frac{4 b \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right ) \left (a+b \text{csch}^{-1}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 6316
Rule 6280
Rule 5452
Rule 4182
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \left (a+b \text{csch}^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \text{csch}^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \coth (x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d}\\ &=\frac{(c+d x) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{d}-\frac{(2 b) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d}\\ &=\frac{(c+d x) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{d}+\frac{4 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text{csch}^{-1}(c+d x)\right )}{d}\\ &=\frac{(c+d x) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{d}+\frac{4 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\text{csch}^{-1}(c+d x)}\right )}{d}\\ &=\frac{(c+d x) \left (a+b \text{csch}^{-1}(c+d x)\right )^2}{d}+\frac{4 b \left (a+b \text{csch}^{-1}(c+d x)\right ) \tanh ^{-1}\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d}+\frac{2 b^2 \text{Li}_2\left (-e^{\text{csch}^{-1}(c+d x)}\right )}{d}-\frac{2 b^2 \text{Li}_2\left (e^{\text{csch}^{-1}(c+d x)}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.248536, size = 160, normalized size = 1.88 \[ \frac{-2 b^2 \text{PolyLog}\left (2,-e^{-\text{csch}^{-1}(c+d x)}\right )+2 b^2 \text{PolyLog}\left (2,e^{-\text{csch}^{-1}(c+d x)}\right )+a^2 c+a^2 d x+2 a b (c+d x) \text{csch}^{-1}(c+d x)-2 a b \log \left (\tanh \left (\frac{1}{2} \text{csch}^{-1}(c+d x)\right )\right )+b^2 c \text{csch}^{-1}(c+d x)^2+b^2 d x \text{csch}^{-1}(c+d x)^2-2 b^2 \text{csch}^{-1}(c+d x) \log \left (1-e^{-\text{csch}^{-1}(c+d x)}\right )+2 b^2 \text{csch}^{-1}(c+d x) \log \left (e^{-\text{csch}^{-1}(c+d x)}+1\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.218, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccsch} \left (dx+c\right ) \right ) ^{2}\, 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*}{\left (x \log \left (\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right )^{2} - \int -\frac{{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac{3}{2}} \log \left (d x + c\right )^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \log \left (d x + c\right )^{2} - 2 \,{\left ({\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \log \left (d x + c\right ) + \sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1}{\left (d^{2} x^{2} + c d x +{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )} \log \left (d x + c\right )\right )}\right )} \log \left (\sqrt{d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} +{\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}^{\frac{3}{2}} + 1}\,{d x}\right )} b^{2} + a^{2} x + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{arcsch}\left (d x + c\right ) + \log \left (\sqrt{\frac{1}{{\left (d x + c\right )}^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{{\left (d x + c\right )}^{2}} + 1} - 1\right )\right )} a b}{d} \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 (b^{2} \operatorname{arcsch}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcsch}\left (d x + c\right ) + a^{2}, 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*} \int \left (a + b \operatorname{acsch}{\left (c + d x \right )}\right )^{2}\, 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{\left (b \operatorname{arcsch}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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