Optimal. Leaf size=97 \[ -\frac{b^2 \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{2 b \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d} \]
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Rubi [A] time = 0.11621, antiderivative size = 97, 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 = {6104, 5911, 5985, 5919, 2402, 2315} \[ -\frac{b^2 \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{2 b \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Rule 6104
Rule 5911
Rule 5985
Rule 5919
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d}-\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-c-d x}\right )}{d}\\ &=\frac{\left (a+b \coth ^{-1}(c+d x)\right )^2}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d}-\frac{2 b \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac{2}{1-c-d x}\right )}{d}-\frac{b^2 \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.148008, size = 111, normalized size = 1.14 \[ \frac{b^2 \text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )+a \left (a c+a d x-2 b \log \left (\frac{1}{(c+d x) \sqrt{1-\frac{1}{(c+d x)^2}}}\right )\right )+2 b \coth ^{-1}(c+d x) \left (a c+a d x-b \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+b^2 (c+d x-1) \coth ^{-1}(c+d x)^2}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.111, size = 226, normalized size = 2.3 \begin{align*} \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}x{b}^{2}+{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}{b}^{2}c}{d}}+2\,{\rm arccoth} \left (dx+c\right )xab-2\,{\frac{{\rm arccoth} \left (dx+c\right ){b}^{2}}{d}\ln \left ( 1-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }-2\,{\frac{{\rm arccoth} \left (dx+c\right ){b}^{2}}{d}\ln \left ( 1+{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}{b}^{2}}{d}}+2\,{\frac{{\rm arccoth} \left (dx+c\right )abc}{d}}+{a}^{2}x-2\,{\frac{{b}^{2}}{d}{\it polylog} \left ( 2,{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }-2\,{\frac{{b}^{2}}{d}{\it polylog} \left ( 2,-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+{\frac{ab\ln \left ( \left ( dx+c \right ) ^{2}-1 \right ) }{d}}+{\frac{{a}^{2}c}{d}} \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*} a^{2} x + \frac{1}{4} \, b^{2}{\left (\frac{d x \log \left (d x + c - 1\right )^{2} +{\left (d x + c + 1\right )} \log \left (d x + c + 1\right )^{2} - 2 \,{\left (d x + c - 1\right )} \log \left (d x + c + 1\right ) \log \left (d x + c - 1\right )}{d} + \int \frac{2 \,{\left (c^{2} +{\left (c d - 3 \, d\right )} x - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\,{d x}\right )} + \frac{{\left (2 \,{\left (d x + c\right )} \operatorname{arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 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{arcoth}\left (d x + c\right )^{2} + 2 \, a b \operatorname{arcoth}\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{acoth}{\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{arcoth}\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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