Optimal. Leaf size=132 \[ -\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{-c-d x+1}\right )}{2 d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d} \]
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Rubi [A] time = 0.227334, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6104, 5911, 5985, 5919, 5949, 6059, 6610} \[ -\frac{3 b^2 \text{PolyLog}\left (2,1-\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d}+\frac{3 b^3 \text{PolyLog}\left (3,1-\frac{2}{-c-d x+1}\right )}{2 d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d} \]
Antiderivative was successfully verified.
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Rule 6104
Rule 5911
Rule 5985
Rule 5919
Rule 5949
Rule 6059
Rule 6610
Rubi steps
\begin{align*} \int \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+b \coth ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{x \left (a+b \coth ^{-1}(x)\right )^2}{1-x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{(3 b) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right )^2}{1-x} \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b \coth ^{-1}(c+d x)\right )^3}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1-c-d x}\right )}{d}+\frac{\left (6 b^2\right ) \operatorname{Subst}\left (\int \frac{\left (a+b \coth ^{-1}(x)\right ) \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 )^3}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1-c-d x}\right )}{d}-\frac{3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{d}+\frac{\left (3 b^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (1-\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 )^3}{d}+\frac{(c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}-\frac{3 b \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac{2}{1-c-d x}\right )}{d}-\frac{3 b^2 \left (a+b \coth ^{-1}(c+d x)\right ) \text{Li}_2\left (1-\frac{2}{1-c-d x}\right )}{d}+\frac{3 b^3 \text{Li}_3\left (1-\frac{2}{1-c-d x}\right )}{2 d}\\ \end{align*}
Mathematica [C] time = 0.319898, size = 208, normalized size = 1.58 \[ \frac{6 a b^2 \left (\text{PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )+\coth ^{-1}(c+d x) \left ((c+d x-1) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )\right )+2 b^3 \left (-3 \coth ^{-1}(c+d x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac{3}{2} \text{PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )+(c+d x) \coth ^{-1}(c+d x)^3+\coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-\frac{i \pi ^3}{8}\right )+3 a^2 b \log \left (1-(c+d x)^2\right )+6 a^2 b (c+d x) \coth ^{-1}(c+d x)+2 a^3 (c+d x)}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.14, size = 485, normalized size = 3.7 \begin{align*} x{a}^{3}+{\frac{{a}^{3}c}{d}}+ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{3}x{b}^{3}+{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{3}{b}^{3}c}{d}}-3\,{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}{b}^{3}}{d}\ln \left ( 1+{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }-3\,{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}{b}^{3}}{d}\ln \left ( 1-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{3}{b}^{3}}{d}}-6\,{\frac{{\rm arccoth} \left (dx+c\right ){b}^{3}}{d}{\it polylog} \left ( 2,{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }-6\,{\frac{{\rm arccoth} \left (dx+c\right ){b}^{3}}{d}{\it polylog} \left ( 2,-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+6\,{\frac{{b}^{3}}{d}{\it polylog} \left ( 3,-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+6\,{\frac{{b}^{3}}{d}{\it polylog} \left ( 3,{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+3\, \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}xa{b}^{2}+3\,{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}a{b}^{2}c}{d}}-6\,{\frac{{\rm arccoth} \left (dx+c\right )a{b}^{2}}{d}\ln \left ( 1+{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }-6\,{\frac{{\rm arccoth} \left (dx+c\right )a{b}^{2}}{d}\ln \left ( 1-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+3\,{\frac{ \left ({\rm arccoth} \left (dx+c\right ) \right ) ^{2}a{b}^{2}}{d}}-6\,{\frac{a{b}^{2}}{d}{\it polylog} \left ( 2,{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }-6\,{\frac{a{b}^{2}}{d}{\it polylog} \left ( 2,-{\frac{1}{\sqrt{{\frac{dx+c-1}{dx+c+1}}}}} \right ) }+3\,{\rm arccoth} \left (dx+c\right )x{a}^{2}b+3\,{\frac{{\rm arccoth} \left (dx+c\right ){a}^{2}bc}{d}}+{\frac{3\,{a}^{2}b\ln \left ( \left ( dx+c \right ) ^{2}-1 \right ) }{2\,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^{3} x + \frac{3 \,{\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^{2} b}{2 \, d} + \frac{{\left (b^{3} d x + b^{3}{\left (c + 1\right )}\right )} \log \left (d x + c + 1\right )^{3} + 3 \,{\left (2 \, a b^{2} d x -{\left (b^{3} d x + b^{3}{\left (c - 1\right )}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )^{2}}{8 \, d} + \int -\frac{{\left (b^{3} d x + b^{3}{\left (c + 1\right )}\right )} \log \left (d x + c - 1\right )^{3} - 6 \,{\left (a b^{2} d x + a b^{2}{\left (c + 1\right )}\right )} \log \left (d x + c - 1\right )^{2} + 3 \,{\left (4 \, a b^{2} d x -{\left (b^{3} d x + b^{3}{\left (c + 1\right )}\right )} \log \left (d x + c - 1\right )^{2} + 2 \,{\left (2 \, a b^{2}{\left (c + 1\right )} - b^{3}{\left (c - 1\right )} +{\left (2 \, a b^{2} d - b^{3} d\right )} x\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right )}{8 \,{\left (d x + c + 1\right )}}\,{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 (b^{3} \operatorname{arcoth}\left (d x + c\right )^{3} + 3 \, a b^{2} \operatorname{arcoth}\left (d x + c\right )^{2} + 3 \, a^{2} b \operatorname{arcoth}\left (d x + c\right ) + a^{3}, 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 )^{3}\, 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 )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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