Optimal. Leaf size=148 \[ \frac{1}{2} \text{PolyLog}\left (3,1-\frac{2}{a+b x+1}\right )-\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 b x}{(1-a) (a+b x+1)}\right )+\coth ^{-1}(a+b x) \text{PolyLog}\left (2,1-\frac{2}{a+b x+1}\right )-\coth ^{-1}(a+b x) \text{PolyLog}\left (2,1-\frac{2 b x}{(1-a) (a+b x+1)}\right )-\log \left (\frac{2}{a+b x+1}\right ) \coth ^{-1}(a+b x)^2+\log \left (\frac{2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)^2 \]
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Rubi [A] time = 0.0913941, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6112, 5923} \[ \frac{1}{2} \text{PolyLog}\left (3,1-\frac{2}{a+b x+1}\right )-\frac{1}{2} \text{PolyLog}\left (3,1-\frac{2 b x}{(1-a) (a+b x+1)}\right )+\coth ^{-1}(a+b x) \text{PolyLog}\left (2,1-\frac{2}{a+b x+1}\right )-\coth ^{-1}(a+b x) \text{PolyLog}\left (2,1-\frac{2 b x}{(1-a) (a+b x+1)}\right )-\log \left (\frac{2}{a+b x+1}\right ) \coth ^{-1}(a+b x)^2+\log \left (\frac{2 b x}{(1-a) (a+b x+1)}\right ) \coth ^{-1}(a+b x)^2 \]
Antiderivative was successfully verified.
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Rule 6112
Rule 5923
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a+b x)^2}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)^2}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\coth ^{-1}(a+b x)^2 \log \left (\frac{2}{1+a+b x}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac{2 b x}{(1-a) (1+a+b x)}\right )+\coth ^{-1}(a+b x) \text{Li}_2\left (1-\frac{2}{1+a+b x}\right )-\coth ^{-1}(a+b x) \text{Li}_2\left (1-\frac{2 b x}{(1-a) (1+a+b x)}\right )+\frac{1}{2} \text{Li}_3\left (1-\frac{2}{1+a+b x}\right )-\frac{1}{2} \text{Li}_3\left (1-\frac{2 b x}{(1-a) (1+a+b x)}\right )\\ \end{align*}
Mathematica [C] time = 3.20549, size = 714, normalized size = 4.82 \[ 2 \coth ^{-1}(a+b x) \text{PolyLog}\left (2,-\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )+2 \coth ^{-1}(a+b x) \text{PolyLog}\left (2,\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x) \text{PolyLog}\left (2,\frac{(a-1) e^{2 \coth ^{-1}(a+b x)}}{a+1}\right )-2 \text{PolyLog}\left (3,-\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )-2 \text{PolyLog}\left (3,\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,e^{2 \coth ^{-1}(a+b x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,\frac{(a-1) e^{2 \coth ^{-1}(a+b x)}}{a+1}\right )+\coth ^{-1}(a+b x) \text{PolyLog}\left (2,e^{2 \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{1}{a}\right )}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{2 \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{1}{a}\right )}\right )+\frac{2}{3} \sqrt{1-\frac{1}{a^2}} a e^{\tanh ^{-1}\left (\frac{1}{a}\right )} \coth ^{-1}(a+b x)^3-\frac{2}{3} a \coth ^{-1}(a+b x)^3-\frac{2}{3} \coth ^{-1}(a+b x)^3+\coth ^{-1}(a+b x)^2 \log \left (1-\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}\right )+\coth ^{-1}(a+b x)^2 \log \left (\sqrt{\frac{a-1}{a+1}} e^{\coth ^{-1}(a+b x)}+1\right )-\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)}\right )-\coth ^{-1}(a+b x)^2 \log \left (1-\frac{(a-1) e^{2 \coth ^{-1}(a+b x)}}{a+1}\right )+\coth ^{-1}(a+b x)^2 \log \left (\frac{1}{2} e^{-\coth ^{-1}(a+b x)} \left (a \left (e^{2 \coth ^{-1}(a+b x)}-1\right )-e^{2 \coth ^{-1}(a+b x)}-1\right )\right )-\log \left (-\frac{b x}{(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}\right ) \coth ^{-1}(a+b x)^2-i \pi \coth ^{-1}(a+b x) \log \left (\frac{1}{2} \left (e^{-\coth ^{-1}(a+b x)}+e^{\coth ^{-1}(a+b x)}\right )\right )+i \pi \log \left (\frac{1}{\sqrt{1-\frac{1}{(a+b x)^2}}}\right ) \coth ^{-1}(a+b x)+\coth ^{-1}(a+b x)^2 \log \left (1-e^{2 \coth ^{-1}(a+b x)-2 \tanh ^{-1}\left (\frac{1}{a}\right )}\right )-2 \tanh ^{-1}\left (\frac{1}{a}\right ) \coth ^{-1}(a+b x) \log \left (\frac{1}{2} i \left (e^{\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac{1}{a}\right )}-e^{\tanh ^{-1}\left (\frac{1}{a}\right )-\coth ^{-1}(a+b x)}\right )\right )+2 \tanh ^{-1}\left (\frac{1}{a}\right ) \coth ^{-1}(a+b x) \log \left (i \sinh \left (\coth ^{-1}(a+b x)-\tanh ^{-1}\left (\frac{1}{a}\right )\right )\right )-\frac{i \pi ^3}{24} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 0.543, size = 985, normalized size = 6.7 \begin{align*} \text{result too large to display} \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*} \int \frac{\operatorname{arcoth}\left (b x + a\right )^{2}}{x}\,{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{\operatorname{arcoth}\left (b x + a\right )^{2}}{x}, 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 \frac{\operatorname{acoth}^{2}{\left (a + b x \right )}}{x}\, 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 \frac{\operatorname{arcoth}\left (b x + a\right )^{2}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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