Optimal. Leaf size=35 \[ \frac{\log \left (1-(a+b x)^2\right )}{2 b}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b} \]
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Rubi [A] time = 0.0159401, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6104, 5911, 260} \[ \frac{\log \left (1-(a+b x)^2\right )}{2 b}+\frac{(a+b x) \coth ^{-1}(a+b x)}{b} \]
Antiderivative was successfully verified.
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Rule 6104
Rule 5911
Rule 260
Rubi steps
\begin{align*} \int \coth ^{-1}(a+b x) \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \coth ^{-1}(a+b x)}{b}-\frac{\operatorname{Subst}\left (\int \frac{x}{1-x^2} \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \coth ^{-1}(a+b x)}{b}+\frac{\log \left (1-(a+b x)^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0145124, size = 43, normalized size = 1.23 \[ \frac{(a+1) \log (a+b x+1)-(a-1) \log (-a-b x+1)}{2 b}+x \coth ^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 36, normalized size = 1. \begin{align*} x{\rm arccoth} \left (bx+a\right )+{\frac{{\rm arccoth} \left (bx+a\right )a}{b}}+{\frac{\ln \left ( \left ( bx+a \right ) ^{2}-1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.973925, size = 42, normalized size = 1.2 \begin{align*} \frac{2 \,{\left (b x + a\right )} \operatorname{arcoth}\left (b x + a\right ) + \log \left (-{\left (b x + a\right )}^{2} + 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54404, size = 135, normalized size = 3.86 \begin{align*} \frac{b x \log \left (\frac{b x + a + 1}{b x + a - 1}\right ) +{\left (a + 1\right )} \log \left (b x + a + 1\right ) -{\left (a - 1\right )} \log \left (b x + a - 1\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.8191, size = 41, normalized size = 1.17 \begin{align*} \begin{cases} \frac{a \operatorname{acoth}{\left (a + b x \right )}}{b} + x \operatorname{acoth}{\left (a + b x \right )} + \frac{\log{\left (a + b x + 1 \right )}}{b} - \frac{\operatorname{acoth}{\left (a + b x \right )}}{b} & \text{for}\: b \neq 0 \\x \operatorname{acoth}{\left (a \right )} & \text{otherwise} \end{cases} \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 \operatorname{arcoth}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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