Optimal. Leaf size=48 \[ \frac{\log (a+b x)}{b}-\frac{\log \left (1-(a+b x)^2\right )}{2 b}-\frac{\coth ^{-1}(a+b x)}{b (a+b x)} \]
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Rubi [A] time = 0.0451322, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {6108, 5917, 266, 36, 31, 29} \[ \frac{\log (a+b x)}{b}-\frac{\log \left (1-(a+b x)^2\right )}{2 b}-\frac{\coth ^{-1}(a+b x)}{b (a+b x)} \]
Antiderivative was successfully verified.
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Rule 6108
Rule 5917
Rule 266
Rule 36
Rule 31
Rule 29
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a+b x)}{(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\coth ^{-1}(x)}{x^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\coth ^{-1}(a+b x)}{b (a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (1-x^2\right )} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\coth ^{-1}(a+b x)}{b (a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{(1-x) x} \, dx,x,(a+b x)^2\right )}{2 b}\\ &=-\frac{\coth ^{-1}(a+b x)}{b (a+b x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,(a+b x)^2\right )}{2 b}+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,(a+b x)^2\right )}{2 b}\\ &=-\frac{\coth ^{-1}(a+b x)}{b (a+b x)}+\frac{\log (a+b x)}{b}-\frac{\log \left (1-(a+b x)^2\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0257019, size = 43, normalized size = 0.9 \[ -\frac{-2 \log (a+b x)+\log \left (1-(a+b x)^2\right )+\frac{2 \coth ^{-1}(a+b x)}{a+b x}}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 54, normalized size = 1.1 \begin{align*} -{\frac{{\rm arccoth} \left (bx+a\right )}{b \left ( bx+a \right ) }}-{\frac{\ln \left ( bx+a-1 \right ) }{2\,b}}+{\frac{\ln \left ( bx+a \right ) }{b}}-{\frac{\ln \left ( bx+a+1 \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.97605, size = 72, normalized size = 1.5 \begin{align*} -\frac{\log \left (b x + a + 1\right )}{2 \, b} + \frac{\log \left (b x + a\right )}{b} - \frac{\log \left (b x + a - 1\right )}{2 \, b} - \frac{\operatorname{arcoth}\left (b x + a\right )}{{\left (b x + a\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91232, size = 171, normalized size = 3.56 \begin{align*} -\frac{{\left (b x + a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right ) - 2 \,{\left (b x + a\right )} \log \left (b x + a\right ) + \log \left (\frac{b x + a + 1}{b x + a - 1}\right )}{2 \,{\left (b^{2} x + a b\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.82632, size = 136, normalized size = 2.83 \begin{align*} \begin{cases} \frac{a \log{\left (\frac{a}{b} + x \right )}}{a b + b^{2} x} - \frac{a \log{\left (\frac{a}{b} + x + \frac{1}{b} \right )}}{a b + b^{2} x} + \frac{a \operatorname{acoth}{\left (a + b x \right )}}{a b + b^{2} x} + \frac{b x \log{\left (\frac{a}{b} + x \right )}}{a b + b^{2} x} - \frac{b x \log{\left (\frac{a}{b} + x + \frac{1}{b} \right )}}{a b + b^{2} x} + \frac{b x \operatorname{acoth}{\left (a + b x \right )}}{a b + b^{2} x} - \frac{\operatorname{acoth}{\left (a + b x \right )}}{a b + b^{2} x} & \text{for}\: b \neq 0 \\\frac{x \operatorname{acoth}{\left (a \right )}}{a^{2}} & \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 \frac{\operatorname{arcoth}\left (b x + a\right )}{{\left (b x + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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