Optimal. Leaf size=64 \[ \frac{b \log (x)}{1-a^2}-\frac{b \log (-a-b x+1)}{2 (1-a)}-\frac{b \log (a+b x+1)}{2 (a+1)}-\frac{\coth ^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.0510754, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6110, 371, 706, 31, 633} \[ \frac{b \log (x)}{1-a^2}-\frac{b \log (-a-b x+1)}{2 (1-a)}-\frac{b \log (a+b x+1)}{2 (a+1)}-\frac{\coth ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 6110
Rule 371
Rule 706
Rule 31
Rule 633
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(a+b x)}{x^2} \, dx &=-\frac{\coth ^{-1}(a+b x)}{x}+b \int \frac{1}{x \left (1-(a+b x)^2\right )} \, dx\\ &=-\frac{\coth ^{-1}(a+b x)}{x}+b \operatorname{Subst}\left (\int \frac{1}{(-a+x) \left (1-x^2\right )} \, dx,x,a+b x\right )\\ &=-\frac{\coth ^{-1}(a+b x)}{x}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-a+x} \, dx,x,a+b x\right )}{1-a^2}+\frac{b \operatorname{Subst}\left (\int \frac{a+x}{1-x^2} \, dx,x,a+b x\right )}{1-a^2}\\ &=-\frac{\coth ^{-1}(a+b x)}{x}+\frac{b \log (x)}{1-a^2}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-x} \, dx,x,a+b x\right )}{2 (1-a)}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-1-x} \, dx,x,a+b x\right )}{2 (1+a)}\\ &=-\frac{\coth ^{-1}(a+b x)}{x}+\frac{b \log (x)}{1-a^2}-\frac{b \log (1-a-b x)}{2 (1-a)}-\frac{b \log (1+a+b x)}{2 (1+a)}\\ \end{align*}
Mathematica [A] time = 0.0505785, size = 55, normalized size = 0.86 \[ \frac{b ((a+1) \log (-a-b x+1)-(a-1) \log (a+b x+1)-2 \log (x))}{2 \left (a^2-1\right )}-\frac{\coth ^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 63, normalized size = 1. \begin{align*} -{\frac{{\rm arccoth} \left (bx+a\right )}{x}}+{\frac{b\ln \left ( bx+a-1 \right ) }{2\,a-2}}-{\frac{b\ln \left ( bx \right ) }{ \left ( 1+a \right ) \left ( a-1 \right ) }}-{\frac{b\ln \left ( bx+a+1 \right ) }{2+2\,a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.959147, size = 73, normalized size = 1.14 \begin{align*} -\frac{1}{2} \, b{\left (\frac{\log \left (b x + a + 1\right )}{a + 1} - \frac{\log \left (b x + a - 1\right )}{a - 1} + \frac{2 \, \log \left (x\right )}{a^{2} - 1}\right )} - \frac{\operatorname{arcoth}\left (b x + a\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67592, size = 192, normalized size = 3. \begin{align*} -\frac{{\left (a - 1\right )} b x \log \left (b x + a + 1\right ) -{\left (a + 1\right )} b x \log \left (b x + a - 1\right ) + 2 \, b x \log \left (x\right ) +{\left (a^{2} - 1\right )} \log \left (\frac{b x + a + 1}{b x + a - 1}\right )}{2 \,{\left (a^{2} - 1\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.8849, size = 144, normalized size = 2.25 \begin{align*} \begin{cases} \frac{b \operatorname{acoth}{\left (b x - 1 \right )}}{2} - \frac{\operatorname{acoth}{\left (b x - 1 \right )}}{x} - \frac{1}{2 x} & \text{for}\: a = -1 \\- \frac{b \operatorname{acoth}{\left (b x + 1 \right )}}{2} - \frac{\operatorname{acoth}{\left (b x + 1 \right )}}{x} + \frac{1}{2 x} & \text{for}\: a = 1 \\- \frac{a^{2} \operatorname{acoth}{\left (a + b x \right )}}{a^{2} x - x} - \frac{a b x \operatorname{acoth}{\left (a + b x \right )}}{a^{2} x - x} - \frac{b x \log{\left (x \right )}}{a^{2} x - x} + \frac{b x \log{\left (a + b x + 1 \right )}}{a^{2} x - x} - \frac{b x \operatorname{acoth}{\left (a + b x \right )}}{a^{2} x - x} + \frac{\operatorname{acoth}{\left (a + b x \right )}}{a^{2} x - x} & \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 )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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