Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}(\coth (a+b x))}{2 x^2}-\frac{b}{2 x} \]
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Rubi [A] time = 0.0089536, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2168, 30} \[ -\frac{\tanh ^{-1}(\coth (a+b x))}{2 x^2}-\frac{b}{2 x} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\coth (a+b x))}{x^3} \, dx &=-\frac{\tanh ^{-1}(\coth (a+b x))}{2 x^2}+\frac{1}{2} b \int \frac{1}{x^2} \, dx\\ &=-\frac{b}{2 x}-\frac{\tanh ^{-1}(\coth (a+b x))}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0135759, size = 18, normalized size = 0.78 \[ -\frac{\tanh ^{-1}(\coth (a+b x))+b x}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b}{2\,x}}-{\frac{{\it Artanh} \left ({\rm coth} \left (bx+a\right ) \right ) }{2\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14687, size = 26, normalized size = 1.13 \begin{align*} -\frac{b}{2 \, x} - \frac{\operatorname{artanh}\left (\coth \left (b x + a\right )\right )}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12019, size = 30, normalized size = 1.3 \begin{align*} -\frac{2 \, b x + a}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 142.164, size = 49, normalized size = 2.13 \begin{align*} \begin{cases} \frac{\left \langle - \frac{\pi }{4}, \frac{\pi }{4}\right \rangle i}{x^{2}} & \text{for}\: a = \log{\left (- e^{- b x} \right )} \vee a = \log{\left (e^{- b x} \right )} \\- \frac{b}{2 x} - \frac{\operatorname{atanh}{\left (\frac{1}{\tanh{\left (a + b x \right )}} \right )}}{2 x^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.19043, size = 96, normalized size = 4.17 \begin{align*} -\frac{b}{2 \, x} - \frac{\log \left (-\frac{\frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} + 1}{\frac{e^{\left (2 \, b x + 2 \, a\right )} + 1}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - 1}\right )}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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