Optimal. Leaf size=23 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}-\frac{b}{2 x} \]
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Rubi [A] time = 0.0089049, 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} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}-\frac{b}{2 x} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^3} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}+\frac{1}{2} b \int \frac{1}{x^2} \, dx\\ &=-\frac{b}{2 x}-\frac{\tanh ^{-1}(\tanh (a+b x))}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0139429, size = 18, normalized size = 0.78 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))+b x}{2 x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b}{2\,x}}-{\frac{{\it Artanh} \left ( \tanh \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 = 1.16409, size = 26, normalized size = 1.13 \begin{align*} -\frac{b}{2 \, x} - \frac{\operatorname{artanh}\left (\tanh \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 = 1.66006, 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 = 1.1003, size = 19, normalized size = 0.83 \begin{align*} - \frac{b}{2 x} - \frac{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1388, size = 15, normalized size = 0.65 \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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