Optimal. Leaf size=23 \[ -\frac{\coth ^{-1}(\tanh (a+b x))}{3 x^3}-\frac{b}{6 x^2} \]
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Rubi [A] time = 0.0088794, 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} \[ -\frac{\coth ^{-1}(\tanh (a+b x))}{3 x^3}-\frac{b}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 30
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}(\tanh (a+b x))}{x^4} \, dx &=-\frac{\coth ^{-1}(\tanh (a+b x))}{3 x^3}+\frac{1}{3} b \int \frac{1}{x^3} \, dx\\ &=-\frac{b}{6 x^2}-\frac{\coth ^{-1}(\tanh (a+b x))}{3 x^3}\\ \end{align*}
Mathematica [A] time = 0.0154072, size = 20, normalized size = 0.87 \[ -\frac{2 \coth ^{-1}(\tanh (a+b x))+b x}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 20, normalized size = 0.9 \begin{align*} -{\frac{b}{6\,{x}^{2}}}-{\frac{{\rm arccoth} \left (\tanh \left ( bx+a \right ) \right )}{3\,{x}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17902, size = 26, normalized size = 1.13 \begin{align*} -\frac{b}{6 \, x^{2}} - \frac{\operatorname{arcoth}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.668, size = 32, normalized size = 1.39 \begin{align*} -\frac{3 \, b x + 2 \, a}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.27728, size = 20, normalized size = 0.87 \begin{align*} - \frac{b}{6 x^{2}} - \frac{\operatorname{acoth}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{3}} \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 (\tanh \left (b x + a\right )\right )}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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