Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0126636, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2167} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^4} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0387074, size = 34, normalized size = 1.1 \[ -\frac{b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 38, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{3\,{x}^{3}}}+{\frac{2\,b}{3} \left ( -{\frac{b}{2\,x}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{2\,{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.37101, size = 49, normalized size = 1.58 \begin{align*} -\frac{b^{2}}{3 \, x} - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{2}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52019, size = 51, normalized size = 1.65 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.06786, size = 37, normalized size = 1.19 \begin{align*} - \frac{b^{2}}{3 x} - \frac{b \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{2}} - \frac{\operatorname{atanh}^{2}{\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 [A] time = 1.11319, size = 30, normalized size = 0.97 \begin{align*} -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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