Optimal. Leaf size=42 \[ -\frac{b \tanh ^{-1}(\tanh (a+b x))}{6 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^2}{4 x^4}-\frac{b^2}{12 x^2} \]
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Rubi [A] time = 0.0303805, antiderivative size = 64, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^5} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^4} \, dx}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0289113, size = 37, normalized size = 0.88 \[ -\frac{2 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 38, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{b}{2} \left ( -{\frac{b}{6\,{x}^{2}}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3\,{x}^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36886, size = 49, normalized size = 1.17 \begin{align*} -\frac{b^{2}}{12 \, x^{2}} - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{6 \, x^{3}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46397, size = 55, normalized size = 1.31 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.01543, size = 39, normalized size = 0.93 \begin{align*} - \frac{b^{2}}{12 x^{2}} - \frac{b \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{6 x^{3}} - \frac{\operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17859, size = 32, normalized size = 0.76 \begin{align*} -\frac{6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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