Optimal. Leaf size=31 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.0132947, 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} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^5} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0224993, size = 50, normalized size = 1.61 \[ -\frac{b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+b x \tanh ^{-1}(\tanh (a+b x))^2+\tanh ^{-1}(\tanh (a+b x))^3+b^3 x^3}{4 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 56, normalized size = 1.8 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{4\,{x}^{4}}}+{\frac{3\,b}{4} \left ( -{\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 ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59351, size = 72, normalized size = 2.32 \begin{align*} -\frac{1}{4} \, b{\left (\frac{b^{2}}{x} + \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\right )} - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{3}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42449, size = 73, normalized size = 2.35 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.61478, size = 56, normalized size = 1.81 \begin{align*} - \frac{b^{3}}{4 x} - \frac{b^{2} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{2}} - \frac{b \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{4 x^{3}} - \frac{\operatorname{atanh}^{3}{\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 = 1.1343, size = 45, normalized size = 1.45 \begin{align*} -\frac{4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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