Optimal. Leaf size=74 \[ -\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \log (x) \]
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Rubi [A] time = 0.05442, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2168, 29} \[ -\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \log (x) \]
Antiderivative was successfully verified.
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Rule 2168
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^5} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^4} \, dx\\ &=-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^2 \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^3} \, dx\\ &=-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^3 \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^2} \, dx\\ &=-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \int \frac{1}{x} \, dx\\ &=-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0301265, size = 78, normalized size = 1.05 \[ -\frac{12 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+4 b x \tanh ^{-1}(\tanh (a+b x))^3+3 \tanh ^{-1}(\tanh (a+b x))^4-b^4 x^4 (12 \log (x)+25)}{12 x^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 69, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{x}}-{\frac{{b}^{2} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{2\,{x}^{2}}}-{\frac{b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{3\,{x}^{3}}}-{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{4\,{x}^{4}}}+{b}^{4}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.79604, size = 97, normalized size = 1.31 \begin{align*} \frac{1}{2} \,{\left (2 \,{\left (b^{2} \log \left (x\right ) - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x}\right )} b - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{2}}\right )} b - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{3}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56394, size = 112, normalized size = 1.51 \begin{align*} \frac{12 \, b^{4} x^{4} \log \left (x\right ) - 48 \, a b^{3} x^{3} - 36 \, a^{2} b^{2} x^{2} - 16 \, a^{3} b x - 3 \, a^{4}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.29266, size = 70, normalized size = 0.95 \begin{align*} b^{4} \log{\left (x \right )} - \frac{b^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{x} - \frac{b^{2} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{2 x^{2}} - \frac{b \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{3 x^{3}} - \frac{\operatorname{atanh}^{4}{\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 = 1.12572, size = 62, normalized size = 0.84 \begin{align*} b^{4} \log \left ({\left | x \right |}\right ) - \frac{48 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 16 \, a^{3} b x + 3 \, a^{4}}{12 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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