Optimal. Leaf size=77 \[ -\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-4 b^3 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+4 b^4 x \]
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Rubi [A] time = 0.0582334, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2168, 2158, 29} \[ -\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-4 b^3 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+4 b^4 x \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^4} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\frac{1}{3} (4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^3} \, dx\\ &=-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\left (2 b^2\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^2} \, dx\\ &=-\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\left (4 b^3\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=4 b^4 x-\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-\left (4 b^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=4 b^4 x-\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-4 b^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end{align*}
Mathematica [A] time = 0.0481872, size = 82, normalized size = 1.06 \[ -\frac{6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-2 b^3 x^3 (6 \log (x)+11) \tanh ^{-1}(\tanh (a+b x))+2 b x \tanh ^{-1}(\tanh (a+b x))^3+\tanh ^{-1}(\tanh (a+b x))^4+2 b^4 x^4 (6 \log (x)+5)}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 76, normalized size = 1. \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{3\,{x}^{3}}}-{\frac{2\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{3\,{x}^{2}}}-2\,{\frac{{b}^{2} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{x}}-4\,\ln \left ( x \right ) x{b}^{4}+4\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \ln \left ( x \right ){b}^{3}+4\,{b}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61241, size = 123, normalized size = 1.6 \begin{align*} 2 \,{\left (2 \,{\left (b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) \log \left (x\right ) -{\left (b{\left (x + \frac{a}{b}\right )} \log \left (x\right ) - b{\left (x + \frac{a \log \left (x\right )}{b}\right )}\right )} b\right )} b - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x}\right )} b - \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{2}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54661, size = 105, normalized size = 1.36 \begin{align*} \frac{3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} \log \left (x\right ) - 18 \, a^{2} b^{2} x^{2} - 6 \, a^{3} b x - a^{4}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{4}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13538, size = 57, normalized size = 0.74 \begin{align*} b^{4} x + 4 \, a b^{3} \log \left ({\left | x \right |}\right ) - \frac{18 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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