Optimal. Leaf size=68 \[ -3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}+\frac{3}{2} b \tanh ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
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Rubi [A] time = 0.0412704, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {2168, 2159, 2158, 29} \[ -3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}+\frac{3}{2} b \tanh ^{-1}(\tanh (a+b x))^2+3 b \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2159
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^2} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}+(3 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=\frac{3}{2} b \tanh ^{-1}(\tanh (a+b x))^2-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}-\left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=-3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{3}{2} b \tanh ^{-1}(\tanh (a+b x))^2-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}+\left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2\right ) \int \frac{1}{x} \, dx\\ &=-3 b^2 x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{3}{2} b \tanh ^{-1}(\tanh (a+b x))^2-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}+3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0381505, size = 62, normalized size = 0.91 \[ -6 b^2 x \log (x) \tanh ^{-1}(\tanh (a+b x))-\frac{\tanh ^{-1}(\tanh (a+b x))^3}{x}+3 b (\log (x)+1) \tanh ^{-1}(\tanh (a+b x))^2+\frac{3}{2} b^3 x^2 (2 \log (x)-1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 76, normalized size = 1.1 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{x}}+3\,\ln \left ( x \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}b+3\,{b}^{3}{x}^{2}\ln \left ( x \right ) -{\frac{9\,{b}^{3}{x}^{2}}{2}}-6\,{b}^{2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \ln \left ( x \right ) x+6\,{b}^{2}{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.58652, size = 88, normalized size = 1.29 \begin{align*} 3 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right ) + \frac{3}{2} \,{\left (b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} \log \left (x\right ) - 2 \, \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2} \log \left (x\right )\right )} b - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50572, size = 78, normalized size = 1.15 \begin{align*} \frac{b^{3} x^{3} + 6 \, a b^{2} x^{2} + 6 \, a^{2} b x \log \left (x\right ) - 2 \, a^{3}}{2 \, x} \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}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10957, size = 45, normalized size = 0.66 \begin{align*} \frac{1}{2} \, b^{3} x^{2} + 3 \, a b^{2} x + 3 \, a^{2} b \log \left ({\left | x \right |}\right ) - \frac{a^{3}}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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