Optimal. Leaf size=77 \[ b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]
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Rubi [A] time = 0.0809734, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2159, 2158, 29} \[ b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2158
Rule 29
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x} \, dx &=\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3+\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0576073, size = 104, normalized size = 1.35 \[ (a+b x) \left (a^2-3 a \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )+3 \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )^2\right )+\frac{1}{3} (a+b x)^3-\frac{1}{2} (a+b x)^2 \left (-3 \tanh ^{-1}(\tanh (a+b x))+2 a+3 b x\right )+\log (b x) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 92, normalized size = 1.2 \begin{align*} \ln \left ( x \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}+3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \ln \left ( x \right ){x}^{2}{b}^{2}-{\frac{9\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ){x}^{2}{b}^{2}}{2}}-3\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}\ln \left ( x \right ) x-{b}^{3}{x}^{3}\ln \left ( x \right ) +{\frac{11\,{x}^{3}{b}^{3}}{6}}+3\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.44981, size = 42, normalized size = 0.55 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57665, size = 73, normalized size = 0.95 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15529, size = 43, normalized size = 0.56 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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