Optimal. Leaf size=125 \[ \frac{b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}-\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{2 x^2}-\frac{b}{4 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
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Rubi [A] time = 0.0716427, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2163, 2161} \[ \frac{b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}-\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{2 x^2}-\frac{b}{4 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \]
Antiderivative was successfully verified.
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Rule 2168
Rule 2163
Rule 2161
Rubi steps
\begin{align*} \int \frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x^3} \, dx &=-\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{2 x^2}+\frac{1}{4} b \int \frac{1}{x^2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-\frac{b}{4 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{2 x^2}-\frac{1}{8} b^2 \int \frac{1}{x \tanh ^{-1}(\tanh (a+b x))^{3/2}} \, dx\\ &=-\frac{b}{4 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{2 x^2}-\frac{b^2 \int \frac{1}{x \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx}{8 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right )}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^{3/2}}-\frac{b}{4 x \sqrt{\tanh ^{-1}(\tanh (a+b x))}}+\frac{b^2}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}-\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{2 x^2}\\ \end{align*}
Mathematica [A] time = 0.0970263, size = 89, normalized size = 0.71 \[ \frac{1}{4} \left (\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right )}{\left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^{3/2}}+\frac{\left (\frac{b x}{b x-\tanh ^{-1}(\tanh (a+b x))}-2\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.127, size = 92, normalized size = 0.7 \begin{align*} 2\,{b}^{2} \left ({\frac{1}{{b}^{2}{x}^{2}} \left ( -1/8\,{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3/2}}{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}-1/8\,\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) } \right ) }+1/8\,{\frac{1}{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx \right ) ^{3/2}}{\it Artanh} \left ({\frac{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}}} \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72048, size = 292, normalized size = 2.34 \begin{align*} \left [\frac{\sqrt{a} b^{2} x^{2} \log \left (\frac{b x + 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) - 2 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{b x + a}}{8 \, a^{2} x^{2}}, -\frac{\sqrt{-a} b^{2} x^{2} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (a b x + 2 \, a^{2}\right )} \sqrt{b x + a}}{4 \, a^{2} x^{2}}\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{\sqrt{\operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}}{x^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16984, size = 89, normalized size = 0.71 \begin{align*} -\frac{\frac{b^{3} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{{\left (b x + a\right )}^{\frac{3}{2}} b^{3} + \sqrt{b x + a} a b^{3}}{a b^{2} x^{2}}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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