Optimal. Leaf size=63 \[ 2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}-2 \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
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Rubi [A] time = 0.0595282, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2159, 2161} \[ 2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}-2 \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
Antiderivative was successfully verified.
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Rule 2159
Rule 2161
Rubi steps
\begin{align*} \int \frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x} \, dx &=2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{1}{x \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-2 \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}+2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}\\ \end{align*}
Mathematica [A] time = 0.0733578, size = 61, normalized size = 0.97 \[ 2 \sqrt{\tanh ^{-1}(\tanh (a+b x))}-2 \tanh ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))-b x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.128, size = 54, normalized size = 0.9 \begin{align*} 2\,\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }-2\,\sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) -bx}{\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 ) \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55731, size = 188, normalized size = 2.98 \begin{align*} \left [\sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \, \sqrt{b x + a}, 2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) + 2 \, \sqrt{b x + a}\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}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15686, size = 43, normalized size = 0.68 \begin{align*} \frac{2 \, a \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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