Optimal. Leaf size=81 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-3 b \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0474029, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2168, 2159, 2161} \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-3 b \sqrt{b x-\tanh ^{-1}(\tanh (a+b x))} \tan ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{b x-\tanh ^{-1}(\tanh (a+b x))}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2168
Rule 2159
Rule 2161
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x^2} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+\frac{1}{2} (3 b) \int \frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{x} \, dx\\ &=3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}-\frac{1}{2} \left (3 b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x \sqrt{\tanh ^{-1}(\tanh (a+b x))}} \, dx\\ &=-3 b \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))}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}\\ \end{align*}
Mathematica [A] time = 0.0350054, size = 79, normalized size = 0.98 \[ -\frac{\tanh ^{-1}(\tanh (a+b x))^{3/2}}{x}+3 b \sqrt{\tanh ^{-1}(\tanh (a+b x))}-3 b \tanh ^{-1}\left (\frac{\sqrt{\tanh ^{-1}(\tanh (a+b x))}}{\sqrt{\tanh ^{-1}(\tanh (a+b x))-b x}}\right ) \sqrt{\tanh ^{-1}(\tanh (a+b x))-b x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.118, size = 85, normalized size = 1.1 \begin{align*} 2\,b \left ( \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }+{\frac{ \left ( -1/2\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) +1/2\,bx \right ) \sqrt{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }}{bx}}-3/2\,\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 ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{\frac{3}{2}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.21208, size = 247, normalized size = 3.05 \begin{align*} \left [\frac{3 \, \sqrt{a} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, b x - a\right )} \sqrt{b x + a}}{2 \, x}, \frac{3 \, \sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a} \sqrt{-a}}{a}\right ) +{\left (2 \, b x - a\right )} \sqrt{b x + a}}{x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{\frac{3}{2}}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.15655, size = 93, normalized size = 1.15 \begin{align*} \frac{\sqrt{2}{\left (\frac{3 \, \sqrt{2} a b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{2} \sqrt{b x + a} b^{2} - \frac{\sqrt{2} \sqrt{b x + a} a b}{x}\right )}}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]