Optimal. Leaf size=63 \[ \frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^2} \]
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Rubi [A] time = 0.0644179, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6169, 835, 807, 266, 63, 208} \[ \frac{1}{2} x^2 \sqrt{1-\frac{1}{a^2 x^2}}+\frac{x \sqrt{1-\frac{1}{a^2 x^2}}}{a}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 6169
Rule 835
Rule 807
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} x \, dx &=-\operatorname{Subst}\left (\int \frac{1+\frac{x}{a}}{x^3 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{2} \operatorname{Subst}\left (\int \frac{-\frac{2}{a}-\frac{x}{a^2}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{2 a^2}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )}{4 a^2}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a^2-a^2 x^2} \, dx,x,\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x}{a}+\frac{1}{2} \sqrt{1-\frac{1}{a^2 x^2}} x^2+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0360467, size = 49, normalized size = 0.78 \[ \frac{a x \sqrt{1-\frac{1}{a^2 x^2}} (a x+2)+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )}{2 a^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.123, size = 152, normalized size = 2.4 \begin{align*}{\frac{ax-1}{2\,{a}^{2}} \left ( \sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}} \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) a+2\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }+2\,a\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}}{\frac{1}{\sqrt{{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.982333, size = 173, normalized size = 2.75 \begin{align*} \frac{1}{2} \, a{\left (\frac{2 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 3 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{\frac{2 \,{\left (a x - 1\right )} a^{3}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{3}}{{\left (a x + 1\right )}^{2}} - a^{3}} + \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5686, size = 180, normalized size = 2.86 \begin{align*} \frac{{\left (a^{2} x^{2} + 3 \, a x + 2\right )} \sqrt{\frac{a x - 1}{a x + 1}} + \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{2 \, a^{2}} \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{x}{\sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16286, size = 159, normalized size = 2.52 \begin{align*} \frac{1}{2} \, a{\left (\frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a^{3}} - \frac{\log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{3}} - \frac{2 \,{\left (\frac{{\left (a x - 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - 3 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{3}{\left (\frac{a x - 1}{a x + 1} - 1\right )}^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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