Optimal. Leaf size=46 \[ -\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+\csc ^{-1}(a x) \]
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Rubi [A] time = 0.782756, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {6169, 6742, 216, 651, 266, 63, 208} \[ -\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )+\csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 6742
Rule 216
Rule 651
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{3 \coth ^{-1}(a x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\left (1+\frac{x}{a}\right )^2}{x \left (1-\frac{x}{a}\right ) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\operatorname{Subst}\left (\int \left (-\frac{1}{a \sqrt{1-\frac{x^2}{a^2}}}+\frac{4}{(a-x) \sqrt{1-\frac{x^2}{a^2}}}+\frac{1}{x \sqrt{1-\frac{x^2}{a^2}}}\right ) \, dx,x,\frac{1}{x}\right )\\ &=-\left (4 \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\right )+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}-\operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\csc ^{-1}(a x)-\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-\frac{x}{a^2}}} \, dx,x,\frac{1}{x^2}\right )\\ &=-\frac{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\csc ^{-1}(a x)+a^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{4 a \sqrt{1-\frac{1}{a^2 x^2}}}{a-\frac{1}{x}}+\csc ^{-1}(a x)+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.058624, size = 53, normalized size = 1.15 \[ -\frac{4 a x \sqrt{1-\frac{1}{a^2 x^2}}}{a x-1}+\log \left (x \left (\sqrt{1-\frac{1}{a^2 x^2}}+1\right )\right )+\sin ^{-1}\left (\frac{1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.173, size = 363, normalized size = 7.9 \begin{align*}{\frac{1}{ax+1} \left ( \ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ){x}^{2}{a}^{3}+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) +\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }{x}^{2}{a}^{2}-2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}{\sqrt{{a}^{2}}}} \right ) x{a}^{2}-2\,\sqrt{{a}^{2}}\sqrt{{a}^{2}{x}^{2}-1}xa-2\,ax\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) -2\, \left ( \left ( ax-1 \right ) \left ( ax+1 \right ) \right ) ^{3/2}\sqrt{{a}^{2}}-2\,\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }xa+a\ln \left ({ \left ({a}^{2}x+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}} \right ) +\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \sqrt{{a}^{2}}+\sqrt{{a}^{2}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) } \right ){\frac{1}{\sqrt{{a}^{2}}}}{\frac{1}{\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \right ) }}} \left ({\frac{ax-1}{ax+1}} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4833, size = 122, normalized size = 2.65 \begin{align*} -a{\left (\frac{2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a} + \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right )}{a} + \frac{4}{a \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.86066, size = 262, normalized size = 5.7 \begin{align*} -\frac{2 \,{\left (a x - 1\right )} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) -{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) +{\left (a x - 1\right )} \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\right ) + 4 \,{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a x - 1} \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{1}{x \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.20681, size = 123, normalized size = 2.67 \begin{align*} -a{\left (\frac{2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )}{a} - \frac{\log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right )}{a} + \frac{\log \left ({\left | \sqrt{\frac{a x - 1}{a x + 1}} - 1 \right |}\right )}{a} + \frac{4}{a \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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