Optimal. Leaf size=22 \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0434277, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6169, 844, 216, 266, 63, 208} \[ \tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6169
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{1+\frac{x}{a}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\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 )\\ &=-\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 )\\ &=-\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 )\\ &=-\csc ^{-1}(a x)+\tanh ^{-1}\left (\sqrt{1-\frac{1}{a^2 x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0131496, size = 36, normalized size = 1.64 \[ \log \left (x \left (\sqrt{\frac{a^2 x^2-1}{a^2 x^2}}+1\right )\right )-\sin ^{-1}\left (\frac{1}{a x}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.128, size = 132, normalized size = 6. \begin{align*} -{(ax-1) \left ( \sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}+\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) \sqrt{{a}^{2}}-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}}\sqrt{ \left ( ax-1 \right ) \left ( ax+1 \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.56288, size = 93, normalized size = 4.23 \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}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.53101, size = 150, normalized size = 6.82 \begin{align*} 2 \, \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \log \left (\sqrt{\frac{a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt{\frac{a x - 1}{a x + 1}} - 1\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{1}{x \sqrt{\frac{a x - 1}{a x + 1}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.17373, size = 95, normalized size = 4.32 \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}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]