Optimal. Leaf size=24 \[ a \sqrt{1-\frac{1}{a^2 x^2}}-a \csc ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0247176, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6169, 641, 216} \[ a \sqrt{1-\frac{1}{a^2 x^2}}-a \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6169
Rule 641
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{x^2} \, dx &=-\operatorname{Subst}\left (\int \frac{1+\frac{x}{a}}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=a \sqrt{1-\frac{1}{a^2 x^2}}-\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=a \sqrt{1-\frac{1}{a^2 x^2}}-a \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0192168, size = 27, normalized size = 1.12 \[ a \left (\sqrt{1-\frac{1}{a^2 x^2}}-\sin ^{-1}\left (\frac{1}{a x}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.13, size = 220, normalized size = 9.2 \begin{align*} -{\frac{ax-1}{x} \left ( -\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{2}}+\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 ) x{a}^{2}+ax\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 ) }xa-\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{a}^{2} \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 = 1.54295, size = 72, normalized size = 3. \begin{align*} 2 \, a{\left (\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\frac{a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.63493, size = 112, normalized size = 4.67 \begin{align*} \frac{2 \, a x \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) +{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{x} \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^{2} \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 = 1.15409, size = 72, normalized size = 3. \begin{align*} 2 \, a{\left (\frac{\sqrt{\frac{a x - 1}{a x + 1}}}{\frac{a x - 1}{a x + 1} + 1} + \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]