Optimal. Leaf size=38 \[ \frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{1}{x}\right )-\frac{1}{2} a^2 \csc ^{-1}(a x) \]
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Rubi [A] time = 0.0308692, antiderivative size = 38, 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, 780, 216} \[ \frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{1}{x}\right )-\frac{1}{2} a^2 \csc ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 6169
Rule 780
Rule 216
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{x^3} \, dx &=-\operatorname{Subst}\left (\int \frac{x \left (1+\frac{x}{a}\right )}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{1}{x}\right )-\frac{1}{2} a \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{1}{2} a \sqrt{1-\frac{1}{a^2 x^2}} \left (2 a+\frac{1}{x}\right )-\frac{1}{2} a^2 \csc ^{-1}(a x)\\ \end{align*}
Mathematica [A] time = 0.0419218, size = 42, normalized size = 1.11 \[ \frac{a \left (\sqrt{1-\frac{1}{a^2 x^2}} (2 a x+1)-a x \sin ^{-1}\left (\frac{1}{a x}\right )\right )}{2 x} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.126, size = 257, normalized size = 6.8 \begin{align*} -{\frac{ax-1}{2\,{x}^{2}} \left ( -2\,\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{3}{a}^{3}+2\,\sqrt{{a}^{2}} \left ({a}^{2}{x}^{2}-1 \right ) ^{3/2}xa+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}{x}^{2}{a}^{2}+2\,\ln \left ({\frac{{a}^{2}x+\sqrt{{a}^{2}{x}^{2}-1}\sqrt{{a}^{2}}}{\sqrt{{a}^{2}}}} \right ){x}^{2}{a}^{3}+{a}^{2}{x}^{2}\sqrt{{a}^{2}}\arctan \left ({\frac{1}{\sqrt{{a}^{2}{x}^{2}-1}}} \right ) -2\,\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}^{2}{a}^{3}+ \left ({a}^{2}{x}^{2}-1 \right ) ^{{\frac{3}{2}}}\sqrt{{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.
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Maxima [B] time = 1.52503, size = 123, normalized size = 3.24 \begin{align*}{\left (a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{a \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 3 \, a \sqrt{\frac{a x - 1}{a x + 1}}}{\frac{2 \,{\left (a x - 1\right )}}{a x + 1} + \frac{{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.72788, size = 144, normalized size = 3.79 \begin{align*} \frac{2 \, a^{2} x^{2} \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) +{\left (2 \, a^{2} x^{2} + 3 \, a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{2 \, x^{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{1}{x^{3} \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 = 1.13562, size = 117, normalized size = 3.08 \begin{align*}{\left (a \arctan \left (\sqrt{\frac{a x - 1}{a x + 1}}\right ) + \frac{\frac{{\left (a x - 1\right )} a \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} + 3 \, a \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (\frac{a x - 1}{a x + 1} + 1\right )}^{2}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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