Optimal. Leaf size=47 \[ \frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.246109, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6197, 6193, 37} \[ \frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 a x^2 \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}}}{x^2} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{-1+a x}{x^3} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}{2 a \sqrt{1-\frac{1}{a^2 x^2}} x^2}\\ \end{align*}
Mathematica [A] time = 0.0239413, size = 47, normalized size = 1. \[ \frac{\left (\frac{1}{2 x^2}-\frac{a}{x}\right ) \sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.121, size = 53, normalized size = 1.1 \begin{align*} -{\frac{2\,ax-1}{ \left ( 2\,ax-2 \right ) x}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}} \sqrt{\frac{a x - 1}{a x + 1}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54959, size = 54, normalized size = 1.15 \begin{align*} -\frac{\sqrt{a^{2} c}{\left (2 \, a x - 1\right )}}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a^{2} x^{2}}} \sqrt{\frac{a x - 1}{a x + 1}}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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