Optimal. Leaf size=44 \[ -\frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.221966, antiderivative size = 44, 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 = {6160, 6150, 37} \[ -\frac{(1-a x)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x^3} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1-a x}{x^3} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1-a x)^2}{2 x \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0202038, size = 44, normalized size = 1. \[ \frac{x \left (\frac{a}{x}-\frac{1}{2 x^2}\right ) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.078, size = 57, normalized size = 1.3 \begin{align*} -{\frac{2\,ax-1}{ \left ( 2\,ax+2 \right ) \left ( ax-1 \right ) x}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{-{a}^{2}{x}^{2}+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{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11657, size = 132, normalized size = 3. \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a - 1\right )} x^{2} - 2 \, a x + 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{3} - x\right )}} \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{\sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )}}{x^{2} \left (a x + 1\right )}\, dx \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{-a^{2} x^{2} + 1} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{{\left (a x + 1\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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