Optimal. Leaf size=67 \[ -\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{a x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.230661, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6160, 6150, 43} \[ -\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{a x \log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, 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^2} \, 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^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \left (\frac{1}{x^2}-\frac{a}{x}\right ) \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x \log (x)}{\sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0214352, size = 38, normalized size = 0.57 \[ -\frac{\sqrt{c-\frac{c}{a^2 x^2}} (a x \log (x)+1)}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.141, size = 51, normalized size = 0.8 \begin{align*}{\frac{a\ln \left ( x \right ) x+1}{{a}^{2}{x}^{2}-1}\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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.37658, size = 620, normalized size = 9.25 \begin{align*} \left [\frac{{\left (a^{2} x^{2} - 1\right )} \sqrt{-c} \log \left (\frac{a^{2} c x^{6} + a^{2} c x^{2} - c x^{4} -{\left (a x^{5} - a x\right )} \sqrt{-a^{2} x^{2} + 1} \sqrt{-c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}} - c}{a^{2} x^{4} - x^{2}}\right ) - 2 \, \sqrt{-a^{2} x^{2} + 1}{\left (x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{2} - 1\right )}}, \frac{{\left (a^{2} x^{2} - 1\right )} \sqrt{c} \arctan \left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} + a x\right )} \sqrt{c} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} c x^{4} -{\left (a^{2} + 1\right )} c x^{2} + c}\right ) - \sqrt{-a^{2} x^{2} + 1}{\left (x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{a^{2} x^{2} - 1}\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 \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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