Optimal. Leaf size=70 \[ \frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.146107, antiderivative size = 70, 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 = {6192, 6193, 43} \[ \frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\log (x) \sqrt{c-a^2 c x^2}}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2}}{x} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \frac{-1+a x}{x} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int \left (a-\frac{1}{x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-a^2 c x^2} \log (x)}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ \end{align*}
Mathematica [A] time = 0.0193746, size = 44, normalized size = 0.63 \[ \frac{\sqrt{c-a^2 c x^2} (a x-\log (x))}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.12, size = 46, normalized size = 0.7 \begin{align*} -{\frac{-ax+\ln \left ( x \right ) }{ax-1}\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }\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{-a^{2} c x^{2} + c} \sqrt{\frac{a x - 1}{a x + 1}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63677, size = 42, normalized size = 0.6 \begin{align*} \frac{\sqrt{-a^{2} c}{\left (a x - \log \left (x\right )\right )}}{a} \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{\frac{a x - 1}{a x + 1}} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}{x}\, 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} c x^{2} + c} \sqrt{\frac{a x - 1}{a x + 1}}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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