Optimal. Leaf size=70 \[ \frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.250164, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6197, 6193, 43} \[ \frac{\log (x) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-\frac{1}{a^2 x^2}}}-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a x \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6197
Rule 6193
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x} \, 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} \, dx}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \frac{1+a x}{x^2} \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} \int \left (\frac{1}{x^2}+\frac{a}{x}\right ) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}}}{a \sqrt{1-\frac{1}{a^2 x^2}} x}+\frac{\sqrt{c-\frac{c}{a^2 x^2}} \log (x)}{\sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0235748, size = 43, normalized size = 0.61 \[ \frac{\sqrt{c-\frac{c}{a^2 x^2}} \left (a \log (x)-\frac{1}{x}\right )}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.225, size = 50, normalized size = 0.7 \begin{align*}{\frac{a\ln \left ( x \right ) x-1}{ax+1}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}{\frac{1}{\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}}}}{x \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58486, size = 51, normalized size = 0.73 \begin{align*} \frac{\sqrt{a^{2} c}{\left (a x \log \left (x\right ) - 1\right )}}{a^{2} x} \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}}}}{x \sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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