Optimal. Leaf size=38 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.146069, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {6192, 6193, 31} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{c-a^2 c x^2}} \, dx &=\frac{\left (\sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{e^{\coth ^{-1}(a x)}}{\sqrt{1-\frac{1}{a^2 x^2}} x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\left (a \sqrt{1-\frac{1}{a^2 x^2}} x\right ) \int \frac{1}{-1+a x} \, dx}{\sqrt{c-a^2 c x^2}}\\ &=\frac{\sqrt{1-\frac{1}{a^2 x^2}} x \log (1-a x)}{\sqrt{c-a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0186762, size = 38, normalized size = 1. \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (1-a x)}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.134, size = 51, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( ax-1 \right ) }{ca \left ( ax+1 \right ) }\sqrt{-c \left ({a}^{2}{x}^{2}-1 \right ) }{\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{1}{\sqrt{-a^{2} c x^{2} + c} \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 = 1.46157, size = 49, normalized size = 1.29 \begin{align*} -\frac{\sqrt{-a^{2} c} \log \left (a x - 1\right )}{a^{2} c} \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{1}{\sqrt{-a^{2} c x^{2} + c} \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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