Optimal. Leaf size=37 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}} \]
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Rubi [A] time = 0.152539, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6192, 6193, 31} \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}} \]
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 = 0.0193147, size = 37, normalized size = 1. \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \log (a x+1)}{\sqrt{c-a^2 c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.134, size = 51, normalized size = 1.4 \begin{align*} -{\frac{\ln \left ( ax+1 \right ) }{c \left ( ax-1 \right ) a}\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{\frac{a x - 1}{a x + 1}}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70503, size = 49, normalized size = 1.32 \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] 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 )}}\, 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{\frac{a x - 1}{a x + 1}}}{\sqrt{-a^{2} c x^{2} + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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