Optimal. Leaf size=29 \[ \frac{2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt{c-a c x}} \]
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Rubi [A] time = 0.0378088, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {6174} \[ \frac{2 (a x+1) e^{-\coth ^{-1}(a x)}}{a \sqrt{c-a c x}} \]
Antiderivative was successfully verified.
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Rule 6174
Rubi steps
\begin{align*} \int \frac{e^{-\coth ^{-1}(a x)}}{\sqrt{c-a c x}} \, dx &=\frac{2 e^{-\coth ^{-1}(a x)} (1+a x)}{a \sqrt{c-a c x}}\\ \end{align*}
Mathematica [A] time = 0.0226005, size = 28, normalized size = 0.97 \[ \frac{2 x \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-a c x}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.041, size = 35, normalized size = 1.2 \begin{align*} 2\,{\frac{ax+1}{a\sqrt{-acx+c}}\sqrt{{\frac{ax-1}{ax+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10366, size = 39, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (a \sqrt{-c} x + \sqrt{-c}\right )}}{\sqrt{a x + 1} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.88169, size = 99, normalized size = 3.41 \begin{align*} -\frac{2 \, \sqrt{-a c x + c}{\left (a x + 1\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c x - a 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 )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13755, size = 66, normalized size = 2.28 \begin{align*} -\frac{2 \,{\left (\frac{\sqrt{2} \sqrt{-c}}{a} + \frac{{\left (-a c x - c\right )}^{\frac{3}{2}}}{{\left (a c x + c\right )} a}\right )}{\left | c \right |} \mathrm{sgn}\left (a x + 1\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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