Optimal. Leaf size=68 \[ \frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
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Rubi [A] time = 0.108736, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6192, 6193} \[ \frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Rule 6192
Rule 6193
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{\coth ^{-1}(a x)} \sqrt{1-\frac{1}{a^2 x^2}} x \, dx}{\sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2} \int (1+a x) \, dx}{a \sqrt{1-\frac{1}{a^2 x^2}} x}\\ &=\frac{\sqrt{c-a^2 c x^2}}{a \sqrt{1-\frac{1}{a^2 x^2}}}+\frac{x \sqrt{c-a^2 c x^2}}{2 \sqrt{1-\frac{1}{a^2 x^2}}}\\ \end{align*}
Mathematica [A] time = 0.0192544, size = 41, normalized size = 0.6 \[ \frac{(a x+2) \sqrt{c-a^2 c x^2}}{2 a \sqrt{1-\frac{1}{a^2 x^2}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 44, normalized size = 0.7 \begin{align*}{\frac{x \left ( ax+2 \right ) }{2\,ax+2}\sqrt{-{a}^{2}c{x}^{2}+c}{\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{-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 = 1.55544, size = 46, normalized size = 0.68 \begin{align*} \frac{\sqrt{-a^{2} c}{\left (a x^{2} + 2 \, x\right )}}{2 \, 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{- c \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt{\frac{a x - 1}{a x + 1}}}\, 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}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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