Optimal. Leaf size=69 \[ \frac{x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{a x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.0713259, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {6143, 6140} \[ \frac{x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{a x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6143
Rule 6140
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \sqrt{c-a^2 c x^2} \, dx &=\frac{\sqrt{c-a^2 c x^2} \int e^{-\tanh ^{-1}(a x)} \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-a^2 c x^2} \int (1-a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{x \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}}-\frac{a x^2 \sqrt{c-a^2 c x^2}}{2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0155949, size = 40, normalized size = 0.58 \[ \frac{\left (x-\frac{a x^2}{2}\right ) \sqrt{c-a^2 c x^2}}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 48, normalized size = 0.7 \begin{align*}{\frac{x \left ( ax-2 \right ) }{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) }\sqrt{-{a}^{2}c{x}^{2}+c}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00756, size = 65, normalized size = 0.94 \begin{align*} -\frac{{\left (a^{2} \sqrt{c} x^{2} - 2 \, a \sqrt{c} x + 2 \, \sqrt{c}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}{2 \,{\left (a^{3} x^{2} - a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.42799, size = 99, normalized size = 1.43 \begin{align*} \frac{\sqrt{-a^{2} c x^{2} + c} \sqrt{-a^{2} x^{2} + 1}{\left (a x^{2} - 2 \, x\right )}}{2 \,{\left (a^{2} x^{2} - 1\right )}} \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{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \left (a x - 1\right ) \left (a x + 1\right )}}{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{-a^{2} x^{2} + 1}}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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