Optimal. Leaf size=71 \[ \frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{a x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.131313, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {6160, 6140} \[ \frac{x^2 \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}}-\frac{a x^3 \sqrt{c-\frac{c}{a^2 x^2}}}{2 \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6140
Rubi steps
\begin{align*} \int e^{-\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}} x \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int e^{-\tanh ^{-1}(a x)} \sqrt{1-a^2 x^2} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int (1-a x) \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\sqrt{c-\frac{c}{a^2 x^2}} x^2}{\sqrt{1-a^2 x^2}}-\frac{a \sqrt{c-\frac{c}{a^2 x^2}} x^3}{2 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0191384, size = 41, normalized size = 0.58 \[ \frac{x \left (x-\frac{a x^2}{2}\right ) \sqrt{c-\frac{c}{a^2 x^2}}}{\sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 56, normalized size = 0.8 \begin{align*}{\frac{ \left ( ax-2 \right ){x}^{2}}{ \left ( 2\,ax-2 \right ) \left ( ax+1 \right ) }\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}\sqrt{-{a}^{2}{x}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.12499, size = 69, normalized size = 0.97 \begin{align*} \frac{{\left (i \, a^{2} \sqrt{c} x^{2} - 2 i \, a \sqrt{c} x + 2 i \, \sqrt{c}\right )}{\left (a x + 1\right )}{\left (a x - 1\right )}}{2 \,{\left (a^{4} x^{2} - a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09807, size = 116, normalized size = 1.63 \begin{align*} \frac{\sqrt{-a^{2} x^{2} + 1}{\left (a x^{3} - 2 \, x^{2}\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{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{x \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )} \sqrt{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\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} x^{2} + 1} \sqrt{c - \frac{c}{a^{2} x^{2}}} x}{a x + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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