Optimal. Leaf size=43 \[ -\frac{(a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}} \]
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Rubi [A] time = 0.21235, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {6160, 6150, 37} \[ -\frac{(a x+1)^2 \sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 6160
Rule 6150
Rule 37
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} \sqrt{c-\frac{c}{a^2 x^2}}}{x^2} \, dx &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{e^{\tanh ^{-1}(a x)} \sqrt{1-a^2 x^2}}{x^3} \, dx}{\sqrt{1-a^2 x^2}}\\ &=\frac{\left (\sqrt{c-\frac{c}{a^2 x^2}} x\right ) \int \frac{1+a x}{x^3} \, dx}{\sqrt{1-a^2 x^2}}\\ &=-\frac{\sqrt{c-\frac{c}{a^2 x^2}} (1+a x)^2}{2 x \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0185749, size = 42, normalized size = 0.98 \[ -\frac{(2 a x+1) \sqrt{c-\frac{c}{a^2 x^2}}}{2 x \sqrt{1-a^2 x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.091, size = 43, normalized size = 1. \begin{align*} -{\frac{2\,ax+1}{2\,x}\sqrt{{\frac{c \left ({a}^{2}{x}^{2}-1 \right ) }{{a}^{2}{x}^{2}}}}{\frac{1}{\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 = 1.15945, size = 27, normalized size = 0.63 \begin{align*} \frac{i \, \sqrt{c}}{x} + \frac{i \, \sqrt{c}}{2 \, a x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.8587, size = 134, normalized size = 3.12 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}{\left ({\left (2 \, a + 1\right )} x^{2} - 2 \, a x - 1\right )} \sqrt{\frac{a^{2} c x^{2} - c}{a^{2} x^{2}}}}{2 \,{\left (a^{2} x^{3} - x\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{- c \left (-1 + \frac{1}{a x}\right ) \left (1 + \frac{1}{a x}\right )} \left (a x + 1\right )}{x^{2} \sqrt{- \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{{\left (a x + 1\right )} \sqrt{c - \frac{c}{a^{2} x^{2}}}}{\sqrt{-a^{2} x^{2} + 1} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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