Optimal. Leaf size=12 \[ -\frac{x e^{\text{sech}^{-1}(a x)}}{a} \]
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Rubi [B] time = 1.05158, antiderivative size = 26, normalized size of antiderivative = 2.17, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6725, 260, 6341, 1956, 74} \[ -\frac{\sqrt{1-a x}}{a^2 \sqrt{\frac{1}{a x+1}}} \]
Warning: Unable to verify antiderivative.
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Rule 6725
Rule 260
Rule 6341
Rule 1956
Rule 74
Rubi steps
\begin{align*} \int \frac{x \left (-1+a e^{\text{sech}^{-1}(a x)} x\right )}{1-a^2 x^2} \, dx &=\int \left (\frac{x}{-1+a^2 x^2}-\frac{a e^{\text{sech}^{-1}(a x)} x^2}{-1+a^2 x^2}\right ) \, dx\\ &=-\left (a \int \frac{e^{\text{sech}^{-1}(a x)} x^2}{-1+a^2 x^2} \, dx\right )+\int \frac{x}{-1+a^2 x^2} \, dx\\ &=\frac{\log \left (1-a^2 x^2\right )}{2 a^2}+\int \frac{x \sqrt{\frac{1}{1+a x}}}{\sqrt{1-a x}} \, dx-\int \frac{x}{-1+a^2 x^2} \, dx\\ &=\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{x}{\sqrt{1-a x} \sqrt{1+a x}} \, dx\\ &=-\frac{\sqrt{1-a x}}{a^2 \sqrt{\frac{1}{1+a x}}}\\ \end{align*}
Mathematica [B] time = 0.243834, size = 28, normalized size = 2.33 \[ -\frac{\sqrt{\frac{1-a x}{a x+1}} (a x+1)}{a^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.15, size = 36, normalized size = 3. \begin{align*} -{\frac{x}{a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}}} \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{{\left (a x{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04182, size = 69, normalized size = 5.75 \begin{align*} -\frac{x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}}{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*} - a \int \frac{x^{2} \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{a^{2} x^{2} - 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{{\left (a x{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )} - 1\right )} x}{a^{2} x^{2} - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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