Optimal. Leaf size=26 \[ -\frac{\sqrt{1-a^2 x^2}}{a (a x+1)} \]
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Rubi [A] time = 0.0095005, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {651} \[ -\frac{\sqrt{1-a^2 x^2}}{a (a x+1)} \]
Antiderivative was successfully verified.
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Rule 651
Rubi steps
\begin{align*} \int \frac{1}{(1+a x) \sqrt{1-a^2 x^2}} \, dx &=-\frac{\sqrt{1-a^2 x^2}}{a (1+a x)}\\ \end{align*}
Mathematica [A] time = 0.0061963, size = 25, normalized size = 0.96 \[ -\frac{\sqrt{1-a^2 x^2}}{a^2 x+a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 22, normalized size = 0.9 \begin{align*}{\frac{ax-1}{a}{\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 [A] time = 1.47253, size = 31, normalized size = 1.19 \begin{align*} -\frac{\sqrt{-a^{2} x^{2} + 1}}{a^{2} x + a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54348, size = 61, normalized size = 2.35 \begin{align*} -\frac{a x + \sqrt{-a^{2} x^{2} + 1} + 1}{a^{2} 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*} \int \frac{1}{\sqrt{- \left (a x - 1\right ) \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 [A] time = 1.25429, size = 46, normalized size = 1.77 \begin{align*} \frac{2}{{\left (\frac{\sqrt{-a^{2} x^{2} + 1}{\left | a \right |} + a}{a^{2} x} + 1\right )}{\left | a \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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