Optimal. Leaf size=27 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]
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Rubi [A] time = 0.0081858, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {63, 206} \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{1-a x} (1+a x)} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1-a x}\right )}{a}\\ &=-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a}\\ \end{align*}
Mathematica [A] time = 0.0069099, size = 27, normalized size = 1. \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-a x}}{\sqrt{2}}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 23, normalized size = 0.9 \begin{align*} -{\frac{\sqrt{2}}{a}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-ax+1}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.67271, size = 53, normalized size = 1.96 \begin{align*} \frac{\sqrt{2} \log \left (-\frac{\sqrt{2} - \sqrt{-a x + 1}}{\sqrt{2} + \sqrt{-a x + 1}}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78686, size = 90, normalized size = 3.33 \begin{align*} \frac{\sqrt{2} \log \left (\frac{a x + 2 \, \sqrt{2} \sqrt{-a x + 1} - 3}{a x + 1}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.7172, size = 65, normalized size = 2.41 \begin{align*} \begin{cases} \frac{2 \left (\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2}}{\sqrt{- a x + 1}} \right )}}{2} & \text{for}\: \frac{1}{- a x + 1} > \frac{1}{2} \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2}}{\sqrt{- a x + 1}} \right )}}{2} & \text{for}\: \frac{1}{- a x + 1} < \frac{1}{2} \end{cases}\right )}{a} & \text{for}\: a \neq 0 \\x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14819, size = 57, normalized size = 2.11 \begin{align*} \frac{\sqrt{2} \log \left (\frac{{\left | -2 \, \sqrt{2} + 2 \, \sqrt{-a x + 1} \right |}}{2 \,{\left (\sqrt{2} + \sqrt{-a x + 1}\right )}}\right )}{2 \, a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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