Optimal. Leaf size=26 \[ -\frac {\sqrt {1-a^2 x^2}}{a (a x+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a (a x+1)} \end {gather*}
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*}
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Mathematica [A] time = 0.01, size = 25, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a^2 x+a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.32, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {1-a^2 x^2}}{a (a x+1)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 28, normalized size = 1.08 \begin {gather*} -\frac {a x + \sqrt {-a^{2} x^{2} + 1} + 1}{a^{2} x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 34, normalized size = 1.31 \begin {gather*} \frac {2}{{\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} + 1\right )} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 22, normalized size = 0.85 \begin {gather*} \frac {a x -1}{\sqrt {-a^{2} x^{2}+1}\, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.97, size = 23, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1}}{a^{2} x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 23, normalized size = 0.88 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}}{x\,a^2+a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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