Optimal. Leaf size=20 \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]
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Rubi [A] time = 0.0074811, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {31} \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a+\sqrt{-a} x} \, dx &=\frac{\log \left (a+\sqrt{-a} x\right )}{\sqrt{-a}}\\ \end{align*}
Mathematica [A] time = 0.0061207, size = 20, normalized size = 1. \[ \frac{\log \left (\sqrt{-a} x+a\right )}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 17, normalized size = 0.9 \begin{align*}{\ln \left ( a+x\sqrt{-a} \right ){\frac{1}{\sqrt{-a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06086, size = 22, normalized size = 1.1 \begin{align*} \frac{\log \left (\sqrt{-a} x + a\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42402, size = 42, normalized size = 2.1 \begin{align*} -\frac{\sqrt{-a} \log \left (x - \sqrt{-a}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.070181, size = 17, normalized size = 0.85 \begin{align*} \frac{\log{\left (a + x \sqrt{- a} \right )}}{\sqrt{- a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17921, size = 23, normalized size = 1.15 \begin{align*} \frac{\log \left ({\left | \sqrt{-a} x + a \right |}\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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