Optimal. Leaf size=22 \[ \frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}} \]
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Rubi [A] time = 0.0023876, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {31} \[ \frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Rule 31
Rubi steps
\begin{align*} \int \frac{1}{a^2+\sqrt{-a} x} \, dx &=\frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}}\\ \end{align*}
Mathematica [A] time = 0.0036597, size = 22, normalized size = 1. \[ \frac{\log \left (a^2+\sqrt{-a} x\right )}{\sqrt{-a}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 19, normalized size = 0.9 \begin{align*}{\ln \left ({a}^{2}+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 = 1.02064, size = 24, normalized size = 1.09 \begin{align*} \frac{\log \left (a^{2} + \sqrt{-a} x\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47102, size = 46, normalized size = 2.09 \begin{align*} -\frac{\sqrt{-a} \log \left (-\sqrt{-a} a + x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.09407, size = 19, normalized size = 0.86 \begin{align*} \frac{\log{\left (a^{2} + 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 = 1.20031, size = 26, normalized size = 1.18 \begin{align*} \frac{\log \left ({\left | a^{2} + \sqrt{-a} x \right |}\right )}{\sqrt{-a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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