Optimal. Leaf size=15 \[ -\frac{2 \sqrt{1-a x}}{a} \]
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Rubi [C] time = 0.0558809, antiderivative size = 52, normalized size of antiderivative = 3.47, number of steps used = 5, number of rules used = 2, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.054, Rules used = {12, 2295} \[ \frac{\sqrt{a x-1} \log (a x-1)}{\pi a}-\frac{2 \sqrt{a x-1} \log \left (-\sqrt{a x-1}\right )}{\pi a} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2295
Rubi steps
\begin{align*} \int \frac{-2 \log \left (-\sqrt{-1+a x}\right )+\log (-1+a x)}{2 \pi \sqrt{-1+a x}} \, dx &=\frac{\int \frac{-2 \log \left (-\sqrt{-1+a x}\right )+\log (-1+a x)}{\sqrt{-1+a x}} \, dx}{2 \pi }\\ &=\frac{\operatorname{Subst}\left (\int \left (-2 \log (-x)+\log \left (x^2\right )\right ) \, dx,x,\sqrt{-1+a x}\right )}{a \pi }\\ &=\frac{\operatorname{Subst}\left (\int \log \left (x^2\right ) \, dx,x,\sqrt{-1+a x}\right )}{a \pi }-\frac{2 \operatorname{Subst}\left (\int \log (-x) \, dx,x,\sqrt{-1+a x}\right )}{a \pi }\\ &=-\frac{2 \sqrt{-1+a x} \log \left (-\sqrt{-1+a x}\right )}{a \pi }+\frac{\sqrt{-1+a x} \log (-1+a x)}{a \pi }\\ \end{align*}
Mathematica [C] time = 0.025668, size = 37, normalized size = 2.47 \[ \frac{\sqrt{a x-1} \left (\log (a x-1)-2 \log \left (-\sqrt{a x-1}\right )\right )}{\pi a} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.003, size = 34, normalized size = 2.3 \begin{align*}{\frac{1}{a\pi }\sqrt{ax-1} \left ( \ln \left ( ax-1 \right ) -2\,\ln \left ( -\sqrt{ax-1} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.947855, size = 55, normalized size = 3.67 \begin{align*} \frac{\sqrt{a x - 1} \log \left (a x - 1\right ) - 2 \, \sqrt{a x - 1} \log \left (-\sqrt{a x - 1}\right )}{\pi a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.14124, size = 4, normalized size = 0.27 \begin{align*} 0 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 57.8506, size = 42, normalized size = 2.8 \begin{align*} \frac{\begin{cases} \frac{- 2 \sqrt{a x - 1} \log{\left (- \sqrt{a x - 1} \right )} + \sqrt{a x - 1} \log{\left (a x - 1 \right )}}{a} & \text{for}\: a \neq 0 \\\pi x & \text{otherwise} \end{cases}}{\pi } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.07061, size = 55, normalized size = 3.67 \begin{align*} \frac{\sqrt{a x - 1} \log \left (a x - 1\right ) - 2 \, \sqrt{a x - 1} \log \left (-\sqrt{a x - 1}\right )}{\pi a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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