Optimal. Leaf size=15 \[ -\frac {2 \sqrt {1-a x}}{a} \]
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Rubi [C] time = 0.06, 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*}
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Mathematica [C] time = 0.03, 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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fricas [A] time = 0.85, size = 1, normalized size = 0.07 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.06, size = 41, normalized size = 2.73 \[ \frac {\sqrt {a x - 1} \log \left (a x - 1\right ) - 2 \, \sqrt {a x - 1} \log \left (-\sqrt {a x - 1}\right )}{\pi a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 34, normalized size = 2.27 \[ \frac {\sqrt {a x -1}\, \left (-2 \ln \left (-\sqrt {a x -1}\right )+\ln \left (a x -1\right )\right )}{\pi a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 41, normalized size = 2.73 \[ \frac {\sqrt {a x - 1} \log \left (a x - 1\right ) - 2 \, \sqrt {a x - 1} \log \left (-\sqrt {a x - 1}\right )}{\pi a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 43, normalized size = 2.87 \[ -\frac {2\,\ln \left (-\sqrt {a\,x-1}\right )\,\sqrt {a\,x-1}-\ln \left (a\,x-1\right )\,\sqrt {a\,x-1}}{\Pi \,a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 59.58, size = 42, normalized size = 2.80 \[ \frac {\begin {cases} \frac {- 2 \sqrt {a x - 1} \log {\left (- \sqrt {a x - 1} \right )} + \sqrt {a x - 1} \log {\left (a x - 1 \right )}}{a} & \text {for}\: a \neq 0 \\\pi x & \text {otherwise} \end {cases}}{\pi } \]
Verification of antiderivative is not currently implemented for this CAS.
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