Optimal. Leaf size=15 \[ -\frac {2 \sqrt {1-a x}}{a} \]
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Rubi [C] Result contains higher order function than in optimal. Order 3 vs. order 2 in
optimal.
time = 0.20, 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, 2332}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2332
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 {\text {Subst}\left (\int \left (-2 \log (-x)+\log \left (x^2\right )\right ) \, dx,x,\sqrt {-1+a x}\right )}{a \pi }\\ &=\frac {\text {Subst}\left (\int \log \left (x^2\right ) \, dx,x,\sqrt {-1+a x}\right )}{a \pi }-\frac {2 \text {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] Result contains higher order function than in optimal. Order 3 vs. order 2 in
optimal.
time = 0.02, size = 37, normalized size = 2.47 \begin {gather*} \frac {\sqrt {-1+a x} \left (-2 \log \left (-\sqrt {-1+a x}\right )+\log (-1+a x)\right )}{a \pi } \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.10, size = 42, normalized size = 2.80
method | result | size |
gosper | \(\frac {\sqrt {a x -1}\, \left (\ln \left (a x -1\right )-2 \ln \left (-\sqrt {a x -1}\right )\right )}{a \pi }\) | \(34\) |
derivativedivides | \(\frac {-2 \ln \left (-\sqrt {a x -1}\right ) \sqrt {a x -1}+\sqrt {a x -1}\, \ln \left (a x -1\right )}{\pi a}\) | \(42\) |
default | \(\frac {-2 \ln \left (-\sqrt {a x -1}\right ) \sqrt {a x -1}+\sqrt {a x -1}\, \ln \left (a x -1\right )}{\pi a}\) | \(42\) |
meijerg | \(\frac {i \sqrt {-\mathrm {signum}\left (a x -1\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {-a x +1}\right )}{\sqrt {\pi }\, \sqrt {\mathrm {signum}\left (a x -1\right )}\, a}\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (13) = 26\).
time = 0.26, size = 41, normalized size = 2.73 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 29.28, size = 42, normalized size = 2.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (13) = 26\).
time = 0.76, size = 41, normalized size = 2.73 \begin {gather*} \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 {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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