Optimal. Leaf size=12 \[ -\frac {\cosh ^{-1}\left (-\frac {b x}{2}\right )}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {54}
\begin {gather*} -\frac {\cosh ^{-1}\left (-\frac {b x}{2}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 54
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-2-b x} \sqrt {2-b x}} \, dx &=-\frac {\cosh ^{-1}\left (-\frac {b x}{2}\right )}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(12)=24\).
time = 0.04, size = 27, normalized size = 2.25 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {2-b x}}{\sqrt {-2-b x}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(60\) vs.
\(2(10)=20\).
time = 0.16, size = 61, normalized size = 5.08
method | result | size |
default | \(\frac {\sqrt {\left (-b x -2\right ) \left (-b x +2\right )}\, \ln \left (\frac {b^{2} x}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}-4}\right )}{\sqrt {-b x -2}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(61\) |
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. 26 vs.
\(2 (10) = 20\).
time = 0.27, size = 26, normalized size = 2.17 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 4} b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (10) = 20\).
time = 1.31, size = 28, normalized size = 2.33 \begin {gather*} -\frac {\log \left (-b x + \sqrt {-b x + 2} \sqrt {-b x - 2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 14.38, size = 78, normalized size = 6.50 \begin {gather*} - \frac {{G_{6, 6}^{6, 2}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} & \frac {1}{2}, \frac {1}{2}, 1, 1 \\0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 0 & \end {matrix} \middle | {\frac {4}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b} - \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 1 & \\- \frac {1}{4}, \frac {1}{4} & - \frac {1}{2}, 0, 0, 0 \end {matrix} \middle | {\frac {4 e^{- 2 i \pi }}{b^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} b} \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. 25 vs.
\(2 (10) = 20\).
time = 1.41, size = 25, normalized size = 2.08 \begin {gather*} \frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x - 2}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.29, size = 52, normalized size = 4.33 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {-b\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {2-b\,x}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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