Optimal. Leaf size=11 \[ \frac {\cosh ^{-1}\left (\frac {b x}{2}\right )}{b} \]
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Rubi [A] time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {52} \begin {gather*} \frac {\cosh ^{-1}\left (\frac {b x}{2}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
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] time = 0.00, size = 25, normalized size = 2.27 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x-2}}{\sqrt {b x+2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [B] time = 0.00, size = 25, normalized size = 2.27 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+2}}{\sqrt {b x-2}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.87, size = 26, normalized size = 2.36 \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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giac [B] time = 0.96, size = 23, normalized size = 2.09 \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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maple [B] time = 0.00, size = 57, normalized size = 5.18 \begin {gather*} \frac {\sqrt {\left (b x -2\right ) \left (b x +2\right )}\, \ln \left (\frac {b^{2} x}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}-4}\right )}{\sqrt {b x -2}\, \sqrt {b x +2}\, \sqrt {b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.36, size = 26, normalized size = 2.36 \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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mupad [B] time = 0.00, size = 50, normalized size = 4.55 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (-\sqrt {b\,x-2}+\sqrt {2}\,1{}\mathrm {i}\right )}{\left (\sqrt {2}-\sqrt {b\,x+2}\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 4.28, size = 75, normalized size = 6.82 \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 e^{2 i \pi }}{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}{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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