Optimal. Leaf size=19 \[ \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {b x-1}\right )}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {63, 215} \begin {gather*} \frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {b x-1}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 215
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+b x} \sqrt {3+b x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {4+x^2}} \, dx,x,\sqrt {-1+b x}\right )}{b}\\ &=\frac {2 \sinh ^{-1}\left (\frac {1}{2} \sqrt {-1+b x}\right )}{b}\\ \end {align*}
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Mathematica [B] time = 0.01, size = 39, normalized size = 2.05 \begin {gather*} \frac {2 \sqrt {b x-1} \sin ^{-1}\left (\frac {1}{2} \sqrt {1-b x}\right )}{b \sqrt {1-b x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 25, normalized size = 1.32 \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b x+3}}{\sqrt {b x-1}}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 27, normalized size = 1.42 \begin {gather*} -\frac {\log \left (-b x + \sqrt {b x + 3} \sqrt {b x - 1} - 1\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 23, normalized size = 1.21 \begin {gather*} -\frac {2 \, \log \left (\sqrt {b x + 3} - \sqrt {b x - 1}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 64, normalized size = 3.37 \begin {gather*} \frac {\sqrt {\left (b x -1\right ) \left (b x +3\right )}\, \ln \left (\frac {b^{2} x +b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 b x -3}\right )}{\sqrt {b x -1}\, \sqrt {b x +3}\, \sqrt {b^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.32, size = 33, normalized size = 1.74 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} + 2 \, b x - 3} b + 2 \, b\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 44, normalized size = 2.32 \begin {gather*} \frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {b\,x-1}-\mathrm {i}\right )}{\left (\sqrt {3}-\sqrt {b\,x+3}\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 [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b x - 1} \sqrt {b x + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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