Optimal. Leaf size=16 \[ -\frac {2 \sinh ^{-1}\left (\sqrt {1-b x}\right )}{b} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {65, 221}
\begin {gather*} -\frac {2 \sinh ^{-1}\left (\sqrt {1-b x}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 221
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-b x} \sqrt {2-b x}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {1-b x}\right )}{b}\\ &=-\frac {2 \sinh ^{-1}\left (\sqrt {1-b x}\right )}{b}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(16)=32\).
time = 0.04, size = 59, normalized size = 3.69 \begin {gather*} \frac {\log \left (-1+\frac {\sqrt {2-b x}}{\sqrt {1-b x}}\right )}{b}-\frac {\log \left (b+\frac {b \sqrt {2-b x}}{\sqrt {1-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. \(69\) vs.
\(2(14)=28\).
time = 0.16, size = 70, normalized size = 4.38
method | result | size |
default | \(\frac {\sqrt {\left (-b x +1\right ) \left (-b x +2\right )}\, \ln \left (\frac {-\frac {3}{2} b +b^{2} x}{\sqrt {b^{2}}}+\sqrt {x^{2} b^{2}-3 b x +2}\right )}{\sqrt {-b x +1}\, \sqrt {-b x +2}\, \sqrt {b^{2}}}\) | \(70\) |
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. 33 vs.
\(2 (14) = 28\).
time = 0.28, size = 33, normalized size = 2.06 \begin {gather*} \frac {\log \left (2 \, b^{2} x + 2 \, \sqrt {b^{2} x^{2} - 3 \, b x + 2} b - 3 \, 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. 30 vs.
\(2 (14) = 28\).
time = 1.49, size = 30, normalized size = 1.88 \begin {gather*} -\frac {\log \left (-2 \, b x + 2 \, \sqrt {-b x + 2} \sqrt {-b x + 1} + 3\right )}{b} \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 + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.07, size = 25, normalized size = 1.56 \begin {gather*} \frac {2 \, \log \left (\sqrt {-b x + 2} - \sqrt {-b x + 1}\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 45, normalized size = 2.81 \begin {gather*} -\frac {4\,\mathrm {atan}\left (\frac {b\,\left (\sqrt {2}-\sqrt {2-b\,x}\right )}{\left (\sqrt {1-b\,x}-1\right )\,\sqrt {-b^2}}\right )}{\sqrt {-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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