Optimal. Leaf size=25 \[ \frac {2^x \sqrt {a-2^{-2 x} b}}{a \log (2)} \]
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Rubi [A]
time = 0.04, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2281, 197}
\begin {gather*} \frac {2^x \sqrt {a-b 2^{-2 x}}}{a \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 2281
Rubi steps
\begin {align*} \int \frac {2^x}{\sqrt {a-2^{-2 x} b}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b}{x^2}}} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {2^x \sqrt {a-2^{-2 x} b}}{a \log (2)}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 38, normalized size = 1.52 \begin {gather*} \frac {2^{-x} \left (2^{2 x} a-b\right )}{a \sqrt {a-2^{-2 x} b} \log (2)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 44, normalized size = 1.76
method | result | size |
risch | \(\frac {\left (a 2^{2 x}-b \right ) 2^{-x}}{\sqrt {\left (a 2^{2 x}-b \right ) 2^{-2 x}}\, a \ln \left (2\right )}\) | \(44\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 25, normalized size = 1.00 \begin {gather*} \frac {2^{x} \sqrt {a - \frac {b}{2^{2 \, x}}}}{a \log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 32, normalized size = 1.28 \begin {gather*} \frac {2^{x} \sqrt {\frac {2^{2 \, x} a - b}{2^{2 \, x}}}}{a \log \left (2\right )} \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 {2^{x}}{\sqrt {a - 2^{- 2 x} b}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 33, normalized size = 1.32 \begin {gather*} \frac {\frac {\sqrt {2^{2 \, x} a - b}}{a} - \frac {\sqrt {-b}}{a}}{\log \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.51, size = 25, normalized size = 1.00 \begin {gather*} \frac {2^x\,\sqrt {a-\frac {b}{2^{2\,x}}}}{a\,\ln \left (2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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