Optimal. Leaf size=25 \[ \frac {2^x \sqrt {a-b 2^{-2 x}}}{a \log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, 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 = {2249, 191} \[ \frac {2^x \sqrt {a-b 2^{-2 x}}}{a \log (2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 191
Rule 2249
Rubi steps
\begin {align*} \int \frac {2^x}{\sqrt {a-2^{-2 x} b}} \, dx &=\frac {\operatorname {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*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 38, normalized size = 1.52 \[ \frac {2^{-x} \left (a 2^{2 x}-b\right )}{a \log (2) \sqrt {a-b 2^{-2 x}}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.42, size = 32, normalized size = 1.28 \[ \frac {2^{x} \sqrt {\frac {2^{2 \, x} a - b}{2^{2 \, x}}}}{a \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.41, size = 33, normalized size = 1.32 \[ \frac {\frac {\sqrt {2^{2 \, x} a - b}}{a} - \frac {\sqrt {-b}}{a}}{\log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 44, normalized size = 1.76 \[ \frac {\left (a 2^{2 x}-b \right ) 2^{-x}}{\sqrt {\left (a 2^{2 x}-b \right ) 2^{-2 x}}\, \ln \relax (2) a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.16, size = 25, normalized size = 1.00 \[ \frac {2^{x} \sqrt {a - \frac {b}{2^{2 \, x}}}}{a \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.51, size = 25, normalized size = 1.00 \[ \frac {2^x\,\sqrt {a-\frac {b}{2^{2\,x}}}}{a\,\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {2^{x}}{\sqrt {a - 2^{- 2 x} b}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________