Integrand size = 18, antiderivative size = 25 \[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\frac {2^x \sqrt {a-2^{-2 x} b}}{a \log (2)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2281, 197} \[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\frac {2^x \sqrt {a-b 2^{-2 x}}}{a \log (2)} \]
[In]
[Out]
Rule 197
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\frac {2^{-x} \left (2^{2 x} a-b\right )}{a \sqrt {a-2^{-2 x} b} \log (2)} \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76
method | result | size |
risch | \(\frac {\left (a 2^{2 x}-b \right ) 2^{-x}}{\sqrt {\left (a 2^{2 x}-b \right ) 2^{-2 x}}\, a \ln \left (2\right )}\) | \(44\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\frac {2^{x} \sqrt {\frac {2^{2 \, x} a - b}{2^{2 \, x}}}}{a \log \left (2\right )} \]
[In]
[Out]
\[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\int \frac {2^{x}}{\sqrt {a - 4^{- x} b}}\, dx \]
[In]
[Out]
none
Time = 0.31 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\frac {\sqrt {2^{2 \, x} a - b}}{a \log \left (2\right )} \]
[In]
[Out]
\[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\int { \frac {2^{x}}{\sqrt {a - \frac {b}{4^{x}}}} \,d x } \]
[In]
[Out]
Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {2^x}{\sqrt {a-4^{-x} b}} \, dx=\frac {2^x\,\sqrt {a-\frac {b}{2^{2\,x}}}}{a\,\ln \left (2\right )} \]
[In]
[Out]