Integrand size = 16, antiderivative size = 30 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=\frac {\text {arctanh}\left (\frac {2^x \sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2281, 214} \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {b} 2^x}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \]
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Rule 214
Rule 2281
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\tanh ^{-1}\left (\frac {2^x \sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=\frac {\text {arctanh}\left (\frac {2^x \sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \log (2)} \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\operatorname {arctanh}\left (\frac {b 2^{x}}{\sqrt {a b}}\right )}{\ln \left (2\right ) \sqrt {a b}}\) | \(22\) |
default | \(\frac {\operatorname {arctanh}\left (\frac {b 2^{x}}{\sqrt {a b}}\right )}{\ln \left (2\right ) \sqrt {a b}}\) | \(22\) |
risch | \(\frac {\ln \left (2^{x}+\frac {a}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, \ln \left (2\right )}-\frac {\ln \left (2^{x}-\frac {a}{\sqrt {a b}}\right )}{2 \sqrt {a b}\, \ln \left (2\right )}\) | \(49\) |
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none
Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.87 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=\left [\frac {\sqrt {a b} \log \left (\frac {2^{2 \, x} b + 2 \, \sqrt {a b} 2^{x} + a}{2^{2 \, x} b - a}\right )}{2 \, a b \log \left (2\right )}, -\frac {\sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{2^{x} b}\right )}{a b \log \left (2\right )}\right ] \]
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Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=\frac {\operatorname {RootSum} {\left (4 z^{2} a b - 1, \left ( i \mapsto i \log {\left (2^{x} + 2 i a \right )} \right )\right )}}{\log {\left (2 \right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=-\frac {\log \left (\frac {2^{x + 1} b - 2 \, \sqrt {a b}}{2^{x + 1} b + 2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} \log \left (2\right )} \]
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none
Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=-\frac {\arctan \left (\frac {2^{x} b}{\sqrt {-a b}}\right )}{\sqrt {-a b} \log \left (2\right )} \]
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Time = 0.42 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {2^x}{a-2^{2 x} b} \, dx=\frac {\mathrm {atanh}\left (\frac {2^x\,\sqrt {b}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}\,\ln \left (2\right )} \]
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