Integrand size = 13, antiderivative size = 30 \[ \int \frac {4^x}{a+2^x b} \, dx=\frac {2^x}{b \log (2)}-\frac {a \log \left (a+2^x b\right )}{b^2 \log (2)} \]
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Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2280, 45} \[ \int \frac {4^x}{a+2^x b} \, dx=\frac {2^x}{b \log (2)}-\frac {a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
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Rule 45
Rule 2280
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x}{a+b x} \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {\text {Subst}\left (\int \left (\frac {1}{b}-\frac {a}{b (a+b x)}\right ) \, dx,x,2^x\right )}{\log (2)} \\ & = \frac {2^x}{b \log (2)}-\frac {a \log \left (a+2^x b\right )}{b^2 \log (2)} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {4^x}{a+2^x b} \, dx=\frac {\frac {2^x}{b}-\frac {a \log \left (a+2^x b\right )}{b^2}}{\log (2)} \]
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Time = 0.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {2^{x}}{b \ln \left (2\right )}-\frac {a \ln \left (2^{x}+\frac {a}{b}\right )}{\ln \left (2\right ) b^{2}}\) | \(33\) |
norman | \(\frac {{\mathrm e}^{x \ln \left (2\right )}}{\ln \left (2\right ) b}-\frac {a \ln \left (a +{\mathrm e}^{x \ln \left (2\right )} b \right )}{\ln \left (2\right ) b^{2}}\) | \(35\) |
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none
Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83 \[ \int \frac {4^x}{a+2^x b} \, dx=\frac {2^{x} b - a \log \left (2^{x} b + a\right )}{b^{2} \log \left (2\right )} \]
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Time = 0.15 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {4^x}{a+2^x b} \, dx=- \frac {a \log {\left (\frac {a}{b} + e^{\frac {x \log {\left (4 \right )}}{2}} \right )}}{b^{2} \log {\left (2 \right )}} + \begin {cases} \frac {e^{\frac {x \log {\left (4 \right )}}{2}}}{b \log {\left (2 \right )}} & \text {for}\: b \log {\left (2 \right )} \neq 0 \\\frac {x}{b} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {4^x}{a+2^x b} \, dx=\frac {2^{x}}{b \log \left (2\right )} - \frac {a \log \left (2^{x} b + a\right )}{b^{2} \log \left (2\right )} \]
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\[ \int \frac {4^x}{a+2^x b} \, dx=\int { \frac {4^{x}}{2^{x} b + a} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \frac {4^x}{a+2^x b} \, dx=-\frac {a\,\ln \left (a+2^x\,b\right )-2^x\,b}{b^2\,\ln \left (2\right )} \]
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