Optimal. Leaf size=32 \[ -\frac{a \log \left (a-b 2^x\right )}{b^2 \log (2)}-\frac{2^x}{b \log (2)} \]
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Rubi [A] time = 0.0354631, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2248, 43} \[ -\frac{a \log \left (a-b 2^x\right )}{b^2 \log (2)}-\frac{2^x}{b \log (2)} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{2^{2 x}}{a-2^x b} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{a-b x} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac{\operatorname{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*}
Mathematica [A] time = 0.0130584, size = 26, normalized size = 0.81 \[ -\frac{a \log \left (a-b 2^x\right )+b 2^x}{b^2 \log (2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 37, normalized size = 1.2 \begin{align*} -{\frac{{{\rm e}^{x\ln \left ( 2 \right ) }}}{\ln \left ( 2 \right ) b}}-{\frac{a\ln \left ( a-{{\rm e}^{x\ln \left ( 2 \right ) }}b \right ) }{\ln \left ( 2 \right ){b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967473, size = 45, normalized size = 1.41 \begin{align*} -\frac{2^{x}}{b \log \left (2\right )} - \frac{a \log \left (2^{x} b - a\right )}{b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59681, size = 57, normalized size = 1.78 \begin{align*} -\frac{2^{x} b + a \log \left (2^{x} b - a\right )}{b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.155308, size = 34, normalized size = 1.06 \begin{align*} - \frac{a \log{\left (2^{x} - \frac{a}{b} \right )}}{b^{2} \log{\left (2 \right )}} + \begin{cases} - \frac{2^{x}}{b \log{\left (2 \right )}} & \text{for}\: b \log{\left (2 \right )} \neq 0 \\- \frac{x}{b} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1736, size = 46, normalized size = 1.44 \begin{align*} -\frac{2^{x}}{b \log \left (2\right )} - \frac{a \log \left ({\left | 2^{x} b - a \right |}\right )}{b^{2} \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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