Optimal. Leaf size=30 \[ \frac{2^x}{b \log (2)}-\frac{a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
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Rubi [A] time = 0.0372135, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2248, 43} \[ \frac{2^x}{b \log (2)}-\frac{a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
Antiderivative was successfully verified.
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Rule 2248
Rule 43
Rubi steps
\begin{align*} \int \frac{4^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 = 0.0186802, size = 27, normalized size = 0.9 \[ \frac{\frac{2^x}{b}-\frac{a \log \left (a+b 2^x\right )}{b^2}}{\log (2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 35, 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 = 1.43986, size = 41, normalized size = 1.37 \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 = 1.56517, size = 55, normalized size = 1.83 \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 = 0.310819, size = 41, normalized size = 1.37 \begin{align*} - \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4^{x}}{2^{x} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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