Optimal. Leaf size=34 \[ \frac{2 \log \left (2^x+1\right )}{\log (2)}-\frac{2^x}{\log (2)}+\frac{2^{2 x-1}}{\log (2)} \]
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Rubi [A] time = 0.0312803, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2282, 697} \[ \frac{2 \log \left (2^x+1\right )}{\log (2)}-\frac{2^x}{\log (2)}+\frac{2^{2 x-1}}{\log (2)} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 697
Rubi steps
\begin{align*} \int \frac{1+4^x}{1+2^{-x}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1+x^2}{1+x} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac{\operatorname{Subst}\left (\int \left (-1+x+\frac{2}{1+x}\right ) \, dx,x,2^x\right )}{\log (2)}\\ &=-\frac{2^x}{\log (2)}+\frac{2^{-1+2 x}}{\log (2)}+\frac{2 \log \left (1+2^x\right )}{\log (2)}\\ \end{align*}
Mathematica [A] time = 0.0214097, size = 23, normalized size = 0.68 \[ \frac{2^x \left (2^x-2\right )+4 \log \left (2^x+1\right )}{\log (4)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 40, normalized size = 1.2 \begin{align*} -{\frac{{{\rm e}^{x\ln \left ( 2 \right ) }}}{\ln \left ( 2 \right ) }}+{\frac{ \left ({{\rm e}^{x\ln \left ( 2 \right ) }} \right ) ^{2}}{2\,\ln \left ( 2 \right ) }}+2\,{\frac{\ln \left ( 1+{{\rm e}^{x\ln \left ( 2 \right ) }} \right ) }{\ln \left ( 2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45459, size = 54, normalized size = 1.59 \begin{align*} 2 \, x - \frac{2^{2 \, x - 1}{\left (2^{-x + 1} - 1\right )}}{\log \left (2\right )} + \frac{2 \, \log \left (\frac{1}{2^{x}} + 1\right )}{\log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.843412, size = 63, normalized size = 1.85 \begin{align*} \frac{2^{2 \, x} - 2 \cdot 2^{x} + 4 \, \log \left (2^{x} + 1\right )}{2 \, \log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.316153, size = 42, normalized size = 1.24 \begin{align*} \frac{4^{x} \log{\left (2 \right )} - 2 e^{\frac{x \log{\left (4 \right )}}{2}} \log{\left (2 \right )}}{2 \log{\left (2 \right )}^{2}} + \frac{2 \log{\left (e^{\frac{x \log{\left (4 \right )}}{2}} + 1 \right )}}{\log{\left (2 \right )}} \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} + 1}{\frac{1}{2^{x}} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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