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 = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 697
Rule 2282
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.40, size = 25, normalized size = 0.74 \[ \frac {2^{2 \, x} - 2 \cdot 2^{x} + 4 \, \log \left (2^{x} + 1\right )}{2 \, \log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {4^{x} + 1}{\frac {1}{2^{x}} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 40, normalized size = 1.18 \[ -\frac {{\mathrm e}^{\ln \relax (2) x}}{\ln \relax (2)}+\frac {{\mathrm e}^{2 \ln \relax (2) x}}{2 \ln \relax (2)}+\frac {2 \ln \left ({\mathrm e}^{\ln \relax (2) x}+1\right )}{\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.14, size = 40, normalized size = 1.18 \[ 2 \, x - \frac {2^{2 \, x - 1} {\left (2^{-x + 1} - 1\right )}}{\log \relax (2)} + \frac {2 \, \log \left (\frac {1}{2^{x}} + 1\right )}{\log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {4^x+1}{\frac {1}{2^x}+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.30, size = 39, normalized size = 1.15 \[ 2 x + \frac {2^{2 x} \log {\relax (2 )} - 2 \cdot 2^{x} \log {\relax (2 )}}{2 \log {\relax (2 )}^{2}} + \frac {2 \log {\left (1 + 2^{- x} \right )}}{\log {\relax (2 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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