Optimal. Leaf size=22 \[ x-\frac {2 \log \left (2^x+1\right )}{\log (2)}+\frac {2^x}{\log (2)} \]
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Rubi [A] time = 0.03, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {2282, 894} \[ x-\frac {2 \log \left (2^x+1\right )}{\log (2)}+\frac {2^x}{\log (2)} \]
Antiderivative was successfully verified.
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Rule 894
Rule 2282
Rubi steps
\begin {align*} \int \frac {1+4^x}{1+2^x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1+x^2}{x (1+x)} \, dx,x,2^x\right )}{\log (2)}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {1}{x}-\frac {2}{1+x}\right ) \, dx,x,2^x\right )}{\log (2)}\\ &=x+\frac {2^x}{\log (2)}-\frac {2 \log \left (1+2^x\right )}{\log (2)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 21, normalized size = 0.95 \[ \frac {2^x+x \log (2)-2 \log \left (2^x+1\right )}{\log (2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 21, normalized size = 0.95 \[ \frac {x \log \relax (2) + 2^{x} - 2 \, \log \left (2^{x} + 1\right )}{\log \relax (2)} \]
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 \[ \int \frac {4^{x} + 1}{2^{x} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 27, normalized size = 1.23 \[ x +\frac {{\mathrm e}^{\ln \relax (2) x}}{\ln \relax (2)}-\frac {2 \ln \left ({\mathrm e}^{\ln \relax (2) x}+1\right )}{\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.91, size = 22, normalized size = 1.00 \[ x + \frac {2^{x}}{\log \relax (2)} - \frac {2 \, \log \left (2^{x} + 1\right )}{\log \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.48, size = 21, normalized size = 0.95 \[ \frac {x\,\ln \relax (2)-2\,\ln \left (2^x+1\right )+2^x}{\ln \relax (2)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 29, normalized size = 1.32 \[ x + \frac {e^{\frac {x \log {\relax (4 )}}{2}}}{\log {\relax (2 )}} - \frac {2 \log {\left (e^{\frac {x \log {\relax (4 )}}{2}} + 1 \right )}}{\log {\relax (2 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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