Optimal. Leaf size=22 \[ x-\frac{2 \log \left (2^x+1\right )}{\log (2)}+\frac{2^x}{\log (2)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0267354, antiderivative size = 22, 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 = {2282, 894} \[ x-\frac{2 \log \left (2^x+1\right )}{\log (2)}+\frac{2^x}{\log (2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2282
Rule 894
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*}
Mathematica [A] time = 0.0122895, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.027, size = 27, normalized size = 1.2 \begin{align*} x+{\frac{{{\rm e}^{x\ln \left ( 2 \right ) }}}{\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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.45288, size = 30, normalized size = 1.36 \begin{align*} x + \frac{2^{x}}{\log \left (2\right )} - \frac{2 \, \log \left (2^{x} + 1\right )}{\log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.651464, size = 57, normalized size = 2.59 \begin{align*} \frac{x \log \left (2\right ) + 2^{x} - 2 \, \log \left (2^{x} + 1\right )}{\log \left (2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.289959, size = 29, normalized size = 1.32 \begin{align*} x + \frac{e^{\frac{x \log{\left (4 \right )}}{2}}}{\log{\left (2 \right )}} - \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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{4^{x} + 1}{2^{x} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]