Optimal. Leaf size=30 \[ \frac{2^x}{b \log (2)}-\frac{a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0637345, 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 \[ \frac{2^x}{b \log (2)}-\frac{a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]
Antiderivative was successfully verified.
[In] Int[4^x/(a + 2^x*b),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a \log{\left (2^{x} b + a \right )}}{b^{2} \log{\left (2 \right )}} + \frac{\int ^{2^{x}} \frac{1}{b}\, dx}{\log{\left (2 \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(4**x/(a+2**x*b),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0303491, size = 28, normalized size = 0.93 \[ \frac{b 2^x-a \log \left (\frac{b 2^x}{a}+1\right )}{b^2 \log (2)} \]
Antiderivative was successfully verified.
[In] Integrate[4^x/(a + 2^x*b),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.023, size = 35, normalized size = 1.2 \[{\frac{{{\rm e}^{x\ln \left ( 2 \right ) }}}{\ln \left ( 2 \right ) b}}-{\frac{a\ln \left ( a+{{\rm e}^{x\ln \left ( 2 \right ) }}b \right ) }{\ln \left ( 2 \right ){b}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(4^x/(a+2^x*b),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.855279, size = 41, normalized size = 1.37 \[ \frac{2^{x}}{b \log \left (2\right )} - \frac{a \log \left (2^{x} b + a\right )}{b^{2} \log \left (2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4^x/(2^x*b + a),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.251235, size = 34, normalized size = 1.13 \[ \frac{2^{x} b - a \log \left (2^{x} b + a\right )}{b^{2} \log \left (2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4^x/(2^x*b + a),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.168714, size = 31, normalized size = 1.03 \[ - \frac{a \log{\left (2^{x} + \frac{a}{b} \right )}}{b^{2} \log{\left (2 \right )}} + \begin{cases} \frac{2^{x}}{b \log{\left (2 \right )}} & \text{for}\: b \log{\left (2 \right )} \neq 0 \\\frac{x}{b} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4**x/(a+2**x*b),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{4^{x}}{2^{x} b + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(4^x/(2^x*b + a),x, algorithm="giac")
[Out]