3.479 \(\int \frac{4^x}{a-2^{-x} b} \, dx\)

Optimal. Leaf size=58 \[ \frac{b^2 x}{a^3}+\frac{b^2 \log \left (a-b 2^{-x}\right )}{a^3 \log (2)}+\frac{b 2^x}{a^2 \log (2)}+\frac{2^{2 x-1}}{a \log (2)} \]

[Out]

(b^2*x)/a^3 + 2^(-1 + 2*x)/(a*Log[2]) + (2^x*b)/(a^2*Log[2]) + (b^2*Log[a - b/2^
x])/(a^3*Log[2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0963789, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{b^2 x}{a^3}+\frac{b^2 \log \left (a-b 2^{-x}\right )}{a^3 \log (2)}+\frac{b 2^x}{a^2 \log (2)}+\frac{2^{2 x-1}}{a \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[4^x/(a - b/2^x),x]

[Out]

(b^2*x)/a^3 + 2^(-1 + 2*x)/(a*Log[2]) + (2^x*b)/(a^2*Log[2]) + (b^2*Log[a - b/2^
x])/(a^3*Log[2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.5821, size = 58, normalized size = 1. \[ \frac{2^{2 x}}{2 a \log{\left (2 \right )}} + \frac{2^{x} b}{a^{2} \log{\left (2 \right )}} - \frac{b^{2} \log{\left (2^{- x} \right )}}{a^{3} \log{\left (2 \right )}} + \frac{b^{2} \log{\left (a - 2^{- x} b \right )}}{a^{3} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(4**x/(a-b/(2**x)),x)

[Out]

2**(2*x)/(2*a*log(2)) + 2**x*b/(a**2*log(2)) - b**2*log(2**(-x))/(a**3*log(2)) +
b**2*log(a - 2**(-x)*b)/(a**3*log(2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0461032, size = 40, normalized size = 0.69 \[ \frac{2 b^2 \log \left (1-\frac{a 2^x}{b}\right )+a 2^x \left (a 2^x+2 b\right )}{a^3 \log (4)} \]

Antiderivative was successfully verified.

[In]  Integrate[4^x/(a - b/2^x),x]

[Out]

(2^x*a*(2^x*a + 2*b) + 2*b^2*Log[1 - (2^x*a)/b])/(a^3*Log[4])

_______________________________________________________________________________________

Maple [A]  time = 0.016, size = 55, normalized size = 1. \[{\frac{{{\rm e}^{x\ln \left ( 2 \right ) }}b}{\ln \left ( 2 \right ){a}^{2}}}+{\frac{ \left ({{\rm e}^{x\ln \left ( 2 \right ) }} \right ) ^{2}}{2\,a\ln \left ( 2 \right ) }}+{\frac{{b}^{2}\ln \left ( a{{\rm e}^{x\ln \left ( 2 \right ) }}-b \right ) }{{a}^{3}\ln \left ( 2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(4^x/(a-b/(2^x)),x)

[Out]

1/a^2/ln(2)*b*exp(x*ln(2))+1/2/a/ln(2)*exp(x*ln(2))^2+1/a^3/ln(2)*b^2*ln(a*exp(x
*ln(2))-b)

_______________________________________________________________________________________

Maxima [A]  time = 0.873051, size = 78, normalized size = 1.34 \[ \frac{b^{2} x}{a^{3}} + \frac{{\left (2^{-x + 1} b + a\right )} 2^{2 \, x - 1}}{a^{2} \log \left (2\right )} + \frac{b^{2} \log \left (-a + \frac{b}{2^{x}}\right )}{a^{3} \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4^x/(a - b/2^x),x, algorithm="maxima")

[Out]

b^2*x/a^3 + (2^(-x + 1)*b + a)*2^(2*x - 1)/(a^2*log(2)) + b^2*log(-a + b/2^x)/(a
^3*log(2))

_______________________________________________________________________________________

Fricas [A]  time = 0.255121, size = 55, normalized size = 0.95 \[ \frac{2^{2 \, x} a^{2} + 2 \cdot 2^{x} a b + 2 \, b^{2} \log \left (2^{x} a - b\right )}{2 \, a^{3} \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4^x/(a - b/2^x),x, algorithm="fricas")

[Out]

1/2*(2^(2*x)*a^2 + 2*2^x*a*b + 2*b^2*log(2^x*a - b))/(a^3*log(2))

_______________________________________________________________________________________

Sympy [A]  time = 0.700311, size = 92, normalized size = 1.59 \[ \begin{cases} \frac{2^{2 x} a^{2} \log{\left (2 \right )} + 2 \cdot 2^{x} a b \log{\left (2 \right )}}{2 a^{3} \log{\left (2 \right )}^{2}} & \text{for}\: 2 a^{3} \log{\left (2 \right )}^{2} \neq 0 \\x \left (- \frac{b^{2}}{a^{3}} + \frac{a^{2} + a b + b^{2}}{a^{3}}\right ) & \text{otherwise} \end{cases} + \frac{b^{2} x}{a^{3}} + \frac{b^{2} \log{\left (- \frac{a}{b} + 2^{- x} \right )}}{a^{3} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4**x/(a-b/(2**x)),x)

[Out]

Piecewise(((2**(2*x)*a**2*log(2) + 2*2**x*a*b*log(2))/(2*a**3*log(2)**2), Ne(2*a
**3*log(2)**2, 0)), (x*(-b**2/a**3 + (a**2 + a*b + b**2)/a**3), True)) + b**2*x/
a**3 + b**2*log(-a/b + 2**(-x))/(a**3*log(2))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{4^{x}}{a - \frac{b}{2^{x}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(4^x/(a - b/2^x),x, algorithm="giac")

[Out]

integrate(4^x/(a - b/2^x), x)