3.474 \(\int \frac{2^{2 x}}{a+2^x b} \, dx\)

Optimal. Leaf size=30 \[ \frac{2^x}{b \log (2)}-\frac{a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]

[Out]

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

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Rubi [A]  time = 0.0576705, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{2^x}{b \log (2)}-\frac{a \log \left (a+b 2^x\right )}{b^2 \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^(2*x)/(a + 2^x*b),x]

[Out]

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

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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(2**(2*x)/(a+2**x*b),x)

[Out]

-a*log(2**x*b + a)/(b**2*log(2)) + Integral(1/b, (x, 2**x))/log(2)

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Mathematica [A]  time = 0.00762424, 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[2^(2*x)/(a + 2^x*b),x]

[Out]

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

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Maple [A]  time = 0.012, 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(2^(2*x)/(a+2^x*b),x)

[Out]

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

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Maxima [A]  time = 0.851628, 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(2^(2*x)/(2^x*b + a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.259091, 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(2^(2*x)/(2^x*b + a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.169948, 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(2**(2*x)/(a+2**x*b),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.240235, size = 42, normalized size = 1.4 \[ \frac{2^{x}}{b{\rm ln}\left (2\right )} - \frac{a{\rm ln}\left ({\left | 2^{x} b + a \right |}\right )}{b^{2}{\rm ln}\left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2^x/(b*ln(2)) - a*ln(abs(2^x*b + a))/(b^2*ln(2))