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

Optimal. Leaf size=32 \[ -\frac{a \log \left (a-b 2^x\right )}{b^2 \log (2)}-\frac{2^x}{b \log (2)} \]

[Out]

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

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Rubi [A]  time = 0.0605875, antiderivative size = 32, 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 \[ -\frac{a \log \left (a-b 2^x\right )}{b^2 \log (2)}-\frac{2^x}{b \log (2)} \]

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 = 0.00929551, size = 29, normalized size = 0.91 \[ -\frac{a \log \left (1-\frac{b 2^x}{a}\right )+b 2^x}{b^2 \log (2)} \]

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 = 0.015, size = 37, 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 = 0.875119, size = 45, normalized size = 1.41 \[ -\frac{2^{x}}{b \log \left (2\right )} - \frac{a \log \left (2^{x} b - a\right )}{b^{2} \log \left (2\right )} \]

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 = 0.268435, size = 36, normalized size = 1.12 \[ -\frac{2^{x} b + a \log \left (2^{x} b - a\right )}{b^{2} \log \left (2\right )} \]

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 = 0.18783, size = 34, normalized size = 1.06 \[ - \frac{a \log{\left (2^{x} - \frac{a}{b} \right )}}{b^{2} \log{\left (2 \right )}} + \begin{cases} - \frac{2^{x}}{b \log{\left (2 \right )}} & \text{for}\: b \log{\left (2 \right )} \neq 0 \\- \frac{x}{b} & \text{otherwise} \end{cases} \]

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 = 0.250169, size = 46, normalized size = 1.44 \[ -\frac{2^{x}}{b{\rm ln}\left (2\right )} - \frac{a{\rm ln}\left ({\left | 2^{x} b - a \right |}\right )}{b^{2}{\rm ln}\left (2\right )} \]

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))