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

Optimal. Leaf size=43 \[ \frac{2^x}{a \log (2)}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{a} 2^x}{\sqrt{b}}\right )}{a^{3/2} \log (2)} \]

[Out]

2^x/(a*Log[2]) - (Sqrt[b]*ArcTanh[(2^x*Sqrt[a])/Sqrt[b]])/(a^(3/2)*Log[2])

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Rubi [A]  time = 0.076921, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2^x}{a \log (2)}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{a} 2^x}{\sqrt{b}}\right )}{a^{3/2} \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^x/(a - b/4^x),x]

[Out]

2^x/(a*Log[2]) - (Sqrt[b]*ArcTanh[(2^x*Sqrt[a])/Sqrt[b]])/(a^(3/2)*Log[2])

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Rubi in Sympy [A]  time = 12.5044, size = 36, normalized size = 0.84 \[ \frac{2^{x}}{a \log{\left (2 \right )}} - \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{2^{x} \sqrt{a}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} \log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**x/(a*log(2)) - sqrt(b)*atanh(2**x*sqrt(a)/sqrt(b))/(a**(3/2)*log(2))

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Mathematica [C]  time = 0.0329019, size = 36, normalized size = 0.84 \[ -\frac{8^x \text{Hypergeometric2F1}\left (1,\frac{\log (8)}{\log (4)},\frac{\log (32)}{\log (4)},\frac{a 4^x}{b}\right )}{b \log (8)} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[2^x/(a - b/4^x),x]

[Out]

-((8^x*Hypergeometric2F1[1, Log[8]/Log[4], Log[32]/Log[4], (4^x*a)/b])/(b*Log[8]
))

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Maple [A]  time = 0.042, size = 70, normalized size = 1.6 \[{\frac{{2}^{x}}{a\ln \left ( 2 \right ) }}+{\frac{1}{2\,{a}^{2}\ln \left ( 2 \right ) }\sqrt{ab}\ln \left ({2}^{x}-{\frac{1}{a}\sqrt{ab}} \right ) }-{\frac{1}{2\,{a}^{2}\ln \left ( 2 \right ) }\sqrt{ab}\ln \left ({2}^{x}+{\frac{1}{a}\sqrt{ab}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(2^x/(a-b/(4^x)),x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.272439, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{\frac{b}{a}} \log \left (-\frac{2 \cdot 2^{x} a \sqrt{\frac{b}{a}} - 2^{2 \, x} a - b}{2^{2 \, x} a - b}\right ) + 2 \cdot 2^{x}}{2 \, a \log \left (2\right )}, -\frac{\sqrt{-\frac{b}{a}} \arctan \left (\frac{2^{x}}{\sqrt{-\frac{b}{a}}}\right ) - 2^{x}}{a \log \left (2\right )}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(b/a)*log(-(2*2^x*a*sqrt(b/a) - 2^(2*x)*a - b)/(2^(2*x)*a - b)) + 2*2^
x)/(a*log(2)), -(sqrt(-b/a)*arctan(2^x/sqrt(-b/a)) - 2^x)/(a*log(2))]

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Sympy [A]  time = 0.259781, size = 39, normalized size = 0.91 \[ \begin{cases} \frac{2^{x}}{a \log{\left (2 \right )}} & \text{for}\: a \log{\left (2 \right )} \neq 0 \\\frac{x}{a} & \text{otherwise} \end{cases} + \frac{\operatorname{RootSum}{\left (4 z^{2} a^{3} - b, \left ( i \mapsto i \log{\left (2^{x} - 2 i a \right )} \right )\right )}}{\log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2**x/(a-b/(4**x)),x)

[Out]

Piecewise((2**x/(a*log(2)), Ne(a*log(2), 0)), (x/a, True)) + RootSum(4*_z**2*a**
3 - b, Lambda(_i, _i*log(2**x - 2*_i*a)))/log(2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(2^x/(a - b/4^x), x)