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

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

[Out]

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

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Rubi [A]  time = 0.0763111, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2^x}{a \log (2)}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} 2^x}{\sqrt{b}}\right )}{a^{3/2} \log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^x/(a + b/4^x),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.4097, size = 36, normalized size = 0.84 \[ \frac{2^{x}}{a \log{\left (2 \right )}} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{2^{x} \sqrt{a}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}} \log{\left (2 \right )}} \]

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)*atan(2**x*sqrt(a)/sqrt(b))/(a**(3/2)*log(2))

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Mathematica [C]  time = 0.0339928, 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 [B]  time = 0.045, size = 74, normalized size = 1.7 \[{\frac{{2}^{x}}{a\ln \left ( 2 \right ) }}+{\frac{1}{2\,{a}^{2}\ln \left ( 2 \right ) }\sqrt{-ab}\ln \left ({2}^{x}-{\frac{1}{a}\sqrt{-ab}} \right ) }-{\frac{1}{2\,{a}^{2}\ln \left ( 2 \right ) }\sqrt{-ab}\ln \left ({2}^{x}+{\frac{1}{a}\sqrt{-ab}} \right ) } \]

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 = 0.265477, 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 = 0.250115, 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)