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