3.723 \(\int \left (2^{-x}+2^x\right ) \, dx\)

Optimal. Leaf size=20 \[ \frac{2^x}{\log (2)}-\frac{2^{-x}}{\log (2)} \]

[Out]

-(1/(2^x*Log[2])) + 2^x/Log[2]

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Rubi [A]  time = 0.0129935, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2^x}{\log (2)}-\frac{2^{-x}}{\log (2)} \]

Antiderivative was successfully verified.

[In]  Int[2^(-x) + 2^x,x]

[Out]

-(1/(2^x*Log[2])) + 2^x/Log[2]

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Rubi in Sympy [A]  time = 1.00278, size = 14, normalized size = 0.7 \[ \frac{2^{x}}{\log{\left (2 \right )}} - \frac{2^{- x}}{\log{\left (2 \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(2**x)+2**x,x)

[Out]

2**x/log(2) - 2**(-x)/log(2)

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Mathematica [A]  time = 0.00626431, size = 20, normalized size = 1. \[ \frac{2^x}{\log (2)}-\frac{2^{-x}}{\log (2)} \]

Antiderivative was successfully verified.

[In]  Integrate[2^(-x) + 2^x,x]

[Out]

-(1/(2^x*Log[2])) + 2^x/Log[2]

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Maple [A]  time = 0.002, size = 21, normalized size = 1.1 \[ -{\frac{1}{{2}^{x}\ln \left ( 2 \right ) }}+{\frac{{2}^{x}}{\ln \left ( 2 \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(2^x)+2^x,x)

[Out]

-1/(2^x)/ln(2)+2^x/ln(2)

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Maxima [A]  time = 0.811416, size = 27, normalized size = 1.35 \[ \frac{2^{x}}{\log \left (2\right )} - \frac{1}{2^{x} \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2^x + 1/2^x,x, algorithm="maxima")

[Out]

2^x/log(2) - 1/(2^x*log(2))

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Fricas [A]  time = 0.255443, size = 23, normalized size = 1.15 \[ \frac{2^{2 \, x} - 1}{2^{x} \log \left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2^x + 1/2^x,x, algorithm="fricas")

[Out]

(2^(2*x) - 1)/(2^x*log(2))

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Sympy [A]  time = 0.104302, size = 17, normalized size = 0.85 \[ \frac{2^{x} \log{\left (2 \right )} - 2^{- x} \log{\left (2 \right )}}{\log{\left (2 \right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(2**x)+2**x,x)

[Out]

(2**x*log(2) - 2**(-x)*log(2))/log(2)**2

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GIAC/XCAS [A]  time = 0.229181, size = 27, normalized size = 1.35 \[ \frac{2^{x}}{{\rm ln}\left (2\right )} - \frac{1}{2^{x}{\rm ln}\left (2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(2^x + 1/2^x,x, algorithm="giac")

[Out]

2^x/ln(2) - 1/(2^x*ln(2))