3.740 \(\int (2^{-x}+2^x) \, dx\)

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 = 0.0062042, 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, Rules used = {2194} \[ \frac{2^x}{\log (2)}-\frac{2^{-x}}{\log (2)} \]

Antiderivative was successfully verified.

[In]

Int[2^(-x) + 2^x,x]

[Out]

-(1/(2^x*Log[2])) + 2^x/Log[2]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps

\begin{align*} \int \left (2^{-x}+2^x\right ) \, dx &=\int 2^{-x} \, dx+\int 2^x \, dx\\ &=-\frac{2^{-x}}{\log (2)}+\frac{2^x}{\log (2)}\\ \end{align*}

Mathematica [A]  time = 0.0031962, 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 = 0.018, size = 21, normalized size = 1.1 \begin{align*} -{\frac{1}{{2}^{x}\ln \left ( 2 \right ) }}+{\frac{{2}^{x}}{\ln \left ( 2 \right ) }} \end{align*}

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 = 0.979951, size = 27, normalized size = 1.35 \begin{align*} \frac{2^{x}}{\log \left (2\right )} - \frac{1}{2^{x} \log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2^x/log(2) - 1/(2^x*log(2))

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Fricas [A]  time = 0.795417, size = 38, normalized size = 1.9 \begin{align*} \frac{2^{2 \, x} - 1}{2^{x} \log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2^(2*x) - 1)/(2^x*log(2))

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Sympy [A]  time = 0.113027, size = 17, normalized size = 0.85 \begin{align*} \frac{2^{x} \log{\left (2 \right )} - 2^{- x} \log{\left (2 \right )}}{\log{\left (2 \right )}^{2}} \end{align*}

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 [A]  time = 1.20912, size = 27, normalized size = 1.35 \begin{align*} \frac{2^{x}}{\log \left (2\right )} - \frac{1}{2^{x} \log \left (2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2^x/log(2) - 1/(2^x*log(2))