3.22.38 \(\int \frac {1}{9} (9-2^{5-x} \log (2)) \, dx\) [2138]

Optimal. Leaf size=13 \[ \frac {2^{5-x}}{9}+x \]

[Out]

x+2/9*exp((4-x)*ln(2))

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {12, 2225} \begin {gather*} x+\frac {2^{5-x}}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(9 - 2^(5 - x)*Log[2])/9,x]

[Out]

2^(5 - x)/9 + x

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2225

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 {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \left (9-2^{5-x} \log (2)\right ) \, dx\\ &=x-\frac {1}{9} \log (2) \int 2^{5-x} \, dx\\ &=\frac {2^{5-x}}{9}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.03, size = 13, normalized size = 1.00 \begin {gather*} \frac {2^{5-x}}{9}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 - 2^(5 - x)*Log[2])/9,x]

[Out]

2^(5 - x)/9 + x

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Maple [A]
time = 0.05, size = 14, normalized size = 1.08

method	result	size
risch	\(x +\frac {2 \,2^{-x +4}}{9}\)	\(12\)
default	\(x +\frac {2 \,{\mathrm e}^{\left (-x +4\right ) \ln \left (2\right )}}{9}\)	\(14\)
norman	\(x +\frac {2 \,{\mathrm e}^{\left (-x +4\right ) \ln \left (2\right )}}{9}\)	\(14\)
derivativedivides	\(-\frac {-2 \ln \left (2\right ) {\mathrm e}^{\left (-x +4\right ) \ln \left (2\right )}+9 \ln \left ({\mathrm e}^{\left (-x +4\right ) \ln \left (2\right )}\right )}{9 \ln \left (2\right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-2/9*ln(2)*exp((-x+4)*ln(2))+1,x,method=_RETURNVERBOSE)

[Out]

x+2/9*exp((-x+4)*ln(2))

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Maxima [A]
time = 0.27, size = 11, normalized size = 0.85 \begin {gather*} \frac {1}{9} \cdot 2^{-x + 5} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/9*log(2)*exp((-x+4)*log(2))+1,x, algorithm="maxima")

[Out]

1/9*2^(-x + 5) + x

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Fricas [A]
time = 0.34, size = 11, normalized size = 0.85 \begin {gather*} \frac {2}{9} \cdot 2^{-x + 4} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/9*log(2)*exp((-x+4)*log(2))+1,x, algorithm="fricas")

[Out]

2/9*2^(-x + 4) + x

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Sympy [A]
time = 0.03, size = 12, normalized size = 0.92 \begin {gather*} x + \frac {2 e^{\left (4 - x\right ) \log {\left (2 \right )}}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/9*ln(2)*exp((-x+4)*ln(2))+1,x)

[Out]

x + 2*exp((4 - x)*log(2))/9

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Giac [A]
time = 0.42, size = 11, normalized size = 0.85 \begin {gather*} \frac {2}{9} \cdot 2^{-x + 4} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-2/9*log(2)*exp((-x+4)*log(2))+1,x, algorithm="giac")

[Out]

2/9*2^(-x + 4) + x

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Mupad [B]
time = 1.23, size = 9, normalized size = 0.69 \begin {gather*} x+\frac {32}{9\,2^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1 - (2*exp(-log(2)*(x - 4))*log(2))/9,x)

[Out]

x + 32/(9*2^x)

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