3.14.19 \(\int \frac {256}{1+256 x} \, dx\) [1319]

Optimal. Leaf size=8 \[ \log \left (e \left (\frac {1}{256}+x\right )\right ) \]

[Out]

ln((1/256+x)*exp(1))

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Rubi [A]
time = 0.00, antiderivative size = 6, normalized size of antiderivative = 0.75, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {12, 31} \begin {gather*} \log (256 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[256/(1 + 256*x),x]

[Out]

Log[1 + 256*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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=256 \int \frac {1}{1+256 x} \, dx\\ &=\log (1+256 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 6, normalized size = 0.75 \begin {gather*} \log (1+256 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[256/(1 + 256*x),x]

[Out]

Log[1 + 256*x]

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Maple [A]
time = 0.11, size = 7, normalized size = 0.88

method	result	size
default	\(\ln \left (256 x +1\right )\)	\(7\)
norman	\(\ln \left (256 x +1\right )\)	\(7\)
meijerg	\(\ln \left (256 x +1\right )\)	\(7\)
risch	\(\ln \left (256 x +1\right )\)	\(7\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(256/(256*x+1),x,method=_RETURNVERBOSE)

[Out]

ln(256*x+1)

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Maxima [A]
time = 0.26, size = 6, normalized size = 0.75 \begin {gather*} \log \left (256 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256/(256*x+1),x, algorithm="maxima")

[Out]

log(256*x + 1)

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Fricas [A]
time = 0.34, size = 6, normalized size = 0.75 \begin {gather*} \log \left (256 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256/(256*x+1),x, algorithm="fricas")

[Out]

log(256*x + 1)

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Sympy [A]
time = 0.01, size = 5, normalized size = 0.62 \begin {gather*} \log {\left (256 x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256/(256*x+1),x)

[Out]

log(256*x + 1)

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Giac [A]
time = 0.37, size = 7, normalized size = 0.88 \begin {gather*} \log \left ({\left | 256 \, x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(256/(256*x+1),x, algorithm="giac")

[Out]

log(abs(256*x + 1))

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Mupad [B]
time = 0.06, size = 4, normalized size = 0.50 \begin {gather*} \ln \left (x+\frac {1}{256}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(256/(256*x + 1),x)

[Out]

log(x + 1/256)

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