3.79.7 \(\int \frac {4+x}{8 x} \, dx\)

Optimal. Leaf size=13 \[ \frac {1}{8} (5+x+4 (8+\log (x))) \]

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {12, 43} \begin {gather*} \frac {x}{8}+\frac {\log (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + x)/(8*x),x]

[Out]

x/8 + Log[x]/2

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {4+x}{x} \, dx\\ &=\frac {1}{8} \int \left (1+\frac {4}{x}\right ) \, dx\\ &=\frac {x}{8}+\frac {\log (x)}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.77 \begin {gather*} \frac {1}{8} (x+4 \log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + x)/(8*x),x]

[Out]

(x + 4*Log[x])/8

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fricas [A]  time = 1.36, size = 8, normalized size = 0.62 \begin {gather*} \frac {1}{8} \, x + \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(4+x)/x,x, algorithm="fricas")

[Out]

1/8*x + 1/2*log(x)

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giac [A]  time = 0.36, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{8} \, x + \frac {1}{2} \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(4+x)/x,x, algorithm="giac")

[Out]

1/8*x + 1/2*log(abs(x))

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maple [A]  time = 0.02, size = 9, normalized size = 0.69

method	result	size
default	\(\frac {\ln \relax (x )}{2}+\frac {x}{8}\)	\(9\)
norman	\(\frac {\ln \relax (x )}{2}+\frac {x}{8}\)	\(9\)
risch	\(\frac {\ln \relax (x )}{2}+\frac {x}{8}\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*(4+x)/x,x,method=_RETURNVERBOSE)

[Out]

1/2*ln(x)+1/8*x

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maxima [A]  time = 0.36, size = 8, normalized size = 0.62 \begin {gather*} \frac {1}{8} \, x + \frac {1}{2} \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(4+x)/x,x, algorithm="maxima")

[Out]

1/8*x + 1/2*log(x)

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mupad [B]  time = 4.99, size = 8, normalized size = 0.62 \begin {gather*} \frac {x}{8}+\frac {\ln \relax (x)}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x/8 + 1/2)/x,x)

[Out]

x/8 + log(x)/2

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sympy [A]  time = 0.07, size = 7, normalized size = 0.54 \begin {gather*} \frac {x}{8} + \frac {\log {\relax (x )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*(4+x)/x,x)

[Out]

x/8 + log(x)/2

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