3.93.78 \(\int \frac {2+2 x}{x} \, dx\)

Optimal. Leaf size=20 \[ \log (2)+\log \left (\frac {4}{25} e^{\frac {2}{5}+2 x} x^2\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 8, normalized size of antiderivative = 0.40, number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {43} \begin {gather*} 2 x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 2*x)/x,x]

[Out]

2*x + 2*Log[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} &=\int \left (2+\frac {2}{x}\right ) \, dx\\ &=2 x+2 \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 8, normalized size = 0.40 \begin {gather*} 2 x+2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 2*x)/x,x]

[Out]

2*x + 2*Log[x]

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fricas [A]  time = 0.91, size = 8, normalized size = 0.40 \begin {gather*} 2 \, x + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 2*log(x)

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giac [A]  time = 0.15, size = 9, normalized size = 0.45 \begin {gather*} 2 \, x + 2 \, \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 2*log(abs(x))

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maple [A]  time = 0.01, size = 9, normalized size = 0.45

method	result	size
default	\(2 x +2 \ln \relax (x )\)	\(9\)
norman	\(2 x +2 \ln \relax (x )\)	\(9\)
risch	\(2 x +2 \ln \relax (x )\)	\(9\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+2)/x,x,method=_RETURNVERBOSE)

[Out]

2*x+2*ln(x)

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maxima [A]  time = 0.36, size = 8, normalized size = 0.40 \begin {gather*} 2 \, x + 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x + 2*log(x)

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mupad [B]  time = 0.02, size = 8, normalized size = 0.40 \begin {gather*} 2\,x+2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 2)/x,x)

[Out]

2*x + 2*log(x)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.35 \begin {gather*} 2 x + 2 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x+2)/x,x)

[Out]

2*x + 2*log(x)

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