3.47.22 \(\int \frac {-5+\log (\frac {3}{x})}{x^2} \, dx\) [4622]

Optimal. Leaf size=14 \[ \frac {6-\log \left (\frac {3}{x}\right )}{x} \]

[Out]

5/x*(6/5-1/5*ln(3/x))

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.29, number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2341} \begin {gather*} \frac {1}{x}+\frac {5-\log \left (\frac {3}{x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + Log[3/x])/x^2,x]

[Out]

x^(-1) + (5 - Log[3/x])/x

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{x}+\frac {5-\log \left (\frac {3}{x}\right )}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 17, normalized size = 1.21 \begin {gather*} \frac {6}{x}-\frac {\log \left (\frac {3}{x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + Log[3/x])/x^2,x]

[Out]

6/x - Log[3/x]/x

________________________________________________________________________________________

Maple [A]
time = 0.20, size = 18, normalized size = 1.29

method	result	size
norman	\(\frac {6-\ln \left (\frac {3}{x}\right )}{x}\)	\(15\)
derivativedivides	\(\frac {6}{x}-\frac {\ln \left (\frac {3}{x}\right )}{x}\)	\(18\)
default	\(\frac {6}{x}-\frac {\ln \left (\frac {3}{x}\right )}{x}\)	\(18\)
risch	\(\frac {6}{x}-\frac {\ln \left (\frac {3}{x}\right )}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(3/x)-5)/x^2,x,method=_RETURNVERBOSE)

[Out]

6/x-1/x*ln(3/x)

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 17, normalized size = 1.21 \begin {gather*} -\frac {\log \left (\frac {3}{x}\right )}{x} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3/x)-5)/x^2,x, algorithm="maxima")

[Out]

-log(3/x)/x + 6/x

________________________________________________________________________________________

Fricas [A]
time = 0.39, size = 13, normalized size = 0.93 \begin {gather*} -\frac {\log \left (\frac {3}{x}\right ) - 6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3/x)-5)/x^2,x, algorithm="fricas")

[Out]

-(log(3/x) - 6)/x

________________________________________________________________________________________

Sympy [A]
time = 0.04, size = 8, normalized size = 0.57 \begin {gather*} - \frac {\log {\left (\frac {3}{x} \right )}}{x} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(3/x)-5)/x**2,x)

[Out]

-log(3/x)/x + 6/x

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 17, normalized size = 1.21 \begin {gather*} -\frac {\log \left (\frac {3}{x}\right )}{x} + \frac {6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(3/x)-5)/x^2,x, algorithm="giac")

[Out]

-log(3/x)/x + 6/x

________________________________________________________________________________________

Mupad [B]
time = 3.35, size = 13, normalized size = 0.93 \begin {gather*} -\frac {\ln \left (\frac {3}{x}\right )-6}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(3/x) - 5)/x^2,x)

[Out]

-(log(3/x) - 6)/x

________________________________________________________________________________________