3.48.62 \(\int \frac {4+\log (x)}{x^2} \, dx\) [4762]

Optimal. Leaf size=11 \[ \frac {-5+x-\log (x)}{x} \]

[Out]

(-ln(x)+x-5)/x

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.36, number of steps used = 1, number of rules used = 1, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2341} \begin {gather*} -\frac {1}{x}-\frac {\log (x)+4}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4 + Log[x])/x^2,x]

[Out]

-x^(-1) - (4 + Log[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 {4+\log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 13, normalized size = 1.18 \begin {gather*} -\frac {5}{x}-\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + Log[x])/x^2,x]

[Out]

-5/x - Log[x]/x

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Maple [A]
time = 0.15, size = 14, normalized size = 1.27

method	result	size
norman	\(\frac {-5-\ln \left (x \right )}{x}\)	\(11\)
default	\(-\frac {\ln \left (x \right )}{x}-\frac {5}{x}\)	\(14\)
risch	\(-\frac {\ln \left (x \right )}{x}-\frac {5}{x}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)+4)/x^2,x,method=_RETURNVERBOSE)

[Out]

-ln(x)/x-5/x

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Maxima [A]
time = 0.29, size = 15, normalized size = 1.36 \begin {gather*} -\frac {\log \left (x\right ) + 1}{x} - \frac {4}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)+4)/x^2,x, algorithm="maxima")

[Out]

-(log(x) + 1)/x - 4/x

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Fricas [A]
time = 0.34, size = 9, normalized size = 0.82 \begin {gather*} -\frac {\log \left (x\right ) + 5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)+4)/x^2,x, algorithm="fricas")

[Out]

-(log(x) + 5)/x

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Sympy [A]
time = 0.03, size = 8, normalized size = 0.73 \begin {gather*} - \frac {\log {\left (x \right )}}{x} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)+4)/x**2,x)

[Out]

-log(x)/x - 5/x

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Giac [A]
time = 0.41, size = 13, normalized size = 1.18 \begin {gather*} -\frac {\log \left (x\right )}{x} - \frac {5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)+4)/x^2,x, algorithm="giac")

[Out]

-log(x)/x - 5/x

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Mupad [B]
time = 3.55, size = 9, normalized size = 0.82 \begin {gather*} -\frac {\ln \left (x\right )+5}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x) + 4)/x^2,x)

[Out]

-(log(x) + 5)/x

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