3.103.97 \(\int \frac {2 \log (4)-\log (4) \log (x)}{x^2} \, dx\)

Optimal. Leaf size=13 \[ \frac {\log (4) (-1-6 x+\log (x))}{x} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.77, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2304} \begin {gather*} \frac {\log (4)}{x}-\frac {2 \log (4)-\log (4) \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2*Log[4] - Log[4]*Log[x])/x^2,x]

[Out]

Log[4]/x - (2*Log[4] - Log[4]*Log[x])/x

Rule 2304

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 {\log (4)}{x}-\frac {2 \log (4)-\log (4) \log (x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 1.23 \begin {gather*} -\frac {\log (4)}{x}+\frac {\log (4) \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*Log[4] - Log[4]*Log[x])/x^2,x]

[Out]

-(Log[4]/x) + (Log[4]*Log[x])/x

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fricas [A]  time = 0.59, size = 15, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (\log \relax (2) \log \relax (x) - \log \relax (2)\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(log(2)*log(x) - log(2))/x

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giac [A]  time = 0.20, size = 17, normalized size = 1.31 \begin {gather*} \frac {2 \, \log \relax (2) \log \relax (x)}{x} - \frac {2 \, \log \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(2)*log(x)/x - 2*log(2)/x

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maple [A]  time = 0.03, size = 16, normalized size = 1.23

method	result	size
norman	\(\frac {2 \ln \relax (2) \ln \relax (x )-2 \ln \relax (2)}{x}\)	\(16\)
risch	\(\frac {2 \ln \relax (2) \ln \relax (x )}{x}-\frac {2 \ln \relax (2)}{x}\)	\(18\)
default	\(-2 \ln \relax (2) \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )-\frac {4 \ln \relax (2)}{x}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*ln(2)*ln(x)-2*ln(2))/x

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maxima [A]  time = 0.35, size = 19, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (\log \relax (x) + 1\right )} \log \relax (2)}{x} - \frac {4 \, \log \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(log(x) + 1)*log(2)/x - 4*log(2)/x

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mupad [B]  time = 6.55, size = 11, normalized size = 0.85 \begin {gather*} \frac {2\,\ln \relax (2)\,\left (\ln \relax (x)-1\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*log(2) - 2*log(2)*log(x))/x^2,x)

[Out]

(2*log(2)*(log(x) - 1))/x

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sympy [A]  time = 0.12, size = 15, normalized size = 1.15 \begin {gather*} \frac {2 \log {\relax (2 )} \log {\relax (x )}}{x} - \frac {2 \log {\relax (2 )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(2)*log(x)/x - 2*log(2)/x

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