3.103.97 \(\int \frac {2 \log (4)-\log (4) \log (x)}{x^2} \, dx\) [10297]

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

[Out]

2*ln(2)/x*(-6*x-1+ln(x))

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Rubi [A]
time = 0.02, 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 = {2341} \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 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 {\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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Maple [A]
time = 0.01, size = 26, normalized size = 2.00

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

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)/x-1/x)-4*ln(2)/x

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Maxima [A]
time = 0.25, size = 19, normalized size = 1.46 \begin {gather*} \frac {2 \, {\left (\log \left (x\right ) + 1\right )} \log \left (2\right )}{x} - \frac {4 \, \log \left (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="maxima")

[Out]

2*(log(x) + 1)*log(2)/x - 4*log(2)/x

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Fricas [A]
time = 0.36, size = 15, normalized size = 1.15 \begin {gather*} \frac {2 \, {\left (\log \left (2\right ) \log \left (x\right ) - \log \left (2\right )\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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Sympy [A]
time = 0.05, size = 15, normalized size = 1.15 \begin {gather*} \frac {2 \log {\left (2 \right )} \log {\left (x \right )}}{x} - \frac {2 \log {\left (2 \right )}}{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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Giac [A]
time = 0.41, size = 17, normalized size = 1.31 \begin {gather*} \frac {2 \, \log \left (2\right ) \log \left (x\right )}{x} - \frac {2 \, \log \left (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="giac")

[Out]

2*log(2)*log(x)/x - 2*log(2)/x

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