3.31.22 \(\int \frac {-\log (5)+\log (5) \log (4 x)}{x^2} \, dx\) [3022]

Optimal. Leaf size=19 \[ 3-\log ^2(4)-\frac {\log (5) \log (4 x)}{x} \]

[Out]

-ln(4*x)*ln(5)/x-4*ln(2)^2+3

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.58, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2340} \begin {gather*} -\frac {\log (5) \log (4 x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-Log[5] + Log[5]*Log[4*x])/x^2,x]

[Out]

-((Log[5]*Log[4*x])/x)

Rule 2340

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[b*(d*x)^(m + 1)*(Log[c*x^n]/(d
*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\log (5) \log (4 x)}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 11, normalized size = 0.58 \begin {gather*} -\frac {\log (5) \log (4 x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-Log[5] + Log[5]*Log[4*x])/x^2,x]

[Out]

-((Log[5]*Log[4*x])/x)

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Maple [A]
time = 0.18, size = 27, normalized size = 1.42

method	result	size
norman	\(-\frac {\ln \left (4 x \right ) \ln \left (5\right )}{x}\)	\(12\)
risch	\(-\frac {\ln \left (4 x \right ) \ln \left (5\right )}{x}\)	\(12\)
derivativedivides	\(4 \ln \left (5\right ) \left (-\frac {\ln \left (4 x \right )}{4 x}-\frac {1}{4 x}\right )+\frac {\ln \left (5\right )}{x}\)	\(27\)
default	\(4 \ln \left (5\right ) \left (-\frac {\ln \left (4 x \right )}{4 x}-\frac {1}{4 x}\right )+\frac {\ln \left (5\right )}{x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(5)*ln(4*x)-ln(5))/x^2,x,method=_RETURNVERBOSE)

[Out]

4*ln(5)*(-1/4*ln(4*x)/x-1/4/x)+ln(5)/x

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Maxima [A]
time = 0.27, size = 20, normalized size = 1.05 \begin {gather*} -\frac {{\left (\log \left (4 \, x\right ) + 1\right )} \log \left (5\right )}{x} + \frac {\log \left (5\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(5)*log(4*x)-log(5))/x^2,x, algorithm="maxima")

[Out]

-(log(4*x) + 1)*log(5)/x + log(5)/x

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Fricas [A]
time = 0.42, size = 11, normalized size = 0.58 \begin {gather*} -\frac {\log \left (5\right ) \log \left (4 \, x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(5)*log(4*x)-log(5))/x^2,x, algorithm="fricas")

[Out]

-log(5)*log(4*x)/x

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Sympy [A]
time = 0.04, size = 10, normalized size = 0.53 \begin {gather*} - \frac {\log {\left (5 \right )} \log {\left (4 x \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(5)*ln(4*x)-ln(5))/x**2,x)

[Out]

-log(5)*log(4*x)/x

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Giac [A]
time = 0.39, size = 11, normalized size = 0.58 \begin {gather*} -\frac {\log \left (5\right ) \log \left (4 \, x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(5)*log(4*x)-log(5))/x^2,x, algorithm="giac")

[Out]

-log(5)*log(4*x)/x

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Mupad [B]
time = 1.77, size = 11, normalized size = 0.58 \begin {gather*} -\frac {\ln \left (4\,x\right )\,\ln \left (5\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5) - log(4*x)*log(5))/x^2,x)

[Out]

-(log(4*x)*log(5))/x

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