3.77.19 \(\int \frac {-2 x \log (4)+\log (4) \log (x)-\log (4) \log (4 x)}{10 \log (2)} \, dx\)

Optimal. Leaf size=21 \[ -\frac {x \log (4) (x-\log (x)+\log (4 x))}{10 \log (2)} \]

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Rubi [A]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 2.00, number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {12, 2295} \begin {gather*} -\frac {x^2 \log (4)}{10 \log (2)}+\frac {x \log (4) \log (x)}{10 \log (2)}-\frac {x \log (4) \log (4 x)}{10 \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2*x*Log[4] + Log[4]*Log[x] - Log[4]*Log[4*x])/(10*Log[2]),x]

[Out]

-1/10*(x^2*Log[4])/Log[2] + (x*Log[4]*Log[x])/(10*Log[2]) - (x*Log[4]*Log[4*x])/(10*Log[2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int (-2 x \log (4)+\log (4) \log (x)-\log (4) \log (4 x)) \, dx}{10 \log (2)}\\ &=-\frac {x^2 \log (4)}{10 \log (2)}+\frac {\log (4) \int \log (x) \, dx}{10 \log (2)}-\frac {\log (4) \int \log (4 x) \, dx}{10 \log (2)}\\ &=-\frac {x^2 \log (4)}{10 \log (2)}+\frac {x \log (4) \log (x)}{10 \log (2)}-\frac {x \log (4) \log (4 x)}{10 \log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.76 \begin {gather*} -\frac {\log (4) \left (x^2+x \log (4)\right )}{\log (1024)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x*Log[4] + Log[4]*Log[x] - Log[4]*Log[4*x])/(10*Log[2]),x]

[Out]

-((Log[4]*(x^2 + x*Log[4]))/Log[1024])

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fricas [A]  time = 1.99, size = 11, normalized size = 0.52 \begin {gather*} -\frac {1}{5} \, x^{2} - \frac {2}{5} \, x \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-2*log(2)*log(4*x)+2*log(2)*log(x)-4*x*log(2))/log(2),x, algorithm="fricas")

[Out]

-1/5*x^2 - 2/5*x*log(2)

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giac [B]  time = 0.20, size = 38, normalized size = 1.81 \begin {gather*} -\frac {x^{2} \log \relax (2) + {\left (x \log \left (4 \, x\right ) - x\right )} \log \relax (2) - {\left (x \log \relax (x) - x\right )} \log \relax (2)}{5 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-2*log(2)*log(4*x)+2*log(2)*log(x)-4*x*log(2))/log(2),x, algorithm="giac")

[Out]

-1/5*(x^2*log(2) + (x*log(4*x) - x)*log(2) - (x*log(x) - x)*log(2))/log(2)

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maple [A]  time = 0.06, size = 12, normalized size = 0.57

method	result	size
risch	\(-\frac {2 x \ln \relax (2)}{5}-\frac {x^{2}}{5}\)	\(12\)
norman	\(-\frac {x^{2}}{5}+\frac {x \ln \relax (x )}{5}-\frac {x \ln \left (4 x \right )}{5}\)	\(19\)
default	\(\frac {x \ln \relax (2) \ln \relax (x )-x^{2} \ln \relax (2)-\ln \relax (2) \ln \left (4 x \right ) x}{5 \ln \relax (2)}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/10*(-2*ln(2)*ln(4*x)+2*ln(2)*ln(x)-4*x*ln(2))/ln(2),x,method=_RETURNVERBOSE)

[Out]

-2/5*x*ln(2)-1/5*x^2

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maxima [B]  time = 0.37, size = 38, normalized size = 1.81 \begin {gather*} -\frac {x^{2} \log \relax (2) + {\left (x \log \left (4 \, x\right ) - x\right )} \log \relax (2) - {\left (x \log \relax (x) - x\right )} \log \relax (2)}{5 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-2*log(2)*log(4*x)+2*log(2)*log(x)-4*x*log(2))/log(2),x, algorithm="maxima")

[Out]

-1/5*(x^2*log(2) + (x*log(4*x) - x)*log(2) - (x*log(x) - x)*log(2))/log(2)

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mupad [B]  time = 4.60, size = 7, normalized size = 0.33 \begin {gather*} -\frac {x\,\left (x+\ln \relax (4)\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(x + log(4)))/5

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sympy [A]  time = 0.06, size = 14, normalized size = 0.67 \begin {gather*} - \frac {x^{2}}{5} - \frac {2 x \log {\relax (2 )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/10*(-2*ln(2)*ln(4*x)+2*ln(2)*ln(x)-4*x*ln(2))/ln(2),x)

[Out]

-x**2/5 - 2*x*log(2)/5

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