3.53.56 \(\int (-45+4 \log (x)+16 \log (\log (4))) \, dx\)

Optimal. Leaf size=19 \[ -x+4 x (\log (x)-4 (3-\log (\log (4)))) \]

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2295} \begin {gather*} -4 x+4 x \log (x)-x (45-16 \log (\log (4))) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-45 + 4*Log[x] + 16*Log[Log[4]],x]

[Out]

-4*x + 4*x*Log[x] - x*(45 - 16*Log[Log[4]])

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} &=-x (45-16 \log (\log (4)))+4 \int \log (x) \, dx\\ &=-4 x+4 x \log (x)-x (45-16 \log (\log (4)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.79 \begin {gather*} -49 x+4 x \log (x)+16 x \log (\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-45 + 4*Log[x] + 16*Log[Log[4]],x]

[Out]

-49*x + 4*x*Log[x] + 16*x*Log[Log[4]]

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fricas [A]  time = 0.54, size = 17, normalized size = 0.89 \begin {gather*} 4 \, x \log \relax (x) + 16 \, x \log \left (2 \, \log \relax (2)\right ) - 49 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*log(2*log(2))+4*log(x)-45,x, algorithm="fricas")

[Out]

4*x*log(x) + 16*x*log(2*log(2)) - 49*x

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giac [A]  time = 0.17, size = 17, normalized size = 0.89 \begin {gather*} 4 \, x \log \relax (x) + 16 \, x \log \left (2 \, \log \relax (2)\right ) - 49 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*log(2*log(2))+4*log(x)-45,x, algorithm="giac")

[Out]

4*x*log(x) + 16*x*log(2*log(2)) - 49*x

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maple [A]  time = 0.03, size = 18, normalized size = 0.95

method	result	size
default	\(-49 x +4 x \ln \relax (x )+16 x \ln \left (2 \ln \relax (2)\right )\)	\(18\)
norman	\(\left (-49+16 \ln \relax (2)+16 \ln \left (\ln \relax (2)\right )\right ) x +4 x \ln \relax (x )\)	\(20\)
risch	\(4 x \ln \relax (x )+16 x \ln \relax (2)+16 x \ln \left (\ln \relax (2)\right )-49 x\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*ln(2*ln(2))+4*ln(x)-45,x,method=_RETURNVERBOSE)

[Out]

-49*x+4*x*ln(x)+16*x*ln(2*ln(2))

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maxima [A]  time = 0.36, size = 17, normalized size = 0.89 \begin {gather*} 4 \, x \log \relax (x) + 16 \, x \log \left (2 \, \log \relax (2)\right ) - 49 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*log(2*log(2))+4*log(x)-45,x, algorithm="maxima")

[Out]

4*x*log(x) + 16*x*log(2*log(2)) - 49*x

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mupad [B]  time = 3.42, size = 15, normalized size = 0.79 \begin {gather*} x\,\left (16\,\ln \left (\ln \relax (4)\right )-49\right )+4\,x\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(16*log(2*log(2)) + 4*log(x) - 45,x)

[Out]

x*(16*log(log(4)) - 49) + 4*x*log(x)

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sympy [A]  time = 0.08, size = 20, normalized size = 1.05 \begin {gather*} 4 x \log {\relax (x )} + x \left (-49 + 16 \log {\left (\log {\relax (2 )} \right )} + 16 \log {\relax (2 )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(16*ln(2*ln(2))+4*ln(x)-45,x)

[Out]

4*x*log(x) + x*(-49 + 16*log(log(2)) + 16*log(2))

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