3.55.24 \(\int (-4 x+4 \log (x \log (4))+2 \log ^2(x \log (4))) \, dx\) [5424]

Optimal. Leaf size=14 \[ 2 x \left (-x+\log ^2(x \log (4))\right ) \]

[Out]

2*(ln(2*x*ln(2))^2-x)*x

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.14, number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2332, 2333} \begin {gather*} 2 x \log ^2(x \log (4))-2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-4*x + 4*Log[x*Log[4]] + 2*Log[x*Log[4]]^2,x]

[Out]

-2*x^2 + 2*x*Log[x*Log[4]]^2

Rule 2332

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

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.14 \begin {gather*} -2 x^2+2 x \log ^2(x \log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-4*x + 4*Log[x*Log[4]] + 2*Log[x*Log[4]]^2,x]

[Out]

-2*x^2 + 2*x*Log[x*Log[4]]^2

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Maple [A]
time = 0.48, size = 18, normalized size = 1.29

method	result	size
default	\(-2 x^{2}+2 \ln \left (2 x \ln \left (2\right )\right )^{2} x\)	\(18\)
norman	\(-2 x^{2}+2 \ln \left (2 x \ln \left (2\right )\right )^{2} x\)	\(18\)
risch	\(-2 x^{2}+2 \ln \left (2 x \ln \left (2\right )\right )^{2} x\)	\(18\)
derivativedivides	\(\frac {2 \ln \left (2 x \ln \left (2\right )\right )^{2} x \ln \left (2\right )-2 x^{2} \ln \left (2\right )}{\ln \left (2\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x^2+2*ln(2*x*ln(2))^2*x

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).
time = 0.28, size = 49, normalized size = 3.50 \begin {gather*} 2 \, {\left (\log \left (2 \, x \log \left (2\right )\right )^{2} - 2 \, \log \left (2 \, x \log \left (2\right )\right ) + 2\right )} x - 2 \, x^{2} + \frac {4 \, {\left (x \log \left (2\right ) \log \left (2 \, x \log \left (2\right )\right ) - x \log \left (2\right )\right )}}{\log \left (2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 17, normalized size = 1.21 \begin {gather*} 2 \, x \log \left (2 \, x \log \left (2\right )\right )^{2} - 2 \, x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x*log(2*x*log(2))^2 - 2*x^2

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Sympy [A]
time = 0.03, size = 17, normalized size = 1.21 \begin {gather*} - 2 x^{2} + 2 x \log {\left (2 x \log {\left (2 \right )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x**2 + 2*x*log(2*x*log(2))**2

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (15) = 30\).
time = 0.38, size = 51, normalized size = 3.64 \begin {gather*} 2 \, x \log \left (2 \, x \log \left (2\right )\right )^{2} - 2 \, x^{2} - 4 \, x \log \left (2 \, x \log \left (2\right )\right ) + 4 \, x + \frac {4 \, {\left (x \log \left (2\right ) \log \left (2 \, x \log \left (2\right )\right ) - x \log \left (2\right )\right )}}{\log \left (2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 3.63, size = 15, normalized size = 1.07 \begin {gather*} -2\,x\,\left (x-{\ln \left (2\,x\,\ln \left (2\right )\right )}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*(x - log(2*x*log(2))^2)

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