3.46.81 \(\int \frac {-3 x-8 \log (\frac {3}{16 x})+(-3 x+4 \log ^2(\frac {3}{16 x})) \log (\frac {1}{4} (-3 x+4 \log ^2(\frac {3}{16 x})))}{-3 x+4 \log ^2(\frac {3}{16 x})} \, dx\)

Optimal. Leaf size=19 \[ x \log \left (-\frac {3 x}{4}+\log ^2\left (\frac {3}{16 x}\right )\right ) \]

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Rubi [F]  time = 0.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-3 x-8 \log \left (\frac {3}{16 x}\right )+\left (-3 x+4 \log ^2\left (\frac {3}{16 x}\right )\right ) \log \left (\frac {1}{4} \left (-3 x+4 \log ^2\left (\frac {3}{16 x}\right )\right )\right )}{-3 x+4 \log ^2\left (\frac {3}{16 x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-3*x - 8*Log[3/(16*x)] + (-3*x + 4*Log[3/(16*x)]^2)*Log[(-3*x + 4*Log[3/(16*x)]^2)/4])/(-3*x + 4*Log[3/(1
6*x)]^2),x]

[Out]

3*Defer[Int][x/(3*x - 4*Log[3/(16*x)]^2), x] - 8*Defer[Int][Log[3/(16*x)]/(-3*x + 4*Log[3/(16*x)]^2), x] + Def
er[Int][Log[(-3*x)/4 + Log[3/(16*x)]^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3 x+8 \log \left (\frac {3}{16 x}\right )}{3 x-4 \log ^2\left (\frac {3}{16 x}\right )}+\log \left (-\frac {3 x}{4}+\log ^2\left (\frac {3}{16 x}\right )\right )\right ) \, dx\\ &=\int \frac {3 x+8 \log \left (\frac {3}{16 x}\right )}{3 x-4 \log ^2\left (\frac {3}{16 x}\right )} \, dx+\int \log \left (-\frac {3 x}{4}+\log ^2\left (\frac {3}{16 x}\right )\right ) \, dx\\ &=\int \left (\frac {3 x}{3 x-4 \log ^2\left (\frac {3}{16 x}\right )}-\frac {8 \log \left (\frac {3}{16 x}\right )}{-3 x+4 \log ^2\left (\frac {3}{16 x}\right )}\right ) \, dx+\int \log \left (-\frac {3 x}{4}+\log ^2\left (\frac {3}{16 x}\right )\right ) \, dx\\ &=3 \int \frac {x}{3 x-4 \log ^2\left (\frac {3}{16 x}\right )} \, dx-8 \int \frac {\log \left (\frac {3}{16 x}\right )}{-3 x+4 \log ^2\left (\frac {3}{16 x}\right )} \, dx+\int \log \left (-\frac {3 x}{4}+\log ^2\left (\frac {3}{16 x}\right )\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 19, normalized size = 1.00 \begin {gather*} x \log \left (-\frac {3 x}{4}+\log ^2\left (\frac {3}{16 x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3*x - 8*Log[3/(16*x)] + (-3*x + 4*Log[3/(16*x)]^2)*Log[(-3*x + 4*Log[3/(16*x)]^2)/4])/(-3*x + 4*Lo
g[3/(16*x)]^2),x]

[Out]

x*Log[(-3*x)/4 + Log[3/(16*x)]^2]

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fricas [A]  time = 0.52, size = 15, normalized size = 0.79 \begin {gather*} x \log \left (\log \left (\frac {3}{16 \, x}\right )^{2} - \frac {3}{4} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(3/16/x)^2-3*x)*log(log(3/16/x)^2-3/4*x)-8*log(3/16/x)-3*x)/(4*log(3/16/x)^2-3*x),x, algorith
m="fricas")

[Out]

x*log(log(3/16/x)^2 - 3/4*x)

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giac [A]  time = 0.42, size = 23, normalized size = 1.21 \begin {gather*} -2 \, x \log \relax (2) + x \log \left (4 \, \log \left (\frac {3}{16 \, x}\right )^{2} - 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(3/16/x)^2-3*x)*log(log(3/16/x)^2-3/4*x)-8*log(3/16/x)-3*x)/(4*log(3/16/x)^2-3*x),x, algorith
m="giac")

[Out]

-2*x*log(2) + x*log(4*log(3/16/x)^2 - 3*x)

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maple [A]  time = 0.12, size = 16, normalized size = 0.84

method	result	size
norman	\(x \ln \left (\ln \left (\frac {3}{16 x}\right )^{2}-\frac {3 x}{4}\right )\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*ln(3/16/x)^2-3*x)*ln(ln(3/16/x)^2-3/4*x)-8*ln(3/16/x)-3*x)/(4*ln(3/16/x)^2-3*x),x,method=_RETURNVERBOS
E)

[Out]

x*ln(ln(3/16/x)^2-3/4*x)

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maxima [B]  time = 0.48, size = 48, normalized size = 2.53 \begin {gather*} -2 \, x \log \relax (2) + x \log \left (4 \, \log \relax (3)^{2} - 32 \, \log \relax (3) \log \relax (2) + 64 \, \log \relax (2)^{2} - 8 \, {\left (\log \relax (3) - 4 \, \log \relax (2)\right )} \log \relax (x) + 4 \, \log \relax (x)^{2} - 3 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(3/16/x)^2-3*x)*log(log(3/16/x)^2-3/4*x)-8*log(3/16/x)-3*x)/(4*log(3/16/x)^2-3*x),x, algorith
m="maxima")

[Out]

-2*x*log(2) + x*log(4*log(3)^2 - 32*log(3)*log(2) + 64*log(2)^2 - 8*(log(3) - 4*log(2))*log(x) + 4*log(x)^2 -
3*x)

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mupad [B]  time = 3.44, size = 15, normalized size = 0.79 \begin {gather*} x\,\ln \left ({\ln \left (\frac {3}{16\,x}\right )}^2-\frac {3\,x}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 8*log(3/(16*x)) + log(log(3/(16*x))^2 - (3*x)/4)*(3*x - 4*log(3/(16*x))^2))/(3*x - 4*log(3/(16*x))^
2),x)

[Out]

x*log(log(3/(16*x))^2 - (3*x)/4)

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sympy [A]  time = 0.41, size = 15, normalized size = 0.79 \begin {gather*} x \log {\left (- \frac {3 x}{4} + \log {\left (\frac {3}{16 x} \right )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*ln(3/16/x)**2-3*x)*ln(ln(3/16/x)**2-3/4*x)-8*ln(3/16/x)-3*x)/(4*ln(3/16/x)**2-3*x),x)

[Out]

x*log(-3*x/4 + log(3/(16*x))**2)

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