3.82.68 \(\int \frac {(-6 x+3 \log (4)) \log (\frac {x}{2})+(-2 x^2+x \log (4)) \log ^2(\frac {x}{2})+(3 x-3 \log (4)+(-x^2+x \log (4)) \log ^2(\frac {x}{2})) \log (-x^2+x \log (4))}{(-x^2+x \log (4)) \log ^2(\frac {x}{2})} \, dx\)

Optimal. Leaf size=22 \[ \left (x+\frac {3}{\log \left (\frac {x}{2}\right )}\right ) \log (x (-x+\log (4))) \]

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Rubi [F]  time = 1.53, 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 {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {(-6 x+3 \log (4)) \log \left (\frac {x}{2}\right )+\left (-2 x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )+\left (3 x-3 \log (4)+\left (-x^2+x \log (4)\right ) \log ^2\left (\frac {x}{2}\right )\right ) \log \left (-x^2+x \log (4)\right )}{x (-x+\log (4)) \log ^2\left (\frac {x}{2}\right )} \, dx\\ &=\int \left (\frac {(2 x-\log (4)) \left (3+x \log \left (\frac {x}{2}\right )\right )}{x (x-\log (4)) \log \left (\frac {x}{2}\right )}+\frac {\left (-3+x \log ^2\left (\frac {x}{2}\right )\right ) \log (x (-x+\log (4)))}{x \log ^2\left (\frac {x}{2}\right )}\right ) \, dx\\ &=\int \frac {(2 x-\log (4)) \left (3+x \log \left (\frac {x}{2}\right )\right )}{x (x-\log (4)) \log \left (\frac {x}{2}\right )} \, dx+\int \frac {\left (-3+x \log ^2\left (\frac {x}{2}\right )\right ) \log (x (-x+\log (4)))}{x \log ^2\left (\frac {x}{2}\right )} \, dx\\ &=\int \left (\frac {2 x-\log (4)}{x-\log (4)}+\frac {3 (2 x-\log (4))}{x (x-\log (4)) \log \left (\frac {x}{2}\right )}\right ) \, dx+\int \left (\log (x (-x+\log (4)))-\frac {3 \log (x (-x+\log (4)))}{x \log ^2\left (\frac {x}{2}\right )}\right ) \, dx\\ &=3 \int \frac {2 x-\log (4)}{x (x-\log (4)) \log \left (\frac {x}{2}\right )} \, dx-3 \int \frac {\log (x (-x+\log (4)))}{x \log ^2\left (\frac {x}{2}\right )} \, dx+\int \frac {2 x-\log (4)}{x-\log (4)} \, dx+\int \log (x (-x+\log (4))) \, dx\\ &=x \log (-x (x-\log (4)))-2 \int 1 \, dx+3 \int \frac {2 x-\log (4)}{x (x-\log (4)) \log \left (\frac {x}{2}\right )} \, dx-3 \int \frac {\log (x (-x+\log (4)))}{x \log ^2\left (\frac {x}{2}\right )} \, dx+\log (4) \int \frac {1}{-x+\log (4)} \, dx+\int \left (2+\frac {\log (4)}{x-\log (4)}\right ) \, dx\\ &=x \log (-x (x-\log (4)))+3 \int \frac {2 x-\log (4)}{x (x-\log (4)) \log \left (\frac {x}{2}\right )} \, dx-3 \int \frac {\log (x (-x+\log (4)))}{x \log ^2\left (\frac {x}{2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 28, normalized size = 1.27 \begin {gather*} \frac {\left (3+x \log \left (\frac {x}{2}\right )\right ) \log (x (-x+\log (4)))}{\log \left (\frac {x}{2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-6*x + 3*Log[4])*Log[x/2] + (-2*x^2 + x*Log[4])*Log[x/2]^2 + (3*x - 3*Log[4] + (-x^2 + x*Log[4])*L
og[x/2]^2)*Log[-x^2 + x*Log[4]])/((-x^2 + x*Log[4])*Log[x/2]^2),x]

[Out]

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

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fricas [A]  time = 0.57, size = 27, normalized size = 1.23 \begin {gather*} \frac {{\left (x \log \left (\frac {1}{2} \, x\right ) + 3\right )} \log \left (-x^{2} + 2 \, x \log \relax (2)\right )}{\log \left (\frac {1}{2} \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2)+(2*x*log(2)-2*x^2)*log(1/2*x)^2+(6
*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x^2)/log(1/2*x)^2,x, algorithm="fricas")

[Out]

(x*log(1/2*x) + 3)*log(-x^2 + 2*x*log(2))/log(1/2*x)

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giac [A]  time = 0.25, size = 41, normalized size = 1.86 \begin {gather*} x \log \relax (x) + {\left (x - \frac {3}{\log \relax (2) - \log \relax (x)}\right )} \log \left (-x + 2 \, \log \relax (2)\right ) - \frac {3 \, \log \relax (2)}{\log \relax (2) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2)+(2*x*log(2)-2*x^2)*log(1/2*x)^2+(6
*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x^2)/log(1/2*x)^2,x, algorithm="giac")

[Out]

x*log(x) + (x - 3/(log(2) - log(x)))*log(-x + 2*log(2)) - 3*log(2)/(log(2) - log(x))

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maple [C]  time = 0.58, size = 382, normalized size = 17.36

method	result	size
risch	\(\frac {\left (-2 i x \ln \relax (2)+2 i x \ln \relax (x )+6 i\right ) \ln \left (\ln \relax (2)-\frac {x}{2}\right )}{-2 i \ln \relax (2)+2 i \ln \relax (x )}+\frac {-4 x \ln \relax (2) \ln \relax (x )+12 \ln \relax (2)+2 i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{2} \ln \relax (x )-2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{2} \mathrm {csgn}\left (i x \right )+2 i \pi x \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{2} \mathrm {csgn}\left (i x \right ) \ln \relax (x )-2 i \pi x \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{3} \ln \relax (x )+4 x \ln \relax (x )^{2}-6 i \pi \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{3}+2 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (i \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \right )-2 i \pi x \,\mathrm {csgn}\left (i \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \right ) \ln \relax (x )+2 i \pi \ln \relax (2) x \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{3}-6 i \pi \,\mathrm {csgn}\left (i \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \right )+6 i \pi \,\mathrm {csgn}\left (i \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{2}-2 i \pi \ln \relax (2) x \,\mathrm {csgn}\left (i \left (\ln \relax (2)-\frac {x}{2}\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{2}+6 i \pi \mathrm {csgn}\left (i x \left (\ln \relax (2)-\frac {x}{2}\right )\right )^{2} \mathrm {csgn}\left (i x \right )}{4 \ln \relax (x )-4 \ln \relax (2)}\)	\(382\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x*ln(2)-x^2)*ln(1/2*x)^2-6*ln(2)+3*x)*ln(2*x*ln(2)-x^2)+(2*x*ln(2)-2*x^2)*ln(1/2*x)^2+(6*ln(2)-6*x)*l
n(1/2*x))/(2*x*ln(2)-x^2)/ln(1/2*x)^2,x,method=_RETURNVERBOSE)

[Out]

(-2*I*x*ln(2)+2*I*x*ln(x)+6*I)/(-2*I*ln(2)+2*I*ln(x))*ln(ln(2)-1/2*x)+1/2*(-4*x*ln(2)*ln(x)+12*ln(2)+2*I*Pi*x*
csgn(I*(ln(2)-1/2*x))*csgn(I*x*(ln(2)-1/2*x))^2*ln(x)-2*I*Pi*ln(2)*x*csgn(I*x*(ln(2)-1/2*x))^2*csgn(I*x)+2*I*P
i*x*csgn(I*x*(ln(2)-1/2*x))^2*csgn(I*x)*ln(x)-2*I*Pi*x*csgn(I*x*(ln(2)-1/2*x))^3*ln(x)+4*x*ln(x)^2-6*I*Pi*csgn
(I*x*(ln(2)-1/2*x))^3+2*I*Pi*ln(2)*x*csgn(I*(ln(2)-1/2*x))*csgn(I*x*(ln(2)-1/2*x))*csgn(I*x)-2*I*Pi*x*csgn(I*(
ln(2)-1/2*x))*csgn(I*x*(ln(2)-1/2*x))*csgn(I*x)*ln(x)+2*I*Pi*ln(2)*x*csgn(I*x*(ln(2)-1/2*x))^3-6*I*Pi*csgn(I*(
ln(2)-1/2*x))*csgn(I*x*(ln(2)-1/2*x))*csgn(I*x)+6*I*Pi*csgn(I*(ln(2)-1/2*x))*csgn(I*x*(ln(2)-1/2*x))^2-2*I*Pi*
ln(2)*x*csgn(I*(ln(2)-1/2*x))*csgn(I*x*(ln(2)-1/2*x))^2+6*I*Pi*csgn(I*x*(ln(2)-1/2*x))^2*csgn(I*x))/(2*ln(x)-2
*ln(2))

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maxima [B]  time = 0.52, size = 49, normalized size = 2.23 \begin {gather*} \frac {x \log \relax (2) \log \relax (x) - x \log \relax (x)^{2} + {\left (x \log \relax (2) - x \log \relax (x) - 3\right )} \log \left (-x + 2 \, \log \relax (2)\right ) - 3 \, \log \relax (2)}{\log \relax (2) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*log(2)-x^2)*log(1/2*x)^2-6*log(2)+3*x)*log(2*x*log(2)-x^2)+(2*x*log(2)-2*x^2)*log(1/2*x)^2+(6
*log(2)-6*x)*log(1/2*x))/(2*x*log(2)-x^2)/log(1/2*x)^2,x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\ln \left (2\,x\,\ln \relax (2)-x^2\right )\,\left (\left (2\,x\,\ln \relax (2)-x^2\right )\,{\ln \left (\frac {x}{2}\right )}^2+3\,x-6\,\ln \relax (2)\right )-\ln \left (\frac {x}{2}\right )\,\left (6\,x-6\,\ln \relax (2)\right )+{\ln \left (\frac {x}{2}\right )}^2\,\left (2\,x\,\ln \relax (2)-2\,x^2\right )}{{\ln \left (\frac {x}{2}\right )}^2\,\left (2\,x\,\ln \relax (2)-x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2*x*log(2) - x^2)*(3*x - 6*log(2) + log(x/2)^2*(2*x*log(2) - x^2)) - log(x/2)*(6*x - 6*log(2)) + log(
x/2)^2*(2*x*log(2) - 2*x^2))/(log(x/2)^2*(2*x*log(2) - x^2)),x)

[Out]

int((log(2*x*log(2) - x^2)*(3*x - 6*log(2) + log(x/2)^2*(2*x*log(2) - x^2)) - log(x/2)*(6*x - 6*log(2)) + log(
x/2)^2*(2*x*log(2) - 2*x^2))/(log(x/2)^2*(2*x*log(2) - x^2)), x)

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sympy [A]  time = 0.46, size = 24, normalized size = 1.09 \begin {gather*} \frac {\left (x \log {\left (\frac {x}{2} \right )} + 3\right ) \log {\left (- x^{2} + 2 x \log {\relax (2 )} \right )}}{\log {\left (\frac {x}{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x*ln(2)-x**2)*ln(1/2*x)**2-6*ln(2)+3*x)*ln(2*x*ln(2)-x**2)+(2*x*ln(2)-2*x**2)*ln(1/2*x)**2+(6*l
n(2)-6*x)*ln(1/2*x))/(2*x*ln(2)-x**2)/ln(1/2*x)**2,x)

[Out]

(x*log(x/2) + 3)*log(-x**2 + 2*x*log(2))/log(x/2)

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