3.104.18 \(\int \frac {(-208 x-60 x^2+108 x^3-36 x^4+4 x^5) \log (2 x)+(-60+164 x-57 x^2-15 x^3+9 x^4-x^5) \log (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4})}{(60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6) \log ^2(2 x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {\log \left (\left (-4+\left (\frac {4}{3-x}+x\right )^2\right )^2\right )}{\log (2 x)} \]

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Rubi [F]  time = 1.73, 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 {\left (-208 x-60 x^2+108 x^3-36 x^4+4 x^5\right ) \log (2 x)+\left (-60+164 x-57 x^2-15 x^3+9 x^4-x^5\right ) \log \left (\frac {400-1920 x+2424 x^2-48 x^3-607 x^4+132 x^5+30 x^6-12 x^7+x^8}{81-108 x+54 x^2-12 x^3+x^4}\right )}{\left (60 x-164 x^2+57 x^3+15 x^4-9 x^5+x^6\right ) \log ^2(2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-208*x - 60*x^2 + 108*x^3 - 36*x^4 + 4*x^5)*Log[2*x] + (-60 + 164*x - 57*x^2 - 15*x^3 + 9*x^4 - x^5)*Log
[(400 - 1920*x + 2424*x^2 - 48*x^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^8)/(81 - 108*x + 54*x^2 - 12*x^3
+ x^4)])/((60*x - 164*x^2 + 57*x^3 + 15*x^4 - 9*x^5 + x^6)*Log[2*x]^2),x]

[Out]

4*Defer[Int][((-4 + x)*(1 + x)*(13 - 6*x + x^2))/((-3 + x)*(2 - 5*x + x^2)*(-10 - x + x^2)*Log[2*x]), x] - Def
er[Int][Log[(-20 + 48*x - 3*x^2 - 6*x^3 + x^4)^2/(-3 + x)^4]/(x*Log[2*x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {4 \left (-52-15 x+27 x^2-9 x^3+x^4\right ) \log (2 x)}{60-164 x+57 x^2+15 x^3-9 x^4+x^5}-\frac {\log \left (\frac {\left (-20+48 x-3 x^2-6 x^3+x^4\right )^2}{(-3+x)^4}\right )}{x}}{\log ^2(2 x)} \, dx\\ &=\int \left (\frac {4 (-4+x) (1+x) \left (13-6 x+x^2\right )}{(-3+x) \left (2-5 x+x^2\right ) \left (-10-x+x^2\right ) \log (2 x)}-\frac {\log \left (\frac {\left (-20+48 x-3 x^2-6 x^3+x^4\right )^2}{(-3+x)^4}\right )}{x \log ^2(2 x)}\right ) \, dx\\ &=4 \int \frac {(-4+x) (1+x) \left (13-6 x+x^2\right )}{(-3+x) \left (2-5 x+x^2\right ) \left (-10-x+x^2\right ) \log (2 x)} \, dx-\int \frac {\log \left (\frac {\left (-20+48 x-3 x^2-6 x^3+x^4\right )^2}{(-3+x)^4}\right )}{x \log ^2(2 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 34, normalized size = 1.36 \begin {gather*} \frac {\log \left (\frac {\left (-20+48 x-3 x^2-6 x^3+x^4\right )^2}{(-3+x)^4}\right )}{\log (2 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-208*x - 60*x^2 + 108*x^3 - 36*x^4 + 4*x^5)*Log[2*x] + (-60 + 164*x - 57*x^2 - 15*x^3 + 9*x^4 - x^
5)*Log[(400 - 1920*x + 2424*x^2 - 48*x^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^8)/(81 - 108*x + 54*x^2 - 1
2*x^3 + x^4)])/((60*x - 164*x^2 + 57*x^3 + 15*x^4 - 9*x^5 + x^6)*Log[2*x]^2),x]

[Out]

Log[(-20 + 48*x - 3*x^2 - 6*x^3 + x^4)^2/(-3 + x)^4]/Log[2*x]

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fricas [B]  time = 0.71, size = 67, normalized size = 2.68 \begin {gather*} \frac {\log \left (\frac {x^{8} - 12 \, x^{7} + 30 \, x^{6} + 132 \, x^{5} - 607 \, x^{4} - 48 \, x^{3} + 2424 \, x^{2} - 1920 \, x + 400}{x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81}\right )}{\log \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*log((x^8-12*x^7+30*x^6+132*x^5-607*x^4-48*x^3+2424*x^2-1920*x+4
00)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x^5-36*x^4+108*x^3-60*x^2-208*x)*log(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x
^2+60*x)/log(2*x)^2,x, algorithm="fricas")

[Out]

log((x^8 - 12*x^7 + 30*x^6 + 132*x^5 - 607*x^4 - 48*x^3 + 2424*x^2 - 1920*x + 400)/(x^4 - 12*x^3 + 54*x^2 - 10
8*x + 81))/log(2*x)

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giac [B]  time = 0.61, size = 74, normalized size = 2.96 \begin {gather*} \frac {\log \left (x^{8} - 12 \, x^{7} + 30 \, x^{6} + 132 \, x^{5} - 607 \, x^{4} - 48 \, x^{3} + 2424 \, x^{2} - 1920 \, x + 400\right )}{\log \left (2 \, x\right )} - \frac {\log \left (x^{4} - 12 \, x^{3} + 54 \, x^{2} - 108 \, x + 81\right )}{\log \left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*log((x^8-12*x^7+30*x^6+132*x^5-607*x^4-48*x^3+2424*x^2-1920*x+4
00)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x^5-36*x^4+108*x^3-60*x^2-208*x)*log(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x
^2+60*x)/log(2*x)^2,x, algorithm="giac")

[Out]

log(x^8 - 12*x^7 + 30*x^6 + 132*x^5 - 607*x^4 - 48*x^3 + 2424*x^2 - 1920*x + 400)/log(2*x) - log(x^4 - 12*x^3
+ 54*x^2 - 108*x + 81)/log(2*x)

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maple [C]  time = 0.65, size = 621, normalized size = 24.84

method	result	size
risch	\(\frac {2 \ln \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )}{\ln \left (2 x \right )}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (x -3\right )^{4}}\right )^{3}+i \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (i \left (x -3\right )^{3}\right )^{2}-i \pi \,\mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (x -3\right )^{4}}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (i \left (x -3\right )^{4}\right )^{2}+i \pi \mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )\right )^{2} \mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right )-2 i \pi \,\mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )\right ) \mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \left (x -3\right )^{2}\right ) \mathrm {csgn}\left (i \left (x -3\right )^{3}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (x -3\right )^{3}\right )^{3}-i \pi \mathrm {csgn}\left (i \left (x -3\right )^{4}\right )^{3}+i \pi \,\mathrm {csgn}\left (i \left (x -3\right )^{3}\right ) \mathrm {csgn}\left (i \left (x -3\right )^{4}\right )^{2}+i \pi \mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right )^{3}-i \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (i \left (x -3\right )^{2}\right ) \mathrm {csgn}\left (i \left (x -3\right )^{3}\right )-i \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (i \left (x -3\right )^{3}\right ) \mathrm {csgn}\left (i \left (x -3\right )^{4}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{\left (x -3\right )^{4}}\right ) \mathrm {csgn}\left (i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (x -3\right )^{4}}\right )-i \pi \mathrm {csgn}\left (i \left (x -3\right )\right )^{2} \mathrm {csgn}\left (i \left (x -3\right )^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i \left (x -3\right )\right ) \mathrm {csgn}\left (i \left (x -3\right )^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i \left (x -3\right )^{2}\right )^{3}-i \pi \,\mathrm {csgn}\left (\frac {i}{\left (x -3\right )^{4}}\right ) \mathrm {csgn}\left (\frac {i \left (x^{4}-6 x^{3}-3 x^{2}+48 x -20\right )^{2}}{\left (x -3\right )^{4}}\right )^{2}+8 \ln \left (x -3\right )}{2 \ln \left (2 x \right )}\)	\(621\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*ln((x^8-12*x^7+30*x^6+132*x^5-607*x^4-48*x^3+2424*x^2-1920*x+400)/(x^
4-12*x^3+54*x^2-108*x+81))+(4*x^5-36*x^4+108*x^3-60*x^2-208*x)*ln(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x^2+60*x)
/ln(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

2/ln(2*x)*ln(x^4-6*x^3-3*x^2+48*x-20)-1/2*(I*Pi*csgn(I/(x-3)^4*(x^4-6*x^3-3*x^2+48*x-20)^2)^3+I*Pi*csgn(I*(x-3
))*csgn(I*(x-3)^3)^2-I*Pi*csgn(I*(x^4-6*x^3-3*x^2+48*x-20)^2)*csgn(I/(x-3)^4*(x^4-6*x^3-3*x^2+48*x-20)^2)^2+I*
Pi*csgn(I*(x-3))*csgn(I*(x-3)^4)^2+I*Pi*csgn(I*(x^4-6*x^3-3*x^2+48*x-20))^2*csgn(I*(x^4-6*x^3-3*x^2+48*x-20)^2
)-2*I*Pi*csgn(I*(x^4-6*x^3-3*x^2+48*x-20))*csgn(I*(x^4-6*x^3-3*x^2+48*x-20)^2)^2+I*Pi*csgn(I*(x-3)^2)*csgn(I*(
x-3)^3)^2-I*Pi*csgn(I*(x-3)^3)^3-I*Pi*csgn(I*(x-3)^4)^3+I*Pi*csgn(I*(x-3)^3)*csgn(I*(x-3)^4)^2+I*Pi*csgn(I*(x^
4-6*x^3-3*x^2+48*x-20)^2)^3-I*Pi*csgn(I*(x-3))*csgn(I*(x-3)^2)*csgn(I*(x-3)^3)-I*Pi*csgn(I*(x-3))*csgn(I*(x-3)
^3)*csgn(I*(x-3)^4)+I*Pi*csgn(I/(x-3)^4)*csgn(I*(x^4-6*x^3-3*x^2+48*x-20)^2)*csgn(I/(x-3)^4*(x^4-6*x^3-3*x^2+4
8*x-20)^2)-I*Pi*csgn(I*(x-3))^2*csgn(I*(x-3)^2)+2*I*Pi*csgn(I*(x-3))*csgn(I*(x-3)^2)^2-I*Pi*csgn(I*(x-3)^2)^3-
I*Pi*csgn(I/(x-3)^4)*csgn(I/(x-3)^4*(x^4-6*x^3-3*x^2+48*x-20)^2)^2+8*ln(x-3))/ln(2*x)

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maxima [A]  time = 0.50, size = 34, normalized size = 1.36 \begin {gather*} \frac {2 \, {\left (\log \left (x^{2} - x - 10\right ) + \log \left (x^{2} - 5 \, x + 2\right ) - 2 \, \log \left (x - 3\right )\right )}}{\log \relax (2) + \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^5+9*x^4-15*x^3-57*x^2+164*x-60)*log((x^8-12*x^7+30*x^6+132*x^5-607*x^4-48*x^3+2424*x^2-1920*x+4
00)/(x^4-12*x^3+54*x^2-108*x+81))+(4*x^5-36*x^4+108*x^3-60*x^2-208*x)*log(2*x))/(x^6-9*x^5+15*x^4+57*x^3-164*x
^2+60*x)/log(2*x)^2,x, algorithm="maxima")

[Out]

2*(log(x^2 - x - 10) + log(x^2 - 5*x + 2) - 2*log(x - 3))/(log(2) + log(x))

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mupad [B]  time = 7.22, size = 67, normalized size = 2.68 \begin {gather*} \frac {\ln \left (\frac {x^8-12\,x^7+30\,x^6+132\,x^5-607\,x^4-48\,x^3+2424\,x^2-1920\,x+400}{x^4-12\,x^3+54\,x^2-108\,x+81}\right )}{\ln \left (2\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log((2424*x^2 - 1920*x - 48*x^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^8 + 400)/(54*x^2 - 108*x - 12*
x^3 + x^4 + 81))*(57*x^2 - 164*x + 15*x^3 - 9*x^4 + x^5 + 60) + log(2*x)*(208*x + 60*x^2 - 108*x^3 + 36*x^4 -
4*x^5))/(log(2*x)^2*(60*x - 164*x^2 + 57*x^3 + 15*x^4 - 9*x^5 + x^6)),x)

[Out]

log((2424*x^2 - 1920*x - 48*x^3 - 607*x^4 + 132*x^5 + 30*x^6 - 12*x^7 + x^8 + 400)/(54*x^2 - 108*x - 12*x^3 +
x^4 + 81))/log(2*x)

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sympy [B]  time = 0.62, size = 63, normalized size = 2.52 \begin {gather*} \frac {\log {\left (\frac {x^{8} - 12 x^{7} + 30 x^{6} + 132 x^{5} - 607 x^{4} - 48 x^{3} + 2424 x^{2} - 1920 x + 400}{x^{4} - 12 x^{3} + 54 x^{2} - 108 x + 81} \right )}}{\log {\left (2 x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**5+9*x**4-15*x**3-57*x**2+164*x-60)*ln((x**8-12*x**7+30*x**6+132*x**5-607*x**4-48*x**3+2424*x**
2-1920*x+400)/(x**4-12*x**3+54*x**2-108*x+81))+(4*x**5-36*x**4+108*x**3-60*x**2-208*x)*ln(2*x))/(x**6-9*x**5+1
5*x**4+57*x**3-164*x**2+60*x)/ln(2*x)**2,x)

[Out]

log((x**8 - 12*x**7 + 30*x**6 + 132*x**5 - 607*x**4 - 48*x**3 + 2424*x**2 - 1920*x + 400)/(x**4 - 12*x**3 + 54
*x**2 - 108*x + 81))/log(2*x)

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