3.94.90 \(\int \frac {-500-80 x^2+(-250+100 x-40 x^2+16 x^3) \log (x)}{(-5000 x+5600 x^2-2160 x^3+288 x^4) \log (2)+(-500 x+400 x^2-80 x^3) \log (2) \log (x)+(-125 x+150 x^2-60 x^3+8 x^4) \log (2) \log ^2(x)+(-5000 x+5600 x^2-2160 x^3+288 x^4+(-500 x+400 x^2-80 x^3) \log (x)+(-125 x+150 x^2-60 x^3+8 x^4) \log ^2(x)) \log (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+(25-20 x+4 x^2) \log ^2(x)}{100-80 x+16 x^2})} \, dx\)

Optimal. Leaf size=25 \[ \log \left (\log (2)+\log \left (9+\left (\frac {1}{1-\frac {2 x}{5}}+\frac {\log (x)}{2}\right )^2\right )\right ) \]

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Rubi [F]  time = 13.31, 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 {-500-80 x^2+\left (-250+100 x-40 x^2+16 x^3\right ) \log (x)}{\left (-5000 x+5600 x^2-2160 x^3+288 x^4\right ) \log (2)+\left (-500 x+400 x^2-80 x^3\right ) \log (2) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log (2) \log ^2(x)+\left (-5000 x+5600 x^2-2160 x^3+288 x^4+\left (-500 x+400 x^2-80 x^3\right ) \log (x)+\left (-125 x+150 x^2-60 x^3+8 x^4\right ) \log ^2(x)\right ) \log \left (\frac {1000-720 x+144 x^2+(100-40 x) \log (x)+\left (25-20 x+4 x^2\right ) \log ^2(x)}{100-80 x+16 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-500 - 80*x^2 + (-250 + 100*x - 40*x^2 + 16*x^3)*Log[x])/((-5000*x + 5600*x^2 - 2160*x^3 + 288*x^4)*Log[2
] + (-500*x + 400*x^2 - 80*x^3)*Log[2]*Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x^4)*Log[2]*Log[x]^2 + (-5000*x
+ 5600*x^2 - 2160*x^3 + 288*x^4 + (-500*x + 400*x^2 - 80*x^3)*Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x^4)*Lo
g[x]^2)*Log[(1000 - 720*x + 144*x^2 + (100 - 40*x)*Log[x] + (25 - 20*x + 4*x^2)*Log[x]^2)/(100 - 80*x + 16*x^2
)]),x]

[Out]

-40*Defer[Int][1/((1000 - 720*x + 144*x^2 + 100*Log[x] - 40*x*Log[x] + 25*Log[x]^2 - 20*x*Log[x]^2 + 4*x^2*Log
[x]^2)*(Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2*x) + Log[x]^2/4])), x] + 100*De
fer[Int][1/(x*(1000 - 720*x + 144*x^2 + 100*Log[x] - 40*x*Log[x] + 25*Log[x]^2 - 20*x*Log[x]^2 + 4*x^2*Log[x]^
2)*(Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2*x) + Log[x]^2/4])), x] - 400*Defer[
Int][1/((-5 + 2*x)*(1000 - 720*x + 144*x^2 + 100*Log[x] - 40*x*Log[x] + 25*Log[x]^2 - 20*x*Log[x]^2 + 4*x^2*Lo
g[x]^2)*(Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2*x) + Log[x]^2/4])), x] + 50*De
fer[Int][Log[x]/(x*(1000 - 720*x + 144*x^2 + 100*Log[x] - 40*x*Log[x] + 25*Log[x]^2 - 20*x*Log[x]^2 + 4*x^2*Lo
g[x]^2)*(Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2*x) + Log[x]^2/4])), x] + 8*Def
er[Int][(x*Log[x])/((1000 - 720*x + 144*x^2 + 100*Log[x] - 40*x*Log[x] + 25*Log[x]^2 - 20*x*Log[x]^2 + 4*x^2*L
og[x]^2)*(Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2*x) + Log[x]^2/4])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (25+4 x^2\right ) (10-(-5+2 x) \log (x))}{(5-2 x) x \left (8 \left (125-90 x+18 x^2\right )-20 (-5+2 x) \log (x)+(5-2 x)^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )} \, dx\\ &=2 \int \frac {\left (25+4 x^2\right ) (10-(-5+2 x) \log (x))}{(5-2 x) x \left (8 \left (125-90 x+18 x^2\right )-20 (-5+2 x) \log (x)+(5-2 x)^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )} \, dx\\ &=2 \int \left (\frac {2 (-10-5 \log (x)+2 x \log (x))}{\left (1000-720 x+144 x^2+100 \log (x)-40 x \log (x)+25 \log ^2(x)-20 x \log ^2(x)+4 x^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )}-\frac {5 (-10-5 \log (x)+2 x \log (x))}{x \left (1000-720 x+144 x^2+100 \log (x)-40 x \log (x)+25 \log ^2(x)-20 x \log ^2(x)+4 x^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )}+\frac {20 (-10-5 \log (x)+2 x \log (x))}{(-5+2 x) \left (1000-720 x+144 x^2+100 \log (x)-40 x \log (x)+25 \log ^2(x)-20 x \log ^2(x)+4 x^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )}\right ) \, dx\\ &=4 \int \frac {-10-5 \log (x)+2 x \log (x)}{\left (1000-720 x+144 x^2+100 \log (x)-40 x \log (x)+25 \log ^2(x)-20 x \log ^2(x)+4 x^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )} \, dx-10 \int \frac {-10-5 \log (x)+2 x \log (x)}{x \left (1000-720 x+144 x^2+100 \log (x)-40 x \log (x)+25 \log ^2(x)-20 x \log ^2(x)+4 x^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )} \, dx+40 \int \frac {-10-5 \log (x)+2 x \log (x)}{(-5+2 x) \left (1000-720 x+144 x^2+100 \log (x)-40 x \log (x)+25 \log ^2(x)-20 x \log ^2(x)+4 x^2 \log ^2(x)\right ) \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 44, normalized size = 1.76 \begin {gather*} \log \left (\log (2)+\log \left (\frac {2 \left (125-90 x+18 x^2\right )}{(5-2 x)^2}+\frac {5 \log (x)}{5-2 x}+\frac {\log ^2(x)}{4}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-500 - 80*x^2 + (-250 + 100*x - 40*x^2 + 16*x^3)*Log[x])/((-5000*x + 5600*x^2 - 2160*x^3 + 288*x^4)
*Log[2] + (-500*x + 400*x^2 - 80*x^3)*Log[2]*Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x^4)*Log[2]*Log[x]^2 + (-
5000*x + 5600*x^2 - 2160*x^3 + 288*x^4 + (-500*x + 400*x^2 - 80*x^3)*Log[x] + (-125*x + 150*x^2 - 60*x^3 + 8*x
^4)*Log[x]^2)*Log[(1000 - 720*x + 144*x^2 + (100 - 40*x)*Log[x] + (25 - 20*x + 4*x^2)*Log[x]^2)/(100 - 80*x +
16*x^2)]),x]

[Out]

Log[Log[2] + Log[(2*(125 - 90*x + 18*x^2))/(5 - 2*x)^2 + (5*Log[x])/(5 - 2*x) + Log[x]^2/4]]

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fricas [B]  time = 0.57, size = 53, normalized size = 2.12 \begin {gather*} \log \left (\log \relax (2) + \log \left (\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} \log \relax (x)^{2} + 144 \, x^{2} - 20 \, {\left (2 \, x - 5\right )} \log \relax (x) - 720 \, x + 1000}{4 \, {\left (4 \, x^{2} - 20 \, x + 25\right )}}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+150*x^2-125*x)*log(x)^2+(-80*x^3+400*x
^2-500*x)*log(x)+288*x^4-2160*x^3+5600*x^2-5000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-72
0*x+1000)/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(2)*log(x
)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log(2)),x, algorithm="fricas")

[Out]

log(log(2) + log(1/4*((4*x^2 - 20*x + 25)*log(x)^2 + 144*x^2 - 20*(2*x - 5)*log(x) - 720*x + 1000)/(4*x^2 - 20
*x + 25)))

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giac [B]  time = 1.21, size = 61, normalized size = 2.44 \begin {gather*} \log \left (-\log \relax (2) + \log \left (4 \, x^{2} \log \relax (x)^{2} - 20 \, x \log \relax (x)^{2} + 144 \, x^{2} - 40 \, x \log \relax (x) + 25 \, \log \relax (x)^{2} - 720 \, x + 100 \, \log \relax (x) + 1000\right ) - \log \left (4 \, x^{2} - 20 \, x + 25\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+150*x^2-125*x)*log(x)^2+(-80*x^3+400*x
^2-500*x)*log(x)+288*x^4-2160*x^3+5600*x^2-5000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-72
0*x+1000)/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(2)*log(x
)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log(2)),x, algorithm="giac")

[Out]

log(-log(2) + log(4*x^2*log(x)^2 - 20*x*log(x)^2 + 144*x^2 - 40*x*log(x) + 25*log(x)^2 - 720*x + 100*log(x) +
1000) - log(4*x^2 - 20*x + 25))

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maple [C]  time = 0.72, size = 406, normalized size = 16.24

method	result	size
risch	\(\ln \left (\ln \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{\left (x -\frac {5}{2}\right )^{2}}\right ) \mathrm {csgn}\left (i \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )}{\left (x -\frac {5}{2}\right )^{2}}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (x -\frac {5}{2}\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )}{\left (x -\frac {5}{2}\right )^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )\right ) \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )}{\left (x -\frac {5}{2}\right )^{2}}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (\left (\ln \relax (x )^{2}+36\right ) x^{2}+\left (-5 \ln \relax (x )^{2}-10 \ln \relax (x )-180\right ) x +\frac {25 \ln \relax (x )^{2}}{4}+25 \ln \relax (x )+250\right )}{\left (x -\frac {5}{2}\right )^{2}}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x -\frac {5}{2}\right )\right )^{2} \mathrm {csgn}\left (i \left (x -\frac {5}{2}\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x -\frac {5}{2}\right )\right ) \mathrm {csgn}\left (i \left (x -\frac {5}{2}\right )^{2}\right )^{2}-\pi \mathrm {csgn}\left (i \left (x -\frac {5}{2}\right )^{2}\right )^{3}-2 i \ln \relax (2)-4 i \ln \left (x -\frac {5}{2}\right )\right )}{2}\right )\)	\(406\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*x^3-40*x^2+100*x-250)*ln(x)-80*x^2-500)/(((8*x^4-60*x^3+150*x^2-125*x)*ln(x)^2+(-80*x^3+400*x^2-500*x
)*ln(x)+288*x^4-2160*x^3+5600*x^2-5000*x)*ln(((4*x^2-20*x+25)*ln(x)^2+(-40*x+100)*ln(x)+144*x^2-720*x+1000)/(1
6*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*ln(2)*ln(x)^2+(-80*x^3+400*x^2-500*x)*ln(2)*ln(x)+(288*x^4-2160*
x^3+5600*x^2-5000*x)*ln(2)),x,method=_RETURNVERBOSE)

[Out]

ln(ln((ln(x)^2+36)*x^2+(-5*ln(x)^2-10*ln(x)-180)*x+25/4*ln(x)^2+25*ln(x)+250)-1/2*I*(Pi*csgn(I/(x-5/2)^2)*csgn
(I*((ln(x)^2+36)*x^2+(-5*ln(x)^2-10*ln(x)-180)*x+25/4*ln(x)^2+25*ln(x)+250))*csgn(I/(x-5/2)^2*((ln(x)^2+36)*x^
2+(-5*ln(x)^2-10*ln(x)-180)*x+25/4*ln(x)^2+25*ln(x)+250))-Pi*csgn(I/(x-5/2)^2)*csgn(I/(x-5/2)^2*((ln(x)^2+36)*
x^2+(-5*ln(x)^2-10*ln(x)-180)*x+25/4*ln(x)^2+25*ln(x)+250))^2-Pi*csgn(I*((ln(x)^2+36)*x^2+(-5*ln(x)^2-10*ln(x)
-180)*x+25/4*ln(x)^2+25*ln(x)+250))*csgn(I/(x-5/2)^2*((ln(x)^2+36)*x^2+(-5*ln(x)^2-10*ln(x)-180)*x+25/4*ln(x)^
2+25*ln(x)+250))^2+Pi*csgn(I/(x-5/2)^2*((ln(x)^2+36)*x^2+(-5*ln(x)^2-10*ln(x)-180)*x+25/4*ln(x)^2+25*ln(x)+250
))^3-Pi*csgn(I*(x-5/2))^2*csgn(I*(x-5/2)^2)+2*Pi*csgn(I*(x-5/2))*csgn(I*(x-5/2)^2)^2-Pi*csgn(I*(x-5/2)^2)^3-2*
I*ln(2)-4*I*ln(x-5/2)))

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maxima [A]  time = 0.58, size = 49, normalized size = 1.96 \begin {gather*} \log \left (-\log \relax (2) + \log \left ({\left (4 \, x^{2} - 20 \, x + 25\right )} \log \relax (x)^{2} + 144 \, x^{2} - 20 \, {\left (2 \, x - 5\right )} \log \relax (x) - 720 \, x + 1000\right ) - 2 \, \log \left (2 \, x - 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x^3-40*x^2+100*x-250)*log(x)-80*x^2-500)/(((8*x^4-60*x^3+150*x^2-125*x)*log(x)^2+(-80*x^3+400*x
^2-500*x)*log(x)+288*x^4-2160*x^3+5600*x^2-5000*x)*log(((4*x^2-20*x+25)*log(x)^2+(-40*x+100)*log(x)+144*x^2-72
0*x+1000)/(16*x^2-80*x+100))+(8*x^4-60*x^3+150*x^2-125*x)*log(2)*log(x)^2+(-80*x^3+400*x^2-500*x)*log(2)*log(x
)+(288*x^4-2160*x^3+5600*x^2-5000*x)*log(2)),x, algorithm="maxima")

[Out]

log(-log(2) + log((4*x^2 - 20*x + 25)*log(x)^2 + 144*x^2 - 20*(2*x - 5)*log(x) - 720*x + 1000) - 2*log(2*x - 5
))

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mupad [B]  time = 9.88, size = 50, normalized size = 2.00 \begin {gather*} \ln \left (\ln \left (\frac {2\,\left ({\ln \relax (x)}^2\,\left (4\,x^2-20\,x+25\right )-720\,x-\ln \relax (x)\,\left (40\,x-100\right )+144\,x^2+1000\right )}{16\,x^2-80\,x+100}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((80*x^2 - log(x)*(100*x - 40*x^2 + 16*x^3 - 250) + 500)/(log((log(x)^2*(4*x^2 - 20*x + 25) - 720*x - log(x
)*(40*x - 100) + 144*x^2 + 1000)/(16*x^2 - 80*x + 100))*(5000*x + log(x)^2*(125*x - 150*x^2 + 60*x^3 - 8*x^4)
- 5600*x^2 + 2160*x^3 - 288*x^4 + log(x)*(500*x - 400*x^2 + 80*x^3)) + log(2)*(5000*x - 5600*x^2 + 2160*x^3 -
288*x^4) + log(2)*log(x)*(500*x - 400*x^2 + 80*x^3) + log(2)*log(x)^2*(125*x - 150*x^2 + 60*x^3 - 8*x^4)),x)

[Out]

log(log((2*(log(x)^2*(4*x^2 - 20*x + 25) - 720*x - log(x)*(40*x - 100) + 144*x^2 + 1000))/(16*x^2 - 80*x + 100
)))

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sympy [B]  time = 3.01, size = 49, normalized size = 1.96 \begin {gather*} \log {\left (\log {\left (\frac {144 x^{2} - 720 x + \left (100 - 40 x\right ) \log {\relax (x )} + \left (4 x^{2} - 20 x + 25\right ) \log {\relax (x )}^{2} + 1000}{16 x^{2} - 80 x + 100} \right )} + \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*x**3-40*x**2+100*x-250)*ln(x)-80*x**2-500)/(((8*x**4-60*x**3+150*x**2-125*x)*ln(x)**2+(-80*x**3
+400*x**2-500*x)*ln(x)+288*x**4-2160*x**3+5600*x**2-5000*x)*ln(((4*x**2-20*x+25)*ln(x)**2+(-40*x+100)*ln(x)+14
4*x**2-720*x+1000)/(16*x**2-80*x+100))+(8*x**4-60*x**3+150*x**2-125*x)*ln(2)*ln(x)**2+(-80*x**3+400*x**2-500*x
)*ln(2)*ln(x)+(288*x**4-2160*x**3+5600*x**2-5000*x)*ln(2)),x)

[Out]

log(log((144*x**2 - 720*x + (100 - 40*x)*log(x) + (4*x**2 - 20*x + 25)*log(x)**2 + 1000)/(16*x**2 - 80*x + 100
)) + log(2))

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