3.3.80 \(\int \frac {4 x+(-4 x+x^3) \log (x)+(-4 x+3 x^2) \log (x) \log (\frac {\log (x)}{x})+3 x \log (x) \log ^2(\frac {\log (x)}{x})+\log (x) \log ^3(\frac {\log (x)}{x})}{(-7 x^3+x^4) \log (x)+(-17 x^2+3 x^3) \log (x) \log (\frac {\log (x)}{x})+(-15 x+3 x^2) \log (x) \log ^2(\frac {\log (x)}{x})+(-5+x) \log (x) \log ^3(\frac {\log (x)}{x})} \, dx\)

Optimal. Leaf size=22 \[ \log \left (5-x+\frac {2 x^2}{\left (x+\log \left (\frac {\log (x)}{x}\right )\right )^2}\right ) \]

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Rubi [F]  time = 45.13, 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 {4 x+\left (-4 x+x^3\right ) \log (x)+\left (-4 x+3 x^2\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )+3 x \log (x) \log ^2\left (\frac {\log (x)}{x}\right )+\log (x) \log ^3\left (\frac {\log (x)}{x}\right )}{\left (-7 x^3+x^4\right ) \log (x)+\left (-17 x^2+3 x^3\right ) \log (x) \log \left (\frac {\log (x)}{x}\right )+\left (-15 x+3 x^2\right ) \log (x) \log ^2\left (\frac {\log (x)}{x}\right )+(-5+x) \log (x) \log ^3\left (\frac {\log (x)}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x + (-4*x + x^3)*Log[x] + (-4*x + 3*x^2)*Log[x]*Log[Log[x]/x] + 3*x*Log[x]*Log[Log[x]/x]^2 + Log[x]*Log
[Log[x]/x]^3)/((-7*x^3 + x^4)*Log[x] + (-17*x^2 + 3*x^3)*Log[x]*Log[Log[x]/x] + (-15*x + 3*x^2)*Log[x]*Log[Log
[x]/x]^2 + (-5 + x)*Log[x]*Log[Log[x]/x]^3),x]

[Out]

Log[5 - x] - 2*Log[x + Log[Log[x]/x]] + 20*Defer[Int][((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Lo
g[Log[x]/x]^2)^(-1), x] + 50*Defer[Int][1/((-5 + x)*((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[
Log[x]/x]^2)), x] - 14*Defer[Int][x/((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2), x]
+ 2*Defer[Int][x^2/((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2), x] - 10*Defer[Int]
[1/(Log[x]*((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2)), x] + 2*Defer[Int][x/(Log[x
]*((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2)), x] - 12*Defer[Int][Log[Log[x]/x]/((
-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2), x] + 10*Defer[Int][Log[Log[x]/x]/(x*((-7
+ x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2)), x] + 2*Defer[Int][(x*Log[Log[x]/x])/((-7
+ x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2), x] + 2*Defer[Int][Log[Log[x]/x]/(Log[x]*((-
7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2)), x] - 10*Defer[Int][Log[Log[x]/x]/(x*Log[
x]*((-7 + x)*x^2 + 2*(-5 + x)*x*Log[Log[x]/x] + (-5 + x)*Log[Log[x]/x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{-5+x}-\frac {2 (1-\log (x)+x \log (x))}{x \log (x) \left (x+\log \left (\frac {\log (x)}{x}\right )\right )}+\frac {2 \left (25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )\right )}{(-5+x) x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )}\right ) \, dx\\ &=\log (5-x)-2 \int \frac {1-\log (x)+x \log (x)}{x \log (x) \left (x+\log \left (\frac {\log (x)}{x}\right )\right )} \, dx+2 \int \frac {25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{(-5+x) x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+2 \int \frac {-(-5+x)^2 \left (x+\log \left (\frac {\log (x)}{x}\right )\right )-\log (x) \left (x \left (-25+45 x-12 x^2+x^3\right )+(-5+x)^2 (-1+x) \log \left (\frac {\log (x)}{x}\right )\right )}{(5-x) x \log (x) \left ((-7+x) x^2+2 (-5+x) x \log \left (\frac {\log (x)}{x}\right )+(-5+x) \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+2 \int \left (\frac {-25 x+10 x^2-x^3+25 x \log (x)-45 x^2 \log (x)+12 x^3 \log (x)-x^4 \log (x)-25 \log \left (\frac {\log (x)}{x}\right )+10 x \log \left (\frac {\log (x)}{x}\right )-x^2 \log \left (\frac {\log (x)}{x}\right )+25 \log (x) \log \left (\frac {\log (x)}{x}\right )-35 x \log (x) \log \left (\frac {\log (x)}{x}\right )+11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )-x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{5 x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )}+\frac {25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{5 (-5+x) \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )}\right ) \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+\frac {2}{5} \int \frac {-25 x+10 x^2-x^3+25 x \log (x)-45 x^2 \log (x)+12 x^3 \log (x)-x^4 \log (x)-25 \log \left (\frac {\log (x)}{x}\right )+10 x \log \left (\frac {\log (x)}{x}\right )-x^2 \log \left (\frac {\log (x)}{x}\right )+25 \log (x) \log \left (\frac {\log (x)}{x}\right )-35 x \log (x) \log \left (\frac {\log (x)}{x}\right )+11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )-x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{x \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx+\frac {2}{5} \int \frac {25 x-10 x^2+x^3-25 x \log (x)+45 x^2 \log (x)-12 x^3 \log (x)+x^4 \log (x)+25 \log \left (\frac {\log (x)}{x}\right )-10 x \log \left (\frac {\log (x)}{x}\right )+x^2 \log \left (\frac {\log (x)}{x}\right )-25 \log (x) \log \left (\frac {\log (x)}{x}\right )+35 x \log (x) \log \left (\frac {\log (x)}{x}\right )-11 x^2 \log (x) \log \left (\frac {\log (x)}{x}\right )+x^3 \log (x) \log \left (\frac {\log (x)}{x}\right )}{(-5+x) \log (x) \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\log (5-x)-2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+\frac {2}{5} \int \frac {-(-5+x)^2 \left (x+\log \left (\frac {\log (x)}{x}\right )\right )-\log (x) \left (x \left (-25+45 x-12 x^2+x^3\right )+(-5+x)^2 (-1+x) \log \left (\frac {\log (x)}{x}\right )\right )}{(5-x) \log (x) \left ((-7+x) x^2+2 (-5+x) x \log \left (\frac {\log (x)}{x}\right )+(-5+x) \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx+\frac {2}{5} \int \frac {-(-5+x)^2 \left (x+\log \left (\frac {\log (x)}{x}\right )\right )-\log (x) \left (x \left (-25+45 x-12 x^2+x^3\right )+(-5+x)^2 (-1+x) \log \left (\frac {\log (x)}{x}\right )\right )}{x \log (x) \left ((-7+x) x^2+2 (-5+x) x \log \left (\frac {\log (x)}{x}\right )+(-5+x) \log ^2\left (\frac {\log (x)}{x}\right )\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.19, size = 67, normalized size = 3.05 \begin {gather*} -2 \log \left (x+\log \left (\frac {\log (x)}{x}\right )\right )+\log \left (-7 x^2+x^3-10 x \log \left (\frac {\log (x)}{x}\right )+2 x^2 \log \left (\frac {\log (x)}{x}\right )-5 \log ^2\left (\frac {\log (x)}{x}\right )+x \log ^2\left (\frac {\log (x)}{x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x + (-4*x + x^3)*Log[x] + (-4*x + 3*x^2)*Log[x]*Log[Log[x]/x] + 3*x*Log[x]*Log[Log[x]/x]^2 + Log[
x]*Log[Log[x]/x]^3)/((-7*x^3 + x^4)*Log[x] + (-17*x^2 + 3*x^3)*Log[x]*Log[Log[x]/x] + (-15*x + 3*x^2)*Log[x]*L
og[Log[x]/x]^2 + (-5 + x)*Log[x]*Log[Log[x]/x]^3),x]

[Out]

-2*Log[x + Log[Log[x]/x]] + Log[-7*x^2 + x^3 - 10*x*Log[Log[x]/x] + 2*x^2*Log[Log[x]/x] - 5*Log[Log[x]/x]^2 +
x*Log[Log[x]/x]^2]

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fricas [B]  time = 0.94, size = 62, normalized size = 2.82 \begin {gather*} -2 \, \log \left (x + \log \left (\frac {\log \relax (x)}{x}\right )\right ) + \log \left (x - 5\right ) + \log \left (\frac {x^{3} + {\left (x - 5\right )} \log \left (\frac {\log \relax (x)}{x}\right )^{2} - 7 \, x^{2} + 2 \, {\left (x^{2} - 5 \, x\right )} \log \left (\frac {\log \relax (x)}{x}\right )}{x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x)/x)^3+3*x*log(x)*log(log(x)/x)^2+(3*x^2-4*x)*log(x)*log(log(x)/x)+(x^3-4*x)*log(x)
+4*x)/((x-5)*log(x)*log(log(x)/x)^3+(3*x^2-15*x)*log(x)*log(log(x)/x)^2+(3*x^3-17*x^2)*log(x)*log(log(x)/x)+(x
^4-7*x^3)*log(x)),x, algorithm="fricas")

[Out]

-2*log(x + log(log(x)/x)) + log(x - 5) + log((x^3 + (x - 5)*log(log(x)/x)^2 - 7*x^2 + 2*(x^2 - 5*x)*log(log(x)
/x))/(x - 5))

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giac [B]  time = 5.54, size = 92, normalized size = 4.18 \begin {gather*} \log \left (x^{3} - 2 \, x^{2} \log \relax (x) + x \log \relax (x)^{2} + 2 \, x^{2} \log \left (\log \relax (x)\right ) - 2 \, x \log \relax (x) \log \left (\log \relax (x)\right ) + x \log \left (\log \relax (x)\right )^{2} - 7 \, x^{2} + 10 \, x \log \relax (x) - 5 \, \log \relax (x)^{2} - 10 \, x \log \left (\log \relax (x)\right ) + 10 \, \log \relax (x) \log \left (\log \relax (x)\right ) - 5 \, \log \left (\log \relax (x)\right )^{2}\right ) - 2 \, \log \left (-x + \log \relax (x) - \log \left (\log \relax (x)\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x)/x)^3+3*x*log(x)*log(log(x)/x)^2+(3*x^2-4*x)*log(x)*log(log(x)/x)+(x^3-4*x)*log(x)
+4*x)/((x-5)*log(x)*log(log(x)/x)^3+(3*x^2-15*x)*log(x)*log(log(x)/x)^2+(3*x^3-17*x^2)*log(x)*log(log(x)/x)+(x
^4-7*x^3)*log(x)),x, algorithm="giac")

[Out]

log(x^3 - 2*x^2*log(x) + x*log(x)^2 + 2*x^2*log(log(x)) - 2*x*log(x)*log(log(x)) + x*log(log(x))^2 - 7*x^2 + 1
0*x*log(x) - 5*log(x)^2 - 10*x*log(log(x)) + 10*log(x)*log(log(x)) - 5*log(log(x))^2) - 2*log(-x + log(x) - lo
g(log(x)))

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maple [C]  time = 0.98, size = 1095, normalized size = 49.77

method	result	size
risch	\(\ln \left (x -5\right )-2 \ln \left (\ln \left (\ln \relax (x )\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+2 i x -2 i \ln \relax (x )\right )}{2}\right )+\ln \left (\ln \left (\ln \relax (x )\right )^{2}+\left (-i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )+i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+i \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+2 x -2 \ln \relax (x )\right ) \ln \left (\ln \relax (x )\right )-\frac {8 x^{2} \ln \relax (x )-2 \pi ^{2} x \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+20 \ln \relax (x )^{2}-4 x^{3}+28 x^{2}-4 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-4 i \pi \,x^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+20 i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+20 i \pi x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-20 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-20 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-4 i \ln \relax (x ) \pi x \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}-2 \pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+4 \pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{4}+\pi ^{2} x \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}-4 x \ln \relax (x )^{2}-40 x \ln \relax (x )-5 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+10 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+10 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}-20 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{4}+\pi ^{2} x \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{4}-2 \pi ^{2} x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{5}+\pi ^{2} x \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{4}-2 \pi ^{2} x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{5}+20 i \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}-20 i \pi x \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+4 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{3}+\pi ^{2} x \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{6}-5 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x}\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{4}+10 \pi ^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{5}-5 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{4}+10 \pi ^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{5}-5 \pi ^{2} \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{6}-20 i \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )+4 i \pi \,x^{2} \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )+4 i \ln \relax (x ) \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+4 i \ln \relax (x ) \pi x \,\mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )^{2}+20 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )-4 i \ln \relax (x ) \pi x \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \mathrm {csgn}\left (\frac {i \ln \relax (x )}{x}\right )}{4 \left (x -5\right )}\right )\)	\(1095\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((ln(x)*ln(ln(x)/x)^3+3*x*ln(x)*ln(ln(x)/x)^2+(3*x^2-4*x)*ln(x)*ln(ln(x)/x)+(x^3-4*x)*ln(x)+4*x)/((x-5)*ln(
x)*ln(ln(x)/x)^3+(3*x^2-15*x)*ln(x)*ln(ln(x)/x)^2+(3*x^3-17*x^2)*ln(x)*ln(ln(x)/x)+(x^4-7*x^3)*ln(x)),x,method
=_RETURNVERBOSE)

[Out]

ln(x-5)-2*ln(ln(ln(x))-1/2*I*(Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))-Pi*csgn(I/x)*csgn(I/x*ln(x))^2-Pi*csg
n(I*ln(x))*csgn(I/x*ln(x))^2+Pi*csgn(I/x*ln(x))^3+2*I*x-2*I*ln(x)))+ln(ln(ln(x))^2+(-I*Pi*csgn(I/x)*csgn(I*ln(
x))*csgn(I/x*ln(x))+I*Pi*csgn(I/x)*csgn(I/x*ln(x))^2+I*Pi*csgn(I*ln(x))*csgn(I/x*ln(x))^2-I*Pi*csgn(I/x*ln(x))
^3+2*x-2*ln(x))*ln(ln(x))-1/4*(8*x^2*ln(x)+20*ln(x)^2-4*x^3+28*x^2-4*x*ln(x)^2-40*x*ln(x)-5*Pi^2*csgn(I/x)^2*c
sgn(I*ln(x))^2*csgn(I/x*ln(x))^2+10*Pi^2*csgn(I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^3+10*Pi^2*csgn(I/x)*csgn(I*
ln(x))^2*csgn(I/x*ln(x))^3-20*Pi^2*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))^4+Pi^2*x*csgn(I/x)^2*csgn(I/x*ln(x)
)^4-2*Pi^2*x*csgn(I/x)*csgn(I/x*ln(x))^5+Pi^2*x*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4-2*Pi^2*x*csgn(I*ln(x))*csgn(
I/x*ln(x))^5+20*I*ln(x)*Pi*csgn(I/x*ln(x))^3-20*I*Pi*x*csgn(I/x*ln(x))^3+4*I*Pi*x^2*csgn(I/x*ln(x))^3-5*Pi^2*c
sgn(I/x*ln(x))^6+Pi^2*x*csgn(I/x*ln(x))^6-5*Pi^2*csgn(I/x)^2*csgn(I/x*ln(x))^4+10*Pi^2*csgn(I/x)*csgn(I/x*ln(x
))^5-5*Pi^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^4+10*Pi^2*csgn(I*ln(x))*csgn(I/x*ln(x))^5-20*I*Pi*x*csgn(I/x)*csgn
(I*ln(x))*csgn(I/x*ln(x))+4*I*Pi*x^2*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))+4*I*ln(x)*Pi*x*csgn(I/x)*csgn(I/x
*ln(x))^2+4*I*ln(x)*Pi*x*csgn(I*ln(x))*csgn(I/x*ln(x))^2+20*I*ln(x)*Pi*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))
-4*I*Pi*x^2*csgn(I/x)*csgn(I/x*ln(x))^2-4*I*Pi*x^2*csgn(I*ln(x))*csgn(I/x*ln(x))^2+20*I*Pi*x*csgn(I/x)*csgn(I/
x*ln(x))^2+20*I*Pi*x*csgn(I*ln(x))*csgn(I/x*ln(x))^2-20*I*ln(x)*Pi*csgn(I/x)*csgn(I/x*ln(x))^2-20*I*ln(x)*Pi*c
sgn(I*ln(x))*csgn(I/x*ln(x))^2-4*I*ln(x)*Pi*x*csgn(I/x*ln(x))^3-2*Pi^2*x*csgn(I/x)*csgn(I*ln(x))^2*csgn(I/x*ln
(x))^3+4*Pi^2*x*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x))^4+Pi^2*x*csgn(I/x)^2*csgn(I*ln(x))^2*csgn(I/x*ln(x))^2
-2*Pi^2*x*csgn(I/x)^2*csgn(I*ln(x))*csgn(I/x*ln(x))^3-4*I*ln(x)*Pi*x*csgn(I/x)*csgn(I*ln(x))*csgn(I/x*ln(x)))/
(x-5))

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maxima [B]  time = 0.53, size = 80, normalized size = 3.64 \begin {gather*} -2 \, \log \left (x - \log \relax (x) + \log \left (\log \relax (x)\right )\right ) + \log \left (x - 5\right ) + \log \left (\frac {x^{3} + {\left (x - 5\right )} \log \relax (x)^{2} + {\left (x - 5\right )} \log \left (\log \relax (x)\right )^{2} - 7 \, x^{2} - 2 \, {\left (x^{2} - 5 \, x\right )} \log \relax (x) + 2 \, {\left (x^{2} - {\left (x - 5\right )} \log \relax (x) - 5 \, x\right )} \log \left (\log \relax (x)\right )}{x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((log(x)*log(log(x)/x)^3+3*x*log(x)*log(log(x)/x)^2+(3*x^2-4*x)*log(x)*log(log(x)/x)+(x^3-4*x)*log(x)
+4*x)/((x-5)*log(x)*log(log(x)/x)^3+(3*x^2-15*x)*log(x)*log(log(x)/x)^2+(3*x^3-17*x^2)*log(x)*log(log(x)/x)+(x
^4-7*x^3)*log(x)),x, algorithm="maxima")

[Out]

-2*log(x - log(x) + log(log(x))) + log(x - 5) + log((x^3 + (x - 5)*log(x)^2 + (x - 5)*log(log(x))^2 - 7*x^2 -
2*(x^2 - 5*x)*log(x) + 2*(x^2 - (x - 5)*log(x) - 5*x)*log(log(x)))/(x - 5))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int -\frac {\ln \relax (x)\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^3+3\,x\,\ln \relax (x)\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2-\ln \relax (x)\,\left (4\,x-3\,x^2\right )\,\ln \left (\frac {\ln \relax (x)}{x}\right )+4\,x-\ln \relax (x)\,\left (4\,x-x^3\right )}{-\ln \relax (x)\,\left (x-5\right )\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^3+\ln \relax (x)\,\left (15\,x-3\,x^2\right )\,{\ln \left (\frac {\ln \relax (x)}{x}\right )}^2+\ln \relax (x)\,\left (17\,x^2-3\,x^3\right )\,\ln \left (\frac {\ln \relax (x)}{x}\right )+\ln \relax (x)\,\left (7\,x^3-x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + log(log(x)/x)^3*log(x) - log(x)*(4*x - x^3) - log(log(x)/x)*log(x)*(4*x - 3*x^2) + 3*x*log(log(x)/
x)^2*log(x))/(log(x)*(7*x^3 - x^4) - log(log(x)/x)^3*log(x)*(x - 5) + log(log(x)/x)^2*log(x)*(15*x - 3*x^2) +
log(log(x)/x)*log(x)*(17*x^2 - 3*x^3)),x)

[Out]

int(-(4*x + log(log(x)/x)^3*log(x) - log(x)*(4*x - x^3) - log(log(x)/x)*log(x)*(4*x - 3*x^2) + 3*x*log(log(x)/
x)^2*log(x))/(log(x)*(7*x^3 - x^4) - log(log(x)/x)^3*log(x)*(x - 5) + log(log(x)/x)^2*log(x)*(15*x - 3*x^2) +
log(log(x)/x)*log(x)*(17*x^2 - 3*x^3)), x)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: PolynomialError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((ln(x)*ln(ln(x)/x)**3+3*x*ln(x)*ln(ln(x)/x)**2+(3*x**2-4*x)*ln(x)*ln(ln(x)/x)+(x**3-4*x)*ln(x)+4*x)/
((x-5)*ln(x)*ln(ln(x)/x)**3+(3*x**2-15*x)*ln(x)*ln(ln(x)/x)**2+(3*x**3-17*x**2)*ln(x)*ln(ln(x)/x)+(x**4-7*x**3
)*ln(x)),x)

[Out]

Exception raised: PolynomialError

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