3.86.16 \(\int \frac {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+(-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))) \log (\frac {x^2+5 \log (6-\log (x))}{x})}{(-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))) \log ^2(\frac {x^2+5 \log (6-\log (x))}{x})} \, dx\)

Optimal. Leaf size=21 \[ \frac {x^2}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \]

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Rubi [F]  time = 4.54, 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 {-5 x+6 x^3-x^3 \log (x)+(-30 x+5 x \log (x)) \log (6-\log (x))+\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{\left (-6 x^2+x^2 \log (x)+(-30+5 \log (x)) \log (6-\log (x))\right ) \log ^2\left (\frac {x^2+5 \log (6-\log (x))}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5*x + 6*x^3 - x^3*Log[x] + (-30*x + 5*x*Log[x])*Log[6 - Log[x]] + (-12*x^3 + 2*x^3*Log[x] + (-60*x + 10*
x*Log[x])*Log[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x])/((-6*x^2 + x^2*Log[x] + (-30 + 5*Log[x])*Log[6 -
Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x]^2),x]

[Out]

-5*Defer[Int][x/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] + 6*Defer[Int][
x^3/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] - Defer[Int][(x^3*Log[x])/(
(-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] - 30*Defer[Int][(x*Log[6 - Log[x
]])/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] + 5*Defer[Int][(x*Log[x]*Lo
g[6 - Log[x]])/((-6 + Log[x])*(x^2 + 5*Log[6 - Log[x]])*Log[x + (5*Log[6 - Log[x]])/x]^2), x] + 2*Defer[Int][x
/Log[x + (5*Log[6 - Log[x]])/x], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 x-6 x^3+x^3 \log (x)-(-30 x+5 x \log (x)) \log (6-\log (x))-\left (-12 x^3+2 x^3 \log (x)+(-60 x+10 x \log (x)) \log (6-\log (x))\right ) \log \left (\frac {x^2+5 \log (6-\log (x))}{x}\right )}{(6-\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx\\ &=\int \frac {x \left (5-6 x^2+x^2 \log (x)-5 (-6+\log (x)) \log (6-\log (x))-2 (-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log \left (x+\frac {5 \log (6-\log (x))}{x}\right )\right )}{(6-\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx\\ &=\int \left (-\frac {x \left (5-6 x^2+x^2 \log (x)+30 \log (6-\log (x))-5 \log (x) \log (6-\log (x))\right )}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {2 x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )}\right ) \, dx\\ &=2 \int \frac {x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-\int \frac {x \left (5-6 x^2+x^2 \log (x)+30 \log (6-\log (x))-5 \log (x) \log (6-\log (x))\right )}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx\\ &=2 \int \frac {x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-\int \left (\frac {5 x}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {6 x^3}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {x^3 \log (x)}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}+\frac {30 x \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}-\frac {5 x \log (x) \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )}\right ) \, dx\\ &=2 \int \frac {x}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-5 \int \frac {x}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx+5 \int \frac {x \log (x) \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx+6 \int \frac {x^3}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-30 \int \frac {x \log (6-\log (x))}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx-\int \frac {x^3 \log (x)}{(-6+\log (x)) \left (x^2+5 \log (6-\log (x))\right ) \log ^2\left (x+\frac {5 \log (6-\log (x))}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 21, normalized size = 1.00 \begin {gather*} \frac {x^2}{\log \left (x+\frac {5 \log (6-\log (x))}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x + 6*x^3 - x^3*Log[x] + (-30*x + 5*x*Log[x])*Log[6 - Log[x]] + (-12*x^3 + 2*x^3*Log[x] + (-60*x
+ 10*x*Log[x])*Log[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x])/((-6*x^2 + x^2*Log[x] + (-30 + 5*Log[x])*Lo
g[6 - Log[x]])*Log[(x^2 + 5*Log[6 - Log[x]])/x]^2),x]

[Out]

x^2/Log[x + (5*Log[6 - Log[x]])/x]

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fricas [A]  time = 0.59, size = 24, normalized size = 1.14 \begin {gather*} \frac {x^{2}}{\log \left (\frac {x^{2} + 5 \, \log \left (-\log \relax (x) + 6\right )}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5*log(-log(x)+6)+x^2)/x)+(5*x*log(x)-3
0*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5*x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6
)+x^2)/x)^2,x, algorithm="fricas")

[Out]

x^2/log((x^2 + 5*log(-log(x) + 6))/x)

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giac [A]  time = 0.39, size = 25, normalized size = 1.19 \begin {gather*} \frac {x^{2}}{\log \left (x^{2} + 5 \, \log \left (-\log \relax (x) + 6\right )\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5*log(-log(x)+6)+x^2)/x)+(5*x*log(x)-3
0*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5*x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6
)+x^2)/x)^2,x, algorithm="giac")

[Out]

x^2/(log(x^2 + 5*log(-log(x) + 6)) - log(x))

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maple [C]  time = 0.12, size = 176, normalized size = 8.38

method	result	size
risch	\(\frac {2 i x^{2}}{\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )}{x}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )\right ) \mathrm {csgn}\left (\frac {i \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )}{x}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )}{x}\right )^{3}-2 i \ln \relax (x )+2 i \ln \left (5 \ln \left (-\ln \relax (x )+6\right )+x^{2}\right )}\)	\(176\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((10*x*ln(x)-60*x)*ln(-ln(x)+6)+2*x^3*ln(x)-12*x^3)*ln((5*ln(-ln(x)+6)+x^2)/x)+(5*x*ln(x)-30*x)*ln(-ln(x)
+6)-x^3*ln(x)+6*x^3-5*x)/((5*ln(x)-30)*ln(-ln(x)+6)+x^2*ln(x)-6*x^2)/ln((5*ln(-ln(x)+6)+x^2)/x)^2,x,method=_RE
TURNVERBOSE)

[Out]

2*I*x^2/(Pi*csgn(I/x)*csgn(I*(5*ln(-ln(x)+6)+x^2))*csgn(I/x*(5*ln(-ln(x)+6)+x^2))-Pi*csgn(I/x)*csgn(I/x*(5*ln(
-ln(x)+6)+x^2))^2-Pi*csgn(I*(5*ln(-ln(x)+6)+x^2))*csgn(I/x*(5*ln(-ln(x)+6)+x^2))^2+Pi*csgn(I/x*(5*ln(-ln(x)+6)
+x^2))^3-2*I*ln(x)+2*I*ln(5*ln(-ln(x)+6)+x^2))

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maxima [A]  time = 0.41, size = 25, normalized size = 1.19 \begin {gather*} \frac {x^{2}}{\log \left (x^{2} + 5 \, \log \left (-\log \relax (x) + 6\right )\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x*log(x)-60*x)*log(-log(x)+6)+2*x^3*log(x)-12*x^3)*log((5*log(-log(x)+6)+x^2)/x)+(5*x*log(x)-3
0*x)*log(-log(x)+6)-x^3*log(x)+6*x^3-5*x)/((5*log(x)-30)*log(-log(x)+6)+x^2*log(x)-6*x^2)/log((5*log(-log(x)+6
)+x^2)/x)^2,x, algorithm="maxima")

[Out]

x^2/(log(x^2 + 5*log(-log(x) + 6)) - log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {5\,x+x^3\,\ln \relax (x)+\ln \left (6-\ln \relax (x)\right )\,\left (30\,x-5\,x\,\ln \relax (x)\right )+\ln \left (\frac {5\,\ln \left (6-\ln \relax (x)\right )+x^2}{x}\right )\,\left (\ln \left (6-\ln \relax (x)\right )\,\left (60\,x-10\,x\,\ln \relax (x)\right )-2\,x^3\,\ln \relax (x)+12\,x^3\right )-6\,x^3}{{\ln \left (\frac {5\,\ln \left (6-\ln \relax (x)\right )+x^2}{x}\right )}^2\,\left (\ln \left (6-\ln \relax (x)\right )\,\left (5\,\ln \relax (x)-30\right )+x^2\,\ln \relax (x)-6\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + x^3*log(x) + log(6 - log(x))*(30*x - 5*x*log(x)) + log((5*log(6 - log(x)) + x^2)/x)*(log(6 - log(x
))*(60*x - 10*x*log(x)) - 2*x^3*log(x) + 12*x^3) - 6*x^3)/(log((5*log(6 - log(x)) + x^2)/x)^2*(log(6 - log(x))
*(5*log(x) - 30) + x^2*log(x) - 6*x^2)),x)

[Out]

-int((5*x + x^3*log(x) + log(6 - log(x))*(30*x - 5*x*log(x)) + log((5*log(6 - log(x)) + x^2)/x)*(log(6 - log(x
))*(60*x - 10*x*log(x)) - 2*x^3*log(x) + 12*x^3) - 6*x^3)/(log((5*log(6 - log(x)) + x^2)/x)^2*(log(6 - log(x))
*(5*log(x) - 30) + x^2*log(x) - 6*x^2)), x)

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sympy [A]  time = 1.17, size = 17, normalized size = 0.81 \begin {gather*} \frac {x^{2}}{\log {\left (\frac {x^{2} + 5 \log {\left (6 - \log {\relax (x )} \right )}}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((10*x*ln(x)-60*x)*ln(-ln(x)+6)+2*x**3*ln(x)-12*x**3)*ln((5*ln(-ln(x)+6)+x**2)/x)+(5*x*ln(x)-30*x)*
ln(-ln(x)+6)-x**3*ln(x)+6*x**3-5*x)/((5*ln(x)-30)*ln(-ln(x)+6)+x**2*ln(x)-6*x**2)/ln((5*ln(-ln(x)+6)+x**2)/x)*
*2,x)

[Out]

x**2/log((x**2 + 5*log(6 - log(x)))/x)

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