3.100.43 \(\int \frac {((-5 x^2+x^3) \log (4)+(5-x) \log ^2(4) \log (5-x)+((15 x^2-3 x^3) \log (4)+x \log ^2(4)+(-5+x) \log ^2(4) \log (5-x)) \log (x)) \log (\log (2))}{-5 x^6+x^7+(10 x^4-2 x^5) \log (4) \log (5-x)+(-5 x^2+x^3) \log ^2(4) \log ^2(5-x)} \, dx\)

Optimal. Leaf size=31 \[ \left (6-\frac {\log (x)}{x \left (-\frac {x^2}{\log (4)}+\log (5-x)\right )}\right ) \log (\log (2)) \]

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Rubi [F]  time = 15.01, 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 (\left (-5 x^2+x^3\right ) \log (4)+(5-x) \log ^2(4) \log (5-x)+\left (\left (15 x^2-3 x^3\right ) \log (4)+x \log ^2(4)+(-5+x) \log ^2(4) \log (5-x)\right ) \log (x)\right ) \log (\log (2))}{-5 x^6+x^7+\left (10 x^4-2 x^5\right ) \log (4) \log (5-x)+\left (-5 x^2+x^3\right ) \log ^2(4) \log ^2(5-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(((-5*x^2 + x^3)*Log[4] + (5 - x)*Log[4]^2*Log[5 - x] + ((15*x^2 - 3*x^3)*Log[4] + x*Log[4]^2 + (-5 + x)*L
og[4]^2*Log[5 - x])*Log[x])*Log[Log[2]])/(-5*x^6 + x^7 + (10*x^4 - 2*x^5)*Log[4]*Log[5 - x] + (-5*x^2 + x^3)*L
og[4]^2*Log[5 - x]^2),x]

[Out]

Log[4]*Log[Log[2]]*Defer[Int][1/(x^2*(x^2 - Log[4]*Log[5 - x])), x] - 3*Log[4]*Log[Log[2]]*Defer[Int][Log[x]/(
x^2 - Log[4]*Log[5 - x])^2, x] + (Log[4]^2*Log[Log[2]]*Defer[Int][Log[x]/((-5 + x)*(x^2 - Log[4]*Log[5 - x])^2
), x])/5 - (Log[4]^2*Log[Log[2]]*Defer[Int][Log[x]/(x*(x^2 - Log[4]*Log[5 - x])^2), x])/5 + (Log[4]^2*Log[Log[
2]]*Defer[Int][(Log[5 - x]*Log[x])/(x^2 - Log[4]*Log[5 - x])^2, x])/25 - (Log[4]^2*Log[Log[2]]*Defer[Int][(Log
[5 - x]*Log[x])/(-x^2 + Log[4]*Log[5 - x])^2, x])/25 + Log[4]^2*Log[Log[2]]*Defer[Int][(Log[5 - x]*Log[x])/(x^
2*(-x^2 + Log[4]*Log[5 - x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (\log (2)) \int \frac {\left (-5 x^2+x^3\right ) \log (4)+(5-x) \log ^2(4) \log (5-x)+\left (\left (15 x^2-3 x^3\right ) \log (4)+x \log ^2(4)+(-5+x) \log ^2(4) \log (5-x)\right ) \log (x)}{-5 x^6+x^7+\left (10 x^4-2 x^5\right ) \log (4) \log (5-x)+\left (-5 x^2+x^3\right ) \log ^2(4) \log ^2(5-x)} \, dx\\ &=\log (\log (2)) \int \frac {\log (4) \left (-((-5+x) \log (4) \log (5-x) (-1+\log (x)))-x \left ((-5+x) x+\left (15 x-3 x^2+\log (4)\right ) \log (x)\right )\right )}{(5-x) \left (x^3-x \log (4) \log (5-x)\right )^2} \, dx\\ &=(\log (4) \log (\log (2))) \int \frac {-((-5+x) \log (4) \log (5-x) (-1+\log (x)))-x \left ((-5+x) x+\left (15 x-3 x^2+\log (4)\right ) \log (x)\right )}{(5-x) \left (x^3-x \log (4) \log (5-x)\right )^2} \, dx\\ &=(\log (4) \log (\log (2))) \int \left (\frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )}-\frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{(-5+x) x^2 \left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx\\ &=(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx-(\log (4) \log (\log (2))) \int \frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{(-5+x) x^2 \left (x^2-\log (4) \log (5-x)\right )^2} \, dx\\ &=(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx-(\log (4) \log (\log (2))) \int \left (\frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{25 (-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}-\frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{5 x^2 \left (x^2-\log (4) \log (5-x)\right )^2}-\frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{25 x \left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{25} (\log (4) \log (\log (2))) \int \frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx\right )+\frac {1}{25} (\log (4) \log (\log (2))) \int \frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{5} (\log (4) \log (\log (2))) \int \frac {\left (-15 x^2+3 x^3-x \log (4)+5 \log (4) \log (5-x)-x \log (4) \log (5-x)\right ) \log (x)}{x^2 \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx\\ &=-\left (\frac {1}{25} (\log (4) \log (\log (2))) \int \left (-\frac {15 x^2 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}+\frac {3 x^3 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}-\frac {x \log (4) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}+\frac {5 \log (4) \log (5-x) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}-\frac {x \log (4) \log (5-x) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx\right )+\frac {1}{25} (\log (4) \log (\log (2))) \int \left (-\frac {15 x \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {3 x^2 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}-\frac {\log (4) \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {5 \log (4) \log (5-x) \log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2}-\frac {\log (4) \log (5-x) \log (x)}{\left (-x^2+\log (4) \log (5-x)\right )^2}\right ) \, dx+\frac {1}{5} (\log (4) \log (\log (2))) \int \left (-\frac {15 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {3 x \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}-\frac {\log (4) \log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2}-\frac {\log (4) \log (5-x) \log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2}+\frac {5 \log (4) \log (5-x) \log (x)}{x^2 \left (-x^2+\log (4) \log (5-x)\right )^2}\right ) \, dx+(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx\\ &=\frac {1}{25} (3 \log (4) \log (\log (2))) \int \frac {x^2 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} (3 \log (4) \log (\log (2))) \int \frac {x^3 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{5} (3 \log (4) \log (\log (2))) \int \frac {x^2 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx-(3 \log (4) \log (\log (2))) \int \frac {\log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {x \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {x \log (5-x) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{\left (-x^2+\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{5} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{5} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{x^2 \left (-x^2+\log (4) \log (5-x)\right )^2} \, dx\\ &=\frac {1}{25} (3 \log (4) \log (\log (2))) \int \frac {x^2 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} (3 \log (4) \log (\log (2))) \int \left (\frac {25 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {125 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}+\frac {5 x \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {x^2 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx+\frac {1}{5} (3 \log (4) \log (\log (2))) \int \left (\frac {5 \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {25 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}+\frac {x \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx+(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx-(3 \log (4) \log (\log (2))) \int \frac {\log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{\left (-x^2+\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \left (\frac {\log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {5 \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx+\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \left (\frac {\log (5-x) \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2}+\frac {5 \log (5-x) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2}\right ) \, dx-\frac {1}{5} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{5} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{x^2 \left (-x^2+\log (4) \log (5-x)\right )^2} \, dx\\ &=(\log (4) \log (\log (2))) \int \frac {1}{x^2 \left (x^2-\log (4) \log (5-x)\right )} \, dx-(3 \log (4) \log (\log (2))) \int \frac {\log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{\left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{25} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{\left (-x^2+\log (4) \log (5-x)\right )^2} \, dx+\frac {1}{5} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (x)}{(-5+x) \left (x^2-\log (4) \log (5-x)\right )^2} \, dx-\frac {1}{5} \left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (x)}{x \left (x^2-\log (4) \log (5-x)\right )^2} \, dx+\left (\log ^2(4) \log (\log (2))\right ) \int \frac {\log (5-x) \log (x)}{x^2 \left (-x^2+\log (4) \log (5-x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.05, size = 25, normalized size = 0.81 \begin {gather*} \frac {\log (4) \log (x) \log (\log (2))}{x^3-x \log (4) \log (5-x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(((-5*x^2 + x^3)*Log[4] + (5 - x)*Log[4]^2*Log[5 - x] + ((15*x^2 - 3*x^3)*Log[4] + x*Log[4]^2 + (-5
+ x)*Log[4]^2*Log[5 - x])*Log[x])*Log[Log[2]])/(-5*x^6 + x^7 + (10*x^4 - 2*x^5)*Log[4]*Log[5 - x] + (-5*x^2 +
x^3)*Log[4]^2*Log[5 - x]^2),x]

[Out]

(Log[4]*Log[x]*Log[Log[2]])/(x^3 - x*Log[4]*Log[5 - x])

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fricas [A]  time = 0.52, size = 26, normalized size = 0.84 \begin {gather*} \frac {2 \, \log \relax (2) \log \relax (x) \log \left (\log \relax (2)\right )}{x^{3} - 2 \, x \log \relax (2) \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-5)*log(2)^2*log(5-x)+4*x*log(2)^2+2*(-3*x^3+15*x^2)*log(2))*log(x)+4*(5-x)*log(2)^2*log(5-x)+
2*(x^3-5*x^2)*log(2))*log(log(2))/(4*(x^3-5*x^2)*log(2)^2*log(5-x)^2+2*(-2*x^5+10*x^4)*log(2)*log(5-x)+x^7-5*x
^6),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.30, size = 26, normalized size = 0.84 \begin {gather*} \frac {2 \, \log \relax (2) \log \relax (x) \log \left (\log \relax (2)\right )}{x^{3} - 2 \, x \log \relax (2) \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-5)*log(2)^2*log(5-x)+4*x*log(2)^2+2*(-3*x^3+15*x^2)*log(2))*log(x)+4*(5-x)*log(2)^2*log(5-x)+
2*(x^3-5*x^2)*log(2))*log(log(2))/(4*(x^3-5*x^2)*log(2)^2*log(5-x)^2+2*(-2*x^5+10*x^4)*log(2)*log(5-x)+x^7-5*x
^6),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.06, size = 31, normalized size = 1.00

method	result	size
risch	\(-\frac {2 \ln \left (\ln \relax (2)\right ) \ln \relax (2) \ln \relax (x )}{x \left (2 \ln \relax (2) \ln \left (5-x \right )-x^{2}\right )}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*(x-5)*ln(2)^2*ln(5-x)+4*x*ln(2)^2+2*(-3*x^3+15*x^2)*ln(2))*ln(x)+4*(5-x)*ln(2)^2*ln(5-x)+2*(x^3-5*x^2)
*ln(2))*ln(ln(2))/(4*(x^3-5*x^2)*ln(2)^2*ln(5-x)^2+2*(-2*x^5+10*x^4)*ln(2)*ln(5-x)+x^7-5*x^6),x,method=_RETURN
VERBOSE)

[Out]

-2*ln(ln(2))*ln(2)/x*ln(x)/(2*ln(2)*ln(5-x)-x^2)

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maxima [A]  time = 0.49, size = 26, normalized size = 0.84 \begin {gather*} \frac {2 \, \log \relax (2) \log \relax (x) \log \left (\log \relax (2)\right )}{x^{3} - 2 \, x \log \relax (2) \log \left (-x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-5)*log(2)^2*log(5-x)+4*x*log(2)^2+2*(-3*x^3+15*x^2)*log(2))*log(x)+4*(5-x)*log(2)^2*log(5-x)+
2*(x^3-5*x^2)*log(2))*log(log(2))/(4*(x^3-5*x^2)*log(2)^2*log(5-x)^2+2*(-2*x^5+10*x^4)*log(2)*log(5-x)+x^7-5*x
^6),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\ln \left (\ln \relax (2)\right )\,\left (2\,\ln \relax (2)\,\left (5\,x^2-x^3\right )-\ln \relax (x)\,\left (2\,\ln \relax (2)\,\left (15\,x^2-3\,x^3\right )+4\,x\,{\ln \relax (2)}^2+4\,{\ln \relax (2)}^2\,\ln \left (5-x\right )\,\left (x-5\right )\right )+4\,{\ln \relax (2)}^2\,\ln \left (5-x\right )\,\left (x-5\right )\right )}{5\,x^6-x^7-2\,\ln \relax (2)\,\ln \left (5-x\right )\,\left (10\,x^4-2\,x^5\right )+4\,{\ln \relax (2)}^2\,{\ln \left (5-x\right )}^2\,\left (5\,x^2-x^3\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(2))*(2*log(2)*(5*x^2 - x^3) - log(x)*(2*log(2)*(15*x^2 - 3*x^3) + 4*x*log(2)^2 + 4*log(2)^2*log(5
- x)*(x - 5)) + 4*log(2)^2*log(5 - x)*(x - 5)))/(5*x^6 - x^7 - 2*log(2)*log(5 - x)*(10*x^4 - 2*x^5) + 4*log(2
)^2*log(5 - x)^2*(5*x^2 - x^3)),x)

[Out]

int((log(log(2))*(2*log(2)*(5*x^2 - x^3) - log(x)*(2*log(2)*(15*x^2 - 3*x^3) + 4*x*log(2)^2 + 4*log(2)^2*log(5
- x)*(x - 5)) + 4*log(2)^2*log(5 - x)*(x - 5)))/(5*x^6 - x^7 - 2*log(2)*log(5 - x)*(10*x^4 - 2*x^5) + 4*log(2
)^2*log(5 - x)^2*(5*x^2 - x^3)), x)

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sympy [A]  time = 0.36, size = 29, normalized size = 0.94 \begin {gather*} - \frac {2 \log {\relax (2 )} \log {\relax (x )} \log {\left (\log {\relax (2 )} \right )}}{- x^{3} + 2 x \log {\relax (2 )} \log {\left (5 - x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*(x-5)*ln(2)**2*ln(5-x)+4*x*ln(2)**2+2*(-3*x**3+15*x**2)*ln(2))*ln(x)+4*(5-x)*ln(2)**2*ln(5-x)+2*
(x**3-5*x**2)*ln(2))*ln(ln(2))/(4*(x**3-5*x**2)*ln(2)**2*ln(5-x)**2+2*(-2*x**5+10*x**4)*ln(2)*ln(5-x)+x**7-5*x
**6),x)

[Out]

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

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