3.31.87 \(\int \frac {108 x^3+30 x^4+2 x^5+(-108 x^3-21 x^4-x^5) \log (x)+(27 x^3+3 x^4) \log ^2(x)+((432 x^3+105 x^4+7 x^5) \log (x)+(-432 x^3-87 x^4-4 x^5) \log ^2(x)+(108 x^3+15 x^4) \log ^3(x)) \log (\log (x))}{(36+12 x+x^2) \log (x)+(-36-6 x) \log ^2(x)+9 \log ^3(x)} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^4 (9+x) \log (\log (x))}{3-\frac {x}{-2+\log (x)}} \]

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Rubi [F]  time = 3.97, 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 {108 x^3+30 x^4+2 x^5+\left (-108 x^3-21 x^4-x^5\right ) \log (x)+\left (27 x^3+3 x^4\right ) \log ^2(x)+\left (\left (432 x^3+105 x^4+7 x^5\right ) \log (x)+\left (-432 x^3-87 x^4-4 x^5\right ) \log ^2(x)+\left (108 x^3+15 x^4\right ) \log ^3(x)\right ) \log (\log (x))}{\left (36+12 x+x^2\right ) \log (x)+(-36-6 x) \log ^2(x)+9 \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(108*x^3 + 30*x^4 + 2*x^5 + (-108*x^3 - 21*x^4 - x^5)*Log[x] + (27*x^3 + 3*x^4)*Log[x]^2 + ((432*x^3 + 105
*x^4 + 7*x^5)*Log[x] + (-432*x^3 - 87*x^4 - 4*x^5)*Log[x]^2 + (108*x^3 + 15*x^4)*Log[x]^3)*Log[Log[x]])/((36 +
12*x + x^2)*Log[x] + (-36 - 6*x)*Log[x]^2 + 9*Log[x]^3),x]

[Out]

648*Defer[Int][(6 + x - 3*Log[x])^(-1), x] - 108*Defer[Int][x/(6 + x - 3*Log[x]), x] + 18*Defer[Int][x^2/(6 +
x - 3*Log[x]), x] - 3*Defer[Int][x^3/(6 + x - 3*Log[x]), x] - Defer[Int][x^4/(6 + x - 3*Log[x]), x] - 3888*Def
er[Int][1/((6 + x)*(6 + x - 3*Log[x])), x] + 2*Defer[Int][(x^3*(9 + x))/((6 + x)*Log[x]), x] + 432*Defer[Int][
(x^3*Log[Log[x]])/(6 + x - 3*Log[x])^2, x] + 105*Defer[Int][(x^4*Log[Log[x]])/(6 + x - 3*Log[x])^2, x] + 7*Def
er[Int][(x^5*Log[Log[x]])/(6 + x - 3*Log[x])^2, x] - 432*Defer[Int][(x^3*Log[x]*Log[Log[x]])/(6 + x - 3*Log[x]
)^2, x] - 87*Defer[Int][(x^4*Log[x]*Log[Log[x]])/(6 + x - 3*Log[x])^2, x] - 4*Defer[Int][(x^5*Log[x]*Log[Log[x
]])/(6 + x - 3*Log[x])^2, x] + 108*Defer[Int][(x^3*Log[x]^2*Log[Log[x]])/(6 + x - 3*Log[x])^2, x] + 15*Defer[I
nt][(x^4*Log[x]^2*Log[Log[x]])/(6 + x - 3*Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x^3 \left (2 \left (54+15 x+x^2\right )+3 (36+5 x) \log ^3(x) \log (\log (x))+\log ^2(x) \left (3 (9+x)-\left (432+87 x+4 x^2\right ) \log (\log (x))\right )+\log (x) \left (-108-21 x-x^2+\left (432+105 x+7 x^2\right ) \log (\log (x))\right )\right )}{(6+x-3 \log (x))^2 \log (x)} \, dx\\ &=\int \left (-\frac {108 x^3}{(6+x-3 \log (x))^2}-\frac {21 x^4}{(6+x-3 \log (x))^2}-\frac {x^5}{(6+x-3 \log (x))^2}+\frac {2 x^3 (6+x) (9+x)}{(6+x-3 \log (x))^2 \log (x)}+\frac {3 x^3 (9+x) \log (x)}{(6+x-3 \log (x))^2}-\frac {x^3 \left (-432-105 x-7 x^2+432 \log (x)+87 x \log (x)+4 x^2 \log (x)-108 \log ^2(x)-15 x \log ^2(x)\right ) \log (\log (x))}{(6+x-3 \log (x))^2}\right ) \, dx\\ &=2 \int \frac {x^3 (6+x) (9+x)}{(6+x-3 \log (x))^2 \log (x)} \, dx+3 \int \frac {x^3 (9+x) \log (x)}{(6+x-3 \log (x))^2} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^3 \left (-432-105 x-7 x^2+432 \log (x)+87 x \log (x)+4 x^2 \log (x)-108 \log ^2(x)-15 x \log ^2(x)\right ) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx\\ &=2 \int \left (\frac {3 x^3 (9+x)}{(6+x-3 \log (x))^2}+\frac {3 x^3 (9+x)}{(6+x) (6+x-3 \log (x))}+\frac {x^3 (9+x)}{(6+x) \log (x)}\right ) \, dx+3 \int \left (\frac {x^3 \left (54+15 x+x^2\right )}{3 (6+x-3 \log (x))^2}-\frac {x^3 (9+x)}{3 (6+x-3 \log (x))}\right ) \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx-\int \left (-\frac {432 x^3 \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {105 x^4 \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {7 x^5 \log (\log (x))}{(6+x-3 \log (x))^2}+\frac {432 x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2}+\frac {87 x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2}+\frac {4 x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {108 x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2}-\frac {15 x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2}\right ) \, dx\\ &=2 \int \frac {x^3 (9+x)}{(6+x) \log (x)} \, dx-4 \int \frac {x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^3 (9+x)}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^3 (9+x)}{(6+x) (6+x-3 \log (x))} \, dx+7 \int \frac {x^5 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+15 \int \frac {x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-87 \int \frac {x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+105 \int \frac {x^4 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx+108 \int \frac {x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+432 \int \frac {x^3 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-432 \int \frac {x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx+\int \frac {x^3 \left (54+15 x+x^2\right )}{(6+x-3 \log (x))^2} \, dx-\int \frac {x^3 (9+x)}{6+x-3 \log (x)} \, dx\\ &=2 \int \frac {x^3 (9+x)}{(6+x) \log (x)} \, dx-4 \int \frac {x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+6 \int \left (\frac {9 x^3}{(6+x-3 \log (x))^2}+\frac {x^4}{(6+x-3 \log (x))^2}\right ) \, dx+6 \int \left (\frac {108}{6+x-3 \log (x)}-\frac {18 x}{6+x-3 \log (x)}+\frac {3 x^2}{6+x-3 \log (x)}+\frac {x^3}{6+x-3 \log (x)}-\frac {648}{(6+x) (6+x-3 \log (x))}\right ) \, dx+7 \int \frac {x^5 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+15 \int \frac {x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx-87 \int \frac {x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+105 \int \frac {x^4 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx+108 \int \frac {x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+432 \int \frac {x^3 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-432 \int \frac {x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+\int \left (\frac {54 x^3}{(6+x-3 \log (x))^2}+\frac {15 x^4}{(6+x-3 \log (x))^2}+\frac {x^5}{(6+x-3 \log (x))^2}\right ) \, dx-\int \left (\frac {9 x^3}{6+x-3 \log (x)}+\frac {x^4}{6+x-3 \log (x)}\right ) \, dx-\int \frac {x^5}{(6+x-3 \log (x))^2} \, dx\\ &=2 \int \frac {x^3 (9+x)}{(6+x) \log (x)} \, dx-4 \int \frac {x^5 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx+6 \int \frac {x^3}{6+x-3 \log (x)} \, dx+7 \int \frac {x^5 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-9 \int \frac {x^3}{6+x-3 \log (x)} \, dx+15 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx+15 \int \frac {x^4 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+18 \int \frac {x^2}{6+x-3 \log (x)} \, dx-21 \int \frac {x^4}{(6+x-3 \log (x))^2} \, dx+2 \left (54 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx\right )-87 \int \frac {x^4 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+105 \int \frac {x^4 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x^3}{(6+x-3 \log (x))^2} \, dx-108 \int \frac {x}{6+x-3 \log (x)} \, dx+108 \int \frac {x^3 \log ^2(x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+432 \int \frac {x^3 \log (\log (x))}{(6+x-3 \log (x))^2} \, dx-432 \int \frac {x^3 \log (x) \log (\log (x))}{(6+x-3 \log (x))^2} \, dx+648 \int \frac {1}{6+x-3 \log (x)} \, dx-3888 \int \frac {1}{(6+x) (6+x-3 \log (x))} \, dx-\int \frac {x^4}{6+x-3 \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 24, normalized size = 1.04 \begin {gather*} -\frac {x^4 (9+x) (-2+\log (x)) \log (\log (x))}{6+x-3 \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(108*x^3 + 30*x^4 + 2*x^5 + (-108*x^3 - 21*x^4 - x^5)*Log[x] + (27*x^3 + 3*x^4)*Log[x]^2 + ((432*x^3
+ 105*x^4 + 7*x^5)*Log[x] + (-432*x^3 - 87*x^4 - 4*x^5)*Log[x]^2 + (108*x^3 + 15*x^4)*Log[x]^3)*Log[Log[x]])/
((36 + 12*x + x^2)*Log[x] + (-36 - 6*x)*Log[x]^2 + 9*Log[x]^3),x]

[Out]

-((x^4*(9 + x)*(-2 + Log[x])*Log[Log[x]])/(6 + x - 3*Log[x]))

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fricas [A]  time = 0.96, size = 37, normalized size = 1.61 \begin {gather*} \frac {{\left (2 \, x^{5} + 18 \, x^{4} - {\left (x^{5} + 9 \, x^{4}\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right )}{x - 3 \, \log \relax (x) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x^4+108*x^3)*log(x)^3+(-4*x^5-87*x^4-432*x^3)*log(x)^2+(7*x^5+105*x^4+432*x^3)*log(x))*log(log
(x))+(3*x^4+27*x^3)*log(x)^2+(-x^5-21*x^4-108*x^3)*log(x)+2*x^5+30*x^4+108*x^3)/(9*log(x)^3+(-6*x-36)*log(x)^2
+(x^2+12*x+36)*log(x)),x, algorithm="fricas")

[Out]

(2*x^5 + 18*x^4 - (x^5 + 9*x^4)*log(x))*log(log(x))/(x - 3*log(x) + 6)

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giac [A]  time = 0.31, size = 34, normalized size = 1.48 \begin {gather*} \frac {1}{3} \, {\left (x^{5} + 9 \, x^{4} - \frac {x^{6} + 9 \, x^{5}}{x - 3 \, \log \relax (x) + 6}\right )} \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x^4+108*x^3)*log(x)^3+(-4*x^5-87*x^4-432*x^3)*log(x)^2+(7*x^5+105*x^4+432*x^3)*log(x))*log(log
(x))+(3*x^4+27*x^3)*log(x)^2+(-x^5-21*x^4-108*x^3)*log(x)+2*x^5+30*x^4+108*x^3)/(9*log(x)^3+(-6*x-36)*log(x)^2
+(x^2+12*x+36)*log(x)),x, algorithm="giac")

[Out]

1/3*(x^5 + 9*x^4 - (x^6 + 9*x^5)/(x - 3*log(x) + 6))*log(log(x))

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maple [A]  time = 0.05, size = 31, normalized size = 1.35

method	result	size
risch	\(-\frac {x^{4} \left (x \ln \relax (x )+9 \ln \relax (x )-2 x -18\right ) \ln \left (\ln \relax (x )\right )}{x -3 \ln \relax (x )+6}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((15*x^4+108*x^3)*ln(x)^3+(-4*x^5-87*x^4-432*x^3)*ln(x)^2+(7*x^5+105*x^4+432*x^3)*ln(x))*ln(ln(x))+(3*x^4
+27*x^3)*ln(x)^2+(-x^5-21*x^4-108*x^3)*ln(x)+2*x^5+30*x^4+108*x^3)/(9*ln(x)^3+(-6*x-36)*ln(x)^2+(x^2+12*x+36)*
ln(x)),x,method=_RETURNVERBOSE)

[Out]

-x^4*(x*ln(x)+9*ln(x)-2*x-18)/(x-3*ln(x)+6)*ln(ln(x))

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maxima [A]  time = 0.48, size = 37, normalized size = 1.61 \begin {gather*} \frac {{\left (2 \, x^{5} + 18 \, x^{4} - {\left (x^{5} + 9 \, x^{4}\right )} \log \relax (x)\right )} \log \left (\log \relax (x)\right )}{x - 3 \, \log \relax (x) + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x^4+108*x^3)*log(x)^3+(-4*x^5-87*x^4-432*x^3)*log(x)^2+(7*x^5+105*x^4+432*x^3)*log(x))*log(log
(x))+(3*x^4+27*x^3)*log(x)^2+(-x^5-21*x^4-108*x^3)*log(x)+2*x^5+30*x^4+108*x^3)/(9*log(x)^3+(-6*x-36)*log(x)^2
+(x^2+12*x+36)*log(x)),x, algorithm="maxima")

[Out]

(2*x^5 + 18*x^4 - (x^5 + 9*x^4)*log(x))*log(log(x))/(x - 3*log(x) + 6)

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mupad [B]  time = 2.33, size = 394, normalized size = 17.13 \begin {gather*} \frac {\ln \left (\ln \relax (x)\right )\,\left (\left (x+6\right )\,\left (\frac {\frac {25\,x^5}{4}+\frac {117\,x^4}{4}-144\,x^3}{x-3}-\frac {31\,x^5+33\,x^4-648\,x^3}{x-3}-\frac {5\,x^5+16\,x^4-\frac {339\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x}{x-3}+\frac {\frac {155\,x^5}{4}+\frac {331\,x^4}{4}-\frac {1347\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x}{x-3}\right )+\ln \relax (x)\,\left (\frac {3\,\left (31\,x^5+33\,x^4-648\,x^3\right )}{x-3}-\frac {3\,\left (\frac {25\,x^5}{4}+\frac {117\,x^4}{4}-144\,x^3\right )}{x-3}+\frac {3\,\left (5\,x^5+16\,x^4-\frac {339\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x\right )}{x-3}-\frac {3\,\left (\frac {155\,x^5}{4}+\frac {331\,x^4}{4}-\frac {1347\,x^3}{2}-\frac {459\,x^2}{2}+1377\,x\right )}{x-3}+\left (\frac {20\,x^5+33\,x^4-432\,x^3}{x-3}-\frac {25\,x^5+69\,x^4-432\,x^3}{x-3}\right )\,\left (x+6\right )+\frac {x^4\,\left (4\,x^2+87\,x+432\right )}{x-3}\right )-{\ln \relax (x)}^2\,\left (\frac {3\,\left (20\,x^5+33\,x^4-432\,x^3\right )}{x-3}-\frac {3\,\left (25\,x^5+69\,x^4-432\,x^3\right )}{x-3}+\frac {3\,x^4\,\left (5\,x+36\right )}{x-3}\right )-\frac {x^4\,\left (7\,x^2+105\,x+432\right )}{x-3}\right )}{x-3\,\ln \relax (x)+6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x))*(log(x)*(432*x^3 + 105*x^4 + 7*x^5) + log(x)^3*(108*x^3 + 15*x^4) - log(x)^2*(432*x^3 + 87*x^
4 + 4*x^5)) + log(x)^2*(27*x^3 + 3*x^4) + 108*x^3 + 30*x^4 + 2*x^5 - log(x)*(108*x^3 + 21*x^4 + x^5))/(9*log(x
)^3 + log(x)*(12*x + x^2 + 36) - log(x)^2*(6*x + 36)),x)

[Out]

(log(log(x))*((x + 6)*(((117*x^4)/4 - 144*x^3 + (25*x^5)/4)/(x - 3) - (33*x^4 - 648*x^3 + 31*x^5)/(x - 3) - (1
377*x - (459*x^2)/2 - (339*x^3)/2 + 16*x^4 + 5*x^5)/(x - 3) + (1377*x - (459*x^2)/2 - (1347*x^3)/2 + (331*x^4)
/4 + (155*x^5)/4)/(x - 3)) + log(x)*((3*(33*x^4 - 648*x^3 + 31*x^5))/(x - 3) - (3*((117*x^4)/4 - 144*x^3 + (25
*x^5)/4))/(x - 3) + (3*(1377*x - (459*x^2)/2 - (339*x^3)/2 + 16*x^4 + 5*x^5))/(x - 3) - (3*(1377*x - (459*x^2)
/2 - (1347*x^3)/2 + (331*x^4)/4 + (155*x^5)/4))/(x - 3) + ((33*x^4 - 432*x^3 + 20*x^5)/(x - 3) - (69*x^4 - 432
*x^3 + 25*x^5)/(x - 3))*(x + 6) + (x^4*(87*x + 4*x^2 + 432))/(x - 3)) - log(x)^2*((3*(33*x^4 - 432*x^3 + 20*x^
5))/(x - 3) - (3*(69*x^4 - 432*x^3 + 25*x^5))/(x - 3) + (3*x^4*(5*x + 36))/(x - 3)) - (x^4*(105*x + 7*x^2 + 43
2))/(x - 3)))/(x - 3*log(x) + 6)

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sympy [A]  time = 0.57, size = 37, normalized size = 1.61 \begin {gather*} \frac {\left (- x^{5} \log {\relax (x )} + 2 x^{5} - 9 x^{4} \log {\relax (x )} + 18 x^{4}\right ) \log {\left (\log {\relax (x )} \right )}}{x - 3 \log {\relax (x )} + 6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x**4+108*x**3)*ln(x)**3+(-4*x**5-87*x**4-432*x**3)*ln(x)**2+(7*x**5+105*x**4+432*x**3)*ln(x))*
ln(ln(x))+(3*x**4+27*x**3)*ln(x)**2+(-x**5-21*x**4-108*x**3)*ln(x)+2*x**5+30*x**4+108*x**3)/(9*ln(x)**3+(-6*x-
36)*ln(x)**2+(x**2+12*x+36)*ln(x)),x)

[Out]

(-x**5*log(x) + 2*x**5 - 9*x**4*log(x) + 18*x**4)*log(log(x))/(x - 3*log(x) + 6)

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