3.19.64 \(\int \frac {16 x^2+(-4608+1440 x-72 x^2-54 x^3+18 x^4) \log (x)+(-1152 x+288 x^2-72 x^3) \log ^2(x)}{(2304 x-288 x^2-279 x^3+18 x^4+9 x^5) \log (x)+(1152 x^2-72 x^3-72 x^4) \log ^2(x)+144 x^3 \log ^3(x)+16 x^3 \log (x) \log (\log (x))} \, dx\)

Optimal. Leaf size=26 \[ \log \left (9 \left (\frac {1}{4} (-1-x)+\frac {4}{x}+\log (x)\right )^2+\log (\log (x))\right ) \]

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Rubi [F]  time = 4.96, 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 {16 x^2+\left (-4608+1440 x-72 x^2-54 x^3+18 x^4\right ) \log (x)+\left (-1152 x+288 x^2-72 x^3\right ) \log ^2(x)}{\left (2304 x-288 x^2-279 x^3+18 x^4+9 x^5\right ) \log (x)+\left (1152 x^2-72 x^3-72 x^4\right ) \log ^2(x)+144 x^3 \log ^3(x)+16 x^3 \log (x) \log (\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*x^2 + (-4608 + 1440*x - 72*x^2 - 54*x^3 + 18*x^4)*Log[x] + (-1152*x + 288*x^2 - 72*x^3)*Log[x]^2)/((23
04*x - 288*x^2 - 279*x^3 + 18*x^4 + 9*x^5)*Log[x] + (1152*x^2 - 72*x^3 - 72*x^4)*Log[x]^2 + 144*x^3*Log[x]^3 +
16*x^3*Log[x]*Log[Log[x]]),x]

[Out]

1440*Defer[Int][(9*(-16 + x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x] + 144*x^2*Log[x]^2 + 16*x^2*Log[Log[x]])^(-
1), x] - 4608*Defer[Int][1/(x*(9*(-16 + x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x] + 144*x^2*Log[x]^2 + 16*x^2*L
og[Log[x]])), x] - 72*Defer[Int][x/(9*(-16 + x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x] + 144*x^2*Log[x]^2 + 16*
x^2*Log[Log[x]]), x] - 54*Defer[Int][x^2/(9*(-16 + x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x] + 144*x^2*Log[x]^2
+ 16*x^2*Log[Log[x]]), x] + 18*Defer[Int][x^3/(9*(-16 + x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x] + 144*x^2*Lo
g[x]^2 + 16*x^2*Log[Log[x]]), x] + 16*Defer[Int][x/(Log[x]*(9*(-16 + x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x]
+ 144*x^2*Log[x]^2 + 16*x^2*Log[Log[x]])), x] - 1152*Defer[Int][Log[x]/(9*(-16 + x + x^2)^2 - 72*x*(-16 + x +
x^2)*Log[x] + 144*x^2*Log[x]^2 + 16*x^2*Log[Log[x]]), x] + 288*Defer[Int][(x*Log[x])/(9*(-16 + x + x^2)^2 - 72
*x*(-16 + x + x^2)*Log[x] + 144*x^2*Log[x]^2 + 16*x^2*Log[Log[x]]), x] - 72*Defer[Int][(x^2*Log[x])/(9*(-16 +
x + x^2)^2 - 72*x*(-16 + x + x^2)*Log[x] + 144*x^2*Log[x]^2 + 16*x^2*Log[Log[x]]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 x^2+\left (-4608+1440 x-72 x^2-54 x^3+18 x^4\right ) \log (x)+\left (-1152 x+288 x^2-72 x^3\right ) \log ^2(x)}{x \log (x) \left (2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )} \, dx\\ &=\int \left (\frac {1440}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}-\frac {4608}{x \left (2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )}-\frac {72 x}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}-\frac {54 x^2}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}+\frac {18 x^3}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}+\frac {16 x}{\log (x) \left (2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )}-\frac {1152 \log (x)}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}+\frac {288 x \log (x)}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}-\frac {72 x^2 \log (x)}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))}\right ) \, dx\\ &=16 \int \frac {x}{\log (x) \left (2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )} \, dx+18 \int \frac {x^3}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-54 \int \frac {x^2}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-72 \int \frac {x}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-72 \int \frac {x^2 \log (x)}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx+288 \int \frac {x \log (x)}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-1152 \int \frac {\log (x)}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx+1440 \int \frac {1}{2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-4608 \int \frac {1}{x \left (2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )} \, dx\\ &=16 \int \frac {x}{\log (x) \left (9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )} \, dx+18 \int \frac {x^3}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-54 \int \frac {x^2}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-72 \int \frac {x}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-72 \int \frac {x^2 \log (x)}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx+288 \int \frac {x \log (x)}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-1152 \int \frac {\log (x)}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx+1440 \int \frac {1}{9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))} \, dx-4608 \int \frac {1}{x \left (9 \left (-16+x+x^2\right )^2-72 x \left (-16+x+x^2\right ) \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 68, normalized size = 2.62 \begin {gather*} 2 \left (-\log (x)+\frac {1}{2} \log \left (2304-288 x-279 x^2+18 x^3+9 x^4+1152 x \log (x)-72 x^2 \log (x)-72 x^3 \log (x)+144 x^2 \log ^2(x)+16 x^2 \log (\log (x))\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x^2 + (-4608 + 1440*x - 72*x^2 - 54*x^3 + 18*x^4)*Log[x] + (-1152*x + 288*x^2 - 72*x^3)*Log[x]^2
)/((2304*x - 288*x^2 - 279*x^3 + 18*x^4 + 9*x^5)*Log[x] + (1152*x^2 - 72*x^3 - 72*x^4)*Log[x]^2 + 144*x^3*Log[
x]^3 + 16*x^3*Log[x]*Log[Log[x]]),x]

[Out]

2*(-Log[x] + Log[2304 - 288*x - 279*x^2 + 18*x^3 + 9*x^4 + 1152*x*Log[x] - 72*x^2*Log[x] - 72*x^3*Log[x] + 144
*x^2*Log[x]^2 + 16*x^2*Log[Log[x]]]/2)

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fricas [B]  time = 0.82, size = 56, normalized size = 2.15 \begin {gather*} \log \left (\frac {9 \, x^{4} + 144 \, x^{2} \log \relax (x)^{2} + 18 \, x^{3} + 16 \, x^{2} \log \left (\log \relax (x)\right ) - 279 \, x^{2} - 72 \, {\left (x^{3} + x^{2} - 16 \, x\right )} \log \relax (x) - 288 \, x + 2304}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^3+288*x^2-1152*x)*log(x)^2+(18*x^4-54*x^3-72*x^2+1440*x-4608)*log(x)+16*x^2)/(16*x^3*log(x)*
log(log(x))+144*x^3*log(x)^3+(-72*x^4-72*x^3+1152*x^2)*log(x)^2+(9*x^5+18*x^4-279*x^3-288*x^2+2304*x)*log(x)),
x, algorithm="fricas")

[Out]

log((9*x^4 + 144*x^2*log(x)^2 + 18*x^3 + 16*x^2*log(log(x)) - 279*x^2 - 72*(x^3 + x^2 - 16*x)*log(x) - 288*x +
2304)/x^2)

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giac [B]  time = 0.27, size = 62, normalized size = 2.38 \begin {gather*} \log \left (9 \, x^{4} - 72 \, x^{3} \log \relax (x) + 144 \, x^{2} \log \relax (x)^{2} + 18 \, x^{3} - 72 \, x^{2} \log \relax (x) + 16 \, x^{2} \log \left (\log \relax (x)\right ) - 279 \, x^{2} + 1152 \, x \log \relax (x) - 288 \, x + 2304\right ) - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^3+288*x^2-1152*x)*log(x)^2+(18*x^4-54*x^3-72*x^2+1440*x-4608)*log(x)+16*x^2)/(16*x^3*log(x)*
log(log(x))+144*x^3*log(x)^3+(-72*x^4-72*x^3+1152*x^2)*log(x)^2+(9*x^5+18*x^4-279*x^3-288*x^2+2304*x)*log(x)),
x, algorithm="giac")

[Out]

log(9*x^4 - 72*x^3*log(x) + 144*x^2*log(x)^2 + 18*x^3 - 72*x^2*log(x) + 16*x^2*log(log(x)) - 279*x^2 + 1152*x*
log(x) - 288*x + 2304) - 2*log(x)

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maple [A]  time = 0.02, size = 57, normalized size = 2.19

method	result	size
risch	\(\ln \left (\ln \left (\ln \relax (x )\right )+\frac {\frac {9 x^{4}}{16}-\frac {9 x^{3} \ln \relax (x )}{2}+9 x^{2} \ln \relax (x )^{2}+\frac {9 x^{3}}{8}-\frac {9 x^{2} \ln \relax (x )}{2}-\frac {279 x^{2}}{16}+72 x \ln \relax (x )-18 x +144}{x^{2}}\right )\)	\(57\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-72*x^3+288*x^2-1152*x)*ln(x)^2+(18*x^4-54*x^3-72*x^2+1440*x-4608)*ln(x)+16*x^2)/(16*x^3*ln(x)*ln(ln(x))
+144*x^3*ln(x)^3+(-72*x^4-72*x^3+1152*x^2)*ln(x)^2+(9*x^5+18*x^4-279*x^3-288*x^2+2304*x)*ln(x)),x,method=_RETU
RNVERBOSE)

[Out]

ln(ln(ln(x))+9/16*(x^4-8*x^3*ln(x)+16*x^2*ln(x)^2+2*x^3-8*x^2*ln(x)-31*x^2+128*x*ln(x)-32*x+256)/x^2)

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maxima [B]  time = 0.43, size = 57, normalized size = 2.19 \begin {gather*} \log \left (\frac {9 \, x^{4} + 144 \, x^{2} \log \relax (x)^{2} + 18 \, x^{3} + 16 \, x^{2} \log \left (\log \relax (x)\right ) - 279 \, x^{2} - 72 \, {\left (x^{3} + x^{2} - 16 \, x\right )} \log \relax (x) - 288 \, x + 2304}{16 \, x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^3+288*x^2-1152*x)*log(x)^2+(18*x^4-54*x^3-72*x^2+1440*x-4608)*log(x)+16*x^2)/(16*x^3*log(x)*
log(log(x))+144*x^3*log(x)^3+(-72*x^4-72*x^3+1152*x^2)*log(x)^2+(9*x^5+18*x^4-279*x^3-288*x^2+2304*x)*log(x)),
x, algorithm="maxima")

[Out]

log(1/16*(9*x^4 + 144*x^2*log(x)^2 + 18*x^3 + 16*x^2*log(log(x)) - 279*x^2 - 72*(x^3 + x^2 - 16*x)*log(x) - 28
8*x + 2304)/x^2)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\ln \relax (x)\,\left (-18\,x^4+54\,x^3+72\,x^2-1440\,x+4608\right )+{\ln \relax (x)}^2\,\left (72\,x^3-288\,x^2+1152\,x\right )-16\,x^2}{\ln \relax (x)\,\left (9\,x^5+18\,x^4-279\,x^3-288\,x^2+2304\,x\right )+144\,x^3\,{\ln \relax (x)}^3-{\ln \relax (x)}^2\,\left (72\,x^4+72\,x^3-1152\,x^2\right )+16\,x^3\,\ln \left (\ln \relax (x)\right )\,\ln \relax (x)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(72*x^2 - 1440*x + 54*x^3 - 18*x^4 + 4608) + log(x)^2*(1152*x - 288*x^2 + 72*x^3) - 16*x^2)/(log(
x)*(2304*x - 288*x^2 - 279*x^3 + 18*x^4 + 9*x^5) + 144*x^3*log(x)^3 - log(x)^2*(72*x^3 - 1152*x^2 + 72*x^4) +
16*x^3*log(log(x))*log(x)),x)

[Out]

int(-(log(x)*(72*x^2 - 1440*x + 54*x^3 - 18*x^4 + 4608) + log(x)^2*(1152*x - 288*x^2 + 72*x^3) - 16*x^2)/(log(
x)*(2304*x - 288*x^2 - 279*x^3 + 18*x^4 + 9*x^5) + 144*x^3*log(x)^3 - log(x)^2*(72*x^3 - 1152*x^2 + 72*x^4) +
16*x^3*log(log(x))*log(x)), x)

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sympy [B]  time = 0.59, size = 65, normalized size = 2.50 \begin {gather*} \log {\left (\log {\left (\log {\relax (x )} \right )} + \frac {9 x^{4} - 72 x^{3} \log {\relax (x )} + 18 x^{3} + 144 x^{2} \log {\relax (x )}^{2} - 72 x^{2} \log {\relax (x )} - 279 x^{2} + 1152 x \log {\relax (x )} - 288 x + 2304}{16 x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x**3+288*x**2-1152*x)*ln(x)**2+(18*x**4-54*x**3-72*x**2+1440*x-4608)*ln(x)+16*x**2)/(16*x**3*l
n(x)*ln(ln(x))+144*x**3*ln(x)**3+(-72*x**4-72*x**3+1152*x**2)*ln(x)**2+(9*x**5+18*x**4-279*x**3-288*x**2+2304*
x)*ln(x)),x)

[Out]

log(log(log(x)) + (9*x**4 - 72*x**3*log(x) + 18*x**3 + 144*x**2*log(x)**2 - 72*x**2*log(x) - 279*x**2 + 1152*x
*log(x) - 288*x + 2304)/(16*x**2))

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