3.69.7 $\int \frac{(48600x+20250e^{2+2x}x)\log (\log (x))+(48600x+e^{2+2x}(20250x-81000x^2))\log (x){\log }^2(\log (x))}{(248832+518400e^{2+2x}+432000e^{4+4x}+180000e^{6+6x}+37500e^{8+8x}+3125e^{10+10x})\log (x)}dx$

Optimal. Leaf size=25 $$\frac{25x^2{\log }^2(\log (x))}{{(4+\frac{5}{3}{e}^{2+2x})}^4}$$

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Rubi [F]  time = 2.85, 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 = {} $$\begin{array}{c}\int \frac{(48600x+20250e^{2+2x}x)\log (\log (x))+(48600x+e^{2+2x}(20250x-81000x^2))\log (x){\log }^2(\log (x))}{(248832+518400e^{2+2x}+432000e^{4+4x}+180000e^{6+6x}+37500e^{8+8x}+3125e^{10+10x})\log (x)}dx\end{array}$$

Verification is not applicable to the result.

[In]

Int[((48600*x + 20250*E^(2 + 2*x)*x)*Log[Log[x]] + (48600*x + E^(2 + 2*x)*(20250*x - 81000*x^2))*Log[x]*Log[Lo
g[x]]^2)/((248832 + 518400*E^(2 + 2*x) + 432000*E^(4 + 4*x) + 180000*E^(6 + 6*x) + 37500*E^(8 + 8*x) + 3125*E^
(10 + 10*x))*Log[x]),x]

[Out]

4050*Defer[Int][(x*Log[Log[x]])/((12 + 5*E^(2 + 2*x))^4*Log[x]), x] + 4050*Defer[Int][(x*Log[Log[x]]^2)/(12 +
5*E^(2 + 2*x))^4, x] + 194400*Defer[Int][(x^2*Log[Log[x]]^2)/(12 + 5*E^(2 + 2*x))^5, x] - 16200*Defer[Int][(x^
2*Log[Log[x]]^2)/(12 + 5*E^(2 + 2*x))^4, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{4050x\log (\log (x))(12+5e^{2+2x}-(-12+5e^{2+2x}(-1+4x))\log (x)\log (\log (x)))}{{(12+5e^{2+2x})}^5\log (x)}dx\\ & =4050\int \frac{x\log (\log (x))(12+5e^{2+2x}-(-12+5e^{2+2x}(-1+4x))\log (x)\log (\log (x)))}{{(12+5e^{2+2x})}^5\log (x)}dx\\ & =4050\int (\frac{48x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^5}-\frac{x\log (\log (x))(-1-\log (x)\log (\log (x))+4x\log (x)\log (\log (x)))}{{(12+5e^{2+2x})}^4\log (x)})dx\\ & =-(4050\int \frac{x\log (\log (x))(-1-\log (x)\log (\log (x))+4x\log (x)\log (\log (x)))}{{(12+5e^{2+2x})}^4\log (x)}dx)+194400\int \frac{x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^5}dx\\ & =-(4050\int (-\frac{x\log (\log (x))}{{(12+5e^{2+2x})}^4\log (x)}-\frac{x{\log }^2(\log (x))}{{(12+5e^{2+2x})}^4}+\frac{4x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^4})dx)+194400\int \frac{x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^5}dx\\ & =4050\int \frac{x\log (\log (x))}{{(12+5e^{2+2x})}^4\log (x)}dx+4050\int \frac{x{\log }^2(\log (x))}{{(12+5e^{2+2x})}^4}dx-16200\int \frac{x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^4}dx+194400\int \frac{x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^5}dx\end{array}\end{array}$$

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Mathematica [A]  time = 0.40, size = 23, normalized size = 0.92 $$\begin{array}{c}\frac{2025x^2{\log }^2(\log (x))}{{(12+5e^{2+2x})}^4}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[((48600*x + 20250*E^(2 + 2*x)*x)*Log[Log[x]] + (48600*x + E^(2 + 2*x)*(20250*x - 81000*x^2))*Log[x]*
Log[Log[x]]^2)/((248832 + 518400*E^(2 + 2*x) + 432000*E^(4 + 4*x) + 180000*E^(6 + 6*x) + 37500*E^(8 + 8*x) + 3
125*E^(10 + 10*x))*Log[x]),x]

[Out]

(2025*x^2*Log[Log[x]]^2)/(12 + 5*E^(2 + 2*x))^4

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fricas [B]  time = 0.68, size = 46, normalized size = 1.84 $$\begin{array}{c}\frac{2025x^2\log {(\log (x))}^2}{625e^{(8x+8)}+6000e^{(6x+6)}+21600e^{(4x+4)}+34560e^{(2x+2)}+20736}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-81000*x^2+20250*x)*exp(x+1)^2+48600*x)*log(x)*log(log(x))^2+(20250*x*exp(x+1)^2+48600*x)*log(log
(x)))/(3125*exp(x+1)^10+37500*exp(x+1)^8+180000*exp(x+1)^6+432000*exp(x+1)^4+518400*exp(x+1)^2+248832)/log(x),
x, algorithm="fricas")

[Out]

2025*x^2*log(log(x))^2/(625*e^(8*x + 8) + 6000*e^(6*x + 6) + 21600*e^(4*x + 4) + 34560*e^(2*x + 2) + 20736)

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giac [B]  time = 0.17, size = 46, normalized size = 1.84 $$\begin{array}{c}\frac{2025x^2\log {(\log (x))}^2}{625e^{(8x+8)}+6000e^{(6x+6)}+21600e^{(4x+4)}+34560e^{(2x+2)}+20736}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-81000*x^2+20250*x)*exp(x+1)^2+48600*x)*log(x)*log(log(x))^2+(20250*x*exp(x+1)^2+48600*x)*log(log
(x)))/(3125*exp(x+1)^10+37500*exp(x+1)^8+180000*exp(x+1)^6+432000*exp(x+1)^4+518400*exp(x+1)^2+248832)/log(x),
x, algorithm="giac")

[Out]

2025*x^2*log(log(x))^2/(625*e^(8*x + 8) + 6000*e^(6*x + 6) + 21600*e^(4*x + 4) + 34560*e^(2*x + 2) + 20736)

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maple [A]  time = 0.04, size = 23, normalized size = 0.92

method	result	size
risch	$\frac{2025x^2\ln {(\ln (x))}^2}{{(5{\mathrm{e}}^{2x+2}+12)}^4}$	$23$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-81000*x^2+20250*x)*exp(x+1)^2+48600*x)*ln(x)*ln(ln(x))^2+(20250*x*exp(x+1)^2+48600*x)*ln(ln(x)))/(3125
*exp(x+1)^10+37500*exp(x+1)^8+180000*exp(x+1)^6+432000*exp(x+1)^4+518400*exp(x+1)^2+248832)/ln(x),x,method=_RE
TURNVERBOSE)

[Out]

2025*x^2/(5*exp(2*x+2)+12)^4*ln(ln(x))^2

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maxima [B]  time = 0.47, size = 46, normalized size = 1.84 $$\begin{array}{c}\frac{2025x^2\log {(\log (x))}^2}{625e^{(8x+8)}+6000e^{(6x+6)}+21600e^{(4x+4)}+34560e^{(2x+2)}+20736}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-81000*x^2+20250*x)*exp(x+1)^2+48600*x)*log(x)*log(log(x))^2+(20250*x*exp(x+1)^2+48600*x)*log(log
(x)))/(3125*exp(x+1)^10+37500*exp(x+1)^8+180000*exp(x+1)^6+432000*exp(x+1)^4+518400*exp(x+1)^2+248832)/log(x),
x, algorithm="maxima")

[Out]

2025*x^2*log(log(x))^2/(625*e^(8*x + 8) + 6000*e^(6*x + 6) + 21600*e^(4*x + 4) + 34560*e^(2*x + 2) + 20736)

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mupad [B]  time = 4.36, size = 46, normalized size = 1.84 $$\begin{array}{c}\frac{81x^2{\ln (\ln (x))}^2}{25(\frac{6912{\mathrm{e}}^{2x+2}}{125}+\frac{864{\mathrm{e}}^{4x+4}}{25}+\frac{48{\mathrm{e}}^{6x+6}}{5}+{\mathrm{e}}^{8x+8}+\frac{20736}{625})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(x))*(48600*x + 20250*x*exp(2*x + 2)) + log(log(x))^2*log(x)*(48600*x + exp(2*x + 2)*(20250*x - 81
000*x^2)))/(log(x)*(518400*exp(2*x + 2) + 432000*exp(4*x + 4) + 180000*exp(6*x + 6) + 37500*exp(8*x + 8) + 312
5*exp(10*x + 10) + 248832)),x)

[Out]

(81*x^2*log(log(x))^2)/(25*((6912*exp(2*x + 2))/125 + (864*exp(4*x + 4))/25 + (48*exp(6*x + 6))/5 + exp(8*x +
8) + 20736/625))

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sympy [B]  time = 0.37, size = 49, normalized size = 1.96 $$\begin{array}{c}\frac{81x^2\log {(\log (x))}^2}{\frac{6912e^{2x+2}}{5}+864e^{4x+4}+240e^{6x+6}+25e^{8x+8}+\frac{20736}{25}}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-81000*x**2+20250*x)*exp(x+1)**2+48600*x)*ln(x)*ln(ln(x))**2+(20250*x*exp(x+1)**2+48600*x)*ln(ln(
x)))/(3125*exp(x+1)**10+37500*exp(x+1)**8+180000*exp(x+1)**6+432000*exp(x+1)**4+518400*exp(x+1)**2+248832)/ln(
x),x)

[Out]

81*x**2*log(log(x))**2/(6912*exp(2*x + 2)/5 + 864*exp(4*x + 4) + 240*exp(6*x + 6) + 25*exp(8*x + 8) + 20736/25
)

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