3.102.11 \(\int \frac {32 e^{8 x} x+2 x^4+(-120 x^3-6 x^4+e^{8 x} (3840-15168 x-768 x^2)+(40 x^3+2 x^4+e^{8 x} (-1280+5056 x+256 x^2)) \log (\frac {3}{20+x})) \log (3-\log (\frac {3}{20+x}))}{(-60 x^4-3 x^5+e^{8 x} (-960 x-48 x^2)+(20 x^4+x^5+e^{8 x} (320 x+16 x^2)) \log (\frac {3}{20+x})) \log (3-\log (\frac {3}{20+x}))} \, dx\)

Optimal. Leaf size=31 \[ 2 \log \left (\frac {\frac {16 e^{8 x}}{x^2}+x}{\log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \]

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Rubi [F]  time = 7.84, 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 {32 e^{8 x} x+2 x^4+\left (-120 x^3-6 x^4+e^{8 x} \left (3840-15168 x-768 x^2\right )+\left (40 x^3+2 x^4+e^{8 x} \left (-1280+5056 x+256 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{\left (-60 x^4-3 x^5+e^{8 x} \left (-960 x-48 x^2\right )+\left (20 x^4+x^5+e^{8 x} \left (320 x+16 x^2\right )\right ) \log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(32*E^(8*x)*x + 2*x^4 + (-120*x^3 - 6*x^4 + E^(8*x)*(3840 - 15168*x - 768*x^2) + (40*x^3 + 2*x^4 + E^(8*x)
*(-1280 + 5056*x + 256*x^2))*Log[3/(20 + x)])*Log[3 - Log[3/(20 + x)]])/((-60*x^4 - 3*x^5 + E^(8*x)*(-960*x -
48*x^2) + (20*x^4 + x^5 + E^(8*x)*(320*x + 16*x^2))*Log[3/(20 + x)])*Log[3 - Log[3/(20 + x)]]),x]

[Out]

16*x - 4*Log[x] - 2*Log[Log[3 - Log[3/(20 + x)]]] + 6*Defer[Int][x^2/(16*E^(8*x) + x^3), x] - 16*Defer[Int][x^
3/(16*E^(8*x) + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-16 e^{8 x} x-x^4-(20+x) \left (x^3+32 e^{8 x} (-1+4 x)\right ) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )}{x (20+x) \left (16 e^{8 x}+x^3\right ) \left (3-\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx\\ &=2 \int \frac {-16 e^{8 x} x-x^4-(20+x) \left (x^3+32 e^{8 x} (-1+4 x)\right ) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{x (20+x) \left (16 e^{8 x}+x^3\right ) \left (3-\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx\\ &=2 \int \left (-\frac {x^2 (-3+8 x)}{16 e^{8 x}+x^3}+\frac {x+120 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-474 x \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-24 x^2 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-40 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+158 x \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+8 x^2 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{x (20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \, dx\\ &=-\left (2 \int \frac {x^2 (-3+8 x)}{16 e^{8 x}+x^3} \, dx\right )+2 \int \frac {x+120 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-474 x \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-24 x^2 \log \left (3-\log \left (\frac {3}{20+x}\right )\right )-40 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+158 x \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )+8 x^2 \log \left (\frac {3}{20+x}\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}{x (20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx\\ &=-\left (2 \int \left (-\frac {3 x^2}{16 e^{8 x}+x^3}+\frac {8 x^3}{16 e^{8 x}+x^3}\right ) \, dx\right )+2 \int \frac {158-\frac {40}{x}+8 x+\frac {1}{\left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}}{20+x} \, dx\\ &=2 \int \left (\frac {2 (-1+4 x)}{x}+\frac {1}{(20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )}\right ) \, dx+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ &=2 \int \frac {1}{(20+x) \left (-3+\log \left (\frac {3}{20+x}\right )\right ) \log \left (3-\log \left (\frac {3}{20+x}\right )\right )} \, dx+4 \int \frac {-1+4 x}{x} \, dx+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ &=-2 \log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )+4 \int \left (4-\frac {1}{x}\right ) \, dx+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ &=16 x-4 \log (x)-2 \log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )+6 \int \frac {x^2}{16 e^{8 x}+x^3} \, dx-16 \int \frac {x^3}{16 e^{8 x}+x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 35, normalized size = 1.13 \begin {gather*} 2 \left (-2 \log (x)+\log \left (16 e^{8 x}+x^3\right )-\log \left (\log \left (3-\log \left (\frac {3}{20+x}\right )\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32*E^(8*x)*x + 2*x^4 + (-120*x^3 - 6*x^4 + E^(8*x)*(3840 - 15168*x - 768*x^2) + (40*x^3 + 2*x^4 + E
^(8*x)*(-1280 + 5056*x + 256*x^2))*Log[3/(20 + x)])*Log[3 - Log[3/(20 + x)]])/((-60*x^4 - 3*x^5 + E^(8*x)*(-96
0*x - 48*x^2) + (20*x^4 + x^5 + E^(8*x)*(320*x + 16*x^2))*Log[3/(20 + x)])*Log[3 - Log[3/(20 + x)]]),x]

[Out]

2*(-2*Log[x] + Log[16*E^(8*x) + x^3] - Log[Log[3 - Log[3/(20 + x)]]])

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fricas [A]  time = 0.59, size = 34, normalized size = 1.10 \begin {gather*} 2 \, \log \left (x^{3} + 16 \, e^{\left (8 \, x\right )}\right ) - 4 \, \log \relax (x) - 2 \, \log \left (\log \left (-\log \left (\frac {3}{x + 20}\right ) + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*log(3/(20+x))+(-768*x^2-15168*x+3840)*exp(4*x)^2-6
*x^4-120*x^3)*log(-log(3/(20+x))+3)+32*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*log(3/(20+x
))+(-48*x^2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/log(-log(3/(20+x))+3),x, algorithm="fricas")

[Out]

2*log(x^3 + 16*e^(8*x)) - 4*log(x) - 2*log(log(-log(3/(x + 20)) + 3))

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giac [A]  time = 0.64, size = 34, normalized size = 1.10 \begin {gather*} 2 \, \log \left (-x^{3} - 16 \, e^{\left (8 \, x\right )}\right ) - 4 \, \log \relax (x) - 2 \, \log \left (\log \left (-\log \relax (3) + \log \left (x + 20\right ) + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*log(3/(20+x))+(-768*x^2-15168*x+3840)*exp(4*x)^2-6
*x^4-120*x^3)*log(-log(3/(20+x))+3)+32*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*log(3/(20+x
))+(-48*x^2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/log(-log(3/(20+x))+3),x, algorithm="giac")

[Out]

2*log(-x^3 - 16*e^(8*x)) - 4*log(x) - 2*log(log(-log(3) + log(x + 20) + 3))

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maple [A]  time = 0.19, size = 33, normalized size = 1.06

method	result	size
risch	\(-4 \ln \relax (x )+2 \ln \left (\frac {x^{3}}{16}+{\mathrm e}^{8 x}\right )-2 \ln \left (\ln \left (-\ln \relax (3)+\ln \left (20+x \right )+3\right )\right )\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*ln(3/(20+x))+(-768*x^2-15168*x+3840)*exp(4*x)^2-6*x^4-12
0*x^3)*ln(-ln(3/(20+x))+3)+32*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*ln(3/(20+x))+(-48*x^
2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/ln(-ln(3/(20+x))+3),x,method=_RETURNVERBOSE)

[Out]

-4*ln(x)+2*ln(1/16*x^3+exp(8*x))-2*ln(ln(-ln(3)+ln(20+x)+3))

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maxima [A]  time = 0.51, size = 32, normalized size = 1.03 \begin {gather*} 2 \, \log \left (\frac {1}{16} \, x^{3} + e^{\left (8 \, x\right )}\right ) - 4 \, \log \relax (x) - 2 \, \log \left (\log \left (-\log \relax (3) + \log \left (x + 20\right ) + 3\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((256*x^2+5056*x-1280)*exp(4*x)^2+2*x^4+40*x^3)*log(3/(20+x))+(-768*x^2-15168*x+3840)*exp(4*x)^2-6
*x^4-120*x^3)*log(-log(3/(20+x))+3)+32*x*exp(4*x)^2+2*x^4)/(((16*x^2+320*x)*exp(4*x)^2+x^5+20*x^4)*log(3/(20+x
))+(-48*x^2-960*x)*exp(4*x)^2-3*x^5-60*x^4)/log(-log(3/(20+x))+3),x, algorithm="maxima")

[Out]

2*log(1/16*x^3 + e^(8*x)) - 4*log(x) - 2*log(log(-log(3) + log(x + 20) + 3))

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mupad [B]  time = 1.08, size = 34, normalized size = 1.10 \begin {gather*} 2\,\ln \left ({\mathrm {e}}^{8\,x}+\frac {x^3}{16}\right )-2\,\ln \left (\ln \left (3-\ln \left (\frac {3}{x+20}\right )\right )\right )-4\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x*exp(8*x) - log(3 - log(3/(x + 20)))*(exp(8*x)*(15168*x + 768*x^2 - 3840) - log(3/(x + 20))*(exp(8*x
)*(5056*x + 256*x^2 - 1280) + 40*x^3 + 2*x^4) + 120*x^3 + 6*x^4) + 2*x^4)/(log(3 - log(3/(x + 20)))*(exp(8*x)*
(960*x + 48*x^2) + 60*x^4 + 3*x^5 - log(3/(x + 20))*(exp(8*x)*(320*x + 16*x^2) + 20*x^4 + x^5))),x)

[Out]

2*log(exp(8*x) + x^3/16) - 2*log(log(3 - log(3/(x + 20)))) - 4*log(x)

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sympy [A]  time = 0.68, size = 31, normalized size = 1.00 \begin {gather*} - 4 \log {\relax (x )} + 2 \log {\left (\frac {x^{3}}{16} + e^{8 x} \right )} - 2 \log {\left (\log {\left (3 - \log {\left (\frac {3}{x + 20} \right )} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((256*x**2+5056*x-1280)*exp(4*x)**2+2*x**4+40*x**3)*ln(3/(20+x))+(-768*x**2-15168*x+3840)*exp(4*x)
**2-6*x**4-120*x**3)*ln(-ln(3/(20+x))+3)+32*x*exp(4*x)**2+2*x**4)/(((16*x**2+320*x)*exp(4*x)**2+x**5+20*x**4)*
ln(3/(20+x))+(-48*x**2-960*x)*exp(4*x)**2-3*x**5-60*x**4)/ln(-ln(3/(20+x))+3),x)

[Out]

-4*log(x) + 2*log(x**3/16 + exp(8*x)) - 2*log(log(3 - log(3/(x + 20))))

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