3.73.87 \(\int \frac {-400 x+(-2800+e^5 (1200-300 x)+700 x) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{(-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 (4200+1830 x-720 x^2)) \log (-4+x)+(-5600+e^5 (2400-600 x)-2440 x+960 x^2) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx\)

Optimal. Leaf size=27 \[ \frac {x}{1+\frac {3}{4} \left (1-e^5\right )+\frac {6 x}{5}+\log (\log (-4+x))} \]

________________________________________________________________________________________

Rubi [F]  time = 2.10, 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 {-400 x+\left (-2800+e^5 (1200-300 x)+700 x\right ) \log (-4+x)+(-1600+400 x) \log (-4+x) \log (\log (-4+x))}{\left (-4900-5495 x-624 x^2+576 x^3+e^{10} (-900+225 x)+e^5 \left (4200+1830 x-720 x^2\right )\right ) \log (-4+x)+\left (-5600+e^5 (2400-600 x)-2440 x+960 x^2\right ) \log (-4+x) \log (\log (-4+x))+(-1600+400 x) \log (-4+x) \log ^2(\log (-4+x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-400*x + (-2800 + E^5*(1200 - 300*x) + 700*x)*Log[-4 + x] + (-1600 + 400*x)*Log[-4 + x]*Log[Log[-4 + x]])
/((-4900 - 5495*x - 624*x^2 + 576*x^3 + E^10*(-900 + 225*x) + E^5*(4200 + 1830*x - 720*x^2))*Log[-4 + x] + (-5
600 + E^5*(2400 - 600*x) - 2440*x + 960*x^2)*Log[-4 + x]*Log[Log[-4 + x]] + (-1600 + 400*x)*Log[-4 + x]*Log[Lo
g[-4 + x]]^2),x]

[Out]

80/(5*(7 - 3*E^5) + 24*x + 20*Log[Log[-4 + x]]) + 1920*Defer[Int][(35*(1 - (3*E^5)/7) + 24*x + 20*Log[Log[-4 +
x]])^(-2), x] - 480*Defer[Int][x/(35*(1 - (3*E^5)/7) + 24*x + 20*Log[Log[-4 + x]])^2, x] - 400*Defer[Int][1/(
Log[-4 + x]*(35*(1 - (3*E^5)/7) + 24*x + 20*Log[Log[-4 + x]])^2), x] + 20*Defer[Int][(35*(1 - (3*E^5)/7) + 24*
x + 20*Log[Log[-4 + x]])^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100 \left (4 x+(-4+x) \log (-4+x) \left (-7+3 e^5-4 \log (\log (-4+x))\right )\right )}{(4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ &=100 \int \frac {4 x+(-4+x) \log (-4+x) \left (-7+3 e^5-4 \log (\log (-4+x))\right )}{(4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ &=100 \int \left (\frac {4 x (5-24 \log (-4+x)+6 x \log (-4+x))}{5 (4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2}+\frac {1}{5 \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )}\right ) \, dx\\ &=20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx+80 \int \frac {x (5-24 \log (-4+x)+6 x \log (-4+x))}{(4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ &=20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx+80 \int \frac {x (5+6 (-4+x) \log (-4+x))}{(4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ &=20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx+80 \int \left (\frac {-5+24 \log (-4+x)-6 x \log (-4+x)}{\log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2}+\frac {4 (5-24 \log (-4+x)+6 x \log (-4+x))}{(4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2}\right ) \, dx\\ &=20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx+80 \int \frac {-5+24 \log (-4+x)-6 x \log (-4+x)}{\log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx+320 \int \frac {5-24 \log (-4+x)+6 x \log (-4+x)}{(4-x) \log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ &=\frac {80}{5 \left (7-3 e^5\right )+24 x+20 \log (\log (-4+x))}+20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx+80 \int \frac {-5-6 (-4+x) \log (-4+x)}{\log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ &=\frac {80}{5 \left (7-3 e^5\right )+24 x+20 \log (\log (-4+x))}+20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx+80 \int \left (\frac {24}{\left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2}-\frac {6 x}{\left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2}-\frac {5}{\log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2}\right ) \, dx\\ &=\frac {80}{5 \left (7-3 e^5\right )+24 x+20 \log (\log (-4+x))}+20 \int \frac {1}{35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))} \, dx-400 \int \frac {1}{\log (-4+x) \left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx-480 \int \frac {x}{\left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx+1920 \int \frac {1}{\left (35 \left (1-\frac {3 e^5}{7}\right )+24 x+20 \log (\log (-4+x))\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.13, size = 22, normalized size = 0.81 \begin {gather*} \frac {100 x}{175-75 e^5+120 x+100 \log (\log (-4+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-400*x + (-2800 + E^5*(1200 - 300*x) + 700*x)*Log[-4 + x] + (-1600 + 400*x)*Log[-4 + x]*Log[Log[-4
+ x]])/((-4900 - 5495*x - 624*x^2 + 576*x^3 + E^10*(-900 + 225*x) + E^5*(4200 + 1830*x - 720*x^2))*Log[-4 + x]
+ (-5600 + E^5*(2400 - 600*x) - 2440*x + 960*x^2)*Log[-4 + x]*Log[Log[-4 + x]] + (-1600 + 400*x)*Log[-4 + x]*
Log[Log[-4 + x]]^2),x]

[Out]

(100*x)/(175 - 75*E^5 + 120*x + 100*Log[Log[-4 + x]])

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 21, normalized size = 0.78 \begin {gather*} \frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x-1600)*log(x-4)*log(log(x-4))+((-300*x+1200)*exp(5)+700*x-2800)*log(x-4)-400*x)/((400*x-1600)
*log(x-4)*log(log(x-4))^2+((-600*x+2400)*exp(5)+960*x^2-2440*x-5600)*log(x-4)*log(log(x-4))+((225*x-900)*exp(5
)^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(x-4)),x, algorithm="fricas")

[Out]

20*x/(24*x - 15*e^5 + 20*log(log(x - 4)) + 35)

________________________________________________________________________________________

giac [A]  time = 0.26, size = 21, normalized size = 0.78 \begin {gather*} \frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x-1600)*log(x-4)*log(log(x-4))+((-300*x+1200)*exp(5)+700*x-2800)*log(x-4)-400*x)/((400*x-1600)
*log(x-4)*log(log(x-4))^2+((-600*x+2400)*exp(5)+960*x^2-2440*x-5600)*log(x-4)*log(log(x-4))+((225*x-900)*exp(5
)^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(x-4)),x, algorithm="giac")

[Out]

20*x/(24*x - 15*e^5 + 20*log(log(x - 4)) + 35)

________________________________________________________________________________________

maple [A]  time = 0.34, size = 22, normalized size = 0.81

method	result	size
risch	\(-\frac {20 x}{15 \,{\mathrm e}^{5}-20 \ln \left (\ln \left (x -4\right )\right )-24 x -35}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((400*x-1600)*ln(x-4)*ln(ln(x-4))+((-300*x+1200)*exp(5)+700*x-2800)*ln(x-4)-400*x)/((400*x-1600)*ln(x-4)*l
n(ln(x-4))^2+((-600*x+2400)*exp(5)+960*x^2-2440*x-5600)*ln(x-4)*ln(ln(x-4))+((225*x-900)*exp(5)^2+(-720*x^2+18
30*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*ln(x-4)),x,method=_RETURNVERBOSE)

[Out]

-20*x/(15*exp(5)-20*ln(ln(x-4))-24*x-35)

________________________________________________________________________________________

maxima [A]  time = 0.42, size = 21, normalized size = 0.78 \begin {gather*} \frac {20 \, x}{24 \, x - 15 \, e^{5} + 20 \, \log \left (\log \left (x - 4\right )\right ) + 35} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x-1600)*log(x-4)*log(log(x-4))+((-300*x+1200)*exp(5)+700*x-2800)*log(x-4)-400*x)/((400*x-1600)
*log(x-4)*log(log(x-4))^2+((-600*x+2400)*exp(5)+960*x^2-2440*x-5600)*log(x-4)*log(log(x-4))+((225*x-900)*exp(5
)^2+(-720*x^2+1830*x+4200)*exp(5)+576*x^3-624*x^2-5495*x-4900)*log(x-4)),x, algorithm="maxima")

[Out]

20*x/(24*x - 15*e^5 + 20*log(log(x - 4)) + 35)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {400\,x+\ln \left (x-4\right )\,\left ({\mathrm {e}}^5\,\left (300\,x-1200\right )-700\,x+2800\right )-\ln \left (x-4\right )\,\ln \left (\ln \left (x-4\right )\right )\,\left (400\,x-1600\right )}{-\ln \left (x-4\right )\,\left (400\,x-1600\right )\,{\ln \left (\ln \left (x-4\right )\right )}^2+\ln \left (x-4\right )\,\left (2440\,x-960\,x^2+{\mathrm {e}}^5\,\left (600\,x-2400\right )+5600\right )\,\ln \left (\ln \left (x-4\right )\right )+\ln \left (x-4\right )\,\left (5495\,x-{\mathrm {e}}^5\,\left (-720\,x^2+1830\,x+4200\right )+624\,x^2-576\,x^3-{\mathrm {e}}^{10}\,\left (225\,x-900\right )+4900\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((400*x + log(x - 4)*(exp(5)*(300*x - 1200) - 700*x + 2800) - log(x - 4)*log(log(x - 4))*(400*x - 1600))/(l
og(x - 4)*(5495*x - exp(5)*(1830*x - 720*x^2 + 4200) + 624*x^2 - 576*x^3 - exp(10)*(225*x - 900) + 4900) + log
(x - 4)*log(log(x - 4))*(2440*x - 960*x^2 + exp(5)*(600*x - 2400) + 5600) - log(x - 4)*log(log(x - 4))^2*(400*
x - 1600)),x)

[Out]

int((400*x + log(x - 4)*(exp(5)*(300*x - 1200) - 700*x + 2800) - log(x - 4)*log(log(x - 4))*(400*x - 1600))/(l
og(x - 4)*(5495*x - exp(5)*(1830*x - 720*x^2 + 4200) + 624*x^2 - 576*x^3 - exp(10)*(225*x - 900) + 4900) + log
(x - 4)*log(log(x - 4))*(2440*x - 960*x^2 + exp(5)*(600*x - 2400) + 5600) - log(x - 4)*log(log(x - 4))^2*(400*
x - 1600)), x)

________________________________________________________________________________________

sympy [A]  time = 0.41, size = 22, normalized size = 0.81 \begin {gather*} \frac {x}{\frac {6 x}{5} + \log {\left (\log {\left (x - 4 \right )} \right )} - \frac {3 e^{5}}{4} + \frac {7}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((400*x-1600)*ln(x-4)*ln(ln(x-4))+((-300*x+1200)*exp(5)+700*x-2800)*ln(x-4)-400*x)/((400*x-1600)*ln(
x-4)*ln(ln(x-4))**2+((-600*x+2400)*exp(5)+960*x**2-2440*x-5600)*ln(x-4)*ln(ln(x-4))+((225*x-900)*exp(5)**2+(-7
20*x**2+1830*x+4200)*exp(5)+576*x**3-624*x**2-5495*x-4900)*ln(x-4)),x)

[Out]

x/(6*x/5 + log(log(x - 4)) - 3*exp(5)/4 + 7/4)

________________________________________________________________________________________