3.17.16 \(\int \frac {-27000 x+5400 x^2+(216000-54000 x-27000 x^2) \log (4+x)+(9000 x-1800 x^2+(-72000+39600 x+14400 x^2) \log (4+x)) \log (\frac {x^2}{\log (4+x)})+(-7200 x-1800 x^2) \log (4+x) \log ^2(\frac {x^2}{\log (4+x)})}{(-500 x+175 x^2+15 x^3-11 x^4+x^5) \log (4+x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {900 \left (3-\log \left (\frac {x^2}{\log (4+x)}\right )\right )^2}{(-5+x)^2} \]

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Rubi [F]  time = 14.75, 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 {-27000 x+5400 x^2+\left (216000-54000 x-27000 x^2\right ) \log (4+x)+\left (9000 x-1800 x^2+\left (-72000+39600 x+14400 x^2\right ) \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )+\left (-7200 x-1800 x^2\right ) \log (4+x) \log ^2\left (\frac {x^2}{\log (4+x)}\right )}{\left (-500 x+175 x^2+15 x^3-11 x^4+x^5\right ) \log (4+x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-27000*x + 5400*x^2 + (216000 - 54000*x - 27000*x^2)*Log[4 + x] + (9000*x - 1800*x^2 + (-72000 + 39600*x
+ 14400*x^2)*Log[4 + x])*Log[x^2/Log[4 + x]] + (-7200*x - 1800*x^2)*Log[4 + x]*Log[x^2/Log[4 + x]]^2)/((-500*x
+ 175*x^2 + 15*x^3 - 11*x^4 + x^5)*Log[4 + x]),x]

[Out]

8100/(5 - x)^2 + 288*Log[5 - x] - 112*Log[x] - (3200*Log[x^2/Log[4 + x]])/(5 - x)^2 - (160*Log[x^2/Log[4 + x]]
)/(5 - x) - (88*x^2*Log[x^2/Log[4 + x]])/(5 - x)^2 - 8*Log[Log[4 + x]] - 80*Defer[Int][1/((-5 + x)*Log[4 + x])
, x] - 144*Defer[Int][Log[x^2/Log[4 + x]]/(-5 + x), x] + 144*Defer[Int][Log[x^2/Log[4 + x]]/x, x] - 200*Defer[
Int][Log[x^2/Log[4 + x]]/((-5 + x)^2*Log[4 + x]), x] + (200*Defer[Int][Log[x^2/Log[4 + x]]/((-5 + x)*Log[4 + x
]), x])/9 - (200*Defer[Int][Log[x^2/Log[4 + x]]/((4 + x)*Log[4 + x]), x])/9 - 1800*Defer[Int][Log[x^2/Log[4 +
x]]^2/(-5 + x)^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1800 \left (3-\log \left (\frac {x^2}{\log (4+x)}\right )\right ) \left (-((-5+x) x)-(4+x) \log (4+x) \left (10-5 x+x \log \left (\frac {x^2}{\log (4+x)}\right )\right )\right )}{(5-x)^3 x (4+x) \log (4+x)} \, dx\\ &=1800 \int \frac {\left (3-\log \left (\frac {x^2}{\log (4+x)}\right )\right ) \left (-((-5+x) x)-(4+x) \log (4+x) \left (10-5 x+x \log \left (\frac {x^2}{\log (4+x)}\right )\right )\right )}{(5-x)^3 x (4+x) \log (4+x)} \, dx\\ &=1800 \int \left (-\frac {15}{(-5+x)^3}+\frac {30}{(-5+x)^3 x}+\frac {3}{(-5+x)^2 (4+x) \log (4+x)}+\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 x (4+x) \log (4+x)}-\frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}\right ) \, dx\\ &=\frac {13500}{(5-x)^2}+1800 \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 x (4+x) \log (4+x)} \, dx-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx+5400 \int \frac {1}{(-5+x)^2 (4+x) \log (4+x)} \, dx+54000 \int \frac {1}{(-5+x)^3 x} \, dx\\ &=\frac {13500}{(5-x)^2}-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx+1800 \int \left (\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{45 (-5+x)^3 \log (4+x)}-\frac {14 \left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{2025 (-5+x)^2 \log (4+x)}+\frac {151 \left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{91125 (-5+x) \log (4+x)}-\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{500 x \log (4+x)}+\frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{2916 (4+x) \log (4+x)}\right ) \, dx+5400 \int \left (\frac {1}{9 (-5+x)^2 \log (4+x)}-\frac {1}{81 (-5+x) \log (4+x)}+\frac {1}{81 (4+x) \log (4+x)}\right ) \, dx+54000 \int \left (\frac {1}{5 (-5+x)^3}-\frac {1}{25 (-5+x)^2}+\frac {1}{125 (-5+x)}-\frac {1}{125 x}\right ) \, dx\\ &=\frac {8100}{(5-x)^2}-\frac {2160}{5-x}+432 \log (5-x)-432 \log (x)+\frac {50}{81} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(4+x) \log (4+x)} \, dx+\frac {1208}{405} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x) \log (4+x)} \, dx-\frac {18}{5} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{x \log (4+x)} \, dx-\frac {112}{9} \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2 \log (4+x)} \, dx+40 \int \frac {\left (5 x-x^2-40 \log (4+x)+22 x \log (4+x)+8 x^2 \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 \log (4+x)} \, dx-\frac {200}{3} \int \frac {1}{(-5+x) \log (4+x)} \, dx+\frac {200}{3} \int \frac {1}{(4+x) \log (4+x)} \, dx+600 \int \frac {1}{(-5+x)^2 \log (4+x)} \, dx-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx\\ &=\frac {8100}{(5-x)^2}-\frac {2160}{5-x}+432 \log (5-x)-432 \log (x)+\frac {50}{81} \int \frac {\left (-((-5+x) x)+\left (-40+22 x+8 x^2\right ) \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{(4+x) \log (4+x)} \, dx+\frac {1208}{405} \int \left (-\frac {40 \log \left (\frac {x^2}{\log (4+x)}\right )}{-5+x}+\frac {22 x \log \left (\frac {x^2}{\log (4+x)}\right )}{-5+x}+\frac {8 x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{-5+x}+\frac {5 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x) \log (4+x)}-\frac {x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x) \log (4+x)}\right ) \, dx-\frac {18}{5} \int \frac {\left (-((-5+x) x)+\left (-40+22 x+8 x^2\right ) \log (4+x)\right ) \log \left (\frac {x^2}{\log (4+x)}\right )}{x \log (4+x)} \, dx-\frac {112}{9} \int \left (-\frac {40 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2}+\frac {22 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2}+\frac {8 x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2}+\frac {5 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2 \log (4+x)}-\frac {x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^2 \log (4+x)}\right ) \, dx+40 \int \left (-\frac {40 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}+\frac {22 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}+\frac {8 x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3}+\frac {5 x \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 \log (4+x)}-\frac {x^2 \log \left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3 \log (4+x)}\right ) \, dx-\frac {200}{3} \int \frac {1}{(-5+x) \log (4+x)} \, dx+\frac {200}{3} \operatorname {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,4+x\right )+600 \int \frac {1}{(-5+x)^2 \log (4+x)} \, dx-1800 \int \frac {\log ^2\left (\frac {x^2}{\log (4+x)}\right )}{(-5+x)^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 22, normalized size = 0.92 \begin {gather*} \frac {900 \left (-3+\log \left (\frac {x^2}{\log (4+x)}\right )\right )^2}{(-5+x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-27000*x + 5400*x^2 + (216000 - 54000*x - 27000*x^2)*Log[4 + x] + (9000*x - 1800*x^2 + (-72000 + 39
600*x + 14400*x^2)*Log[4 + x])*Log[x^2/Log[4 + x]] + (-7200*x - 1800*x^2)*Log[4 + x]*Log[x^2/Log[4 + x]]^2)/((
-500*x + 175*x^2 + 15*x^3 - 11*x^4 + x^5)*Log[4 + x]),x]

[Out]

(900*(-3 + Log[x^2/Log[4 + x]])^2)/(-5 + x)^2

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fricas [A]  time = 0.70, size = 40, normalized size = 1.67 \begin {gather*} \frac {900 \, {\left (\log \left (\frac {x^{2}}{\log \left (x + 4\right )}\right )^{2} - 6 \, \log \left (\frac {x^{2}}{\log \left (x + 4\right )}\right ) + 9\right )}}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1800*x^2-7200*x)*log(4+x)*log(x^2/log(4+x))^2+((14400*x^2+39600*x-72000)*log(4+x)-1800*x^2+9000*x
)*log(x^2/log(4+x))+(-27000*x^2-54000*x+216000)*log(4+x)+5400*x^2-27000*x)/(x^5-11*x^4+15*x^3+175*x^2-500*x)/l
og(4+x),x, algorithm="fricas")

[Out]

900*(log(x^2/log(x + 4))^2 - 6*log(x^2/log(x + 4)) + 9)/(x^2 - 10*x + 25)

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giac [B]  time = 0.76, size = 101, normalized size = 4.21 \begin {gather*} -1800 \, {\left (\frac {\log \left (x^{2}\right )}{x^{2} - 10 \, x + 25} - \frac {3}{x^{2} - 10 \, x + 25}\right )} \log \left (\log \left (x + 4\right )\right ) + \frac {900 \, \log \left (x^{2}\right )^{2}}{x^{2} - 10 \, x + 25} + \frac {900 \, \log \left (\log \left (x + 4\right )\right )^{2}}{x^{2} - 10 \, x + 25} - \frac {5400 \, \log \left (x^{2}\right )}{x^{2} - 10 \, x + 25} + \frac {8100}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1800*x^2-7200*x)*log(4+x)*log(x^2/log(4+x))^2+((14400*x^2+39600*x-72000)*log(4+x)-1800*x^2+9000*x
)*log(x^2/log(4+x))+(-27000*x^2-54000*x+216000)*log(4+x)+5400*x^2-27000*x)/(x^5-11*x^4+15*x^3+175*x^2-500*x)/l
og(4+x),x, algorithm="giac")

[Out]

-1800*(log(x^2)/(x^2 - 10*x + 25) - 3/(x^2 - 10*x + 25))*log(log(x + 4)) + 900*log(x^2)^2/(x^2 - 10*x + 25) +
900*log(log(x + 4))^2/(x^2 - 10*x + 25) - 5400*log(x^2)/(x^2 - 10*x + 25) + 8100/(x^2 - 10*x + 25)

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maple [C]  time = 0.33, size = 1389, normalized size = 57.88

method	result	size
risch	\(\frac {900 \ln \left (\ln \left (4+x \right )\right )^{2}}{x^{2}-10 x +25}-\frac {900 \left (-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}-i \pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}+4 \ln \relax (x )-6\right ) \ln \left (\ln \left (4+x \right )\right )}{x^{2}-10 x +25}+\frac {8100-10800 \ln \relax (x )+3600 \ln \relax (x )^{2}-225 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{6}-225 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right )^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}+450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right )^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}-900 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{4}-1800 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-1800 i \ln \relax (x ) \pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}-2700 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}-2700 i \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+450 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}-450 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}-900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}-450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )+450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}-5400 i \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+2700 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+2700 i \pi \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}-225 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{4}+450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{5}-225 \pi ^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right )^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{4}+450 \pi ^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{5}+450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{4} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}-450 \pi ^{2} \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{3}-225 \pi ^{2} \mathrm {csgn}\left (i x \right )^{4} \mathrm {csgn}\left (i x^{2}\right )^{2}+900 \pi ^{2} \mathrm {csgn}\left (i x \right )^{3} \mathrm {csgn}\left (i x^{2}\right )^{3}-1350 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{4}+900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{5}-225 \pi ^{2} \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{6}+2700 i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+3600 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+1800 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+1800 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+2700 i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )-450 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )+450 \pi ^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}+900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{3} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )-900 \pi ^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )^{2}-1800 i \ln \relax (x ) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-1800 i \ln \relax (x ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (\frac {i}{\ln \left (4+x \right )}\right ) \mathrm {csgn}\left (\frac {i x^{2}}{\ln \left (4+x \right )}\right )}{x^{2}-10 x +25}\)	\(1389\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-1800*x^2-7200*x)*ln(4+x)*ln(x^2/ln(4+x))^2+((14400*x^2+39600*x-72000)*ln(4+x)-1800*x^2+9000*x)*ln(x^2/l
n(4+x))+(-27000*x^2-54000*x+216000)*ln(4+x)+5400*x^2-27000*x)/(x^5-11*x^4+15*x^3+175*x^2-500*x)/ln(4+x),x,meth
od=_RETURNVERBOSE)

[Out]

900/(x^2-10*x+25)*ln(ln(4+x))^2-900*(-I*Pi*csgn(I*x^2)*csgn(I*x)^2+2*I*Pi*csgn(I*x^2)^2*csgn(I*x)-I*Pi*csgn(I*
x^2)^3-I*Pi*csgn(I*x^2)*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))+I*Pi*csgn(I*x^2)*csgn(I*x^2/ln(4+x))^2+I*Pi*csgn(I
/ln(4+x))*csgn(I*x^2/ln(4+x))^2-I*Pi*csgn(I*x^2/ln(4+x))^3+4*ln(x)-6)/(x^2-10*x+25)*ln(ln(4+x))+225*(36-48*ln(
x)+16*ln(x)^2-Pi^2*csgn(I*x^2)^6+2*Pi^2*csgn(I*x)^2*csgn(I*x^2)^2*csgn(I*x^2/ln(4+x))^2-2*Pi^2*csgn(I*x)^2*csg
n(I*x^2)*csgn(I*x^2/ln(4+x))^3-4*Pi^2*csgn(I*x)*csgn(I*x^2)^3*csgn(I*x^2/ln(4+x))^2+4*Pi^2*csgn(I*x)*csgn(I*x^
2)^2*csgn(I*x^2/ln(4+x))^3-2*Pi^2*csgn(I*x^2)^4*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))+2*Pi^2*csgn(I*x^2)^3*csgn(
I/ln(4+x))*csgn(I*x^2/ln(4+x))^2-Pi^2*csgn(I*x^2)^2*csgn(I/ln(4+x))^2*csgn(I*x^2/ln(4+x))^2+2*Pi^2*csgn(I*x^2)
^2*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))^3+2*Pi^2*csgn(I*x^2)*csgn(I/ln(4+x))^2*csgn(I*x^2/ln(4+x))^3-4*Pi^2*csg
n(I*x^2)*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))^4-8*I*ln(x)*Pi*csgn(I*x^2)^3-8*I*ln(x)*Pi*csgn(I*x^2/ln(4+x))^3+1
2*I*Pi*csgn(I*x)^2*csgn(I*x^2)-24*I*Pi*csgn(I*x)*csgn(I*x^2)^2-12*I*Pi*csgn(I*x^2)*csgn(I*x^2/ln(4+x))^2-12*I*
Pi*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))^2-Pi^2*csgn(I*x)^4*csgn(I*x^2)^2+4*Pi^2*csgn(I*x)^3*csgn(I*x^2)^3-6*Pi^
2*csgn(I*x)^2*csgn(I*x^2)^4+4*Pi^2*csgn(I*x)*csgn(I*x^2)^5-Pi^2*csgn(I*x^2/ln(4+x))^6-2*Pi^2*csgn(I*x)^2*csgn(
I*x^2)^2*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))+2*Pi^2*csgn(I*x)^2*csgn(I*x^2)*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x)
)^2+4*Pi^2*csgn(I*x)*csgn(I*x^2)^3*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))-4*Pi^2*csgn(I*x)*csgn(I*x^2)^2*csgn(I/l
n(4+x))*csgn(I*x^2/ln(4+x))^2+12*I*Pi*csgn(I*x^2)*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))-8*I*ln(x)*Pi*csgn(I*x)^2
*csgn(I*x^2)+16*I*ln(x)*Pi*csgn(I*x)*csgn(I*x^2)^2+8*I*ln(x)*Pi*csgn(I*x^2)*csgn(I*x^2/ln(4+x))^2+8*I*ln(x)*Pi
*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))^2+12*I*Pi*csgn(I*x^2/ln(4+x))^3+12*I*Pi*csgn(I*x^2)^3-Pi^2*csgn(I*x^2)^2*
csgn(I*x^2/ln(4+x))^4+2*Pi^2*csgn(I*x^2)*csgn(I*x^2/ln(4+x))^5-Pi^2*csgn(I/ln(4+x))^2*csgn(I*x^2/ln(4+x))^4+2*
Pi^2*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x))^5+2*Pi^2*csgn(I*x^2)^4*csgn(I*x^2/ln(4+x))^2-2*Pi^2*csgn(I*x^2)^3*csg
n(I*x^2/ln(4+x))^3-8*I*ln(x)*Pi*csgn(I*x^2)*csgn(I/ln(4+x))*csgn(I*x^2/ln(4+x)))/(x^2-10*x+25)

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maxima [A]  time = 0.69, size = 44, normalized size = 1.83 \begin {gather*} \frac {900 \, {\left (4 \, \log \relax (x)^{2} - 2 \, {\left (2 \, \log \relax (x) - 3\right )} \log \left (\log \left (x + 4\right )\right ) + \log \left (\log \left (x + 4\right )\right )^{2} - 12 \, \log \relax (x) + 9\right )}}{x^{2} - 10 \, x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1800*x^2-7200*x)*log(4+x)*log(x^2/log(4+x))^2+((14400*x^2+39600*x-72000)*log(4+x)-1800*x^2+9000*x
)*log(x^2/log(4+x))+(-27000*x^2-54000*x+216000)*log(4+x)+5400*x^2-27000*x)/(x^5-11*x^4+15*x^3+175*x^2-500*x)/l
og(4+x),x, algorithm="maxima")

[Out]

900*(4*log(x)^2 - 2*(2*log(x) - 3)*log(log(x + 4)) + log(log(x + 4))^2 - 12*log(x) + 9)/(x^2 - 10*x + 25)

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mupad [B]  time = 1.42, size = 22, normalized size = 0.92 \begin {gather*} \frac {900\,{\left (\ln \left (\frac {x^2}{\ln \left (x+4\right )}\right )-3\right )}^2}{{\left (x-5\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(27000*x + log(x + 4)*(54000*x + 27000*x^2 - 216000) - log(x^2/log(x + 4))*(9000*x + log(x + 4)*(39600*x
+ 14400*x^2 - 72000) - 1800*x^2) - 5400*x^2 + log(x + 4)*log(x^2/log(x + 4))^2*(7200*x + 1800*x^2))/(log(x + 4
)*(175*x^2 - 500*x + 15*x^3 - 11*x^4 + x^5)),x)

[Out]

(900*(log(x^2/log(x + 4)) - 3)^2)/(x - 5)^2

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sympy [B]  time = 0.58, size = 53, normalized size = 2.21 \begin {gather*} \frac {16200}{2 x^{2} - 20 x + 50} + \frac {900 \log {\left (\frac {x^{2}}{\log {\left (x + 4 \right )}} \right )}^{2}}{x^{2} - 10 x + 25} - \frac {5400 \log {\left (\frac {x^{2}}{\log {\left (x + 4 \right )}} \right )}}{x^{2} - 10 x + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-1800*x**2-7200*x)*ln(4+x)*ln(x**2/ln(4+x))**2+((14400*x**2+39600*x-72000)*ln(4+x)-1800*x**2+9000*
x)*ln(x**2/ln(4+x))+(-27000*x**2-54000*x+216000)*ln(4+x)+5400*x**2-27000*x)/(x**5-11*x**4+15*x**3+175*x**2-500
*x)/ln(4+x),x)

[Out]

16200/(2*x**2 - 20*x + 50) + 900*log(x**2/log(x + 4))**2/(x**2 - 10*x + 25) - 5400*log(x**2/log(x + 4))/(x**2
- 10*x + 25)

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