3.101.5 \(\int \frac {(90-45 x) \log (2-x)+(12800 x^5+1600 x^6+50 x^7) \log (\log (2-x))+(-64000 x^4+22400 x^5+4450 x^6+175 x^7) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx\)

Optimal. Leaf size=25 \[ x \left (-5+\frac {25}{9} x^4 (16+x)^2 \log ^2(\log (2-x))\right ) \]

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Rubi [F]  time = 1.13, 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 {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((90 - 45*x)*Log[2 - x] + (12800*x^5 + 1600*x^6 + 50*x^7)*Log[Log[2 - x]] + (-64000*x^4 + 22400*x^5 + 4450
*x^6 + 175*x^7)*Log[2 - x]*Log[Log[2 - x]]^2)/((-18 + 9*x)*Log[2 - x]),x]

[Out]

-5*x + 28800*Log[Log[2 - x]]^2 + 14400*Defer[Int][(x*Log[Log[2 - x]])/Log[2 - x], x] + 7200*Defer[Int][(x^2*Lo
g[Log[2 - x]])/Log[2 - x], x] + 3600*Defer[Int][(x^3*Log[Log[2 - x]])/Log[2 - x], x] + 1800*Defer[Int][(x^4*Lo
g[Log[2 - x]])/Log[2 - x], x] + (1700*Defer[Int][(x^5*Log[Log[2 - x]])/Log[2 - x], x])/9 + (50*Defer[Int][(x^6
*Log[Log[2 - x]])/Log[2 - x], x])/9 + (32000*Defer[Int][x^4*Log[Log[2 - x]]^2, x])/9 + (1600*Defer[Int][x^5*Lo
g[Log[2 - x]]^2, x])/3 + (175*Defer[Int][x^6*Log[Log[2 - x]]^2, x])/9 - 28800*Defer[Subst][Defer[Int][Log[Log[
x]]/Log[x], x], x, 2 - x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5}{9} \left (-9+\frac {10 x^5 (16+x)^2 \log (\log (2-x))}{(-2+x) \log (2-x)}+5 x^4 \left (1280+192 x+7 x^2\right ) \log ^2(\log (2-x))\right ) \, dx\\ &=\frac {5}{9} \int \left (-9+\frac {10 x^5 (16+x)^2 \log (\log (2-x))}{(-2+x) \log (2-x)}+5 x^4 \left (1280+192 x+7 x^2\right ) \log ^2(\log (2-x))\right ) \, dx\\ &=-5 x+\frac {25}{9} \int x^4 \left (1280+192 x+7 x^2\right ) \log ^2(\log (2-x)) \, dx+\frac {50}{9} \int \frac {x^5 (16+x)^2 \log (\log (2-x))}{(-2+x) \log (2-x)} \, dx\\ &=-5 x+\frac {25}{9} \int \left (1280 x^4 \log ^2(\log (2-x))+192 x^5 \log ^2(\log (2-x))+7 x^6 \log ^2(\log (2-x))\right ) \, dx+\frac {50}{9} \int \left (\frac {5184 \log (\log (2-x))}{\log (2-x)}+\frac {10368 \log (\log (2-x))}{(-2+x) \log (2-x)}+\frac {2592 x \log (\log (2-x))}{\log (2-x)}+\frac {1296 x^2 \log (\log (2-x))}{\log (2-x)}+\frac {648 x^3 \log (\log (2-x))}{\log (2-x)}+\frac {324 x^4 \log (\log (2-x))}{\log (2-x)}+\frac {34 x^5 \log (\log (2-x))}{\log (2-x)}+\frac {x^6 \log (\log (2-x))}{\log (2-x)}\right ) \, dx\\ &=-5 x+\frac {50}{9} \int \frac {x^6 \log (\log (2-x))}{\log (2-x)} \, dx+\frac {175}{9} \int x^6 \log ^2(\log (2-x)) \, dx+\frac {1700}{9} \int \frac {x^5 \log (\log (2-x))}{\log (2-x)} \, dx+\frac {1600}{3} \int x^5 \log ^2(\log (2-x)) \, dx+1800 \int \frac {x^4 \log (\log (2-x))}{\log (2-x)} \, dx+\frac {32000}{9} \int x^4 \log ^2(\log (2-x)) \, dx+3600 \int \frac {x^3 \log (\log (2-x))}{\log (2-x)} \, dx+7200 \int \frac {x^2 \log (\log (2-x))}{\log (2-x)} \, dx+14400 \int \frac {x \log (\log (2-x))}{\log (2-x)} \, dx+28800 \int \frac {\log (\log (2-x))}{\log (2-x)} \, dx+57600 \int \frac {\log (\log (2-x))}{(-2+x) \log (2-x)} \, dx\\ &=-5 x+28800 \log ^2(\log (2-x))+\frac {50}{9} \int \frac {x^6 \log (\log (2-x))}{\log (2-x)} \, dx+\frac {175}{9} \int x^6 \log ^2(\log (2-x)) \, dx+\frac {1700}{9} \int \frac {x^5 \log (\log (2-x))}{\log (2-x)} \, dx+\frac {1600}{3} \int x^5 \log ^2(\log (2-x)) \, dx+1800 \int \frac {x^4 \log (\log (2-x))}{\log (2-x)} \, dx+\frac {32000}{9} \int x^4 \log ^2(\log (2-x)) \, dx+3600 \int \frac {x^3 \log (\log (2-x))}{\log (2-x)} \, dx+7200 \int \frac {x^2 \log (\log (2-x))}{\log (2-x)} \, dx+14400 \int \frac {x \log (\log (2-x))}{\log (2-x)} \, dx-28800 \operatorname {Subst}\left (\int \frac {\log (\log (x))}{\log (x)} \, dx,x,2-x\right )\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(90-45 x) \log (2-x)+\left (12800 x^5+1600 x^6+50 x^7\right ) \log (\log (2-x))+\left (-64000 x^4+22400 x^5+4450 x^6+175 x^7\right ) \log (2-x) \log ^2(\log (2-x))}{(-18+9 x) \log (2-x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((90 - 45*x)*Log[2 - x] + (12800*x^5 + 1600*x^6 + 50*x^7)*Log[Log[2 - x]] + (-64000*x^4 + 22400*x^5
+ 4450*x^6 + 175*x^7)*Log[2 - x]*Log[Log[2 - x]]^2)/((-18 + 9*x)*Log[2 - x]),x]

[Out]

Integrate[((90 - 45*x)*Log[2 - x] + (12800*x^5 + 1600*x^6 + 50*x^7)*Log[Log[2 - x]] + (-64000*x^4 + 22400*x^5
+ 4450*x^6 + 175*x^7)*Log[2 - x]*Log[Log[2 - x]]^2)/((-18 + 9*x)*Log[2 - x]), x]

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fricas [A]  time = 1.48, size = 29, normalized size = 1.16 \begin {gather*} \frac {25}{9} \, {\left (x^{7} + 32 \, x^{6} + 256 \, x^{5}\right )} \log \left (\log \left (-x + 2\right )\right )^{2} - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*log(2-x)*log(log(2-x))^2+(50*x^7+1600*x^6+12800*x^5)*log(log
(2-x))+(-45*x+90)*log(2-x))/(9*x-18)/log(2-x),x, algorithm="fricas")

[Out]

25/9*(x^7 + 32*x^6 + 256*x^5)*log(log(-x + 2))^2 - 5*x

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giac [A]  time = 0.22, size = 29, normalized size = 1.16 \begin {gather*} \frac {25}{9} \, {\left (x^{7} + 32 \, x^{6} + 256 \, x^{5}\right )} \log \left (\log \left (-x + 2\right )\right )^{2} - 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*log(2-x)*log(log(2-x))^2+(50*x^7+1600*x^6+12800*x^5)*log(log
(2-x))+(-45*x+90)*log(2-x))/(9*x-18)/log(2-x),x, algorithm="giac")

[Out]

25/9*(x^7 + 32*x^6 + 256*x^5)*log(log(-x + 2))^2 - 5*x

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maple [A]  time = 0.12, size = 31, normalized size = 1.24

method	result	size
risch	\(\left (\frac {25}{9} x^{7}+\frac {800}{9} x^{6}+\frac {6400}{9} x^{5}\right ) \ln \left (\ln \left (2-x \right )\right )^{2}-5 x\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*ln(2-x)*ln(ln(2-x))^2+(50*x^7+1600*x^6+12800*x^5)*ln(ln(2-x))+(-45
*x+90)*ln(2-x))/(9*x-18)/ln(2-x),x,method=_RETURNVERBOSE)

[Out]

(25/9*x^7+800/9*x^6+6400/9*x^5)*ln(ln(2-x))^2-5*x

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maxima [A]  time = 0.54, size = 43, normalized size = 1.72 \begin {gather*} \frac {25}{9} \, {\left (x^{7} + 32 \, x^{6} + 256 \, x^{5}\right )} \log \left (\log \left (-x + 2\right )\right )^{2} - 5 \, x - 10 \, \log \left (x - 2\right ) + 10 \, \log \left (-x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((175*x^7+4450*x^6+22400*x^5-64000*x^4)*log(2-x)*log(log(2-x))^2+(50*x^7+1600*x^6+12800*x^5)*log(log
(2-x))+(-45*x+90)*log(2-x))/(9*x-18)/log(2-x),x, algorithm="maxima")

[Out]

25/9*(x^7 + 32*x^6 + 256*x^5)*log(log(-x + 2))^2 - 5*x - 10*log(x - 2) + 10*log(-x + 2)

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mupad [B]  time = 8.15, size = 30, normalized size = 1.20 \begin {gather*} {\ln \left (\ln \left (2-x\right )\right )}^2\,\left (\frac {25\,x^7}{9}+\frac {800\,x^6}{9}+\frac {6400\,x^5}{9}\right )-5\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(log(2 - x))*(12800*x^5 + 1600*x^6 + 50*x^7) - log(2 - x)*(45*x - 90) + log(log(2 - x))^2*log(2 - x)*(
22400*x^5 - 64000*x^4 + 4450*x^6 + 175*x^7))/(log(2 - x)*(9*x - 18)),x)

[Out]

log(log(2 - x))^2*((6400*x^5)/9 + (800*x^6)/9 + (25*x^7)/9) - 5*x

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sympy [A]  time = 0.64, size = 31, normalized size = 1.24 \begin {gather*} - 5 x + \left (\frac {25 x^{7}}{9} + \frac {800 x^{6}}{9} + \frac {6400 x^{5}}{9}\right ) \log {\left (\log {\left (2 - x \right )} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((175*x**7+4450*x**6+22400*x**5-64000*x**4)*ln(2-x)*ln(ln(2-x))**2+(50*x**7+1600*x**6+12800*x**5)*ln
(ln(2-x))+(-45*x+90)*ln(2-x))/(9*x-18)/ln(2-x),x)

[Out]

-5*x + (25*x**7/9 + 800*x**6/9 + 6400*x**5/9)*log(log(2 - x))**2

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