Integrand size = 42, antiderivative size = 24 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=2-65536 x^8+\log (x)-\log \left (4+x-\frac {x}{5 \log (3)}\right ) \] Output:
2-ln(x-1/5*x/ln(3)+4)+ln(x)-65536*x^8
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-4 \left (16384 x^8-\frac {\log (x)}{4}+\frac {1}{4} \log (-x+20 \log (3)+x \log (243))\right ) \] Input:
Integrate[(524288*x^9 + (20 - 10485760*x^8 - 2621440*x^9)*Log[3])/(-x^2 + (20*x + 5*x^2)*Log[3]),x]
Output:
-4*(16384*x^8 - Log[x]/4 + Log[-x + 20*Log[3] + x*Log[243]]/4)
Time = 0.34 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2026, 2123, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {524288 x^9+\left (-2621440 x^9-10485760 x^8+20\right ) \log (3)}{\left (5 x^2+20 x\right ) \log (3)-x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {524288 x^9+\left (-2621440 x^9-10485760 x^8+20\right ) \log (3)}{x (20 \log (3)-x (1-\log (243)))}dx\) |
\(\Big \downarrow \) 2123 |
\(\displaystyle \int \left (-524288 x^7+\frac {1}{x}+\frac {1-\log (243)}{20 \log (3)-x (1-\log (243))}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -65536 x^8+\log (x)-\log (20 \log (3)-x (1-\log (243)))\) |
Input:
Int[(524288*x^9 + (20 - 10485760*x^8 - 2621440*x^9)*Log[3])/(-x^2 + (20*x + 5*x^2)*Log[3]),x]
Output:
-65536*x^8 + Log[x] - Log[20*Log[3] - x*(1 - Log[243])]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Time = 0.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
method | result | size |
norman | \(-65536 x^{8}-\ln \left (5 x \ln \left (3\right )+20 \ln \left (3\right )-x \right )+\ln \left (x \right )\) | \(25\) |
risch | \(-65536 x^{8}+\ln \left (-x \right )-\ln \left (x \left (5 \ln \left (3\right )-1\right )+20 \ln \left (3\right )\right )\) | \(27\) |
parallelrisch | \(-65536 x^{8}+\ln \left (x \right )-\ln \left (\frac {5 x \ln \left (3\right )+20 \ln \left (3\right )-x}{5 \ln \left (3\right )-1}\right )\) | \(34\) |
default | \(-65536 x^{8}+\ln \left (x \right )+\frac {4 \left (-\frac {5 \ln \left (3\right )}{4}+\frac {1}{4}\right ) \ln \left (5 x \ln \left (3\right )+20 \ln \left (3\right )-x \right )}{5 \ln \left (3\right )-1}\) | \(39\) |
meijerg | \(\ln \left (x \right )-2 \ln \left (2\right )-\ln \left (5\right )-\ln \left (\ln \left (3\right )\right )+\ln \left (5 \ln \left (3\right )-1\right )-\ln \left (1+\frac {x \left (5 \ln \left (3\right )-1\right )}{20 \ln \left (3\right )}\right )+\frac {512000000000 \ln \left (3\right )^{8} \left (-131072 \ln \left (3\right )+\frac {131072}{5}\right ) \left (-\frac {x \left (5 \ln \left (3\right )-1\right ) \left (-\frac {63 x^{7} \left (5 \ln \left (3\right )-1\right )^{7}}{256000000 \ln \left (3\right )^{7}}+\frac {9 x^{6} \left (5 \ln \left (3\right )-1\right )^{6}}{1600000 \ln \left (3\right )^{6}}-\frac {21 x^{5} \left (5 \ln \left (3\right )-1\right )^{5}}{160000 \ln \left (3\right )^{5}}+\frac {63 x^{4} \left (5 \ln \left (3\right )-1\right )^{4}}{20000 \ln \left (3\right )^{4}}-\frac {63 x^{3} \left (5 \ln \left (3\right )-1\right )^{3}}{800 \ln \left (3\right )^{3}}+\frac {21 x^{2} \left (5 \ln \left (3\right )-1\right )^{2}}{10 \ln \left (3\right )^{2}}-\frac {63 x \left (5 \ln \left (3\right )-1\right )}{\ln \left (3\right )}+2520\right )}{50400 \ln \left (3\right )}+\ln \left (1+\frac {x \left (5 \ln \left (3\right )-1\right )}{20 \ln \left (3\right )}\right )\right )}{\left (5 \ln \left (3\right )-1\right )^{9}}-\frac {13421772800000000 \ln \left (3\right )^{8} \left (\frac {x \left (5 \ln \left (3\right )-1\right ) \left (\frac {3 x^{6} \left (5 \ln \left (3\right )-1\right )^{6}}{1600000 \ln \left (3\right )^{6}}-\frac {7 x^{5} \left (5 \ln \left (3\right )-1\right )^{5}}{160000 \ln \left (3\right )^{5}}+\frac {21 x^{4} \left (5 \ln \left (3\right )-1\right )^{4}}{20000 \ln \left (3\right )^{4}}-\frac {21 x^{3} \left (5 \ln \left (3\right )-1\right )^{3}}{800 \ln \left (3\right )^{3}}+\frac {7 x^{2} \left (5 \ln \left (3\right )-1\right )^{2}}{10 \ln \left (3\right )^{2}}-\frac {21 x \left (5 \ln \left (3\right )-1\right )}{\ln \left (3\right )}+840\right )}{16800 \ln \left (3\right )}-\ln \left (1+\frac {x \left (5 \ln \left (3\right )-1\right )}{20 \ln \left (3\right )}\right )\right )}{\left (5 \ln \left (3\right )-1\right )^{8}}\) | \(355\) |
Input:
int(((-2621440*x^9-10485760*x^8+20)*ln(3)+524288*x^9)/((5*x^2+20*x)*ln(3)- x^2),x,method=_RETURNVERBOSE)
Output:
-65536*x^8-ln(5*x*ln(3)+20*ln(3)-x)+ln(x)
Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-65536 \, x^{8} - \log \left (5 \, {\left (x + 4\right )} \log \left (3\right ) - x\right ) + \log \left (x\right ) \] Input:
integrate(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x) *log(3)-x^2),x, algorithm="fricas")
Output:
-65536*x^8 - log(5*(x + 4)*log(3) - x) + log(x)
Time = 0.29 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=- 65536 x^{8} + \log {\left (x \right )} - \log {\left (x + \frac {20 \log {\left (3 \right )}}{-1 + 5 \log {\left (3 \right )}} \right )} \] Input:
integrate(((-2621440*x**9-10485760*x**8+20)*ln(3)+524288*x**9)/((5*x**2+20 *x)*ln(3)-x**2),x)
Output:
-65536*x**8 + log(x) - log(x + 20*log(3)/(-1 + 5*log(3)))
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-65536 \, x^{8} - \log \left (x {\left (5 \, \log \left (3\right ) - 1\right )} + 20 \, \log \left (3\right )\right ) + \log \left (x\right ) \] Input:
integrate(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x) *log(3)-x^2),x, algorithm="maxima")
Output:
-65536*x^8 - log(x*(5*log(3) - 1) + 20*log(3)) + log(x)
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (22) = 44\).
Time = 0.11 (sec) , antiderivative size = 147, normalized size of antiderivative = 6.12 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-\frac {65536 \, {\left (390625 \, x^{8} \log \left (3\right )^{8} - 625000 \, x^{8} \log \left (3\right )^{7} + 437500 \, x^{8} \log \left (3\right )^{6} - 175000 \, x^{8} \log \left (3\right )^{5} + 43750 \, x^{8} \log \left (3\right )^{4} - 7000 \, x^{8} \log \left (3\right )^{3} + 700 \, x^{8} \log \left (3\right )^{2} - 40 \, x^{8} \log \left (3\right ) + x^{8}\right )}}{390625 \, \log \left (3\right )^{8} - 625000 \, \log \left (3\right )^{7} + 437500 \, \log \left (3\right )^{6} - 175000 \, \log \left (3\right )^{5} + 43750 \, \log \left (3\right )^{4} - 7000 \, \log \left (3\right )^{3} + 700 \, \log \left (3\right )^{2} - 40 \, \log \left (3\right ) + 1} - \log \left ({\left | 5 \, x \log \left (3\right ) - x + 20 \, \log \left (3\right ) \right |}\right ) + \log \left ({\left | x \right |}\right ) \] Input:
integrate(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x) *log(3)-x^2),x, algorithm="giac")
Output:
-65536*(390625*x^8*log(3)^8 - 625000*x^8*log(3)^7 + 437500*x^8*log(3)^6 - 175000*x^8*log(3)^5 + 43750*x^8*log(3)^4 - 7000*x^8*log(3)^3 + 700*x^8*log (3)^2 - 40*x^8*log(3) + x^8)/(390625*log(3)^8 - 625000*log(3)^7 + 437500*l og(3)^6 - 175000*log(3)^5 + 43750*log(3)^4 - 7000*log(3)^3 + 700*log(3)^2 - 40*log(3) + 1) - log(abs(5*x*log(3) - x + 20*log(3))) + log(abs(x))
Time = 0.20 (sec) , antiderivative size = 325, normalized size of antiderivative = 13.54 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx =\text {Too large to display} \] Input:
int(-(log(3)*(10485760*x^8 + 2621440*x^9 - 20) - 524288*x^9)/(log(3)*(20*x + 5*x^2) - x^2),x)
Output:
atan((x*(2*log(243) - 2)*1i)/(20*log(3)) + 1i)*2i - x^7*((10485760*log(3)) /(7*(log(243) - 1)) - (20*log(3)*(2621440*log(3) - 524288))/(7*(log(243) - 1)^2)) - (x^8*(2621440*log(3) - 524288))/(8*(log(243) - 1)) + (10*x^6*log (3)*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 52428 8))/(log(243) - 1)^2))/(3*(log(243) - 1)) - (64000000*x*log(3)^6*((1048576 0*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(log(243) - 1)^6 - (80*x^5*log(3)^2*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(log(243) - 1)^2 + (2000*x^4*log(3)^3*((10485760*log(3))/(log(243) - 1) - (20*log(3 )*(2621440*log(3) - 524288))/(log(243) - 1)^2))/(log(243) - 1)^3 - (160000 *x^3*log(3)^4*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log( 3) - 524288))/(log(243) - 1)^2))/(3*(log(243) - 1)^4) + (1600000*x^2*log(3 )^5*((10485760*log(3))/(log(243) - 1) - (20*log(3)*(2621440*log(3) - 52428 8))/(log(243) - 1)^2))/(log(243) - 1)^5
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {524288 x^9+\left (20-10485760 x^8-2621440 x^9\right ) \log (3)}{-x^2+\left (20 x+5 x^2\right ) \log (3)} \, dx=-\mathrm {log}\left (5 \,\mathrm {log}\left (3\right ) x +20 \,\mathrm {log}\left (3\right )-x \right )+\mathrm {log}\left (x \right )-65536 x^{8} \] Input:
int(((-2621440*x^9-10485760*x^8+20)*log(3)+524288*x^9)/((5*x^2+20*x)*log(3 )-x^2),x)
Output:
- log(5*log(3)*x + 20*log(3) - x) + log(x) - 65536*x**8