Integrand size = 201, antiderivative size = 38 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {\left (-2+\frac {1}{2 \left (\frac {5-x}{3}-x\right )}-x^2\right )^2}{1+x+\log ^2(x)} \end {dmath*}
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.97 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {\left (17-16 x+10 x^2-8 x^3\right )^2}{4 (5-4 x)^2 \left (1+x+\log ^2(x)\right )} \end {dmath*}
Integrate[(1853*x - 7252*x^2 + 10252*x^3 - 5952*x^4 + 1508*x^5 + 784*x^6 - 1856*x^7 + 768*x^8 + (2890 - 7752*x + 10312*x^2 - 10688*x^3 + 8296*x^4 - 4448*x^5 + 1920*x^6 - 512*x^7)*Log[x] + (408*x - 3784*x^2 + 8880*x^3 - 948 8*x^4 + 6848*x^5 - 3840*x^6 + 1024*x^7)*Log[x]^2)/(-500*x + 200*x^2 + 940* x^3 - 464*x^4 - 448*x^5 + 256*x^6 + (-1000*x + 1400*x^2 + 480*x^3 - 1408*x ^4 + 512*x^5)*Log[x]^2 + (-500*x + 1200*x^2 - 960*x^3 + 256*x^4)*Log[x]^4) ,x]
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.
\(\displaystyle \int \frac {768 x^8-1856 x^7+784 x^6+1508 x^5-5952 x^4+10252 x^3-7252 x^2+\left (1024 x^7-3840 x^6+6848 x^5-9488 x^4+8880 x^3-3784 x^2+408 x\right ) \log ^2(x)+\left (-512 x^7+1920 x^6-4448 x^5+8296 x^4-10688 x^3+10312 x^2-7752 x+2890\right ) \log (x)+1853 x}{256 x^6-448 x^5-464 x^4+940 x^3+200 x^2+\left (256 x^4-960 x^3+1200 x^2-500 x\right ) \log ^4(x)+\left (512 x^5-1408 x^4+480 x^3+1400 x^2-1000 x\right ) \log ^2(x)-500 x} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {\left (-8 x^3+10 x^2-16 x+17\right ) \left (8 x \left (16 x^3-40 x^2+25 x-3\right ) \log ^2(x)+x \left (96 x^4-112 x^3-234 x^2+324 x-109\right )-2 \left (32 x^4-80 x^3+114 x^2-148 x+85\right ) \log (x)\right )}{4 (5-4 x)^3 x \left (x+\log ^2(x)+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int -\frac {\left (-8 x^3+10 x^2-16 x+17\right ) \left (8 x \left (-16 x^3+40 x^2-25 x+3\right ) \log ^2(x)+2 \left (32 x^4-80 x^3+114 x^2-148 x+85\right ) \log (x)+x \left (-96 x^4+112 x^3+234 x^2-324 x+109\right )\right )}{(5-4 x)^3 x \left (\log ^2(x)+x+1\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{4} \int \frac {\left (-8 x^3+10 x^2-16 x+17\right ) \left (8 x \left (-16 x^3+40 x^2-25 x+3\right ) \log ^2(x)+2 \left (32 x^4-80 x^3+114 x^2-148 x+85\right ) \log (x)+x \left (-96 x^4+112 x^3+234 x^2-324 x+109\right )\right )}{(5-4 x)^3 x \left (\log ^2(x)+x+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{4} \int \left (\frac {\left (8 x^3-10 x^2+16 x-17\right )^2 (x+2 \log (x))}{x (4 x-5)^2 \left (\log ^2(x)+x+1\right )^2}-\frac {8 \left (128 x^6-480 x^5+856 x^4-1186 x^3+1110 x^2-473 x+51\right )}{(4 x-5)^3 \left (\log ^2(x)+x+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (-4 \int \frac {x^4}{\left (\log ^2(x)+x+1\right )^2}dx-8 \int \frac {x^3 \log (x)}{\left (\log ^2(x)+x+1\right )^2}dx+16 \int \frac {x^3}{\log ^2(x)+x+1}dx-16 \int \frac {x^2}{\left (\log ^2(x)+x+1\right )^2}dx-\frac {819}{100} \int \frac {1}{\left (\log ^2(x)+x+1\right )^2}dx-3 \int \frac {x}{\left (\log ^2(x)+x+1\right )^2}dx-9 \int \frac {1}{(4 x-5)^2 \left (\log ^2(x)+x+1\right )^2}dx-\frac {171}{4} \int \frac {1}{(4 x-5) \left (\log ^2(x)+x+1\right )^2}dx-6 \int \frac {\log (x)}{\left (\log ^2(x)+x+1\right )^2}dx-32 \int \frac {x \log (x)}{\left (\log ^2(x)+x+1\right )^2}dx-\frac {72}{5} \int \frac {\log (x)}{(4 x-5)^2 \left (\log ^2(x)+x+1\right )^2}dx-\frac {1638}{25} \int \frac {\log (x)}{(4 x-5) \left (\log ^2(x)+x+1\right )^2}dx+3 \int \frac {1}{\log ^2(x)+x+1}dx+32 \int \frac {x}{\log ^2(x)+x+1}dx-72 \int \frac {1}{(4 x-5)^3 \left (\log ^2(x)+x+1\right )}dx-171 \int \frac {1}{(4 x-5)^2 \left (\log ^2(x)+x+1\right )}dx+\frac {289}{25 \left (x+\log ^2(x)+1\right )}\right )\) |
Int[(1853*x - 7252*x^2 + 10252*x^3 - 5952*x^4 + 1508*x^5 + 784*x^6 - 1856* x^7 + 768*x^8 + (2890 - 7752*x + 10312*x^2 - 10688*x^3 + 8296*x^4 - 4448*x ^5 + 1920*x^6 - 512*x^7)*Log[x] + (408*x - 3784*x^2 + 8880*x^3 - 9488*x^4 + 6848*x^5 - 3840*x^6 + 1024*x^7)*Log[x]^2)/(-500*x + 200*x^2 + 940*x^3 - 464*x^4 - 448*x^5 + 256*x^6 + (-1000*x + 1400*x^2 + 480*x^3 - 1408*x^4 + 5 12*x^5)*Log[x]^2 + (-500*x + 1200*x^2 - 960*x^3 + 256*x^4)*Log[x]^4),x]
3.8.37.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(26)=52\).
Time = 2.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42
method | result | size |
risch | \(\frac {64 x^{6}-160 x^{5}+356 x^{4}-592 x^{3}+596 x^{2}-544 x +289}{4 \left (16 x^{2}-40 x +25\right ) \left (x +\ln \left (x \right )^{2}+1\right )}\) | \(54\) |
default | \(\frac {714-799 x +425 \ln \left (x \right )^{2}+64 x^{6}-320 x^{3}+272 x^{2} \ln \left (x \right )^{2}-680 x \ln \left (x \right )^{2}+188 x^{2}+356 x^{4}-160 x^{5}}{4 \left (-5+4 x \right )^{2} \left (x +\ln \left (x \right )^{2}+1\right )}\) | \(71\) |
parallelrisch | \(\frac {1024 x^{6}-2560 x^{5}+5696 x^{4}-9472 x^{3}+9536 x^{2}-8704 x +4624}{1024 x^{2} \ln \left (x \right )^{2}+1024 x^{3}-2560 x \ln \left (x \right )^{2}-1536 x^{2}+1600 \ln \left (x \right )^{2}-960 x +1600}\) | \(72\) |
int(((1024*x^7-3840*x^6+6848*x^5-9488*x^4+8880*x^3-3784*x^2+408*x)*ln(x)^2 +(-512*x^7+1920*x^6-4448*x^5+8296*x^4-10688*x^3+10312*x^2-7752*x+2890)*ln( x)+768*x^8-1856*x^7+784*x^6+1508*x^5-5952*x^4+10252*x^3-7252*x^2+1853*x)/( (256*x^4-960*x^3+1200*x^2-500*x)*ln(x)^4+(512*x^5-1408*x^4+480*x^3+1400*x^ 2-1000*x)*ln(x)^2+256*x^6-448*x^5-464*x^4+940*x^3+200*x^2-500*x),x,method= _RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 \, x^{6} - 160 \, x^{5} + 356 \, x^{4} - 592 \, x^{3} + 596 \, x^{2} - 544 \, x + 289}{4 \, {\left (16 \, x^{3} + {\left (16 \, x^{2} - 40 \, x + 25\right )} \log \left (x\right )^{2} - 24 \, x^{2} - 15 \, x + 25\right )}} \end {dmath*}
integrate(((1024*x^7-3840*x^6+6848*x^5-9488*x^4+8880*x^3-3784*x^2+408*x)*l og(x)^2+(-512*x^7+1920*x^6-4448*x^5+8296*x^4-10688*x^3+10312*x^2-7752*x+28 90)*log(x)+768*x^8-1856*x^7+784*x^6+1508*x^5-5952*x^4+10252*x^3-7252*x^2+1 853*x)/((256*x^4-960*x^3+1200*x^2-500*x)*log(x)^4+(512*x^5-1408*x^4+480*x^ 3+1400*x^2-1000*x)*log(x)^2+256*x^6-448*x^5-464*x^4+940*x^3+200*x^2-500*x) ,x, algorithm=\
1/4*(64*x^6 - 160*x^5 + 356*x^4 - 592*x^3 + 596*x^2 - 544*x + 289)/(16*x^3 + (16*x^2 - 40*x + 25)*log(x)^2 - 24*x^2 - 15*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (24) = 48\).
Time = 0.14 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 x^{6} - 160 x^{5} + 356 x^{4} - 592 x^{3} + 596 x^{2} - 544 x + 289}{64 x^{3} - 96 x^{2} - 60 x + \left (64 x^{2} - 160 x + 100\right ) \log {\left (x \right )}^{2} + 100} \end {dmath*}
integrate(((1024*x**7-3840*x**6+6848*x**5-9488*x**4+8880*x**3-3784*x**2+40 8*x)*ln(x)**2+(-512*x**7+1920*x**6-4448*x**5+8296*x**4-10688*x**3+10312*x* *2-7752*x+2890)*ln(x)+768*x**8-1856*x**7+784*x**6+1508*x**5-5952*x**4+1025 2*x**3-7252*x**2+1853*x)/((256*x**4-960*x**3+1200*x**2-500*x)*ln(x)**4+(51 2*x**5-1408*x**4+480*x**3+1400*x**2-1000*x)*ln(x)**2+256*x**6-448*x**5-464 *x**4+940*x**3+200*x**2-500*x),x)
(64*x**6 - 160*x**5 + 356*x**4 - 592*x**3 + 596*x**2 - 544*x + 289)/(64*x* *3 - 96*x**2 - 60*x + (64*x**2 - 160*x + 100)*log(x)**2 + 100)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 \, x^{6} - 160 \, x^{5} + 356 \, x^{4} - 592 \, x^{3} + 596 \, x^{2} - 544 \, x + 289}{4 \, {\left (16 \, x^{3} + {\left (16 \, x^{2} - 40 \, x + 25\right )} \log \left (x\right )^{2} - 24 \, x^{2} - 15 \, x + 25\right )}} \end {dmath*}
integrate(((1024*x^7-3840*x^6+6848*x^5-9488*x^4+8880*x^3-3784*x^2+408*x)*l og(x)^2+(-512*x^7+1920*x^6-4448*x^5+8296*x^4-10688*x^3+10312*x^2-7752*x+28 90)*log(x)+768*x^8-1856*x^7+784*x^6+1508*x^5-5952*x^4+10252*x^3-7252*x^2+1 853*x)/((256*x^4-960*x^3+1200*x^2-500*x)*log(x)^4+(512*x^5-1408*x^4+480*x^ 3+1400*x^2-1000*x)*log(x)^2+256*x^6-448*x^5-464*x^4+940*x^3+200*x^2-500*x) ,x, algorithm=\
1/4*(64*x^6 - 160*x^5 + 356*x^4 - 592*x^3 + 596*x^2 - 544*x + 289)/(16*x^3 + (16*x^2 - 40*x + 25)*log(x)^2 - 24*x^2 - 15*x + 25)
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (29) = 58\).
Time = 0.35 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.87 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=\frac {64 \, x^{6} - 160 \, x^{5} + 356 \, x^{4} - 592 \, x^{3} + 596 \, x^{2} - 544 \, x + 289}{4 \, {\left (16 \, x^{2} \log \left (x\right )^{2} + 16 \, x^{3} - 40 \, x \log \left (x\right )^{2} - 24 \, x^{2} + 25 \, \log \left (x\right )^{2} - 15 \, x + 25\right )}} \end {dmath*}
integrate(((1024*x^7-3840*x^6+6848*x^5-9488*x^4+8880*x^3-3784*x^2+408*x)*l og(x)^2+(-512*x^7+1920*x^6-4448*x^5+8296*x^4-10688*x^3+10312*x^2-7752*x+28 90)*log(x)+768*x^8-1856*x^7+784*x^6+1508*x^5-5952*x^4+10252*x^3-7252*x^2+1 853*x)/((256*x^4-960*x^3+1200*x^2-500*x)*log(x)^4+(512*x^5-1408*x^4+480*x^ 3+1400*x^2-1000*x)*log(x)^2+256*x^6-448*x^5-464*x^4+940*x^3+200*x^2-500*x) ,x, algorithm=\
1/4*(64*x^6 - 160*x^5 + 356*x^4 - 592*x^3 + 596*x^2 - 544*x + 289)/(16*x^2 *log(x)^2 + 16*x^3 - 40*x*log(x)^2 - 24*x^2 + 25*log(x)^2 - 15*x + 25)
Time = 14.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 2.13 \begin {dmath*} \int \frac {1853 x-7252 x^2+10252 x^3-5952 x^4+1508 x^5+784 x^6-1856 x^7+768 x^8+\left (2890-7752 x+10312 x^2-10688 x^3+8296 x^4-4448 x^5+1920 x^6-512 x^7\right ) \log (x)+\left (408 x-3784 x^2+8880 x^3-9488 x^4+6848 x^5-3840 x^6+1024 x^7\right ) \log ^2(x)}{-500 x+200 x^2+940 x^3-464 x^4-448 x^5+256 x^6+\left (-1000 x+1400 x^2+480 x^3-1408 x^4+512 x^5\right ) \log ^2(x)+\left (-500 x+1200 x^2-960 x^3+256 x^4\right ) \log ^4(x)} \, dx=-\frac {-64\,x^{10}-16\,x^9+148\,x^8-227\,x^7+588\,x^6+93\,x^5-1157\,x^4+\frac {6565\,x^3}{4}-2431\,x^2+1445\,x}{{\left (4\,x-5\right )}^3\,\left ({\ln \left (x\right )}^2+x+1\right )\,\left (x^3+4\,x^2+4\,x\right )} \end {dmath*}
int(-(1853*x + log(x)^2*(408*x - 3784*x^2 + 8880*x^3 - 9488*x^4 + 6848*x^5 - 3840*x^6 + 1024*x^7) - log(x)*(7752*x - 10312*x^2 + 10688*x^3 - 8296*x^ 4 + 4448*x^5 - 1920*x^6 + 512*x^7 - 2890) - 7252*x^2 + 10252*x^3 - 5952*x^ 4 + 1508*x^5 + 784*x^6 - 1856*x^7 + 768*x^8)/(500*x + log(x)^4*(500*x - 12 00*x^2 + 960*x^3 - 256*x^4) - log(x)^2*(1400*x^2 - 1000*x + 480*x^3 - 1408 *x^4 + 512*x^5) - 200*x^2 - 940*x^3 + 464*x^4 + 448*x^5 - 256*x^6),x)