Integrand size = 112, antiderivative size = 27 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\left (2-3 x-x \left (\frac {x}{4}+\frac {4}{(2+x) \log (x)}\right )\right )^2 \] Output:
(2-3*x-(4/(2+x)/ln(x)+1/4*x)*x)^2
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {1}{16} x \left (-192+128 x+24 x^2+x^3+\frac {256 x}{(2+x)^2 \log ^2(x)}+\frac {32 \left (-8+12 x+x^2\right )}{(2+x) \log (x)}\right ) \] Input:
Integrate[(-256*x - 128*x^2 + (256 + 128*x - 352*x^2 - 128*x^3 - 8*x^4)*Lo g[x] + (-256 + 640*x + 672*x^2 + 176*x^3 + 16*x^4)*Log[x]^2 + (-384 - 64*x + 624*x^2 + 560*x^3 + 184*x^4 + 24*x^5 + x^6)*Log[x]^3)/((32 + 48*x + 24* x^2 + 4*x^3)*Log[x]^3),x]
Output:
(x*(-192 + 128*x + 24*x^2 + x^3 + (256*x)/((2 + x)^2*Log[x]^2) + (32*(-8 + 12*x + x^2))/((2 + x)*Log[x])))/16
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.
\(\displaystyle \int \frac {-128 x^2+\left (16 x^4+176 x^3+672 x^2+640 x-256\right ) \log ^2(x)+\left (-8 x^4-128 x^3-352 x^2+128 x+256\right ) \log (x)+\left (x^6+24 x^5+184 x^4+560 x^3+624 x^2-64 x-384\right ) \log ^3(x)-256 x}{\left (4 x^3+24 x^2+48 x+32\right ) \log ^3(x)} \, dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {-128 x^2+\left (16 x^4+176 x^3+672 x^2+640 x-256\right ) \log ^2(x)+\left (-8 x^4-128 x^3-352 x^2+128 x+256\right ) \log (x)+\left (x^6+24 x^5+184 x^4+560 x^3+624 x^2-64 x-384\right ) \log ^3(x)-256 x}{\left (2^{2/3} x+2\ 2^{2/3}\right )^3 \log ^3(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{4} (x+6) \left (x^2+12 x-8\right )+\frac {4 \left (x^3+9 x^2+24 x-8\right )}{(x+2)^2 \log (x)}-\frac {2 \left (x^4+16 x^3+44 x^2-16 x-32\right )}{(x+2)^3 \log ^2(x)}-\frac {32 x}{(x+2)^2 \log ^3(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \frac {x^3+9 x^2+24 x-8}{(x+2)^2 \log (x)}dx-2 \int \frac {x^4+16 x^3+44 x^2-16 x-32}{(x+2)^3 \log ^2(x)}dx-32 \int \frac {x}{(x+2)^2 \log ^3(x)}dx+\frac {1}{16} \left (-x^2-12 x+8\right )^2\) |
Input:
Int[(-256*x - 128*x^2 + (256 + 128*x - 352*x^2 - 128*x^3 - 8*x^4)*Log[x] + (-256 + 640*x + 672*x^2 + 176*x^3 + 16*x^4)*Log[x]^2 + (-384 - 64*x + 624 *x^2 + 560*x^3 + 184*x^4 + 24*x^5 + x^6)*Log[x]^3)/((32 + 48*x + 24*x^2 + 4*x^3)*Log[x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(25)=50\).
Time = 3.75 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37
method | result | size |
risch | \(\frac {x^{4}}{16}+\frac {3 x^{3}}{2}+8 x^{2}-12 x +4+\frac {2 \left (x^{3} \ln \left (x \right )+14 x^{2} \ln \left (x \right )+16 x \ln \left (x \right )+8 x -16 \ln \left (x \right )\right ) x}{\left (x^{2}+4 x +4\right ) \ln \left (x \right )^{2}}\) | \(64\) |
default | \(\frac {-224 \ln \left (x \right )+64}{4 \ln \left (x \right )^{2}}+\frac {x^{4}}{16}+\frac {3 x^{3}}{2}+\frac {2 \left (4 \ln \left (x \right )+1\right ) x^{2}}{\ln \left (x \right )}-\frac {4 \left (3 \ln \left (x \right )-5\right ) x}{\ln \left (x \right )}+\frac {112 x \ln \left (x \right )+224 \ln \left (x \right )-64 x -64}{\ln \left (x \right )^{2} \left (2+x \right )^{2}}\) | \(77\) |
parallelrisch | \(-\frac {-x^{6} \ln \left (x \right )^{2}-28 x^{5} \ln \left (x \right )^{2}-228 x^{4} \ln \left (x \right )^{2}-32 x^{4} \ln \left (x \right )-416 x^{3} \ln \left (x \right )^{2}-448 x^{3} \ln \left (x \right )+64 x^{2} \ln \left (x \right )^{2}-512 x^{2} \ln \left (x \right )-256 x^{2}+512 x \ln \left (x \right )-768 \ln \left (x \right )^{2}}{16 \ln \left (x \right )^{2} \left (x^{2}+4 x +4\right )}\) | \(100\) |
Input:
int(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*ln(x)^3+(16*x^4+176*x^3 +672*x^2+640*x-256)*ln(x)^2+(-8*x^4-128*x^3-352*x^2+128*x+256)*ln(x)-128*x ^2-256*x)/(4*x^3+24*x^2+48*x+32)/ln(x)^3,x,method=_RETURNVERBOSE)
Output:
1/16*x^4+3/2*x^3+8*x^2-12*x+4+2*(x^3*ln(x)+14*x^2*ln(x)+16*x*ln(x)+8*x-16* ln(x))*x/(x^2+4*x+4)/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {{\left (x^{6} + 28 \, x^{5} + 228 \, x^{4} + 416 \, x^{3} - 256 \, x^{2} - 768 \, x\right )} \log \left (x\right )^{2} + 256 \, x^{2} + 32 \, {\left (x^{4} + 14 \, x^{3} + 16 \, x^{2} - 16 \, x\right )} \log \left (x\right )}{16 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right )^{2}} \] Input:
integrate(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+ 176*x^3+672*x^2+640*x-256)*log(x)^2+(-8*x^4-128*x^3-352*x^2+128*x+256)*log (x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x, algorithm="fricas")
Output:
1/16*((x^6 + 28*x^5 + 228*x^4 + 416*x^3 - 256*x^2 - 768*x)*log(x)^2 + 256* x^2 + 32*(x^4 + 14*x^3 + 16*x^2 - 16*x)*log(x))/((x^2 + 4*x + 4)*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (19) = 38\).
Time = 0.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {x^{4}}{16} + \frac {3 x^{3}}{2} + 8 x^{2} - 12 x + \frac {16 x^{2} + \left (2 x^{4} + 28 x^{3} + 32 x^{2} - 32 x\right ) \log {\left (x \right )}}{\left (x^{2} + 4 x + 4\right ) \log {\left (x \right )}^{2}} \] Input:
integrate(((x**6+24*x**5+184*x**4+560*x**3+624*x**2-64*x-384)*ln(x)**3+(16 *x**4+176*x**3+672*x**2+640*x-256)*ln(x)**2+(-8*x**4-128*x**3-352*x**2+128 *x+256)*ln(x)-128*x**2-256*x)/(4*x**3+24*x**2+48*x+32)/ln(x)**3,x)
Output:
x**4/16 + 3*x**3/2 + 8*x**2 - 12*x + (16*x**2 + (2*x**4 + 28*x**3 + 32*x** 2 - 32*x)*log(x))/((x**2 + 4*x + 4)*log(x)**2)
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {{\left (x^{6} + 28 \, x^{5} + 228 \, x^{4} + 416 \, x^{3} - 256 \, x^{2} - 768 \, x\right )} \log \left (x\right )^{2} + 256 \, x^{2} + 32 \, {\left (x^{4} + 14 \, x^{3} + 16 \, x^{2} - 16 \, x\right )} \log \left (x\right )}{16 \, {\left (x^{2} + 4 \, x + 4\right )} \log \left (x\right )^{2}} \] Input:
integrate(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+ 176*x^3+672*x^2+640*x-256)*log(x)^2+(-8*x^4-128*x^3-352*x^2+128*x+256)*log (x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x, algorithm="maxima")
Output:
1/16*((x^6 + 28*x^5 + 228*x^4 + 416*x^3 - 256*x^2 - 768*x)*log(x)^2 + 256* x^2 + 32*(x^4 + 14*x^3 + 16*x^2 - 16*x)*log(x))/((x^2 + 4*x + 4)*log(x)^2)
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.81 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {1}{16} \, x^{4} + \frac {3}{2} \, x^{3} + 8 \, x^{2} - 12 \, x + \frac {2 \, {\left (x^{4} \log \left (x\right ) + 14 \, x^{3} \log \left (x\right ) + 16 \, x^{2} \log \left (x\right ) + 8 \, x^{2} - 16 \, x \log \left (x\right )\right )}}{x^{2} \log \left (x\right )^{2} + 4 \, x \log \left (x\right )^{2} + 4 \, \log \left (x\right )^{2}} \] Input:
integrate(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+ 176*x^3+672*x^2+640*x-256)*log(x)^2+(-8*x^4-128*x^3-352*x^2+128*x+256)*log (x)-128*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x, algorithm="giac")
Output:
1/16*x^4 + 3/2*x^3 + 8*x^2 - 12*x + 2*(x^4*log(x) + 14*x^3*log(x) + 16*x^2 *log(x) + 8*x^2 - 16*x*log(x))/(x^2*log(x)^2 + 4*x*log(x)^2 + 4*log(x)^2)
Time = 2.96 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {x\,\left (x^5+28\,x^4+228\,x^3+416\,x^2-256\,x-768\right )}{16\,{\left (x+2\right )}^2}+\frac {16\,x^2+\frac {x\,\ln \left (x\right )\,\left (32\,x^3+448\,x^2+512\,x-512\right )}{16}}{{\ln \left (x\right )}^2\,{\left (x+2\right )}^2} \] Input:
int(-(256*x + log(x)*(352*x^2 - 128*x + 128*x^3 + 8*x^4 - 256) - log(x)^2* (640*x + 672*x^2 + 176*x^3 + 16*x^4 - 256) - log(x)^3*(624*x^2 - 64*x + 56 0*x^3 + 184*x^4 + 24*x^5 + x^6 - 384) + 128*x^2)/(log(x)^3*(48*x + 24*x^2 + 4*x^3 + 32)),x)
Output:
(x*(416*x^2 - 256*x + 228*x^3 + 28*x^4 + x^5 - 768))/(16*(x + 2)^2) + (16* x^2 + (x*log(x)*(512*x + 448*x^2 + 32*x^3 - 512))/16)/(log(x)^2*(x + 2)^2)
Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 3.41 \[ \int \frac {-256 x-128 x^2+\left (256+128 x-352 x^2-128 x^3-8 x^4\right ) \log (x)+\left (-256+640 x+672 x^2+176 x^3+16 x^4\right ) \log ^2(x)+\left (-384-64 x+624 x^2+560 x^3+184 x^4+24 x^5+x^6\right ) \log ^3(x)}{\left (32+48 x+24 x^2+4 x^3\right ) \log ^3(x)} \, dx=\frac {x \left (\mathrm {log}\left (x \right )^{2} x^{5}+28 \mathrm {log}\left (x \right )^{2} x^{4}+228 \mathrm {log}\left (x \right )^{2} x^{3}+416 \mathrm {log}\left (x \right )^{2} x^{2}-256 \mathrm {log}\left (x \right )^{2} x -768 \mathrm {log}\left (x \right )^{2}+32 \,\mathrm {log}\left (x \right ) x^{3}+448 \,\mathrm {log}\left (x \right ) x^{2}+512 \,\mathrm {log}\left (x \right ) x -512 \,\mathrm {log}\left (x \right )+256 x \right )}{16 \mathrm {log}\left (x \right )^{2} \left (x^{2}+4 x +4\right )} \] Input:
int(((x^6+24*x^5+184*x^4+560*x^3+624*x^2-64*x-384)*log(x)^3+(16*x^4+176*x^ 3+672*x^2+640*x-256)*log(x)^2+(-8*x^4-128*x^3-352*x^2+128*x+256)*log(x)-12 8*x^2-256*x)/(4*x^3+24*x^2+48*x+32)/log(x)^3,x)
Output:
(x*(log(x)**2*x**5 + 28*log(x)**2*x**4 + 228*log(x)**2*x**3 + 416*log(x)** 2*x**2 - 256*log(x)**2*x - 768*log(x)**2 + 32*log(x)*x**3 + 448*log(x)*x** 2 + 512*log(x)*x - 512*log(x) + 256*x))/(16*log(x)**2*(x**2 + 4*x + 4))