Integrand size = 125, antiderivative size = 26 \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=\frac {4}{16+\frac {2}{x^2 \left (x^2-(3+x) \log (1+x)\right )}} \]
Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=-\frac {1}{4 \left (1+8 x^4-8 x^2 (3+x) \log (1+x)\right )} \]
Integrate[(-6*x^2 + 6*x^3 + 8*x^4 + (-12*x - 18*x^2 - 6*x^3)*Log[1 + x])/( 1 + x + 16*x^4 + 16*x^5 + 64*x^8 + 64*x^9 + (-48*x^2 - 64*x^3 - 16*x^4 - 3 84*x^6 - 512*x^7 - 128*x^8)*Log[1 + x] + (576*x^4 + 960*x^5 + 448*x^6 + 64 *x^7)*Log[1 + x]^2),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.
\(\displaystyle \int \frac {8 x^4+6 x^3-6 x^2+\left (-6 x^3-18 x^2-12 x\right ) \log (x+1)}{64 x^9+64 x^8+16 x^5+16 x^4+\left (64 x^7+448 x^6+960 x^5+576 x^4\right ) \log ^2(x+1)+\left (-128 x^8-512 x^7-384 x^6-16 x^4-64 x^3-48 x^2\right ) \log (x+1)+x+1} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {2 x \left (x \left (4 x^2+3 x-3\right )-3 \left (x^2+3 x+2\right ) \log (x+1)\right )}{(x+1) \left (8 x^4-8 (x+3) x^2 \log (x+1)+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int -\frac {x \left (x \left (-4 x^2-3 x+3\right )+3 \left (x^2+3 x+2\right ) \log (x+1)\right )}{(x+1) \left (8 x^4-8 (x+3) \log (x+1) x^2+1\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \int \frac {x \left (x \left (-4 x^2-3 x+3\right )+3 \left (x^2+3 x+2\right ) \log (x+1)\right )}{(x+1) \left (8 x^4-8 (x+3) \log (x+1) x^2+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -2 \int \left (\frac {-8 x^6-48 x^5+72 x^3+3 x^2+9 x+6}{8 x (x+1) (x+3) \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}-\frac {3 (x+2)}{8 x (x+3) \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 \left (-29 \int \frac {1}{\left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx+\frac {1}{4} \int \frac {1}{x \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx+11 \int \frac {x}{\left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx-2 \int \frac {x^2}{\left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx-\int \frac {x^3}{\left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx+2 \int \frac {1}{(x+1) \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx+\frac {649}{8} \int \frac {1}{(x+3) \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )^2}dx-\frac {1}{4} \int \frac {1}{x \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )}dx-\frac {1}{8} \int \frac {1}{(x+3) \left (8 x^4-8 \log (x+1) x^3-24 \log (x+1) x^2+1\right )}dx\right )\) |
Int[(-6*x^2 + 6*x^3 + 8*x^4 + (-12*x - 18*x^2 - 6*x^3)*Log[1 + x])/(1 + x + 16*x^4 + 16*x^5 + 64*x^8 + 64*x^9 + (-48*x^2 - 64*x^3 - 16*x^4 - 384*x^6 - 512*x^7 - 128*x^8)*Log[1 + x] + (576*x^4 + 960*x^5 + 448*x^6 + 64*x^7)* Log[1 + x]^2),x]
3.28.22.3.1 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]]
Time = 0.73 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15
method | result | size |
risch | \(-\frac {1}{4 \left (8 x^{4}-8 \ln \left (1+x \right ) x^{3}-24 \ln \left (1+x \right ) x^{2}+1\right )}\) | \(30\) |
parallelrisch | \(-\frac {1}{4 \left (8 x^{4}-8 \ln \left (1+x \right ) x^{3}-24 \ln \left (1+x \right ) x^{2}+1\right )}\) | \(30\) |
derivativedivides | \(\frac {1}{32 \ln \left (1+x \right ) \left (1+x \right )^{3}-32 \left (1+x \right )^{4}+128 \left (1+x \right )^{3}-96 \left (1+x \right ) \ln \left (1+x \right )-192 \left (1+x \right )^{2}+64 \ln \left (1+x \right )+92+128 x}\) | \(57\) |
default | \(\frac {1}{32 \ln \left (1+x \right ) \left (1+x \right )^{3}-32 \left (1+x \right )^{4}+128 \left (1+x \right )^{3}-96 \left (1+x \right ) \ln \left (1+x \right )-192 \left (1+x \right )^{2}+64 \ln \left (1+x \right )+92+128 x}\) | \(57\) |
int(((-6*x^3-18*x^2-12*x)*ln(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x^6+960* x^5+576*x^4)*ln(1+x)^2+(-128*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48*x^2)*ln( 1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=-\frac {1}{4 \, {\left (8 \, x^{4} - 8 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x + 1\right ) + 1\right )}} \]
integrate(((-6*x^3-18*x^2-12*x)*log(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x ^6+960*x^5+576*x^4)*log(1+x)^2+(-128*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48* x^2)*log(1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x, algorithm=\
Time = 0.15 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=\frac {1}{- 32 x^{4} + \left (32 x^{3} + 96 x^{2}\right ) \log {\left (x + 1 \right )} - 4} \]
integrate(((-6*x**3-18*x**2-12*x)*ln(1+x)+8*x**4+6*x**3-6*x**2)/((64*x**7+ 448*x**6+960*x**5+576*x**4)*ln(1+x)**2+(-128*x**8-512*x**7-384*x**6-16*x** 4-64*x**3-48*x**2)*ln(1+x)+64*x**9+64*x**8+16*x**5+16*x**4+x+1),x)
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=-\frac {1}{4 \, {\left (8 \, x^{4} - 8 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \left (x + 1\right ) + 1\right )}} \]
integrate(((-6*x^3-18*x^2-12*x)*log(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x ^6+960*x^5+576*x^4)*log(1+x)^2+(-128*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48* x^2)*log(1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=-\frac {1}{4 \, {\left (8 \, x^{4} - 8 \, x^{3} \log \left (x + 1\right ) - 24 \, x^{2} \log \left (x + 1\right ) + 1\right )}} \]
integrate(((-6*x^3-18*x^2-12*x)*log(1+x)+8*x^4+6*x^3-6*x^2)/((64*x^7+448*x ^6+960*x^5+576*x^4)*log(1+x)^2+(-128*x^8-512*x^7-384*x^6-16*x^4-64*x^3-48* x^2)*log(1+x)+64*x^9+64*x^8+16*x^5+16*x^4+x+1),x, algorithm=\
Timed out. \[ \int \frac {-6 x^2+6 x^3+8 x^4+\left (-12 x-18 x^2-6 x^3\right ) \log (1+x)}{1+x+16 x^4+16 x^5+64 x^8+64 x^9+\left (-48 x^2-64 x^3-16 x^4-384 x^6-512 x^7-128 x^8\right ) \log (1+x)+\left (576 x^4+960 x^5+448 x^6+64 x^7\right ) \log ^2(1+x)} \, dx=\int -\frac {\ln \left (x+1\right )\,\left (6\,x^3+18\,x^2+12\,x\right )+6\,x^2-6\,x^3-8\,x^4}{x+{\ln \left (x+1\right )}^2\,\left (64\,x^7+448\,x^6+960\,x^5+576\,x^4\right )-\ln \left (x+1\right )\,\left (128\,x^8+512\,x^7+384\,x^6+16\,x^4+64\,x^3+48\,x^2\right )+16\,x^4+16\,x^5+64\,x^8+64\,x^9+1} \,d x \]
int(-(log(x + 1)*(12*x + 18*x^2 + 6*x^3) + 6*x^2 - 6*x^3 - 8*x^4)/(x + log (x + 1)^2*(576*x^4 + 960*x^5 + 448*x^6 + 64*x^7) - log(x + 1)*(48*x^2 + 64 *x^3 + 16*x^4 + 384*x^6 + 512*x^7 + 128*x^8) + 16*x^4 + 16*x^5 + 64*x^8 + 64*x^9 + 1),x)