Integrand size = 150, antiderivative size = 26 \[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=-5+x+e^{\frac {(16+x)^2}{(2-x)^2 x^6}} \log (1+x) \]
Time = 0.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=x+e^{\frac {(16+x)^2}{(-2+x)^2 x^6}} \log (1+x) \]
Integrate[(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11 + E^((256 + 32*x + x^2)/ (4*x^6 - 4*x^7 + x^8))*(-8*x^7 + 12*x^8 - 6*x^9 + x^10) + E^((256 + 32*x + x^2)/(4*x^6 - 4*x^7 + x^8))*(3072 + 1344*x - 1944*x^2 - 222*x^3 - 6*x^4)* Log[1 + x])/(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11),x]
Leaf count is larger than twice the leaf count of optimal. \(128\) vs. \(2(26)=52\).
Time = 13.54 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.92, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2026, 2463, 7239, 7293, 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.
\(\displaystyle \int \frac {x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7\right )+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (-6 x^4-222 x^3-1944 x^2+1344 x+3072\right ) \log (x+1)}{x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7\right )+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (-6 x^4-222 x^3-1944 x^2+1344 x+3072\right ) \log (x+1)}{x^7 \left (x^4-5 x^3+6 x^2+4 x-8\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (-\frac {x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7\right )+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (-6 x^4-222 x^3-1944 x^2+1344 x+3072\right ) \log (x+1)}{27 x^7 (x+1)}+\frac {x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7\right )+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (-6 x^4-222 x^3-1944 x^2+1344 x+3072\right ) \log (x+1)}{27 (x-2) x^7}-\frac {x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7\right )+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (-6 x^4-222 x^3-1944 x^2+1344 x+3072\right ) \log (x+1)}{9 (x-2)^2 x^7}+\frac {x^{11}-5 x^{10}+6 x^9+4 x^8-8 x^7+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7\right )+e^{\frac {x^2+32 x+256}{x^8-4 x^7+4 x^6}} \left (-6 x^4-222 x^3-1944 x^2+1344 x+3072\right ) \log (x+1)}{3 (x-2)^3 x^7}\right )dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {e^{\frac {(x+16)^2}{(x-2)^2 x^6}}-\frac {6 e^{\frac {(x+16)^2}{(x-2)^2 x^6}} \left (x^4+37 x^3+324 x^2-224 x-512\right ) \log (x+1)}{(x-2)^3 x^7}+x+1}{x+1}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {e^{\frac {(x+16)^2}{(x-2)^2 x^6}} \left (x^{10}-6 x^9+12 x^8-8 x^7-6 x^4 \log (x+1)-222 x^3 \log (x+1)-1944 x^2 \log (x+1)+1344 x \log (x+1)+3072 \log (x+1)\right )}{(x-2)^3 x^7 (x+1)}+1\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x-\frac {3 e^{\frac {(x+16)^2}{(2-x)^2 x^6}} \left (x^4 (-\log (x+1))-37 x^3 \log (x+1)-324 x^2 \log (x+1)+224 x \log (x+1)+512 \log (x+1)\right )}{(2-x)^3 x^7 (x+1) \left (-\frac {3 (x+16)^2}{(2-x)^2 x^7}+\frac {(x+16)^2}{(2-x)^3 x^6}+\frac {x+16}{(2-x)^2 x^6}\right )}\) |
Int[(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11 + E^((256 + 32*x + x^2)/(4*x^6 - 4*x^7 + x^8))*(-8*x^7 + 12*x^8 - 6*x^9 + x^10) + E^((256 + 32*x + x^2)/ (4*x^6 - 4*x^7 + x^8))*(3072 + 1344*x - 1944*x^2 - 222*x^3 - 6*x^4)*Log[1 + x])/(-8*x^7 + 4*x^8 + 6*x^9 - 5*x^10 + x^11),x]
x - (3*E^((16 + x)^2/((2 - x)^2*x^6))*(512*Log[1 + x] + 224*x*Log[1 + x] - 324*x^2*Log[1 + x] - 37*x^3*Log[1 + x] - x^4*Log[1 + x]))/((2 - x)^3*x^7* (1 + x)*((16 + x)/((2 - x)^2*x^6) - (3*(16 + x)^2)/((2 - x)^2*x^7) + (16 + x)^2/((2 - x)^3*x^6)))
3.27.95.3.1 Defintions of rubi rules used
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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 24.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
risch | \({\mathrm e}^{\frac {\left (x +16\right )^{2}}{x^{6} \left (-2+x \right )^{2}}} \ln \left (1+x \right )+x\) | \(23\) |
int(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4*x^7+4 *x^6))*ln(1+x)+(x^10-6*x^9+12*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4*x^7+4*x ^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^7),x,meth od=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=e^{\left (\frac {x^{2} + 32 \, x + 256}{x^{8} - 4 \, x^{7} + 4 \, x^{6}}\right )} \log \left (x + 1\right ) + x \]
integrate(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4 *x^7+4*x^6))*log(1+x)+(x^10-6*x^9+12*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4* x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^7) ,x, algorithm=\
Time = 0.40 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=x + e^{\frac {x^{2} + 32 x + 256}{x^{8} - 4 x^{7} + 4 x^{6}}} \log {\left (x + 1 \right )} \]
integrate(((-6*x**4-222*x**3-1944*x**2+1344*x+3072)*exp((x**2+32*x+256)/(x **8-4*x**7+4*x**6))*ln(1+x)+(x**10-6*x**9+12*x**8-8*x**7)*exp((x**2+32*x+2 56)/(x**8-4*x**7+4*x**6))+x**11-5*x**10+6*x**9+4*x**8-8*x**7)/(x**11-5*x** 10+6*x**9+4*x**8-8*x**7),x)
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (23) = 46\).
Time = 1.18 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.42 \[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=e^{\left (\frac {81}{16 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {117}{8 \, {\left (x - 2\right )}} + \frac {117}{8 \, x} + \frac {387}{16 \, x^{2}} + \frac {153}{4 \, x^{3}} + \frac {225}{4 \, x^{4}} + \frac {72}{x^{5}} + \frac {64}{x^{6}}\right )} \log \left (x + 1\right ) + x - \frac {8 \, {\left (10 \, x - 17\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} + \frac {20 \, {\left (7 \, x - 11\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {4 \, {\left (4 \, x - 5\right )}}{3 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {4 \, {\left (2 \, x - 7\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} - \frac {4 \, {\left (x + 1\right )}}{9 \, {\left (x^{2} - 4 \, x + 4\right )}} \]
integrate(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4 *x^7+4*x^6))*log(1+x)+(x^10-6*x^9+12*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4* x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^7) ,x, algorithm=\
e^(81/16/(x^2 - 4*x + 4) - 117/8/(x - 2) + 117/8/x + 387/16/x^2 + 153/4/x^ 3 + 225/4/x^4 + 72/x^5 + 64/x^6)*log(x + 1) + x - 8/9*(10*x - 17)/(x^2 - 4 *x + 4) + 20/9*(7*x - 11)/(x^2 - 4*x + 4) - 4/3*(4*x - 5)/(x^2 - 4*x + 4) - 4/9*(2*x - 7)/(x^2 - 4*x + 4) - 4/9*(x + 1)/(x^2 - 4*x + 4)
\[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=\int { \frac {x^{11} - 5 \, x^{10} + 6 \, x^{9} + 4 \, x^{8} - 8 \, x^{7} - 6 \, {\left (x^{4} + 37 \, x^{3} + 324 \, x^{2} - 224 \, x - 512\right )} e^{\left (\frac {x^{2} + 32 \, x + 256}{x^{8} - 4 \, x^{7} + 4 \, x^{6}}\right )} \log \left (x + 1\right ) + {\left (x^{10} - 6 \, x^{9} + 12 \, x^{8} - 8 \, x^{7}\right )} e^{\left (\frac {x^{2} + 32 \, x + 256}{x^{8} - 4 \, x^{7} + 4 \, x^{6}}\right )}}{x^{11} - 5 \, x^{10} + 6 \, x^{9} + 4 \, x^{8} - 8 \, x^{7}} \,d x } \]
integrate(((-6*x^4-222*x^3-1944*x^2+1344*x+3072)*exp((x^2+32*x+256)/(x^8-4 *x^7+4*x^6))*log(1+x)+(x^10-6*x^9+12*x^8-8*x^7)*exp((x^2+32*x+256)/(x^8-4* x^7+4*x^6))+x^11-5*x^10+6*x^9+4*x^8-8*x^7)/(x^11-5*x^10+6*x^9+4*x^8-8*x^7) ,x, algorithm=\
integrate((x^11 - 5*x^10 + 6*x^9 + 4*x^8 - 8*x^7 - 6*(x^4 + 37*x^3 + 324*x ^2 - 224*x - 512)*e^((x^2 + 32*x + 256)/(x^8 - 4*x^7 + 4*x^6))*log(x + 1) + (x^10 - 6*x^9 + 12*x^8 - 8*x^7)*e^((x^2 + 32*x + 256)/(x^8 - 4*x^7 + 4*x ^6)))/(x^11 - 5*x^10 + 6*x^9 + 4*x^8 - 8*x^7), x)
Timed out. \[ \int \frac {-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (-8 x^7+12 x^8-6 x^9+x^{10}\right )+e^{\frac {256+32 x+x^2}{4 x^6-4 x^7+x^8}} \left (3072+1344 x-1944 x^2-222 x^3-6 x^4\right ) \log (1+x)}{-8 x^7+4 x^8+6 x^9-5 x^{10}+x^{11}} \, dx=\int -\frac {{\mathrm {e}}^{\frac {x^2+32\,x+256}{x^8-4\,x^7+4\,x^6}}\,\left (-x^{10}+6\,x^9-12\,x^8+8\,x^7\right )+8\,x^7-4\,x^8-6\,x^9+5\,x^{10}-x^{11}+\ln \left (x+1\right )\,{\mathrm {e}}^{\frac {x^2+32\,x+256}{x^8-4\,x^7+4\,x^6}}\,\left (6\,x^4+222\,x^3+1944\,x^2-1344\,x-3072\right )}{x^{11}-5\,x^{10}+6\,x^9+4\,x^8-8\,x^7} \,d x \]
int(-(exp((32*x + x^2 + 256)/(4*x^6 - 4*x^7 + x^8))*(8*x^7 - 12*x^8 + 6*x^ 9 - x^10) + 8*x^7 - 4*x^8 - 6*x^9 + 5*x^10 - x^11 + log(x + 1)*exp((32*x + x^2 + 256)/(4*x^6 - 4*x^7 + x^8))*(1944*x^2 - 1344*x + 222*x^3 + 6*x^4 - 3072))/(4*x^8 - 8*x^7 + 6*x^9 - 5*x^10 + x^11),x)