Integrand size = 117, antiderivative size = 21 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{(4+x)^2 \left (-5-e^9+x\right )^2 \log ^2(5)} \] Output:
9/ln(5)^2/(4+x)^2/(x-5-exp(9))^2
Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{\left (5+e^9-x\right )^2 (4+x)^2 \log ^2(5)} \] Input:
Integrate[(-18 - 18*E^9 + 36*x)/((8000 + 1200*x - 1140*x^2 - 119*x^3 + 57* x^4 + 3*x^5 - x^6 + E^27*(64 + 48*x + 12*x^2 + x^3) + E^18*(960 + 528*x + 36*x^2 - 21*x^3 - 3*x^4) + E^9*(4800 + 1680*x - 348*x^2 - 141*x^3 + 6*x^4 + 3*x^5))*Log[5]^2),x]
Output:
9/((5 + E^9 - x)^2*(4 + x)^2*Log[5]^2)
Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(21)=42\).
Time = 0.61 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.67, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {27, 27, 2462, 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.
\(\displaystyle \int \frac {36 x-18 e^9-18}{\left (-x^6+3 x^5+57 x^4-119 x^3-1140 x^2+e^{27} \left (x^3+12 x^2+48 x+64\right )+e^{18} \left (-3 x^4-21 x^3+36 x^2+528 x+960\right )+e^9 \left (3 x^5+6 x^4-141 x^3-348 x^2+1680 x+4800\right )+1200 x+8000\right ) \log ^2(5)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {18 \left (-2 x+e^9+1\right )}{-x^6+3 x^5+57 x^4-119 x^3-1140 x^2+1200 x+e^{27} \left (x^3+12 x^2+48 x+64\right )+3 e^{18} \left (-x^4-7 x^3+12 x^2+176 x+320\right )+3 e^9 \left (x^5+2 x^4-47 x^3-116 x^2+560 x+1600\right )+8000}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {18 \int \frac {-2 x+e^9+1}{-x^6+3 x^5+57 x^4-119 x^3-1140 x^2+1200 x+e^{27} \left (x^3+12 x^2+48 x+64\right )+3 e^{18} \left (-x^4-7 x^3+12 x^2+176 x+320\right )+3 e^9 \left (x^5+2 x^4-47 x^3-116 x^2+560 x+1600\right )+8000}dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle -\frac {18 \int \left (\frac {1}{\left (9+e^9\right )^3 (x+4)^2}+\frac {1}{\left (9+e^9\right )^2 (x+4)^3}-\frac {1}{\left (9+e^9\right )^3 \left (-x+e^9+5\right )^2}-\frac {1}{\left (9+e^9\right )^2 \left (-x+e^9+5\right )^3}\right )dx}{\log ^2(5)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {18 \left (-\frac {1}{\left (9+e^9\right )^3 (x+4)}-\frac {1}{2 \left (9+e^9\right )^2 (x+4)^2}-\frac {1}{\left (9+e^9\right )^3 \left (-x+e^9+5\right )}-\frac {1}{2 \left (9+e^9\right )^2 \left (-x+e^9+5\right )^2}\right )}{\log ^2(5)}\) |
Input:
Int[(-18 - 18*E^9 + 36*x)/((8000 + 1200*x - 1140*x^2 - 119*x^3 + 57*x^4 + 3*x^5 - x^6 + E^27*(64 + 48*x + 12*x^2 + x^3) + E^18*(960 + 528*x + 36*x^2 - 21*x^3 - 3*x^4) + E^9*(4800 + 1680*x - 348*x^2 - 141*x^3 + 6*x^4 + 3*x^ 5))*Log[5]^2),x]
Output:
(-18*(-1/2*1/((9 + E^9)^2*(5 + E^9 - x)^2) - 1/((9 + E^9)^3*(5 + E^9 - x)) - 1/(2*(9 + E^9)^2*(4 + x)^2) - 1/((9 + E^9)^3*(4 + x))))/Log[5]^2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.46 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
method | result | size |
norman | \(\frac {9}{\ln \left (5\right )^{2} \left (4+x \right )^{2} \left (-x +{\mathrm e}^{9}+5\right )^{2}}\) | \(21\) |
risch | \(\frac {9}{\ln \left (5\right )^{2} \left ({\mathrm e}^{18} x^{2}-2 x^{3} {\mathrm e}^{9}+x^{4}+8 \,{\mathrm e}^{18} x -6 \,{\mathrm e}^{9} x^{2}-2 x^{3}+16 \,{\mathrm e}^{18}+48 x \,{\mathrm e}^{9}-39 x^{2}+160 \,{\mathrm e}^{9}+40 x +400\right )}\) | \(65\) |
gosper | \(\frac {9}{\ln \left (5\right )^{2} \left ({\mathrm e}^{18} x^{2}-2 x^{3} {\mathrm e}^{9}+x^{4}+8 \,{\mathrm e}^{18} x -6 \,{\mathrm e}^{9} x^{2}-2 x^{3}+16 \,{\mathrm e}^{18}+48 x \,{\mathrm e}^{9}-39 x^{2}+160 \,{\mathrm e}^{9}+40 x +400\right )}\) | \(71\) |
parallelrisch | \(\frac {9}{\ln \left (5\right )^{2} \left ({\mathrm e}^{18} x^{2}-2 x^{3} {\mathrm e}^{9}+x^{4}+8 \,{\mathrm e}^{18} x -6 \,{\mathrm e}^{9} x^{2}-2 x^{3}+16 \,{\mathrm e}^{18}+48 x \,{\mathrm e}^{9}-39 x^{2}+160 \,{\mathrm e}^{9}+40 x +400\right )}\) | \(71\) |
default | \(\frac {-\frac {6 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{3}+\left (-3 \,{\mathrm e}^{9}-15\right ) \textit {\_Z}^{2}+\left (3 \,{\mathrm e}^{18}+30 \,{\mathrm e}^{9}+75\right ) \textit {\_Z} -{\mathrm e}^{27}-15 \,{\mathrm e}^{18}-75 \,{\mathrm e}^{9}-125\right )}{\sum }\frac {\left (7440174-531441 \textit {\_R} -98415 \,{\mathrm e}^{18} \textit {\_R} -14580 \,{\mathrm e}^{27} \textit {\_R} +2086398 \,{\mathrm e}^{18}+6022998 \,{\mathrm e}^{9}-354294 \textit {\_R} \,{\mathrm e}^{9}+3186 \,{\mathrm e}^{45}-54 \textit {\_R} \,{\mathrm e}^{45}-1215 \,{\mathrm e}^{36} \textit {\_R} -\textit {\_R} \,{\mathrm e}^{54}+2 \,{\mathrm e}^{63}+122 \,{\mathrm e}^{54}+400950 \,{\mathrm e}^{27}+46170 \,{\mathrm e}^{36}\right ) \ln \left (x -\textit {\_R} \right )}{25-2 \textit {\_R} \,{\mathrm e}^{9}+\textit {\_R}^{2}+10 \,{\mathrm e}^{9}+{\mathrm e}^{18}-10 \textit {\_R}}\right )}{\left (729+243 \,{\mathrm e}^{9}+27 \,{\mathrm e}^{18}+{\mathrm e}^{27}\right )^{3}}-\frac {9 \left (-4782969-1240029 \,{\mathrm e}^{18}-229635 \,{\mathrm e}^{27}-3720087 \,{\mathrm e}^{9}-1701 \,{\mathrm e}^{45}-25515 \,{\mathrm e}^{36}-63 \,{\mathrm e}^{54}-{\mathrm e}^{63}\right )}{\left (729+243 \,{\mathrm e}^{9}+27 \,{\mathrm e}^{18}+{\mathrm e}^{27}\right )^{3} \left (4+x \right )^{2}}-\frac {18 \left (-14580 \,{\mathrm e}^{27}-98415 \,{\mathrm e}^{18}-354294 \,{\mathrm e}^{9}-54 \,{\mathrm e}^{45}-1215 \,{\mathrm e}^{36}-{\mathrm e}^{54}-531441\right )}{\left (729+243 \,{\mathrm e}^{9}+27 \,{\mathrm e}^{18}+{\mathrm e}^{27}\right )^{3} \left (4+x \right )}}{\ln \left (5\right )^{2}}\) | \(256\) |
Input:
int((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x^3+36* x^2+528*x+960)*exp(9)^2+(3*x^5+6*x^4-141*x^3-348*x^2+1680*x+4800)*exp(9)-x ^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/ln(5)^2,x,method=_RETURNVERB OSE)
Output:
9/ln(5)^2/(4+x)^2/(-x+exp(9)+5)^2
Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (20) = 40\).
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.57 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{{\left (x^{4} - 2 \, x^{3} - 39 \, x^{2} + {\left (x^{2} + 8 \, x + 16\right )} e^{18} - 2 \, {\left (x^{3} + 3 \, x^{2} - 24 \, x - 80\right )} e^{9} + 40 \, x + 400\right )} \log \left (5\right )^{2}} \] Input:
integrate((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x ^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5+6*x^4-141*x^3-348*x^2+1680*x+4800)*ex p(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x, algorithm= "fricas")
Output:
9/((x^4 - 2*x^3 - 39*x^2 + (x^2 + 8*x + 16)*e^18 - 2*(x^3 + 3*x^2 - 24*x - 80)*e^9 + 40*x + 400)*log(5)^2)
Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (19) = 38\).
Time = 0.84 (sec) , antiderivative size = 116, normalized size of antiderivative = 5.52 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{x^{4} \log {\left (5 \right )}^{2} + x^{3} \left (- 2 e^{9} \log {\left (5 \right )}^{2} - 2 \log {\left (5 \right )}^{2}\right ) + x^{2} \left (- 6 e^{9} \log {\left (5 \right )}^{2} - 39 \log {\left (5 \right )}^{2} + e^{18} \log {\left (5 \right )}^{2}\right ) + x \left (40 \log {\left (5 \right )}^{2} + 48 e^{9} \log {\left (5 \right )}^{2} + 8 e^{18} \log {\left (5 \right )}^{2}\right ) + 400 \log {\left (5 \right )}^{2} + 160 e^{9} \log {\left (5 \right )}^{2} + 16 e^{18} \log {\left (5 \right )}^{2}} \] Input:
integrate((-18*exp(9)+36*x-18)/((x**3+12*x**2+48*x+64)*exp(9)**3+(-3*x**4- 21*x**3+36*x**2+528*x+960)*exp(9)**2+(3*x**5+6*x**4-141*x**3-348*x**2+1680 *x+4800)*exp(9)-x**6+3*x**5+57*x**4-119*x**3-1140*x**2+1200*x+8000)/ln(5)* *2,x)
Output:
9/(x**4*log(5)**2 + x**3*(-2*exp(9)*log(5)**2 - 2*log(5)**2) + x**2*(-6*ex p(9)*log(5)**2 - 39*log(5)**2 + exp(18)*log(5)**2) + x*(40*log(5)**2 + 48* exp(9)*log(5)**2 + 8*exp(18)*log(5)**2) + 400*log(5)**2 + 160*exp(9)*log(5 )**2 + 16*exp(18)*log(5)**2)
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (20) = 40\).
Time = 0.04 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.52 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{{\left (x^{4} - 2 \, x^{3} {\left (e^{9} + 1\right )} + x^{2} {\left (e^{18} - 6 \, e^{9} - 39\right )} + 8 \, x {\left (e^{18} + 6 \, e^{9} + 5\right )} + 16 \, e^{18} + 160 \, e^{9} + 400\right )} \log \left (5\right )^{2}} \] Input:
integrate((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x ^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5+6*x^4-141*x^3-348*x^2+1680*x+4800)*ex p(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x, algorithm= "maxima")
Output:
9/((x^4 - 2*x^3*(e^9 + 1) + x^2*(e^18 - 6*e^9 - 39) + 8*x*(e^18 + 6*e^9 + 5) + 16*e^18 + 160*e^9 + 400)*log(5)^2)
Time = 0.12 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{{\left (x^{2} - x e^{9} - x - 4 \, e^{9} - 20\right )}^{2} \log \left (5\right )^{2}} \] Input:
integrate((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x ^3+36*x^2+528*x+960)*exp(9)^2+(3*x^5+6*x^4-141*x^3-348*x^2+1680*x+4800)*ex p(9)-x^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x, algorithm= "giac")
Output:
9/((x^2 - x*e^9 - x - 4*e^9 - 20)^2*log(5)^2)
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.71 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{{\ln \left (5\right )}^2\,{\left ({\mathrm {e}}^9+9\right )}^2\,{\left (x+4\right )}^2}+\frac {18}{{\ln \left (5\right )}^2\,{\left ({\mathrm {e}}^9+9\right )}^3\,\left (x+4\right )}+\frac {9\,\left (3\,{\mathrm {e}}^9-2\,x+19\right )}{{\ln \left (5\right )}^2\,{\left ({\mathrm {e}}^9+9\right )}^3\,\left (x^2+\left (-2\,{\mathrm {e}}^9-10\right )\,x+10\,{\mathrm {e}}^9+{\mathrm {e}}^{18}+25\right )} \] Input:
int(-(18*exp(9) - 36*x + 18)/(log(5)^2*(1200*x + exp(27)*(48*x + 12*x^2 + x^3 + 64) + exp(18)*(528*x + 36*x^2 - 21*x^3 - 3*x^4 + 960) + exp(9)*(1680 *x - 348*x^2 - 141*x^3 + 6*x^4 + 3*x^5 + 4800) - 1140*x^2 - 119*x^3 + 57*x ^4 + 3*x^5 - x^6 + 8000)),x)
Output:
9/(log(5)^2*(exp(9) + 9)^2*(x + 4)^2) + 18/(log(5)^2*(exp(9) + 9)^3*(x + 4 )) + (9*(3*exp(9) - 2*x + 19))/(log(5)^2*(exp(9) + 9)^3*(10*exp(9) + exp(1 8) + x^2 - x*(2*exp(9) + 10) + 25))
Time = 0.21 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.38 \[ \int \frac {-18-18 e^9+36 x}{\left (8000+1200 x-1140 x^2-119 x^3+57 x^4+3 x^5-x^6+e^{27} \left (64+48 x+12 x^2+x^3\right )+e^{18} \left (960+528 x+36 x^2-21 x^3-3 x^4\right )+e^9 \left (4800+1680 x-348 x^2-141 x^3+6 x^4+3 x^5\right )\right ) \log ^2(5)} \, dx=\frac {9}{\mathrm {log}\left (5\right )^{2} \left (e^{18} x^{2}+8 e^{18} x +16 e^{18}-2 e^{9} x^{3}-6 e^{9} x^{2}+48 e^{9} x +160 e^{9}+x^{4}-2 x^{3}-39 x^{2}+40 x +400\right )} \] Input:
int((-18*exp(9)+36*x-18)/((x^3+12*x^2+48*x+64)*exp(9)^3+(-3*x^4-21*x^3+36* x^2+528*x+960)*exp(9)^2+(3*x^5+6*x^4-141*x^3-348*x^2+1680*x+4800)*exp(9)-x ^6+3*x^5+57*x^4-119*x^3-1140*x^2+1200*x+8000)/log(5)^2,x)
Output:
9/(log(5)**2*(e**18*x**2 + 8*e**18*x + 16*e**18 - 2*e**9*x**3 - 6*e**9*x** 2 + 48*e**9*x + 160*e**9 + x**4 - 2*x**3 - 39*x**2 + 40*x + 400))