Integrand size = 130, antiderivative size = 27 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=x \left (9+x+5 \left (\frac {e^{\frac {(4+x)^2}{x^4}}}{x}+\log (25)\right )\right )^2 \] Output:
x*(10*ln(5)+5*exp((4+x)^2/x^4)/x+x+9)^2
Time = 0.08 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=\frac {\left (5 e^{\frac {(4+x)^2}{x^4}}+x (9+x+5 \log (25))\right )^2}{x} \] Input:
Integrate[(81*x^6 + 36*x^7 + 3*x^8 + E^((2*(16 + 8*x + x^2))/x^4)*(-3200 - 1200*x - 100*x^2 - 25*x^4) + (90*x^6 + 20*x^7)*Log[25] + 25*x^6*Log[25]^2 + E^((16 + 8*x + x^2)/x^4)*(-5760*x - 2800*x^2 - 420*x^3 - 20*x^4 + 10*x^ 6 + (-3200*x - 1200*x^2 - 100*x^3)*Log[25]))/x^6,x]
Output:
(5*E^((4 + x)^2/x^4) + x*(9 + x + 5*Log[25]))^2/x
Leaf count is larger than twice the leaf count of optimal. \(141\) vs. \(2(27)=54\).
Time = 0.52 (sec) , antiderivative size = 141, normalized size of antiderivative = 5.22, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6, 2010, 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.
| \(\displaystyle \int \frac {3 x^8+36 x^7+81 x^6+25 x^6 \log ^2(25)+\left (20 x^7+90 x^6\right ) \log (25)+e^{\frac {2 \left (x^2+8 x+16\right )}{x^4}} \left (-25 x^4-100 x^2-1200 x-3200\right )+e^{\frac {x^2+8 x+16}{x^4}} \left (10 x^6-20 x^4-420 x^3-2800 x^2+\left (-100 x^3-1200 x^2-3200 x\right ) \log (25)-5760 x\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {3 x^8+36 x^7+x^6 \left (81+25 \log ^2(25)\right )+\left (20 x^7+90 x^6\right ) \log (25)+e^{\frac {2 \left (x^2+8 x+16\right )}{x^4}} \left (-25 x^4-100 x^2-1200 x-3200\right )+e^{\frac {x^2+8 x+16}{x^4}} \left (10 x^6-20 x^4-420 x^3-2800 x^2+\left (-100 x^3-1200 x^2-3200 x\right ) \log (25)-5760 x\right )}{x^6}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (3 x^2-\frac {25 e^{\frac {2 (x+4)^2}{x^4}} \left (x^4+4 x^2+48 x+128\right )}{x^6}+\frac {10 e^{\frac {(x+4)^2}{x^4}} \left (x^5-2 x^3-42 x^2 \left (1+\frac {10 \log (5)}{21}\right )-280 x \left (1+\frac {6 \log (5)}{7}\right )-576 \left (1+\frac {5 \log (25)}{9}\right )\right )}{x^5}+36 x \left (1+\frac {5 \log (25)}{9}\right )+81 \left (1+\frac {10}{81} (9+5 \log (5)) \log (25)\right )\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^3+2 x^2 (9+5 \log (25))-\frac {25 e^{\frac {2 (x+4)^2}{x^4}} \left (x^2+12 x+32\right )}{\left (\frac {x+4}{x^4}-\frac {2 (x+4)^2}{x^5}\right ) x^6}-\frac {10 e^{\frac {(x+4)^2}{x^4}} \left (x^3+x^2 (21+10 \log (5))+20 x (7+6 \log (5))+32 (9+5 \log (25))\right )}{\left (\frac {x+4}{x^4}-\frac {2 (x+4)^2}{x^5}\right ) x^5}+x (81+10 (9+5 \log (5)) \log (25))\) |
Input:
Int[(81*x^6 + 36*x^7 + 3*x^8 + E^((2*(16 + 8*x + x^2))/x^4)*(-3200 - 1200* x - 100*x^2 - 25*x^4) + (90*x^6 + 20*x^7)*Log[25] + 25*x^6*Log[25]^2 + E^( (16 + 8*x + x^2)/x^4)*(-5760*x - 2800*x^2 - 420*x^3 - 20*x^4 + 10*x^6 + (- 3200*x - 1200*x^2 - 100*x^3)*Log[25]))/x^6,x]
Output:
x^3 - (25*E^((2*(4 + x)^2)/x^4)*(32 + 12*x + x^2))/(x^6*((4 + x)/x^4 - (2* (4 + x)^2)/x^5)) + 2*x^2*(9 + 5*Log[25]) + x*(81 + 10*(9 + 5*Log[5])*Log[2 5]) - (10*E^((4 + x)^2/x^4)*(x^3 + 20*x*(7 + 6*Log[5]) + x^2*(21 + 10*Log[ 5]) + 32*(9 + 5*Log[25])))/(x^5*((4 + x)/x^4 - (2*(4 + x)^2)/x^5))
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(26)=52\).
Time = 3.47 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.52
| method | result | size |
| risch | \(x^{3}+20 x^{2} \ln \left (5\right )+18 x^{2}+100 x \ln \left (5\right )^{2}+180 x \ln \left (5\right )+81 x +\frac {25 \,{\mathrm e}^{\frac {2 \left (4+x \right )^{2}}{x^{4}}}}{x}+\left (90+100 \ln \left (5\right )+10 x \right ) {\mathrm e}^{\frac {\left (4+x \right )^{2}}{x^{4}}}\) | \(68\) |
| norman | \(\frac {x^{8}+\left (20 \ln \left (5\right )+18\right ) x^{7}+\left (100 \ln \left (5\right )^{2}+180 \ln \left (5\right )+81\right ) x^{6}+\left (90+100 \ln \left (5\right )\right ) x^{5} {\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}}+10 \,{\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}} x^{6}+25 \,{\mathrm e}^{\frac {2 x^{2}+16 x +32}{x^{4}}} x^{4}}{x^{5}}\) | \(96\) |
| parts | \(x^{3}+18 x^{2}+81 x +100 x \ln \left (5\right )^{2}+20 x^{2} \ln \left (5\right )+\frac {\left (90+100 \ln \left (5\right )\right ) x^{4} {\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}}+10 \,{\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}} x^{5}}{x^{4}}+\frac {25 \,{\mathrm e}^{\frac {2 x^{2}+16 x +32}{x^{4}}}}{x}+180 x \ln \left (5\right )\) | \(98\) |
| parallelrisch | \(\frac {100 x^{2} \ln \left (5\right )^{2}+20 x^{3} \ln \left (5\right )+x^{4}+180 x^{2} \ln \left (5\right )+100 \,{\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}} \ln \left (5\right ) x +18 x^{3}+10 \,{\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}} x^{2}+81 x^{2}+90 x \,{\mathrm e}^{\frac {x^{2}+8 x +16}{x^{4}}}+25 \,{\mathrm e}^{\frac {2 x^{2}+16 x +32}{x^{4}}}}{x}\) | \(111\) |
Input:
int(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100*x^3-12 00*x^2-3200*x)*ln(5)+10*x^6-20*x^4-420*x^3-2800*x^2-5760*x)*exp((x^2+8*x+1 6)/x^4)+100*x^6*ln(5)^2+2*(20*x^7+90*x^6)*ln(5)+3*x^8+36*x^7+81*x^6)/x^6,x ,method=_RETURNVERBOSE)
Output:
x^3+20*x^2*ln(5)+18*x^2+100*x*ln(5)^2+180*x*ln(5)+81*x+25/x*exp(2*(4+x)^2/ x^4)+(90+100*ln(5)+10*x)*exp((4+x)^2/x^4)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (26) = 52\).
Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.07 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=\frac {x^{4} + 100 \, x^{2} \log \left (5\right )^{2} + 18 \, x^{3} + 81 \, x^{2} + 10 \, {\left (x^{2} + 10 \, x \log \left (5\right ) + 9 \, x\right )} e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 20 \, {\left (x^{3} + 9 \, x^{2}\right )} \log \left (5\right ) + 25 \, e^{\left (\frac {2 \, {\left (x^{2} + 8 \, x + 16\right )}}{x^{4}}\right )}}{x} \] Input:
integrate(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100* x^3-1200*x^2-3200*x)*log(5)+10*x^6-20*x^4-420*x^3-2800*x^2-5760*x)*exp((x^ 2+8*x+16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81*x ^6)/x^6,x, algorithm="fricas")
Output:
(x^4 + 100*x^2*log(5)^2 + 18*x^3 + 81*x^2 + 10*(x^2 + 10*x*log(5) + 9*x)*e ^((x^2 + 8*x + 16)/x^4) + 20*(x^3 + 9*x^2)*log(5) + 25*e^(2*(x^2 + 8*x + 1 6)/x^4))/x
Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (24) = 48\).
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.78 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=x^{3} + x^{2} \cdot \left (18 + 20 \log {\left (5 \right )}\right ) + x \left (81 + 100 \log {\left (5 \right )}^{2} + 180 \log {\left (5 \right )}\right ) + \frac {\left (10 x^{2} + 90 x + 100 x \log {\left (5 \right )}\right ) e^{\frac {x^{2} + 8 x + 16}{x^{4}}} + 25 e^{\frac {2 \left (x^{2} + 8 x + 16\right )}{x^{4}}}}{x} \] Input:
integrate(((-25*x**4-100*x**2-1200*x-3200)*exp((x**2+8*x+16)/x**4)**2+(2*( -100*x**3-1200*x**2-3200*x)*ln(5)+10*x**6-20*x**4-420*x**3-2800*x**2-5760* x)*exp((x**2+8*x+16)/x**4)+100*x**6*ln(5)**2+2*(20*x**7+90*x**6)*ln(5)+3*x **8+36*x**7+81*x**6)/x**6,x)
Output:
x**3 + x**2*(18 + 20*log(5)) + x*(81 + 100*log(5)**2 + 180*log(5)) + ((10* x**2 + 90*x + 100*x*log(5))*exp((x**2 + 8*x + 16)/x**4) + 25*exp(2*(x**2 + 8*x + 16)/x**4))/x
\[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=\int { \frac {3 \, x^{8} + 100 \, x^{6} \log \left (5\right )^{2} + 36 \, x^{7} + 81 \, x^{6} - 25 \, {\left (x^{4} + 4 \, x^{2} + 48 \, x + 128\right )} e^{\left (\frac {2 \, {\left (x^{2} + 8 \, x + 16\right )}}{x^{4}}\right )} + 10 \, {\left (x^{6} - 2 \, x^{4} - 42 \, x^{3} - 280 \, x^{2} - 20 \, {\left (x^{3} + 12 \, x^{2} + 32 \, x\right )} \log \left (5\right ) - 576 \, x\right )} e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 20 \, {\left (2 \, x^{7} + 9 \, x^{6}\right )} \log \left (5\right )}{x^{6}} \,d x } \] Input:
integrate(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100* x^3-1200*x^2-3200*x)*log(5)+10*x^6-20*x^4-420*x^3-2800*x^2-5760*x)*exp((x^ 2+8*x+16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81*x ^6)/x^6,x, algorithm="maxima")
Output:
x^3 + 20*x^2*log(5) + 100*x*log(5)^2 + 18*x^2 + 180*x*log(5) + 81*x + 25*e ^(2/x^2 + 16/x^3 + 32/x^4)/x + integrate(10*(x^5 - 2*x^3 - 2*x^2*(10*log(5 ) + 21) - 40*x*(6*log(5) + 7) - 640*log(5) - 576)*e^(1/x^2 + 8/x^3 + 16/x^ 4)/x^5, x)
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (26) = 52\).
Time = 0.15 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.04 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=\frac {x^{4} + 20 \, x^{3} \log \left (5\right ) + 100 \, x^{2} \log \left (5\right )^{2} + 18 \, x^{3} + 10 \, x^{2} e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 180 \, x^{2} \log \left (5\right ) + 100 \, x e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} \log \left (5\right ) + 81 \, x^{2} + 90 \, x e^{\left (\frac {x^{2} + 8 \, x + 16}{x^{4}}\right )} + 25 \, e^{\left (\frac {2 \, {\left (x^{2} + 8 \, x + 16\right )}}{x^{4}}\right )}}{x} \] Input:
integrate(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100* x^3-1200*x^2-3200*x)*log(5)+10*x^6-20*x^4-420*x^3-2800*x^2-5760*x)*exp((x^ 2+8*x+16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81*x ^6)/x^6,x, algorithm="giac")
Output:
(x^4 + 20*x^3*log(5) + 100*x^2*log(5)^2 + 18*x^3 + 10*x^2*e^((x^2 + 8*x + 16)/x^4) + 180*x^2*log(5) + 100*x*e^((x^2 + 8*x + 16)/x^4)*log(5) + 81*x^2 + 90*x*e^((x^2 + 8*x + 16)/x^4) + 25*e^(2*(x^2 + 8*x + 16)/x^4))/x
Time = 2.84 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=\frac {{\left (9\,x+5\,{\mathrm {e}}^{\frac {x^2+8\,x+16}{x^4}}+10\,x\,\ln \left (5\right )+x^2\right )}^2}{x} \] Input:
int((100*x^6*log(5)^2 - exp((8*x + x^2 + 16)/x^4)*(5760*x + 2*log(5)*(3200 *x + 1200*x^2 + 100*x^3) + 2800*x^2 + 420*x^3 + 20*x^4 - 10*x^6) + 2*log(5 )*(90*x^6 + 20*x^7) - exp((2*(8*x + x^2 + 16))/x^4)*(1200*x + 100*x^2 + 25 *x^4 + 3200) + 81*x^6 + 36*x^7 + 3*x^8)/x^6,x)
Output:
(9*x + 5*exp((8*x + x^2 + 16)/x^4) + 10*x*log(5) + x^2)^2/x
Time = 0.38 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.22 \[ \int \frac {81 x^6+36 x^7+3 x^8+e^{\frac {2 \left (16+8 x+x^2\right )}{x^4}} \left (-3200-1200 x-100 x^2-25 x^4\right )+\left (90 x^6+20 x^7\right ) \log (25)+25 x^6 \log ^2(25)+e^{\frac {16+8 x+x^2}{x^4}} \left (-5760 x-2800 x^2-420 x^3-20 x^4+10 x^6+\left (-3200 x-1200 x^2-100 x^3\right ) \log (25)\right )}{x^6} \, dx=\frac {25 e^{\frac {2 x^{2}+16 x +32}{x^{4}}}+100 e^{\frac {x^{2}+8 x +16}{x^{4}}} \mathrm {log}\left (5\right ) x +10 e^{\frac {x^{2}+8 x +16}{x^{4}}} x^{2}+90 e^{\frac {x^{2}+8 x +16}{x^{4}}} x +100 \mathrm {log}\left (5\right )^{2} x^{2}+20 \,\mathrm {log}\left (5\right ) x^{3}+180 \,\mathrm {log}\left (5\right ) x^{2}+x^{4}+18 x^{3}+81 x^{2}}{x} \] Input:
int(((-25*x^4-100*x^2-1200*x-3200)*exp((x^2+8*x+16)/x^4)^2+(2*(-100*x^3-12 00*x^2-3200*x)*log(5)+10*x^6-20*x^4-420*x^3-2800*x^2-5760*x)*exp((x^2+8*x+ 16)/x^4)+100*x^6*log(5)^2+2*(20*x^7+90*x^6)*log(5)+3*x^8+36*x^7+81*x^6)/x^ 6,x)
Output:
(25*e**((2*x**2 + 16*x + 32)/x**4) + 100*e**((x**2 + 8*x + 16)/x**4)*log(5 )*x + 10*e**((x**2 + 8*x + 16)/x**4)*x**2 + 90*e**((x**2 + 8*x + 16)/x**4) *x + 100*log(5)**2*x**2 + 20*log(5)*x**3 + 180*log(5)*x**2 + x**4 + 18*x** 3 + 81*x**2)/x