Integrand size = 291, antiderivative size = 34 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=\frac {5+3 x^2+\frac {\log (4)}{\left (e^x (1-2 x)+\frac {4}{x}-x\right )^2}}{x} \] Output:
(2*ln(2)/(4/x+(1-2*x)*exp(x)-x)^2+3*x^2+5)/x
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.88 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=\frac {5}{x}+x \left (3+\frac {\log (4)}{\left (-4+x^2+e^x x (-1+2 x)\right )^2}\right ) \] Input:
Integrate[(320 - 432*x^2 + 204*x^4 - 41*x^6 + 3*x^8 + E^(3*x)*(5*x^3 - 30* x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) + E^(2*x)*(60*x^2 - 240*x^3 + 189 *x^4 + 204*x^5 - 195*x^6 - 36*x^7 + 36*x^8) + (-4*x^2 - 3*x^4)*Log[4] + E^ x*(240*x - 480*x^2 - 264*x^3 + 528*x^4 + 87*x^5 - 174*x^6 - 9*x^7 + 18*x^8 + (x^3 - 4*x^4 - 4*x^5)*Log[4]))/(-64*x^2 + 48*x^4 - 12*x^6 + x^8 + E^x*( -48*x^3 + 96*x^4 + 24*x^5 - 48*x^6 - 3*x^7 + 6*x^8) + E^(3*x)*(-x^5 + 6*x^ 6 - 12*x^7 + 8*x^8) + E^(2*x)*(-12*x^4 + 48*x^5 - 45*x^6 - 12*x^7 + 12*x^8 )),x]
Output:
5/x + x*(3 + Log[4]/(-4 + x^2 + E^x*x*(-1 + 2*x))^2)
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.
| \(\displaystyle \int \frac {3 x^8-41 x^6+204 x^4-432 x^2+\left (-3 x^4-4 x^2\right ) \log (4)+e^{3 x} \left (24 x^8-36 x^7-22 x^6+57 x^5-30 x^4+5 x^3\right )+e^{2 x} \left (36 x^8-36 x^7-195 x^6+204 x^5+189 x^4-240 x^3+60 x^2\right )+e^x \left (18 x^8-9 x^7-174 x^6+87 x^5+528 x^4-264 x^3-480 x^2+\left (-4 x^5-4 x^4+x^3\right ) \log (4)+240 x\right )+320}{x^8-12 x^6+48 x^4-64 x^2+e^{3 x} \left (8 x^8-12 x^7+6 x^6-x^5\right )+e^{2 x} \left (12 x^8-12 x^7-45 x^6+48 x^5-12 x^4\right )+e^x \left (6 x^8-3 x^7-48 x^6+24 x^5+96 x^4-48 x^3\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-3 x^8+41 x^6+3 x^4 (\log (4)-68)+4 x^2 (108+\log (4))-3 e^{2 x} (1-2 x)^2 \left (3 x^4-17 x^2+20\right ) x^2-e^{3 x} (2 x-1)^3 \left (3 x^2-5\right ) x^3-e^x x \left (18 x^7-9 x^6-174 x^5+x^4 (87-4 \log (4))-4 x^3 (\log (4)-132)+x^2 (\log (4)-264)-480 x+240\right )-320}{x^2 \left (-x^2-e^x (2 x-1) x+4\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3 x^2-5}{x^2}-\frac {\left (4 x^2+4 x-1\right ) \log (4)}{(2 x-1) \left (2 e^x x^2+x^2-e^x x-4\right )^2}+\frac {2 \left (2 x^4-x^3-7 x^2-12 x+4\right ) \log (4)}{(2 x-1) \left (2 e^x x^2+x^2-e^x x-4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {31}{2} \log (4) \int \frac {1}{\left (2 e^x x^2+x^2-e^x x-4\right )^3}dx-7 \log (4) \int \frac {x}{\left (2 e^x x^2+x^2-e^x x-4\right )^3}dx-\frac {15}{2} \log (4) \int \frac {1}{(2 x-1) \left (2 e^x x^2+x^2-e^x x-4\right )^3}dx-3 \log (4) \int \frac {1}{\left (2 e^x x^2+x^2-e^x x-4\right )^2}dx-2 \log (4) \int \frac {x}{\left (2 e^x x^2+x^2-e^x x-4\right )^2}dx-2 \log (4) \int \frac {1}{(2 x-1) \left (2 e^x x^2+x^2-e^x x-4\right )^2}dx+2 \log (4) \int \frac {x^3}{\left (2 e^x x^2+x^2-e^x x-4\right )^3}dx+3 x+\frac {5}{x}\) |
Input:
Int[(320 - 432*x^2 + 204*x^4 - 41*x^6 + 3*x^8 + E^(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) + E^(2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 204*x^5 - 195*x^6 - 36*x^7 + 36*x^8) + (-4*x^2 - 3*x^4)*Log[4] + E^x*(240 *x - 480*x^2 - 264*x^3 + 528*x^4 + 87*x^5 - 174*x^6 - 9*x^7 + 18*x^8 + (x^ 3 - 4*x^4 - 4*x^5)*Log[4]))/(-64*x^2 + 48*x^4 - 12*x^6 + x^8 + E^x*(-48*x^ 3 + 96*x^4 + 24*x^5 - 48*x^6 - 3*x^7 + 6*x^8) + E^(3*x)*(-x^5 + 6*x^6 - 12 *x^7 + 8*x^8) + E^(2*x)*(-12*x^4 + 48*x^5 - 45*x^6 - 12*x^7 + 12*x^8)),x]
Output:
$Aborted
Time = 2.83 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(3 x +\frac {5}{x}+\frac {2 \ln \left (2\right ) x}{\left (2 \,{\mathrm e}^{x} x^{2}-{\mathrm e}^{x} x +x^{2}-4\right )^{2}}\) | \(34\) |
| parallelrisch | \(\frac {320+192 x +48 x^{6} {\mathrm e}^{x}+44 \,{\mathrm e}^{2 x} x^{4}-136 \,{\mathrm e}^{x} x^{4}+20 \,{\mathrm e}^{2 x} x^{2}+8 x^{2} \ln \left (2\right )-224 \,{\mathrm e}^{x} x^{2}-136 \,{\mathrm e}^{x} x^{3}+160 \,{\mathrm e}^{x} x +48 \,{\mathrm e}^{2 x} x^{6}+24 x^{5} {\mathrm e}^{x}-68 \,{\mathrm e}^{2 x} x^{3}-76 x^{4}-96 x^{3}+32 x^{2}+12 x^{6}+12 x^{5}}{4 x \left (4 \,{\mathrm e}^{2 x} x^{4}+4 \,{\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{2 x} x^{3}+x^{4}-2 \,{\mathrm e}^{x} x^{3}+{\mathrm e}^{2 x} x^{2}-16 \,{\mathrm e}^{x} x^{2}-8 x^{2}+8 \,{\mathrm e}^{x} x +16\right )}\) | \(183\) |
Input:
int(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36*x^7-19 5*x^6+204*x^5+189*x^4-240*x^3+60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3)*ln(2) +18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x)+2*(-3*x ^4-4*x^2)*ln(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6*x^6-x^5 )*exp(x)^3+(12*x^8-12*x^7-45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3*x^7-48*x ^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x,method=_RETURN VERBOSE)
Output:
3*x+5/x+2*ln(2)*x/(2*exp(x)*x^2-exp(x)*x+x^2-4)^2
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (32) = 64\).
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 4.32 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=\frac {3 \, x^{6} - 19 \, x^{4} + 2 \, x^{2} \log \left (2\right ) + 8 \, x^{2} + {\left (12 \, x^{6} - 12 \, x^{5} + 23 \, x^{4} - 20 \, x^{3} + 5 \, x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (6 \, x^{6} - 3 \, x^{5} - 14 \, x^{4} + 7 \, x^{3} - 40 \, x^{2} + 20 \, x\right )} e^{x} + 80}{x^{5} - 8 \, x^{3} + {\left (4 \, x^{5} - 4 \, x^{4} + x^{3}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x^{5} - x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} e^{x} + 16 \, x} \] Input:
integrate(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36* x^7-195*x^6+204*x^5+189*x^4-240*x^3+60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3) *log(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x)+ 2*(-3*x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6 *x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3* x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x, algor ithm="fricas")
Output:
(3*x^6 - 19*x^4 + 2*x^2*log(2) + 8*x^2 + (12*x^6 - 12*x^5 + 23*x^4 - 20*x^ 3 + 5*x^2)*e^(2*x) + 2*(6*x^6 - 3*x^5 - 14*x^4 + 7*x^3 - 40*x^2 + 20*x)*e^ x + 80)/(x^5 - 8*x^3 + (4*x^5 - 4*x^4 + x^3)*e^(2*x) + 2*(2*x^5 - x^4 - 8* x^3 + 4*x^2)*e^x + 16*x)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (27) = 54\).
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.85 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=3 x + \frac {2 x \log {\left (2 \right )}}{x^{4} - 8 x^{2} + \left (4 x^{4} - 4 x^{3} + x^{2}\right ) e^{2 x} + \left (4 x^{4} - 2 x^{3} - 16 x^{2} + 8 x\right ) e^{x} + 16} + \frac {5}{x} \] Input:
integrate(((24*x**8-36*x**7-22*x**6+57*x**5-30*x**4+5*x**3)*exp(x)**3+(36* x**8-36*x**7-195*x**6+204*x**5+189*x**4-240*x**3+60*x**2)*exp(x)**2+(2*(-4 *x**5-4*x**4+x**3)*ln(2)+18*x**8-9*x**7-174*x**6+87*x**5+528*x**4-264*x**3 -480*x**2+240*x)*exp(x)+2*(-3*x**4-4*x**2)*ln(2)+3*x**8-41*x**6+204*x**4-4 32*x**2+320)/((8*x**8-12*x**7+6*x**6-x**5)*exp(x)**3+(12*x**8-12*x**7-45*x **6+48*x**5-12*x**4)*exp(x)**2+(6*x**8-3*x**7-48*x**6+24*x**5+96*x**4-48*x **3)*exp(x)+x**8-12*x**6+48*x**4-64*x**2),x)
Output:
3*x + 2*x*log(2)/(x**4 - 8*x**2 + (4*x**4 - 4*x**3 + x**2)*exp(2*x) + (4*x **4 - 2*x**3 - 16*x**2 + 8*x)*exp(x) + 16) + 5/x
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (32) = 64\).
Time = 0.29 (sec) , antiderivative size = 144, normalized size of antiderivative = 4.24 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=\frac {3 \, x^{6} - 19 \, x^{4} + 2 \, x^{2} {\left (\log \left (2\right ) + 4\right )} + {\left (12 \, x^{6} - 12 \, x^{5} + 23 \, x^{4} - 20 \, x^{3} + 5 \, x^{2}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (6 \, x^{6} - 3 \, x^{5} - 14 \, x^{4} + 7 \, x^{3} - 40 \, x^{2} + 20 \, x\right )} e^{x} + 80}{x^{5} - 8 \, x^{3} + {\left (4 \, x^{5} - 4 \, x^{4} + x^{3}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x^{5} - x^{4} - 8 \, x^{3} + 4 \, x^{2}\right )} e^{x} + 16 \, x} \] Input:
integrate(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36* x^7-195*x^6+204*x^5+189*x^4-240*x^3+60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3) *log(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x)+ 2*(-3*x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6 *x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3* x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x, algor ithm="maxima")
Output:
(3*x^6 - 19*x^4 + 2*x^2*(log(2) + 4) + (12*x^6 - 12*x^5 + 23*x^4 - 20*x^3 + 5*x^2)*e^(2*x) + 2*(6*x^6 - 3*x^5 - 14*x^4 + 7*x^3 - 40*x^2 + 20*x)*e^x + 80)/(x^5 - 8*x^3 + (4*x^5 - 4*x^4 + x^3)*e^(2*x) + 2*(2*x^5 - x^4 - 8*x^ 3 + 4*x^2)*e^x + 16*x)
Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (32) = 64\).
Time = 0.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 5.24 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=\frac {12 \, x^{6} e^{\left (2 \, x\right )} + 12 \, x^{6} e^{x} + 3 \, x^{6} - 12 \, x^{5} e^{\left (2 \, x\right )} - 6 \, x^{5} e^{x} + 23 \, x^{4} e^{\left (2 \, x\right )} - 28 \, x^{4} e^{x} - 19 \, x^{4} - 20 \, x^{3} e^{\left (2 \, x\right )} + 14 \, x^{3} e^{x} + 5 \, x^{2} e^{\left (2 \, x\right )} - 80 \, x^{2} e^{x} + 2 \, x^{2} \log \left (2\right ) + 8 \, x^{2} + 40 \, x e^{x} + 80}{4 \, x^{5} e^{\left (2 \, x\right )} + 4 \, x^{5} e^{x} + x^{5} - 4 \, x^{4} e^{\left (2 \, x\right )} - 2 \, x^{4} e^{x} + x^{3} e^{\left (2 \, x\right )} - 16 \, x^{3} e^{x} - 8 \, x^{3} + 8 \, x^{2} e^{x} + 16 \, x} \] Input:
integrate(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36* x^7-195*x^6+204*x^5+189*x^4-240*x^3+60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3) *log(2)+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x)+ 2*(-3*x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6 *x^6-x^5)*exp(x)^3+(12*x^8-12*x^7-45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3* x^7-48*x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x, algor ithm="giac")
Output:
(12*x^6*e^(2*x) + 12*x^6*e^x + 3*x^6 - 12*x^5*e^(2*x) - 6*x^5*e^x + 23*x^4 *e^(2*x) - 28*x^4*e^x - 19*x^4 - 20*x^3*e^(2*x) + 14*x^3*e^x + 5*x^2*e^(2* x) - 80*x^2*e^x + 2*x^2*log(2) + 8*x^2 + 40*x*e^x + 80)/(4*x^5*e^(2*x) + 4 *x^5*e^x + x^5 - 4*x^4*e^(2*x) - 2*x^4*e^x + x^3*e^(2*x) - 16*x^3*e^x - 8* x^3 + 8*x^2*e^x + 16*x)
Timed out. \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=-\int \frac {{\mathrm {e}}^{3\,x}\,\left (24\,x^8-36\,x^7-22\,x^6+57\,x^5-30\,x^4+5\,x^3\right )+{\mathrm {e}}^{2\,x}\,\left (36\,x^8-36\,x^7-195\,x^6+204\,x^5+189\,x^4-240\,x^3+60\,x^2\right )-2\,\ln \left (2\right )\,\left (3\,x^4+4\,x^2\right )-{\mathrm {e}}^x\,\left (2\,\ln \left (2\right )\,\left (4\,x^5+4\,x^4-x^3\right )-240\,x+480\,x^2+264\,x^3-528\,x^4-87\,x^5+174\,x^6+9\,x^7-18\,x^8\right )-432\,x^2+204\,x^4-41\,x^6+3\,x^8+320}{{\mathrm {e}}^{2\,x}\,\left (-12\,x^8+12\,x^7+45\,x^6-48\,x^5+12\,x^4\right )+{\mathrm {e}}^{3\,x}\,\left (-8\,x^8+12\,x^7-6\,x^6+x^5\right )+64\,x^2-48\,x^4+12\,x^6-x^8+{\mathrm {e}}^x\,\left (-6\,x^8+3\,x^7+48\,x^6-24\,x^5-96\,x^4+48\,x^3\right )} \,d x \] Input:
int(-(exp(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) + exp( 2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 204*x^5 - 195*x^6 - 36*x^7 + 36*x^8) - 2*log(2)*(4*x^2 + 3*x^4) - exp(x)*(2*log(2)*(4*x^4 - x^3 + 4*x^5) - 240*x + 480*x^2 + 264*x^3 - 528*x^4 - 87*x^5 + 174*x^6 + 9*x^7 - 18*x^8) - 432*x ^2 + 204*x^4 - 41*x^6 + 3*x^8 + 320)/(exp(2*x)*(12*x^4 - 48*x^5 + 45*x^6 + 12*x^7 - 12*x^8) + exp(3*x)*(x^5 - 6*x^6 + 12*x^7 - 8*x^8) + 64*x^2 - 48* x^4 + 12*x^6 - x^8 + exp(x)*(48*x^3 - 96*x^4 - 24*x^5 + 48*x^6 + 3*x^7 - 6 *x^8)),x)
Output:
-int((exp(3*x)*(5*x^3 - 30*x^4 + 57*x^5 - 22*x^6 - 36*x^7 + 24*x^8) + exp( 2*x)*(60*x^2 - 240*x^3 + 189*x^4 + 204*x^5 - 195*x^6 - 36*x^7 + 36*x^8) - 2*log(2)*(4*x^2 + 3*x^4) - exp(x)*(2*log(2)*(4*x^4 - x^3 + 4*x^5) - 240*x + 480*x^2 + 264*x^3 - 528*x^4 - 87*x^5 + 174*x^6 + 9*x^7 - 18*x^8) - 432*x ^2 + 204*x^4 - 41*x^6 + 3*x^8 + 320)/(exp(2*x)*(12*x^4 - 48*x^5 + 45*x^6 + 12*x^7 - 12*x^8) + exp(3*x)*(x^5 - 6*x^6 + 12*x^7 - 8*x^8) + 64*x^2 - 48* x^4 + 12*x^6 - x^8 + exp(x)*(48*x^3 - 96*x^4 - 24*x^5 + 48*x^6 + 3*x^7 - 6 *x^8)), x)
Time = 0.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 5.82 \[ \int \frac {320-432 x^2+204 x^4-41 x^6+3 x^8+e^{3 x} \left (5 x^3-30 x^4+57 x^5-22 x^6-36 x^7+24 x^8\right )+e^{2 x} \left (60 x^2-240 x^3+189 x^4+204 x^5-195 x^6-36 x^7+36 x^8\right )+\left (-4 x^2-3 x^4\right ) \log (4)+e^x \left (240 x-480 x^2-264 x^3+528 x^4+87 x^5-174 x^6-9 x^7+18 x^8+\left (x^3-4 x^4-4 x^5\right ) \log (4)\right )}{-64 x^2+48 x^4-12 x^6+x^8+e^x \left (-48 x^3+96 x^4+24 x^5-48 x^6-3 x^7+6 x^8\right )+e^{3 x} \left (-x^5+6 x^6-12 x^7+8 x^8\right )+e^{2 x} \left (-12 x^4+48 x^5-45 x^6-12 x^7+12 x^8\right )} \, dx=\frac {12 e^{2 x} x^{6}+68 e^{2 x} x^{5}-57 e^{2 x} x^{4}+5 e^{2 x} x^{2}+12 e^{x} x^{6}+74 e^{x} x^{5}-68 e^{x} x^{4}-306 e^{x} x^{3}+80 e^{x} x^{2}+40 e^{x} x +2 \,\mathrm {log}\left (2\right ) x^{2}+3 x^{6}+20 x^{5}-19 x^{4}-160 x^{3}+8 x^{2}+320 x +80}{x \left (4 e^{2 x} x^{4}-4 e^{2 x} x^{3}+e^{2 x} x^{2}+4 e^{x} x^{4}-2 e^{x} x^{3}-16 e^{x} x^{2}+8 e^{x} x +x^{4}-8 x^{2}+16\right )} \] Input:
int(((24*x^8-36*x^7-22*x^6+57*x^5-30*x^4+5*x^3)*exp(x)^3+(36*x^8-36*x^7-19 5*x^6+204*x^5+189*x^4-240*x^3+60*x^2)*exp(x)^2+(2*(-4*x^5-4*x^4+x^3)*log(2 )+18*x^8-9*x^7-174*x^6+87*x^5+528*x^4-264*x^3-480*x^2+240*x)*exp(x)+2*(-3* x^4-4*x^2)*log(2)+3*x^8-41*x^6+204*x^4-432*x^2+320)/((8*x^8-12*x^7+6*x^6-x ^5)*exp(x)^3+(12*x^8-12*x^7-45*x^6+48*x^5-12*x^4)*exp(x)^2+(6*x^8-3*x^7-48 *x^6+24*x^5+96*x^4-48*x^3)*exp(x)+x^8-12*x^6+48*x^4-64*x^2),x)
Output:
(12*e**(2*x)*x**6 + 68*e**(2*x)*x**5 - 57*e**(2*x)*x**4 + 5*e**(2*x)*x**2 + 12*e**x*x**6 + 74*e**x*x**5 - 68*e**x*x**4 - 306*e**x*x**3 + 80*e**x*x** 2 + 40*e**x*x + 2*log(2)*x**2 + 3*x**6 + 20*x**5 - 19*x**4 - 160*x**3 + 8* x**2 + 320*x + 80)/(x*(4*e**(2*x)*x**4 - 4*e**(2*x)*x**3 + e**(2*x)*x**2 + 4*e**x*x**4 - 2*e**x*x**3 - 16*e**x*x**2 + 8*e**x*x + x**4 - 8*x**2 + 16) )