Integrand size = 151, antiderivative size = 33 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=\left (e^4 \left (3+\frac {4}{x}\right )-x-\left (-e^x+x^2\right ) \left (x+x^2\right )\right )^2 \] Output:
((3+4/x)*exp(4)-(-exp(x)+x^2)*(x^2+x)-x)^2
Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(33)=66\).
Time = 0.32 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.97 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=e^{2 x} x^2 (1+x)^2+\frac {8 e^8 (2+3 x)}{x^2}+2 e^{4+x} \left (4+7 x+3 x^2\right )-2 e^4 x \left (3+4 x+7 x^2+3 x^3\right )+\left (x+x^3+x^4\right )^2-2 e^x x^2 \left (1+x+x^2+2 x^3+x^4\right ) \] Input:
Integrate[(E^8*(-32 - 24*x) + 2*x^4 + 8*x^6 + 10*x^7 + 6*x^8 + 14*x^9 + 8* x^10 + E^4*(-6*x^3 - 16*x^4 - 42*x^5 - 24*x^6) + E^(2*x)*(2*x^4 + 8*x^5 + 8*x^6 + 2*x^7) + E^x*(-4*x^4 - 8*x^5 - 10*x^6 - 22*x^7 - 16*x^8 - 2*x^9 + E^4*(22*x^3 + 26*x^4 + 6*x^5)))/x^3,x]
Output:
E^(2*x)*x^2*(1 + x)^2 + (8*E^8*(2 + 3*x))/x^2 + 2*E^(4 + x)*(4 + 7*x + 3*x ^2) - 2*E^4*x*(3 + 4*x + 7*x^2 + 3*x^3) + (x + x^3 + x^4)^2 - 2*E^x*x^2*(1 + x + x^2 + 2*x^3 + x^4)
Leaf count is larger than twice the leaf count of optimal. \(221\) vs. \(2(33)=66\).
Time = 0.91 (sec) , antiderivative size = 221, normalized size of antiderivative = 6.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {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.
| \(\displaystyle \int \frac {8 x^{10}+14 x^9+6 x^8+10 x^7+8 x^6+2 x^4+e^{2 x} \left (2 x^7+8 x^6+8 x^5+2 x^4\right )+e^4 \left (-24 x^6-42 x^5-16 x^4-6 x^3\right )+e^x \left (-2 x^9-16 x^8-22 x^7-10 x^6-8 x^5-4 x^4+e^4 \left (6 x^5+26 x^4+22 x^3\right )\right )+e^8 (-24 x-32)}{x^3} \, dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (2 e^{2 x} x (x+1) \left (x^2+3 x+1\right )+2 e^x \left (-x^6-8 x^5-11 x^4-5 x^3-4 \left (1-\frac {3 e^4}{4}\right ) x^2-2 \left (1-\frac {13 e^4}{2}\right ) x+11 e^4\right )+\frac {2 \left (4 x^{10}+7 x^9+3 x^8+5 x^7+4 \left (1-3 e^4\right ) x^6-21 e^4 x^5+\left (1-8 e^4\right ) x^4-3 e^4 x^3-12 e^8 x-16 e^8\right )}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x^8+2 x^7-2 e^x x^6+x^6-4 e^x x^5+2 x^5-2 e^x x^4+e^{2 x} x^4+2 \left (1-3 e^4\right ) x^4-2 e^x x^3+2 e^{2 x} x^3-14 e^4 x^3+6 e^x x^2+e^{2 x} x^2-2 \left (4-3 e^4\right ) e^x x^2+\left (1-8 e^4\right ) x^2+\frac {16 e^8}{x^2}-12 e^x x+4 \left (4-3 e^4\right ) e^x x-2 \left (2-13 e^4\right ) e^x x-6 e^4 x+12 e^x+22 e^{x+4}-4 \left (4-3 e^4\right ) e^x+2 \left (2-13 e^4\right ) e^x+\frac {24 e^8}{x}\) |
Input:
Int[(E^8*(-32 - 24*x) + 2*x^4 + 8*x^6 + 10*x^7 + 6*x^8 + 14*x^9 + 8*x^10 + E^4*(-6*x^3 - 16*x^4 - 42*x^5 - 24*x^6) + E^(2*x)*(2*x^4 + 8*x^5 + 8*x^6 + 2*x^7) + E^x*(-4*x^4 - 8*x^5 - 10*x^6 - 22*x^7 - 16*x^8 - 2*x^9 + E^4*(2 2*x^3 + 26*x^4 + 6*x^5)))/x^3,x]
Output:
12*E^x + 22*E^(4 + x) + 2*E^x*(2 - 13*E^4) - 4*E^x*(4 - 3*E^4) + (16*E^8)/ x^2 + (24*E^8)/x - 6*E^4*x - 12*E^x*x - 2*E^x*(2 - 13*E^4)*x + 4*E^x*(4 - 3*E^4)*x + 6*E^x*x^2 + E^(2*x)*x^2 + (1 - 8*E^4)*x^2 - 2*E^x*(4 - 3*E^4)*x ^2 - 14*E^4*x^3 - 2*E^x*x^3 + 2*E^(2*x)*x^3 - 2*E^x*x^4 + E^(2*x)*x^4 + 2* (1 - 3*E^4)*x^4 + 2*x^5 - 4*E^x*x^5 + x^6 - 2*E^x*x^6 + 2*x^7 + x^8
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. \(127\) vs. \(2(31)=62\).
Time = 7.61 (sec) , antiderivative size = 128, normalized size of antiderivative = 3.88
| method | result | size |
| risch | \(x^{8}+2 x^{7}+x^{6}-6 x^{4} {\mathrm e}^{4}+2 x^{5}-14 x^{3} {\mathrm e}^{4}+2 x^{4}-8 x^{2} {\mathrm e}^{4}-6 x \,{\mathrm e}^{4}+x^{2}+\frac {24 \,{\mathrm e}^{8} x +16 \,{\mathrm e}^{8}}{x^{2}}+\left (x^{4}+2 x^{3}+x^{2}\right ) {\mathrm e}^{2 x}+\left (-2 x^{6}-4 x^{5}-2 x^{4}+6 x^{2} {\mathrm e}^{4}-2 x^{3}+14 x \,{\mathrm e}^{4}-2 x^{2}+8 \,{\mathrm e}^{4}\right ) {\mathrm e}^{x}\) | \(128\) |
| norman | \(\frac {x^{8}+x^{10}+{\mathrm e}^{2 x} x^{6}+\left (2-6 \,{\mathrm e}^{4}\right ) x^{6}+\left (-8 \,{\mathrm e}^{4}+1\right ) x^{4}+x^{4} {\mathrm e}^{2 x}+\left (-2+6 \,{\mathrm e}^{4}\right ) x^{4} {\mathrm e}^{x}+2 x^{7}+2 x^{9}+16 \,{\mathrm e}^{8}+24 \,{\mathrm e}^{8} x -6 x^{3} {\mathrm e}^{4}-2 x^{5} {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x} x^{5}-14 \,{\mathrm e}^{4} x^{5}-2 \,{\mathrm e}^{x} x^{6}-4 \,{\mathrm e}^{x} x^{7}-2 \,{\mathrm e}^{x} x^{8}+8 x^{2} {\mathrm e}^{4} {\mathrm e}^{x}+14 \,{\mathrm e}^{4} {\mathrm e}^{x} x^{3}}{x^{2}}\) | \(152\) |
| parallelrisch | \(\frac {x^{10}-2 \,{\mathrm e}^{x} x^{8}+2 x^{9}-4 \,{\mathrm e}^{x} x^{7}+x^{8}-6 x^{6} {\mathrm e}^{4}-2 \,{\mathrm e}^{x} x^{6}+{\mathrm e}^{2 x} x^{6}+2 x^{7}+6 \,{\mathrm e}^{4} {\mathrm e}^{x} x^{4}-14 \,{\mathrm e}^{4} x^{5}-2 x^{5} {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x} x^{5}+2 x^{6}+14 \,{\mathrm e}^{4} {\mathrm e}^{x} x^{3}-8 x^{4} {\mathrm e}^{4}-2 \,{\mathrm e}^{x} x^{4}+x^{4} {\mathrm e}^{2 x}+8 x^{2} {\mathrm e}^{4} {\mathrm e}^{x}-6 x^{3} {\mathrm e}^{4}+x^{4}+24 \,{\mathrm e}^{8} x +16 \,{\mathrm e}^{8}}{x^{2}}\) | \(158\) |
| parts | \(-2 \,{\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x^{3}-2 \,{\mathrm e}^{x} x^{4}-4 x^{5} {\mathrm e}^{x}+22 \,{\mathrm e}^{4} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{6}+26 \,{\mathrm e}^{4} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+6 \,{\mathrm e}^{4} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )+x^{8}+2 x^{7}+x^{6}-6 x^{4} {\mathrm e}^{4}+2 x^{5}-14 x^{3} {\mathrm e}^{4}+2 x^{4}-8 x^{2} {\mathrm e}^{4}-6 x \,{\mathrm e}^{4}+x^{2}+\frac {24 \,{\mathrm e}^{8}}{x}+\frac {16 \,{\mathrm e}^{8}}{x^{2}}+x^{4} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} x^{3}+{\mathrm e}^{2 x} x^{2}\) | \(165\) |
| default | \(-2 \,{\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x^{3}-2 \,{\mathrm e}^{x} x^{4}-4 x^{5} {\mathrm e}^{x}+22 \,{\mathrm e}^{4} {\mathrm e}^{x}-2 \,{\mathrm e}^{x} x^{6}+26 \,{\mathrm e}^{4} \left ({\mathrm e}^{x} x -{\mathrm e}^{x}\right )+6 \,{\mathrm e}^{4} \left ({\mathrm e}^{x} x^{2}-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}\right )+x^{8}+2 x^{7}+x^{6}-6 x^{4} {\mathrm e}^{4}+2 x^{5}-14 x^{3} {\mathrm e}^{4}+2 x^{4}-8 x^{2} {\mathrm e}^{4}-6 x \,{\mathrm e}^{4}+x^{2}+\frac {24 \,{\mathrm e}^{8}}{x}+\frac {16 \,{\mathrm e}^{8}}{x^{2}}+x^{4} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} x^{3}+{\mathrm e}^{2 x} x^{2}\) | \(169\) |
Input:
int(((2*x^7+8*x^6+8*x^5+2*x^4)*exp(x)^2+((6*x^5+26*x^4+22*x^3)*exp(4)-2*x^ 9-16*x^8-22*x^7-10*x^6-8*x^5-4*x^4)*exp(x)+(-24*x-32)*exp(4)^2+(-24*x^6-42 *x^5-16*x^4-6*x^3)*exp(4)+8*x^10+14*x^9+6*x^8+10*x^7+8*x^6+2*x^4)/x^3,x,me thod=_RETURNVERBOSE)
Output:
x^8+2*x^7+x^6-6*x^4*exp(4)+2*x^5-14*x^3*exp(4)+2*x^4-8*x^2*exp(4)-6*x*exp( 4)+x^2+(24*exp(8)*x+16*exp(8))/x^2+(x^4+2*x^3+x^2)*exp(2*x)+(-2*x^6-4*x^5- 2*x^4+6*x^2*exp(4)-2*x^3+14*x*exp(4)-2*x^2+8*exp(4))*exp(x)
Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 122, normalized size of antiderivative = 3.70 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=\frac {x^{10} + 2 \, x^{9} + x^{8} + 2 \, x^{7} + 2 \, x^{6} + x^{4} + 8 \, {\left (3 \, x + 2\right )} e^{8} - 2 \, {\left (3 \, x^{6} + 7 \, x^{5} + 4 \, x^{4} + 3 \, x^{3}\right )} e^{4} + {\left (x^{6} + 2 \, x^{5} + x^{4}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{8} + 2 \, x^{7} + x^{6} + x^{5} + x^{4} - {\left (3 \, x^{4} + 7 \, x^{3} + 4 \, x^{2}\right )} e^{4}\right )} e^{x}}{x^{2}} \] Input:
integrate(((2*x^7+8*x^6+8*x^5+2*x^4)*exp(x)^2+((6*x^5+26*x^4+22*x^3)*exp(4 )-2*x^9-16*x^8-22*x^7-10*x^6-8*x^5-4*x^4)*exp(x)+(-24*x-32)*exp(4)^2+(-24* x^6-42*x^5-16*x^4-6*x^3)*exp(4)+8*x^10+14*x^9+6*x^8+10*x^7+8*x^6+2*x^4)/x^ 3,x, algorithm="fricas")
Output:
(x^10 + 2*x^9 + x^8 + 2*x^7 + 2*x^6 + x^4 + 8*(3*x + 2)*e^8 - 2*(3*x^6 + 7 *x^5 + 4*x^4 + 3*x^3)*e^4 + (x^6 + 2*x^5 + x^4)*e^(2*x) - 2*(x^8 + 2*x^7 + x^6 + x^5 + x^4 - (3*x^4 + 7*x^3 + 4*x^2)*e^4)*e^x)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (22) = 44\).
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.03 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=x^{8} + 2 x^{7} + x^{6} + 2 x^{5} + x^{4} \cdot \left (2 - 6 e^{4}\right ) - 14 x^{3} e^{4} + x^{2} \cdot \left (1 - 8 e^{4}\right ) - 6 x e^{4} + \left (x^{4} + 2 x^{3} + x^{2}\right ) e^{2 x} + \left (- 2 x^{6} - 4 x^{5} - 2 x^{4} - 2 x^{3} - 2 x^{2} + 6 x^{2} e^{4} + 14 x e^{4} + 8 e^{4}\right ) e^{x} + \frac {24 x e^{8} + 16 e^{8}}{x^{2}} \] Input:
integrate(((2*x**7+8*x**6+8*x**5+2*x**4)*exp(x)**2+((6*x**5+26*x**4+22*x** 3)*exp(4)-2*x**9-16*x**8-22*x**7-10*x**6-8*x**5-4*x**4)*exp(x)+(-24*x-32)* exp(4)**2+(-24*x**6-42*x**5-16*x**4-6*x**3)*exp(4)+8*x**10+14*x**9+6*x**8+ 10*x**7+8*x**6+2*x**4)/x**3,x)
Output:
x**8 + 2*x**7 + x**6 + 2*x**5 + x**4*(2 - 6*exp(4)) - 14*x**3*exp(4) + x** 2*(1 - 8*exp(4)) - 6*x*exp(4) + (x**4 + 2*x**3 + x**2)*exp(2*x) + (-2*x**6 - 4*x**5 - 2*x**4 - 2*x**3 - 2*x**2 + 6*x**2*exp(4) + 14*x*exp(4) + 8*exp (4))*exp(x) + (24*x*exp(8) + 16*exp(8))/x**2
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (29) = 58\).
Time = 0.05 (sec) , antiderivative size = 294, normalized size of antiderivative = 8.91 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{5} - 6 \, x^{4} e^{4} + 2 \, x^{4} - 14 \, x^{3} e^{4} - 8 \, x^{2} e^{4} + x^{2} - 6 \, x e^{4} + \frac {1}{2} \, {\left (2 \, x^{4} - 4 \, x^{3} + 6 \, x^{2} - 6 \, x + 3\right )} e^{\left (2 \, x\right )} + {\left (4 \, x^{3} - 6 \, x^{2} + 6 \, x - 3\right )} e^{\left (2 \, x\right )} + 2 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{6} - 6 \, x^{5} + 30 \, x^{4} - 120 \, x^{3} + 360 \, x^{2} - 720 \, x + 720\right )} e^{x} - 16 \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} - 22 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} - 10 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + 6 \, {\left (x^{2} e^{4} - 2 \, x e^{4} + 2 \, e^{4}\right )} e^{x} - 8 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + 26 \, {\left (x e^{4} - e^{4}\right )} e^{x} - 4 \, {\left (x - 1\right )} e^{x} + \frac {24 \, e^{8}}{x} + \frac {16 \, e^{8}}{x^{2}} + 22 \, e^{\left (x + 4\right )} \] Input:
integrate(((2*x^7+8*x^6+8*x^5+2*x^4)*exp(x)^2+((6*x^5+26*x^4+22*x^3)*exp(4 )-2*x^9-16*x^8-22*x^7-10*x^6-8*x^5-4*x^4)*exp(x)+(-24*x-32)*exp(4)^2+(-24* x^6-42*x^5-16*x^4-6*x^3)*exp(4)+8*x^10+14*x^9+6*x^8+10*x^7+8*x^6+2*x^4)/x^ 3,x, algorithm="maxima")
Output:
x^8 + 2*x^7 + x^6 + 2*x^5 - 6*x^4*e^4 + 2*x^4 - 14*x^3*e^4 - 8*x^2*e^4 + x ^2 - 6*x*e^4 + 1/2*(2*x^4 - 4*x^3 + 6*x^2 - 6*x + 3)*e^(2*x) + (4*x^3 - 6* x^2 + 6*x - 3)*e^(2*x) + 2*(2*x^2 - 2*x + 1)*e^(2*x) + 1/2*(2*x - 1)*e^(2* x) - 2*(x^6 - 6*x^5 + 30*x^4 - 120*x^3 + 360*x^2 - 720*x + 720)*e^x - 16*( x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x - 22*(x^4 - 4*x^3 + 12*x^ 2 - 24*x + 24)*e^x - 10*(x^3 - 3*x^2 + 6*x - 6)*e^x + 6*(x^2*e^4 - 2*x*e^4 + 2*e^4)*e^x - 8*(x^2 - 2*x + 2)*e^x + 26*(x*e^4 - e^4)*e^x - 4*(x - 1)*e ^x + 24*e^8/x + 16*e^8/x^2 + 22*e^(x + 4)
Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (29) = 58\).
Time = 0.12 (sec) , antiderivative size = 153, normalized size of antiderivative = 4.64 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=\frac {x^{10} + 2 \, x^{9} - 2 \, x^{8} e^{x} + x^{8} - 4 \, x^{7} e^{x} + 2 \, x^{7} - 6 \, x^{6} e^{4} + x^{6} e^{\left (2 \, x\right )} - 2 \, x^{6} e^{x} + 2 \, x^{6} - 14 \, x^{5} e^{4} + 2 \, x^{5} e^{\left (2 \, x\right )} - 2 \, x^{5} e^{x} - 8 \, x^{4} e^{4} + x^{4} e^{\left (2 \, x\right )} + 6 \, x^{4} e^{\left (x + 4\right )} - 2 \, x^{4} e^{x} + x^{4} - 6 \, x^{3} e^{4} + 14 \, x^{3} e^{\left (x + 4\right )} + 8 \, x^{2} e^{\left (x + 4\right )} + 24 \, x e^{8} + 16 \, e^{8}}{x^{2}} \] Input:
integrate(((2*x^7+8*x^6+8*x^5+2*x^4)*exp(x)^2+((6*x^5+26*x^4+22*x^3)*exp(4 )-2*x^9-16*x^8-22*x^7-10*x^6-8*x^5-4*x^4)*exp(x)+(-24*x-32)*exp(4)^2+(-24* x^6-42*x^5-16*x^4-6*x^3)*exp(4)+8*x^10+14*x^9+6*x^8+10*x^7+8*x^6+2*x^4)/x^ 3,x, algorithm="giac")
Output:
(x^10 + 2*x^9 - 2*x^8*e^x + x^8 - 4*x^7*e^x + 2*x^7 - 6*x^6*e^4 + x^6*e^(2 *x) - 2*x^6*e^x + 2*x^6 - 14*x^5*e^4 + 2*x^5*e^(2*x) - 2*x^5*e^x - 8*x^4*e ^4 + x^4*e^(2*x) + 6*x^4*e^(x + 4) - 2*x^4*e^x + x^4 - 6*x^3*e^4 + 14*x^3* e^(x + 4) + 8*x^2*e^(x + 4) + 24*x*e^8 + 16*e^8)/x^2
Time = 7.37 (sec) , antiderivative size = 143, normalized size of antiderivative = 4.33 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=8\,{\mathrm {e}}^{x+4}+14\,x\,{\mathrm {e}}^{x+4}-2\,x^3\,{\mathrm {e}}^x-2\,x^4\,{\mathrm {e}}^x-4\,x^5\,{\mathrm {e}}^x-2\,x^6\,{\mathrm {e}}^x-6\,x\,{\mathrm {e}}^4-x^2\,\left (8\,{\mathrm {e}}^4-1\right )-x^4\,\left (6\,{\mathrm {e}}^4-2\right )+x^2\,{\mathrm {e}}^{2\,x}+2\,x^3\,{\mathrm {e}}^{2\,x}+x^4\,{\mathrm {e}}^{2\,x}-14\,x^3\,{\mathrm {e}}^4+\frac {24\,{\mathrm {e}}^8}{x}+\frac {16\,{\mathrm {e}}^8}{x^2}+2\,x^5+x^6+2\,x^7+x^8+x^2\,{\mathrm {e}}^x\,\left (6\,{\mathrm {e}}^4-2\right ) \] Input:
int((exp(2*x)*(2*x^4 + 8*x^5 + 8*x^6 + 2*x^7) - exp(x)*(4*x^4 - exp(4)*(22 *x^3 + 26*x^4 + 6*x^5) + 8*x^5 + 10*x^6 + 22*x^7 + 16*x^8 + 2*x^9) + 2*x^4 + 8*x^6 + 10*x^7 + 6*x^8 + 14*x^9 + 8*x^10 - exp(8)*(24*x + 32) - exp(4)* (6*x^3 + 16*x^4 + 42*x^5 + 24*x^6))/x^3,x)
Output:
8*exp(x + 4) + 14*x*exp(x + 4) - 2*x^3*exp(x) - 2*x^4*exp(x) - 4*x^5*exp(x ) - 2*x^6*exp(x) - 6*x*exp(4) - x^2*(8*exp(4) - 1) - x^4*(6*exp(4) - 2) + x^2*exp(2*x) + 2*x^3*exp(2*x) + x^4*exp(2*x) - 14*x^3*exp(4) + (24*exp(8)) /x + (16*exp(8))/x^2 + 2*x^5 + x^6 + 2*x^7 + x^8 + x^2*exp(x)*(6*exp(4) - 2)
Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 5.24 \[ \int \frac {e^8 (-32-24 x)+2 x^4+8 x^6+10 x^7+6 x^8+14 x^9+8 x^{10}+e^4 \left (-6 x^3-16 x^4-42 x^5-24 x^6\right )+e^{2 x} \left (2 x^4+8 x^5+8 x^6+2 x^7\right )+e^x \left (-4 x^4-8 x^5-10 x^6-22 x^7-16 x^8-2 x^9+e^4 \left (22 x^3+26 x^4+6 x^5\right )\right )}{x^3} \, dx=\frac {e^{2 x} x^{6}+2 e^{2 x} x^{5}+e^{2 x} x^{4}+6 e^{x} e^{4} x^{4}+14 e^{x} e^{4} x^{3}+8 e^{x} e^{4} x^{2}-2 e^{x} x^{8}-4 e^{x} x^{7}-2 e^{x} x^{6}-2 e^{x} x^{5}-2 e^{x} x^{4}+24 e^{8} x +16 e^{8}-6 e^{4} x^{6}-14 e^{4} x^{5}-8 e^{4} x^{4}-6 e^{4} x^{3}+x^{10}+2 x^{9}+x^{8}+2 x^{7}+2 x^{6}+x^{4}}{x^{2}} \] Input:
int(((2*x^7+8*x^6+8*x^5+2*x^4)*exp(x)^2+((6*x^5+26*x^4+22*x^3)*exp(4)-2*x^ 9-16*x^8-22*x^7-10*x^6-8*x^5-4*x^4)*exp(x)+(-24*x-32)*exp(4)^2+(-24*x^6-42 *x^5-16*x^4-6*x^3)*exp(4)+8*x^10+14*x^9+6*x^8+10*x^7+8*x^6+2*x^4)/x^3,x)
Output:
(e**(2*x)*x**6 + 2*e**(2*x)*x**5 + e**(2*x)*x**4 + 6*e**x*e**4*x**4 + 14*e **x*e**4*x**3 + 8*e**x*e**4*x**2 - 2*e**x*x**8 - 4*e**x*x**7 - 2*e**x*x**6 - 2*e**x*x**5 - 2*e**x*x**4 + 24*e**8*x + 16*e**8 - 6*e**4*x**6 - 14*e**4 *x**5 - 8*e**4*x**4 - 6*e**4*x**3 + x**10 + 2*x**9 + x**8 + 2*x**7 + 2*x** 6 + x**4)/x**2