Integrand size = 374, antiderivative size = 31 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=x \left (x-\frac {5 e^x x}{2 \left (x^2+\left (-x+\left (e^x+x\right )^2\right )^2\right )}\right ) \] Output:
(x-5/2*exp(x)*x/(((exp(x)+x)^2-x)^2+x^2))*x
Leaf count is larger than twice the leaf count of optimal. \(66\) vs. \(2(31)=62\).
Time = 10.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.13 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=\frac {1}{2} x^2 \left (2-\frac {5 e^x}{e^{4 x}+4 e^{3 x} x+4 e^x (-1+x) x^2+2 e^{2 x} x (-1+3 x)+x^2 \left (2-2 x+x^2\right )}\right ) \] Input:
Integrate[(4*E^(8*x)*x + 32*E^(7*x)*x^2 + 16*x^5 - 32*x^6 + 32*x^7 - 16*x^ 8 + 4*x^9 + E^(6*x)*(-16*x^2 + 112*x^3) + E^(5*x)*(-10*x + 15*x^2 - 96*x^3 + 224*x^4) + E^(4*x)*(-20*x^2 + 72*x^3 - 240*x^4 + 280*x^5) + E^(3*x)*(10 *x^2 - 10*x^3 + 158*x^4 - 320*x^5 + 224*x^6) + E^(2*x)*(-12*x^4 + 192*x^5 - 240*x^6 + 112*x^7) + E^x*(-20*x^4 - 44*x^5 + 123*x^6 - 96*x^7 + 32*x^8)) /(2*E^(8*x) + 16*E^(7*x)*x + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^7 + 2*x^8 + E^( 6*x)*(-8*x + 56*x^2) + E^(5*x)*(-48*x^2 + 112*x^3) + E^(4*x)*(16*x^2 - 120 *x^3 + 140*x^4) + E^(3*x)*(64*x^3 - 160*x^4 + 112*x^5) + E^(2*x)*(-16*x^3 + 96*x^4 - 120*x^5 + 56*x^6) + E^x*(-32*x^4 + 64*x^5 - 48*x^6 + 16*x^7)),x ]
Output:
(x^2*(2 - (5*E^x)/(E^(4*x) + 4*E^(3*x)*x + 4*E^x*(-1 + x)*x^2 + 2*E^(2*x)* x*(-1 + 3*x) + x^2*(2 - 2*x + x^2))))/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.
| \(\displaystyle \int \frac {4 x^9-16 x^8+32 x^7-32 x^6+16 x^5+32 e^{7 x} x^2+e^{6 x} \left (112 x^3-16 x^2\right )+e^{5 x} \left (224 x^4-96 x^3+15 x^2-10 x\right )+e^{2 x} \left (112 x^7-240 x^6+192 x^5-12 x^4\right )+e^{4 x} \left (280 x^5-240 x^4+72 x^3-20 x^2\right )+e^x \left (32 x^8-96 x^7+123 x^6-44 x^5-20 x^4\right )+e^{3 x} \left (224 x^6-320 x^5+158 x^4-10 x^3+10 x^2\right )+4 e^{8 x} x}{2 x^8-8 x^7+16 x^6-16 x^5+8 x^4+e^{6 x} \left (56 x^2-8 x\right )+e^{5 x} \left (112 x^3-48 x^2\right )+e^{3 x} \left (112 x^5-160 x^4+64 x^3\right )+e^{4 x} \left (140 x^4-120 x^3+16 x^2\right )+e^x \left (16 x^7-48 x^6+64 x^5-32 x^4\right )+e^{2 x} \left (56 x^6-120 x^5+96 x^4-16 x^3\right )+16 e^{7 x} x+2 e^{8 x}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {4 x^9-16 x^8+32 x^7-32 x^6+16 x^5+32 e^{7 x} x^2+e^{6 x} \left (112 x^3-16 x^2\right )+e^{5 x} \left (224 x^4-96 x^3+15 x^2-10 x\right )+e^{2 x} \left (112 x^7-240 x^6+192 x^5-12 x^4\right )+e^{4 x} \left (280 x^5-240 x^4+72 x^3-20 x^2\right )+e^x \left (32 x^8-96 x^7+123 x^6-44 x^5-20 x^4\right )+e^{3 x} \left (224 x^6-320 x^5+158 x^4-10 x^3+10 x^2\right )+4 e^{8 x} x}{2 \left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int \frac {4 x^9-16 x^8+32 x^7-32 x^6+16 x^5+32 e^{7 x} x^2+4 e^{8 x} x-16 e^{6 x} \left (x^2-7 x^3\right )-e^{5 x} \left (-224 x^4+96 x^3-15 x^2+10 x\right )-4 e^{4 x} \left (-70 x^5+60 x^4-18 x^3+5 x^2\right )+2 e^{3 x} \left (112 x^6-160 x^5+79 x^4-5 x^3+5 x^2\right )-4 e^{2 x} \left (-28 x^7+60 x^6-48 x^5+3 x^4\right )-e^x \left (-32 x^8+96 x^7-123 x^6+44 x^5+20 x^4\right )}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {10 \left (2 x^5+6 e^x x^4-6 x^4-10 e^x x^3+6 e^{2 x} x^3+8 x^3+e^x x^2-4 e^{2 x} x^2+2 e^{3 x} x^2-4 x^2+2 e^x x-e^{3 x}\right ) x^2}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}-\frac {5 \left (4 x^2-3 e^x x-4 x+2 e^x\right ) x}{x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}}+4 x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (2 x^2-10 \int \frac {e^{3 x} x^2}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+20 \int \frac {e^x x^3}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx-40 \int \frac {x^4}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+10 \int \frac {e^x x^4}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx-40 \int \frac {e^{2 x} x^4}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+20 \int \frac {e^{3 x} x^4}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+80 \int \frac {x^5}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx-100 \int \frac {e^x x^5}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+60 \int \frac {e^{2 x} x^5}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx-60 \int \frac {x^6}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+60 \int \frac {e^x x^6}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx+20 \int \frac {x^7}{\left (x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}\right )^2}dx-10 \int \frac {e^x x}{x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}}dx+20 \int \frac {x^2}{x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}}dx+15 \int \frac {e^x x^2}{x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}}dx-20 \int \frac {x^3}{x^4+4 e^x x^3-2 x^3-4 e^x x^2+6 e^{2 x} x^2+2 x^2-2 e^{2 x} x+4 e^{3 x} x+e^{4 x}}dx\right )\) |
Input:
Int[(4*E^(8*x)*x + 32*E^(7*x)*x^2 + 16*x^5 - 32*x^6 + 32*x^7 - 16*x^8 + 4* x^9 + E^(6*x)*(-16*x^2 + 112*x^3) + E^(5*x)*(-10*x + 15*x^2 - 96*x^3 + 224 *x^4) + E^(4*x)*(-20*x^2 + 72*x^3 - 240*x^4 + 280*x^5) + E^(3*x)*(10*x^2 - 10*x^3 + 158*x^4 - 320*x^5 + 224*x^6) + E^(2*x)*(-12*x^4 + 192*x^5 - 240* x^6 + 112*x^7) + E^x*(-20*x^4 - 44*x^5 + 123*x^6 - 96*x^7 + 32*x^8))/(2*E^ (8*x) + 16*E^(7*x)*x + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^7 + 2*x^8 + E^(6*x)*( -8*x + 56*x^2) + E^(5*x)*(-48*x^2 + 112*x^3) + E^(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + E^(3*x)*(64*x^3 - 160*x^4 + 112*x^5) + E^(2*x)*(-16*x^3 + 96*x ^4 - 120*x^5 + 56*x^6) + E^x*(-32*x^4 + 64*x^5 - 48*x^6 + 16*x^7)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(27)=54\).
Time = 3.96 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.23
| method | result | size |
| risch | \(x^{2}-\frac {5 x^{2} {\mathrm e}^{x}}{2 \left ({\mathrm e}^{4 x}+4 x \,{\mathrm e}^{3 x}+6 \,{\mathrm e}^{2 x} x^{2}+4 \,{\mathrm e}^{x} x^{3}+x^{4}-2 x \,{\mathrm e}^{2 x}-4 \,{\mathrm e}^{x} x^{2}-2 x^{3}+2 x^{2}\right )}\) | \(69\) |
| parallelrisch | \(\frac {2 x^{6}+8 x^{5} {\mathrm e}^{x}+12 x^{4} {\mathrm e}^{2 x}+8 x^{3} {\mathrm e}^{3 x}+2 x^{2} {\mathrm e}^{4 x}-4 x^{5}-8 \,{\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{2 x} x^{3}+4 x^{4}-5 \,{\mathrm e}^{x} x^{2}}{2 \,{\mathrm e}^{4 x}+8 x \,{\mathrm e}^{3 x}+12 \,{\mathrm e}^{2 x} x^{2}+8 \,{\mathrm e}^{x} x^{3}+2 x^{4}-4 x \,{\mathrm e}^{2 x}-8 \,{\mathrm e}^{x} x^{2}-4 x^{3}+4 x^{2}}\) | \(133\) |
Input:
int((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4-96*x^ 3+15*x^2-10*x)*exp(x)^5+(280*x^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(224*x^6- 320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-12*x^4)*e xp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32*x^7-3 2*x^6+16*x^5)/(2*exp(x)^8+16*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x^3-48* x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4+64*x^3)*e xp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^5-32*x ^4)*exp(x)+2*x^8-8*x^7+16*x^6-16*x^5+8*x^4),x,method=_RETURNVERBOSE)
Output:
x^2-5/2*x^2*exp(x)/(exp(x)^4+4*x*exp(x)^3+6*exp(x)^2*x^2+4*exp(x)*x^3+x^4- 2*x*exp(x)^2-4*exp(x)*x^2-2*x^3+2*x^2)
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (30) = 60\).
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.10 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=\frac {2 \, x^{6} - 4 \, x^{5} + 4 \, x^{4} + 8 \, x^{3} e^{\left (3 \, x\right )} + 2 \, x^{2} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{4} - x^{3}\right )} e^{\left (2 \, x\right )} + {\left (8 \, x^{5} - 8 \, x^{4} - 5 \, x^{2}\right )} e^{x}}{2 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} + 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} - x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )} e^{x} + e^{\left (4 \, x\right )}\right )}} \] Input:
integrate((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4 -96*x^3+15*x^2-10*x)*exp(x)^5+(280*x^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(22 4*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-12* x^4)*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32 *x^7-32*x^6+16*x^5)/(2*exp(x)^8+16*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x ^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4+64* x^3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^ 5-32*x^4)*exp(x)+2*x^8-8*x^7+16*x^6-16*x^5+8*x^4),x, algorithm="fricas")
Output:
1/2*(2*x^6 - 4*x^5 + 4*x^4 + 8*x^3*e^(3*x) + 2*x^2*e^(4*x) + 4*(3*x^4 - x^ 3)*e^(2*x) + (8*x^5 - 8*x^4 - 5*x^2)*e^x)/(x^4 - 2*x^3 + 2*x^2 + 4*x*e^(3* x) + 2*(3*x^2 - x)*e^(2*x) + 4*(x^3 - x^2)*e^x + e^(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).
Time = 0.24 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.19 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=x^{2} - \frac {5 x^{2} e^{x}}{2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x e^{3 x} + \left (12 x^{2} - 4 x\right ) e^{2 x} + \left (8 x^{3} - 8 x^{2}\right ) e^{x} + 2 e^{4 x}} \] Input:
integrate((4*x*exp(x)**8+32*x**2*exp(x)**7+(112*x**3-16*x**2)*exp(x)**6+(2 24*x**4-96*x**3+15*x**2-10*x)*exp(x)**5+(280*x**5-240*x**4+72*x**3-20*x**2 )*exp(x)**4+(224*x**6-320*x**5+158*x**4-10*x**3+10*x**2)*exp(x)**3+(112*x* *7-240*x**6+192*x**5-12*x**4)*exp(x)**2+(32*x**8-96*x**7+123*x**6-44*x**5- 20*x**4)*exp(x)+4*x**9-16*x**8+32*x**7-32*x**6+16*x**5)/(2*exp(x)**8+16*x* exp(x)**7+(56*x**2-8*x)*exp(x)**6+(112*x**3-48*x**2)*exp(x)**5+(140*x**4-1 20*x**3+16*x**2)*exp(x)**4+(112*x**5-160*x**4+64*x**3)*exp(x)**3+(56*x**6- 120*x**5+96*x**4-16*x**3)*exp(x)**2+(16*x**7-48*x**6+64*x**5-32*x**4)*exp( x)+2*x**8-8*x**7+16*x**6-16*x**5+8*x**4),x)
Output:
x**2 - 5*x**2*exp(x)/(2*x**4 - 4*x**3 + 4*x**2 + 8*x*exp(3*x) + (12*x**2 - 4*x)*exp(2*x) + (8*x**3 - 8*x**2)*exp(x) + 2*exp(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (30) = 60\).
Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 4.10 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=\frac {2 \, x^{6} - 4 \, x^{5} + 4 \, x^{4} + 8 \, x^{3} e^{\left (3 \, x\right )} + 2 \, x^{2} e^{\left (4 \, x\right )} + 4 \, {\left (3 \, x^{4} - x^{3}\right )} e^{\left (2 \, x\right )} + {\left (8 \, x^{5} - 8 \, x^{4} - 5 \, x^{2}\right )} e^{x}}{2 \, {\left (x^{4} - 2 \, x^{3} + 2 \, x^{2} + 4 \, x e^{\left (3 \, x\right )} + 2 \, {\left (3 \, x^{2} - x\right )} e^{\left (2 \, x\right )} + 4 \, {\left (x^{3} - x^{2}\right )} e^{x} + e^{\left (4 \, x\right )}\right )}} \] Input:
integrate((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4 -96*x^3+15*x^2-10*x)*exp(x)^5+(280*x^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(22 4*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-12* x^4)*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32 *x^7-32*x^6+16*x^5)/(2*exp(x)^8+16*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x ^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4+64* x^3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^ 5-32*x^4)*exp(x)+2*x^8-8*x^7+16*x^6-16*x^5+8*x^4),x, algorithm="maxima")
Output:
1/2*(2*x^6 - 4*x^5 + 4*x^4 + 8*x^3*e^(3*x) + 2*x^2*e^(4*x) + 4*(3*x^4 - x^ 3)*e^(2*x) + (8*x^5 - 8*x^4 - 5*x^2)*e^x)/(x^4 - 2*x^3 + 2*x^2 + 4*x*e^(3* x) + 2*(3*x^2 - x)*e^(2*x) + 4*(x^3 - x^2)*e^x + e^(4*x))
Leaf count of result is larger than twice the leaf count of optimal. 128 vs. \(2 (30) = 60\).
Time = 3.61 (sec) , antiderivative size = 128, normalized size of antiderivative = 4.13 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=\frac {x^{6} + 4 \, x^{5} e^{x} - 2 \, x^{5} + 6 \, x^{4} e^{\left (2 \, x\right )} - 4 \, x^{4} e^{x} + 2 \, x^{4} + 4 \, x^{3} e^{\left (3 \, x\right )} - 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} e^{\left (4 \, x\right )} - 5 \, x^{2} e^{x}}{x^{4} + 4 \, x^{3} e^{x} - 2 \, x^{3} + 6 \, x^{2} e^{\left (2 \, x\right )} - 4 \, x^{2} e^{x} + 2 \, x^{2} + 4 \, x e^{\left (3 \, x\right )} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (4 \, x\right )}} \] Input:
integrate((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4 -96*x^3+15*x^2-10*x)*exp(x)^5+(280*x^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(22 4*x^6-320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-12* x^4)*exp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32 *x^7-32*x^6+16*x^5)/(2*exp(x)^8+16*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x ^3-48*x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4+64* x^3)*exp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^ 5-32*x^4)*exp(x)+2*x^8-8*x^7+16*x^6-16*x^5+8*x^4),x, algorithm="giac")
Output:
(x^6 + 4*x^5*e^x - 2*x^5 + 6*x^4*e^(2*x) - 4*x^4*e^x + 2*x^4 + 4*x^3*e^(3* x) - 2*x^3*e^(2*x) + x^2*e^(4*x) - 5*x^2*e^x)/(x^4 + 4*x^3*e^x - 2*x^3 + 6 *x^2*e^(2*x) - 4*x^2*e^x + 2*x^2 + 4*x*e^(3*x) - 2*x*e^(2*x) + e^(4*x))
Timed out. \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=-\int \frac {{\mathrm {e}}^{6\,x}\,\left (16\,x^2-112\,x^3\right )-4\,x\,{\mathrm {e}}^{8\,x}-{\mathrm {e}}^{3\,x}\,\left (224\,x^6-320\,x^5+158\,x^4-10\,x^3+10\,x^2\right )-32\,x^2\,{\mathrm {e}}^{7\,x}+{\mathrm {e}}^{5\,x}\,\left (-224\,x^4+96\,x^3-15\,x^2+10\,x\right )+{\mathrm {e}}^x\,\left (-32\,x^8+96\,x^7-123\,x^6+44\,x^5+20\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (-112\,x^7+240\,x^6-192\,x^5+12\,x^4\right )+{\mathrm {e}}^{4\,x}\,\left (-280\,x^5+240\,x^4-72\,x^3+20\,x^2\right )-16\,x^5+32\,x^6-32\,x^7+16\,x^8-4\,x^9}{2\,{\mathrm {e}}^{8\,x}-{\mathrm {e}}^{6\,x}\,\left (8\,x-56\,x^2\right )+16\,x\,{\mathrm {e}}^{7\,x}-{\mathrm {e}}^{5\,x}\,\left (48\,x^2-112\,x^3\right )-{\mathrm {e}}^x\,\left (-16\,x^7+48\,x^6-64\,x^5+32\,x^4\right )+{\mathrm {e}}^{4\,x}\,\left (140\,x^4-120\,x^3+16\,x^2\right )+{\mathrm {e}}^{3\,x}\,\left (112\,x^5-160\,x^4+64\,x^3\right )-{\mathrm {e}}^{2\,x}\,\left (-56\,x^6+120\,x^5-96\,x^4+16\,x^3\right )+8\,x^4-16\,x^5+16\,x^6-8\,x^7+2\,x^8} \,d x \] Input:
int(-(exp(6*x)*(16*x^2 - 112*x^3) - 4*x*exp(8*x) - exp(3*x)*(10*x^2 - 10*x ^3 + 158*x^4 - 320*x^5 + 224*x^6) - 32*x^2*exp(7*x) + exp(5*x)*(10*x - 15* x^2 + 96*x^3 - 224*x^4) + exp(x)*(20*x^4 + 44*x^5 - 123*x^6 + 96*x^7 - 32* x^8) + exp(2*x)*(12*x^4 - 192*x^5 + 240*x^6 - 112*x^7) + exp(4*x)*(20*x^2 - 72*x^3 + 240*x^4 - 280*x^5) - 16*x^5 + 32*x^6 - 32*x^7 + 16*x^8 - 4*x^9) /(2*exp(8*x) - exp(6*x)*(8*x - 56*x^2) + 16*x*exp(7*x) - exp(5*x)*(48*x^2 - 112*x^3) - exp(x)*(32*x^4 - 64*x^5 + 48*x^6 - 16*x^7) + exp(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + exp(3*x)*(64*x^3 - 160*x^4 + 112*x^5) - exp(2*x)*( 16*x^3 - 96*x^4 + 120*x^5 - 56*x^6) + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^7 + 2* x^8),x)
Output:
-int((exp(6*x)*(16*x^2 - 112*x^3) - 4*x*exp(8*x) - exp(3*x)*(10*x^2 - 10*x ^3 + 158*x^4 - 320*x^5 + 224*x^6) - 32*x^2*exp(7*x) + exp(5*x)*(10*x - 15* x^2 + 96*x^3 - 224*x^4) + exp(x)*(20*x^4 + 44*x^5 - 123*x^6 + 96*x^7 - 32* x^8) + exp(2*x)*(12*x^4 - 192*x^5 + 240*x^6 - 112*x^7) + exp(4*x)*(20*x^2 - 72*x^3 + 240*x^4 - 280*x^5) - 16*x^5 + 32*x^6 - 32*x^7 + 16*x^8 - 4*x^9) /(2*exp(8*x) - exp(6*x)*(8*x - 56*x^2) + 16*x*exp(7*x) - exp(5*x)*(48*x^2 - 112*x^3) - exp(x)*(32*x^4 - 64*x^5 + 48*x^6 - 16*x^7) + exp(4*x)*(16*x^2 - 120*x^3 + 140*x^4) + exp(3*x)*(64*x^3 - 160*x^4 + 112*x^5) - exp(2*x)*( 16*x^3 - 96*x^4 + 120*x^5 - 56*x^6) + 8*x^4 - 16*x^5 + 16*x^6 - 8*x^7 + 2* x^8), x)
Time = 0.27 (sec) , antiderivative size = 141, normalized size of antiderivative = 4.55 \[ \int \frac {4 e^{8 x} x+32 e^{7 x} x^2+16 x^5-32 x^6+32 x^7-16 x^8+4 x^9+e^{6 x} \left (-16 x^2+112 x^3\right )+e^{5 x} \left (-10 x+15 x^2-96 x^3+224 x^4\right )+e^{4 x} \left (-20 x^2+72 x^3-240 x^4+280 x^5\right )+e^{3 x} \left (10 x^2-10 x^3+158 x^4-320 x^5+224 x^6\right )+e^{2 x} \left (-12 x^4+192 x^5-240 x^6+112 x^7\right )+e^x \left (-20 x^4-44 x^5+123 x^6-96 x^7+32 x^8\right )}{2 e^{8 x}+16 e^{7 x} x+8 x^4-16 x^5+16 x^6-8 x^7+2 x^8+e^{6 x} \left (-8 x+56 x^2\right )+e^{5 x} \left (-48 x^2+112 x^3\right )+e^{4 x} \left (16 x^2-120 x^3+140 x^4\right )+e^{3 x} \left (64 x^3-160 x^4+112 x^5\right )+e^{2 x} \left (-16 x^3+96 x^4-120 x^5+56 x^6\right )+e^x \left (-32 x^4+64 x^5-48 x^6+16 x^7\right )} \, dx=\frac {x^{2} \left (2 e^{4 x}+8 e^{3 x} x +12 e^{2 x} x^{2}-4 e^{2 x} x +8 e^{x} x^{3}-8 e^{x} x^{2}-5 e^{x}+2 x^{4}-4 x^{3}+4 x^{2}\right )}{2 e^{4 x}+8 e^{3 x} x +12 e^{2 x} x^{2}-4 e^{2 x} x +8 e^{x} x^{3}-8 e^{x} x^{2}+2 x^{4}-4 x^{3}+4 x^{2}} \] Input:
int((4*x*exp(x)^8+32*x^2*exp(x)^7+(112*x^3-16*x^2)*exp(x)^6+(224*x^4-96*x^ 3+15*x^2-10*x)*exp(x)^5+(280*x^5-240*x^4+72*x^3-20*x^2)*exp(x)^4+(224*x^6- 320*x^5+158*x^4-10*x^3+10*x^2)*exp(x)^3+(112*x^7-240*x^6+192*x^5-12*x^4)*e xp(x)^2+(32*x^8-96*x^7+123*x^6-44*x^5-20*x^4)*exp(x)+4*x^9-16*x^8+32*x^7-3 2*x^6+16*x^5)/(2*exp(x)^8+16*x*exp(x)^7+(56*x^2-8*x)*exp(x)^6+(112*x^3-48* x^2)*exp(x)^5+(140*x^4-120*x^3+16*x^2)*exp(x)^4+(112*x^5-160*x^4+64*x^3)*e xp(x)^3+(56*x^6-120*x^5+96*x^4-16*x^3)*exp(x)^2+(16*x^7-48*x^6+64*x^5-32*x ^4)*exp(x)+2*x^8-8*x^7+16*x^6-16*x^5+8*x^4),x)
Output:
(x**2*(2*e**(4*x) + 8*e**(3*x)*x + 12*e**(2*x)*x**2 - 4*e**(2*x)*x + 8*e** x*x**3 - 8*e**x*x**2 - 5*e**x + 2*x**4 - 4*x**3 + 4*x**2))/(2*(e**(4*x) + 4*e**(3*x)*x + 6*e**(2*x)*x**2 - 2*e**(2*x)*x + 4*e**x*x**3 - 4*e**x*x**2 + x**4 - 2*x**3 + 2*x**2))