\(\int \frac {e^x (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7))}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8)+e^{\frac {2 x^2}{1+3 x+16 x^3}} (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9)} \, dx\) [3728]

Optimal result

Integrand size = 231, antiderivative size = 32 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {e^x}{\left (e^{\frac {2 x}{3+\frac {1}{x}+16 x^2}}+\frac {x}{4}\right ) x} \]

[Out]

exp(x)/(exp(2*x/(3+1/x+16*x^2))+1/4*x)/x

Rubi [F]

\[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx \]

[In]

Int[(E^x*(-8*x - 44*x^2 - 48*x^3 - 220*x^4 - 640*x^5 + 384*x^6 - 2048*x^7 + 1024*x^8 + E^((2*x^2)/(1 + 3*x + 1
6*x^3))*(-16 - 80*x - 112*x^2 - 464*x^3 - 1024*x^4 + 2048*x^5 - 4096*x^6 + 4096*x^7)))/(x^4 + 6*x^5 + 9*x^6 +
32*x^7 + 96*x^8 + 256*x^10 + E^((4*x^2)/(1 + 3*x + 16*x^3))*(16*x^2 + 96*x^3 + 144*x^4 + 512*x^5 + 1536*x^6 +
4096*x^8) + E^((2*x^2)/(1 + 3*x + 16*x^3))*(8*x^3 + 48*x^4 + 72*x^5 + 256*x^6 + 768*x^7 + 2048*x^9)),x]

[Out]

((-8*I)/3)*Sqrt[5/3]*Defer[Int][E^x/((1 + I*Sqrt[15] - 8*x)*(4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)^2), x] - 4*
Defer[Int][E^x/(x*(4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)^2), x] - (((32*I)/3)*Defer[Int][E^x/((1 + I*Sqrt[15]
- 8*x)*(4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)), x])/Sqrt[15] - 4*Defer[Int][E^x/(x^2*(4*E^((2*x^2)/(1 + 3*x +
16*x^3)) + x)), x] + 4*Defer[Int][E^x/(x*(4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)), x] - (4*Defer[Int][E^x/((4*E
^((2*x^2)/(1 + 3*x + 16*x^3)) + x)^2*(1 + 4*x)^2), x])/3 - (16*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3
)) + x)*(1 + 4*x)^2), x])/3 - (16*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)*(1 + 4*x)), x])/3 + (
16*(15 - I*Sqrt[15])*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)*(-1 - I*Sqrt[15] + 8*x)), x])/45 -
((8*I)/3)*Sqrt[5/3]*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)^2*(-1 + I*Sqrt[15] + 8*x)), x] - (
((32*I)/3)*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)*(-1 + I*Sqrt[15] + 8*x)), x])/Sqrt[15] + (16
*(15 + I*Sqrt[15])*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)*(-1 + I*Sqrt[15] + 8*x)), x])/45 + 3
*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)^2*(1 - x + 4*x^2)^2), x] + Defer[Int][(E^x*x)/((4*E^((
2*x^2)/(1 + 3*x + 16*x^3)) + x)^2*(1 - x + 4*x^2)^2), x] - 4*Defer[Int][E^x/((4*E^((2*x^2)/(1 + 3*x + 16*x^3))
+ x)*(1 - x + 4*x^2)^2), x] + 12*Defer[Int][(E^x*x)/((4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x)*(1 - x + 4*x^2)^2
), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 e^x \left ((-2+x) x \left (1+3 x+16 x^3\right )^2+4 e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-1-5 x-7 x^2-29 x^3-64 x^4+128 x^5-256 x^6+256 x^7\right )\right )}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1+3 x+16 x^3\right )^2} \, dx \\ & = 4 \int \frac {e^x \left ((-2+x) x \left (1+3 x+16 x^3\right )^2+4 e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-1-5 x-7 x^2-29 x^3-64 x^4+128 x^5-256 x^6+256 x^7\right )\right )}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1+3 x+16 x^3\right )^2} \, dx \\ & = 4 \int \left (-\frac {e^x \left (1+6 x+5 x^2+26 x^3+96 x^4+32 x^5+256 x^6\right )}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2 \left (1-x+4 x^2\right )^2}+\frac {e^x \left (-1-5 x-7 x^2-29 x^3-64 x^4+128 x^5-256 x^6+256 x^7\right )}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1+3 x+16 x^3\right )^2}\right ) \, dx \\ & = -\left (4 \int \frac {e^x \left (1+6 x+5 x^2+26 x^3+96 x^4+32 x^5+256 x^6\right )}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2 \left (1-x+4 x^2\right )^2} \, dx\right )+4 \int \frac {e^x \left (-1-5 x-7 x^2-29 x^3-64 x^4+128 x^5-256 x^6+256 x^7\right )}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1+3 x+16 x^3\right )^2} \, dx \\ & = -\left (4 \int \left (\frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2}+\frac {e^x}{3 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2}-\frac {e^x (3+x)}{4 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2}+\frac {5 e^x}{12 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )}\right ) \, dx\right )+4 \int \left (-\frac {e^x}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )}+\frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )}-\frac {4 e^x}{3 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)^2}-\frac {4 e^x}{3 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)}+\frac {e^x (-1+3 x)}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2}+\frac {e^x (-1+4 x)}{3 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )}\right ) \, dx \\ & = -\left (\frac {4}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2} \, dx\right )+\frac {4}{3} \int \frac {e^x (-1+4 x)}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )} \, dx-\frac {5}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )} \, dx-4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \frac {e^x (-1+3 x)}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)} \, dx+\int \frac {e^x (3+x)}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx \\ & = -\left (\frac {4}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2} \, dx\right )+\frac {4}{3} \int \left (-\frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )}+\frac {4 e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )}\right ) \, dx-\frac {5}{3} \int \left (\frac {8 i e^x}{\sqrt {15} \left (1+i \sqrt {15}-8 x\right ) \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2}+\frac {8 i e^x}{\sqrt {15} \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (-1+i \sqrt {15}+8 x\right )}\right ) \, dx-4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \left (-\frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2}+\frac {3 e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2}\right ) \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)} \, dx+\int \left (\frac {3 e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2}+\frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2}\right ) \, dx \\ & = -\left (\frac {4}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2} \, dx\right )-\frac {4}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )} \, dx+3 \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx-4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx-4 \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)} \, dx+\frac {16}{3} \int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )} \, dx+12 \int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {1}{3} \left (8 i \sqrt {\frac {5}{3}}\right ) \int \frac {e^x}{\left (1+i \sqrt {15}-8 x\right ) \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-\frac {1}{3} \left (8 i \sqrt {\frac {5}{3}}\right ) \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (-1+i \sqrt {15}+8 x\right )} \, dx+\int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx \\ & = -\left (\frac {4}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2} \, dx\right )-\frac {4}{3} \int \left (\frac {8 i e^x}{\sqrt {15} \left (1+i \sqrt {15}-8 x\right ) \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )}+\frac {8 i e^x}{\sqrt {15} \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (-1+i \sqrt {15}+8 x\right )}\right ) \, dx+3 \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx-4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx-4 \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)} \, dx+\frac {16}{3} \int \left (\frac {\left (1-\frac {i}{\sqrt {15}}\right ) e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (-1-i \sqrt {15}+8 x\right )}+\frac {\left (1+\frac {i}{\sqrt {15}}\right ) e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (-1+i \sqrt {15}+8 x\right )}\right ) \, dx+12 \int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {1}{3} \left (8 i \sqrt {\frac {5}{3}}\right ) \int \frac {e^x}{\left (1+i \sqrt {15}-8 x\right ) \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-\frac {1}{3} \left (8 i \sqrt {\frac {5}{3}}\right ) \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (-1+i \sqrt {15}+8 x\right )} \, dx+\int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx \\ & = -\left (\frac {4}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 (1+4 x)^2} \, dx\right )+3 \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx-4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-4 \int \frac {e^x}{x^2 \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx+4 \int \frac {e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx-4 \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)^2} \, dx-\frac {16}{3} \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) (1+4 x)} \, dx+12 \int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (1-x+4 x^2\right )^2} \, dx-\frac {1}{3} \left (8 i \sqrt {\frac {5}{3}}\right ) \int \frac {e^x}{\left (1+i \sqrt {15}-8 x\right ) \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2} \, dx-\frac {1}{3} \left (8 i \sqrt {\frac {5}{3}}\right ) \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (-1+i \sqrt {15}+8 x\right )} \, dx-\frac {(32 i) \int \frac {e^x}{\left (1+i \sqrt {15}-8 x\right ) \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \, dx}{3 \sqrt {15}}-\frac {(32 i) \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (-1+i \sqrt {15}+8 x\right )} \, dx}{3 \sqrt {15}}+\frac {1}{45} \left (16 \left (15-i \sqrt {15}\right )\right ) \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (-1-i \sqrt {15}+8 x\right )} \, dx+\frac {1}{45} \left (16 \left (15+i \sqrt {15}\right )\right ) \int \frac {e^x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right ) \left (-1+i \sqrt {15}+8 x\right )} \, dx+\int \frac {e^x x}{\left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )^2 \left (1-x+4 x^2\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 e^x}{x \left (4 e^{\frac {2 x^2}{1+3 x+16 x^3}}+x\right )} \]

[In]

Integrate[(E^x*(-8*x - 44*x^2 - 48*x^3 - 220*x^4 - 640*x^5 + 384*x^6 - 2048*x^7 + 1024*x^8 + E^((2*x^2)/(1 + 3
*x + 16*x^3))*(-16 - 80*x - 112*x^2 - 464*x^3 - 1024*x^4 + 2048*x^5 - 4096*x^6 + 4096*x^7)))/(x^4 + 6*x^5 + 9*
x^6 + 32*x^7 + 96*x^8 + 256*x^10 + E^((4*x^2)/(1 + 3*x + 16*x^3))*(16*x^2 + 96*x^3 + 144*x^4 + 512*x^5 + 1536*
x^6 + 4096*x^8) + E^((2*x^2)/(1 + 3*x + 16*x^3))*(8*x^3 + 48*x^4 + 72*x^5 + 256*x^6 + 768*x^7 + 2048*x^9)),x]

[Out]

(4*E^x)/(x*(4*E^((2*x^2)/(1 + 3*x + 16*x^3)) + x))

Maple [A] (verified)

Time = 407.38 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00

method	result	size
parallelrisch	\(\frac {4 \,{\mathrm e}^{x}}{x \left (4 \,{\mathrm e}^{\frac {2 x^{2}}{16 x^{3}+3 x +1}}+x \right )}\)	\(32\)
risch	\(\frac {4 \,{\mathrm e}^{x}}{x \left (4 \,{\mathrm e}^{\frac {2 x^{2}}{\left (1+4 x \right ) \left (4 x^{2}-x +1\right )}}+x \right )}\)	\(39\)

[In]

int(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*exp(2*x^2/(16*x^3+3*x+1))+1024*x^8-2048*x^7
+384*x^6-640*x^5-220*x^4-48*x^3-44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*exp(2*x^
2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp(2*x^2/(16*x^3+3*x+1))+256*x^10+96*x^8+3
2*x^7+9*x^6+6*x^5+x^4),x,method=_RETURNVERBOSE)

[Out]

4*exp(x)/x/(4*exp(2*x^2/(16*x^3+3*x+1))+x)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 \, e^{x}}{x^{2} + 4 \, x e^{\left (\frac {2 \, x^{2}}{16 \, x^{3} + 3 \, x + 1}\right )}} \]

[In]

integrate(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*exp(2*x^2/(16*x^3+3*x+1))+1024*x^8-20
48*x^7+384*x^6-640*x^5-220*x^4-48*x^3-44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*ex
p(2*x^2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp(2*x^2/(16*x^3+3*x+1))+256*x^10+96
*x^8+32*x^7+9*x^6+6*x^5+x^4),x, algorithm="fricas")

[Out]

4*e^x/(x^2 + 4*x*e^(2*x^2/(16*x^3 + 3*x + 1)))

Sympy [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 e^{x}}{x^{2} + 4 x e^{\frac {2 x^{2}}{16 x^{3} + 3 x + 1}}} \]

[In]

integrate(((4096*x**7-4096*x**6+2048*x**5-1024*x**4-464*x**3-112*x**2-80*x-16)*exp(2*x**2/(16*x**3+3*x+1))+102
4*x**8-2048*x**7+384*x**6-640*x**5-220*x**4-48*x**3-44*x**2-8*x)*exp(x)/((4096*x**8+1536*x**6+512*x**5+144*x**
4+96*x**3+16*x**2)*exp(2*x**2/(16*x**3+3*x+1))**2+(2048*x**9+768*x**7+256*x**6+72*x**5+48*x**4+8*x**3)*exp(2*x
**2/(16*x**3+3*x+1))+256*x**10+96*x**8+32*x**7+9*x**6+6*x**5+x**4),x)

[Out]

4*exp(x)/(x**2 + 4*x*exp(2*x**2/(16*x**3 + 3*x + 1)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (29) = 58\).

Time = 0.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4 \, e^{\left (x + \frac {1}{12 \, {\left (4 \, x^{2} - x + 1\right )}}\right )}}{x^{2} e^{\left (\frac {1}{12 \, {\left (4 \, x^{2} - x + 1\right )}}\right )} + 4 \, x e^{\left (\frac {5 \, x}{12 \, {\left (4 \, x^{2} - x + 1\right )}} + \frac {1}{12 \, {\left (4 \, x + 1\right )}}\right )}} \]

[In]

integrate(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*exp(2*x^2/(16*x^3+3*x+1))+1024*x^8-20
48*x^7+384*x^6-640*x^5-220*x^4-48*x^3-44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*ex
p(2*x^2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp(2*x^2/(16*x^3+3*x+1))+256*x^10+96
*x^8+32*x^7+9*x^6+6*x^5+x^4),x, algorithm="maxima")

[Out]

4*e^(x + 1/12/(4*x^2 - x + 1))/(x^2*e^(1/12/(4*x^2 - x + 1)) + 4*x*e^(5/12*x/(4*x^2 - x + 1) + 1/12/(4*x + 1))
)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1317 vs. \(2 (29) = 58\).

Time = 0.36 (sec) , antiderivative size = 1317, normalized size of antiderivative = 41.16 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\text {Too large to display} \]

[In]

integrate(((4096*x^7-4096*x^6+2048*x^5-1024*x^4-464*x^3-112*x^2-80*x-16)*exp(2*x^2/(16*x^3+3*x+1))+1024*x^8-20
48*x^7+384*x^6-640*x^5-220*x^4-48*x^3-44*x^2-8*x)*exp(x)/((4096*x^8+1536*x^6+512*x^5+144*x^4+96*x^3+16*x^2)*ex
p(2*x^2/(16*x^3+3*x+1))^2+(2048*x^9+768*x^7+256*x^6+72*x^5+48*x^4+8*x^3)*exp(2*x^2/(16*x^3+3*x+1))+256*x^10+96
*x^8+32*x^7+9*x^6+6*x^5+x^4),x, algorithm="giac")

[Out]

4*(256*x^7*e^(2*x) + 1024*x^7*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) - 1024*x^7*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^
3 + 3*x + 1)) + 32*x^6*e^(2*x) - 1024*x^6*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 2048*x^6*e^(x + (16*x^4 + 5*x^2
+ x)/(16*x^3 + 3*x + 1)) + 96*x^5*e^(2*x) + 512*x^5*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) - 384*x^5*e^(x + (16*x
^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 26*x^4*e^(2*x) - 256*x^4*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 640*x^4*e^
(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 5*x^3*e^(2*x) - 116*x^3*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 2
20*x^3*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 6*x^2*e^(2*x) - 28*x^2*e^(2*x + 2*x^2/(16*x^3 + 3*x +
1)) + 48*x^2*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + x*e^(2*x) - 20*x*e^(2*x + 2*x^2/(16*x^3 + 3*x
+ 1)) + 44*x*e^(x + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) - 4*e^(2*x + 2*x^2/(16*x^3 + 3*x + 1)) + 8*e^(x +
(16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)))/(256*x^9*e^x + 1024*x^8*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 32*x^8*e
^x + 1024*x^8*e^((16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 128*x^7*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 96*x^7*
e^x + 4096*x^7*e^(2*x^2/(16*x^3 + 3*x + 1) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 128*x^7*e^((16*x^4 + 5
*x^2 + x)/(16*x^3 + 3*x + 1)) + 384*x^6*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 26*x^6*e^x + 512*x^6*e^(2*x^2/(16*x
^3 + 3*x + 1) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 384*x^6*e^((16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1))
+ 104*x^5*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 5*x^5*e^x + 1536*x^5*e^(2*x^2/(16*x^3 + 3*x + 1) + (16*x^4 + 5*x
^2 + x)/(16*x^3 + 3*x + 1)) + 104*x^5*e^((16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 20*x^4*e^(x + 2*x^2/(16*x^
3 + 3*x + 1)) + 6*x^4*e^x + 416*x^4*e^(2*x^2/(16*x^3 + 3*x + 1) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 2
0*x^4*e^((16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 24*x^3*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + x^3*e^x + 80*x^3
*e^(2*x^2/(16*x^3 + 3*x + 1) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 24*x^3*e^((16*x^4 + 5*x^2 + x)/(16*x
^3 + 3*x + 1)) + 4*x^2*e^(x + 2*x^2/(16*x^3 + 3*x + 1)) + 96*x^2*e^(2*x^2/(16*x^3 + 3*x + 1) + (16*x^4 + 5*x^2
+ x)/(16*x^3 + 3*x + 1)) + 4*x^2*e^((16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)) + 16*x*e^(2*x^2/(16*x^3 + 3*x +
1) + (16*x^4 + 5*x^2 + x)/(16*x^3 + 3*x + 1)))

Mupad [B] (verification not implemented)

Time = 8.91 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {e^x \left (-8 x-44 x^2-48 x^3-220 x^4-640 x^5+384 x^6-2048 x^7+1024 x^8+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (-16-80 x-112 x^2-464 x^3-1024 x^4+2048 x^5-4096 x^6+4096 x^7\right )\right )}{x^4+6 x^5+9 x^6+32 x^7+96 x^8+256 x^{10}+e^{\frac {4 x^2}{1+3 x+16 x^3}} \left (16 x^2+96 x^3+144 x^4+512 x^5+1536 x^6+4096 x^8\right )+e^{\frac {2 x^2}{1+3 x+16 x^3}} \left (8 x^3+48 x^4+72 x^5+256 x^6+768 x^7+2048 x^9\right )} \, dx=\frac {4\,{\mathrm {e}}^x}{4\,x\,{\mathrm {e}}^{\frac {2\,x^2}{16\,x^3+3\,x+1}}+x^2} \]

[In]

int(-(exp(x)*(8*x + exp((2*x^2)/(3*x + 16*x^3 + 1))*(80*x + 112*x^2 + 464*x^3 + 1024*x^4 - 2048*x^5 + 4096*x^6
- 4096*x^7 + 16) + 44*x^2 + 48*x^3 + 220*x^4 + 640*x^5 - 384*x^6 + 2048*x^7 - 1024*x^8))/(exp((2*x^2)/(3*x +
16*x^3 + 1))*(8*x^3 + 48*x^4 + 72*x^5 + 256*x^6 + 768*x^7 + 2048*x^9) + exp((4*x^2)/(3*x + 16*x^3 + 1))*(16*x^
2 + 96*x^3 + 144*x^4 + 512*x^5 + 1536*x^6 + 4096*x^8) + x^4 + 6*x^5 + 9*x^6 + 32*x^7 + 96*x^8 + 256*x^10),x)

[Out]

(4*exp(x))/(4*x*exp((2*x^2)/(3*x + 16*x^3 + 1)) + x^2)