Integrand size = 65, antiderivative size = 25 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=-\frac {2 \left (e^5-x\right )}{\left (4+\frac {e^{4 x}}{x^8}\right ) x} \] Output:
-2/x/(exp(x)^4/x^8+4)*(exp(5)-x)
Time = 8.63 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=\frac {2 \left (-e^5 x^7+x^8\right )}{e^{4 x}+4 x^8} \] Input:
Integrate[(8*E^5*x^14 + E^(4*x)*(16*x^7 - 8*x^8 + E^5*(-14*x^6 + 8*x^7)))/ (E^(8*x) + 8*E^(4*x)*x^8 + 16*x^16),x]
Output:
(2*(-(E^5*x^7) + x^8))/(E^(4*x) + 4*x^8)
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.
\(\displaystyle \int \frac {8 e^5 x^{14}+e^{4 x} \left (-8 x^8+16 x^7+e^5 \left (8 x^7-14 x^6\right )\right )}{16 x^{16}+8 e^{4 x} x^8+e^{8 x}} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {8 e^5 x^{14}+e^{4 x} \left (-8 x^8+16 x^7+e^5 \left (8 x^7-14 x^6\right )\right )}{\left (4 x^8+e^{4 x}\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {\left (e^5-x\right ) (x-2) \left (x^2-e^x\right ) x^4}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}-\frac {\left (e^5-x\right ) (x-2) \left (x^2+e^x\right ) x^4}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}+\frac {\left (e^x-2 x^2\right ) \left (x^2-\left (2+e^5\right ) x+e^5\right )}{4 \left (2 x^4-2 e^x x^2+e^{2 x}\right )}+\frac {\left (-2 x^2-e^x\right ) \left (x^2-\left (2+e^5\right ) x+e^5\right )}{4 \left (2 x^4+2 e^x x^2+e^{2 x}\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 e^5 \int \frac {e^x x^4}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}dx+\left (2+e^5\right ) \int \frac {e^x x^5}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}dx+2 e^5 \int \frac {x^6}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}dx-\int \frac {e^x x^6}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}dx-\left (2+e^5\right ) \int \frac {x^7}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}dx+\int \frac {x^8}{\left (2 x^4-2 e^x x^2+e^{2 x}\right )^2}dx+\frac {1}{4} e^5 \int \frac {e^x}{2 x^4-2 e^x x^2+e^{2 x}}dx-\frac {1}{4} \left (2+e^5\right ) \int \frac {e^x x}{2 x^4-2 e^x x^2+e^{2 x}}dx-\frac {1}{2} e^5 \int \frac {x^2}{2 x^4-2 e^x x^2+e^{2 x}}dx+\frac {1}{4} \int \frac {e^x x^2}{2 x^4-2 e^x x^2+e^{2 x}}dx+\frac {1}{2} \left (2+e^5\right ) \int \frac {x^3}{2 x^4-2 e^x x^2+e^{2 x}}dx-\frac {1}{2} \int \frac {x^4}{2 x^4-2 e^x x^2+e^{2 x}}dx+2 e^5 \int \frac {e^x x^4}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}dx-\left (2+e^5\right ) \int \frac {e^x x^5}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}dx+2 e^5 \int \frac {x^6}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}dx+\int \frac {e^x x^6}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}dx-\left (2+e^5\right ) \int \frac {x^7}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}dx+\int \frac {x^8}{\left (2 x^4+2 e^x x^2+e^{2 x}\right )^2}dx-\frac {1}{4} e^5 \int \frac {e^x}{2 x^4+2 e^x x^2+e^{2 x}}dx+\frac {1}{4} \left (2+e^5\right ) \int \frac {e^x x}{2 x^4+2 e^x x^2+e^{2 x}}dx-\frac {1}{2} e^5 \int \frac {x^2}{2 x^4+2 e^x x^2+e^{2 x}}dx-\frac {1}{4} \int \frac {e^x x^2}{2 x^4+2 e^x x^2+e^{2 x}}dx+\frac {1}{2} \left (2+e^5\right ) \int \frac {x^3}{2 x^4+2 e^x x^2+e^{2 x}}dx-\frac {1}{2} \int \frac {x^4}{2 x^4+2 e^x x^2+e^{2 x}}dx\) |
Input:
Int[(8*E^5*x^14 + E^(4*x)*(16*x^7 - 8*x^8 + E^5*(-14*x^6 + 8*x^7)))/(E^(8* x) + 8*E^(4*x)*x^8 + 16*x^16),x]
Output:
$Aborted
Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
method | result | size |
risch | \(-\frac {2 \left ({\mathrm e}^{5}-x \right ) x^{7}}{4 x^{8}+{\mathrm e}^{4 x}}\) | \(24\) |
parallelrisch | \(-\frac {2 \,{\mathrm e}^{5} x^{7}-2 x^{8}}{4 x^{8}+{\mathrm e}^{4 x}}\) | \(28\) |
Input:
int((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(exp(x)^ 8+8*x^8*exp(x)^4+16*x^16),x,method=_RETURNVERBOSE)
Output:
-2*(exp(5)-x)*x^7/(4*x^8+exp(4*x))
Time = 0.08 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=\frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \] Input:
integrate((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(e xp(x)^8+8*x^8*exp(x)^4+16*x^16),x, algorithm="fricas")
Output:
2*(x^8 - x^7*e^5)/(4*x^8 + e^(4*x))
Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=\frac {2 x^{8} - 2 x^{7} e^{5}}{4 x^{8} + e^{4 x}} \] Input:
integrate((((8*x**7-14*x**6)*exp(5)-8*x**8+16*x**7)*exp(x)**4+8*x**14*exp( 5))/(exp(x)**8+8*x**8*exp(x)**4+16*x**16),x)
Output:
(2*x**8 - 2*x**7*exp(5))/(4*x**8 + exp(4*x))
Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=\frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \] Input:
integrate((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(e xp(x)^8+8*x^8*exp(x)^4+16*x^16),x, algorithm="maxima")
Output:
2*(x^8 - x^7*e^5)/(4*x^8 + e^(4*x))
Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=\frac {2 \, {\left (x^{8} - x^{7} e^{5}\right )}}{4 \, x^{8} + e^{\left (4 \, x\right )}} \] Input:
integrate((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(e xp(x)^8+8*x^8*exp(x)^4+16*x^16),x, algorithm="giac")
Output:
2*(x^8 - x^7*e^5)/(4*x^8 + e^(4*x))
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.00 \[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=-\frac {2\,\left (2\,x^{14}\,{\mathrm {e}}^5-x^{15}\,{\mathrm {e}}^5-2\,x^{15}+x^{16}\right )}{\left (2\,x^7-x^8\right )\,\left ({\mathrm {e}}^{4\,x}+4\,x^8\right )} \] Input:
int((8*x^14*exp(5) - exp(4*x)*(exp(5)*(14*x^6 - 8*x^7) - 16*x^7 + 8*x^8))/ (exp(8*x) + 8*x^8*exp(4*x) + 16*x^16),x)
Output:
-(2*(2*x^14*exp(5) - x^15*exp(5) - 2*x^15 + x^16))/((2*x^7 - x^8)*(exp(4*x ) + 4*x^8))
\[ \int \frac {8 e^5 x^{14}+e^{4 x} \left (16 x^7-8 x^8+e^5 \left (-14 x^6+8 x^7\right )\right )}{e^{8 x}+8 e^{4 x} x^8+16 x^{16}} \, dx=8 \left (\int \frac {x^{14}}{e^{8 x}+8 e^{4 x} x^{8}+16 x^{16}}d x \right ) e^{5}-8 \left (\int \frac {e^{4 x} x^{8}}{e^{8 x}+8 e^{4 x} x^{8}+16 x^{16}}d x \right )+8 \left (\int \frac {e^{4 x} x^{7}}{e^{8 x}+8 e^{4 x} x^{8}+16 x^{16}}d x \right ) e^{5}+16 \left (\int \frac {e^{4 x} x^{7}}{e^{8 x}+8 e^{4 x} x^{8}+16 x^{16}}d x \right )-14 \left (\int \frac {e^{4 x} x^{6}}{e^{8 x}+8 e^{4 x} x^{8}+16 x^{16}}d x \right ) e^{5} \] Input:
int((((8*x^7-14*x^6)*exp(5)-8*x^8+16*x^7)*exp(x)^4+8*x^14*exp(5))/(exp(x)^ 8+8*x^8*exp(x)^4+16*x^16),x)
Output:
2*(4*int(x**14/(e**(8*x) + 8*e**(4*x)*x**8 + 16*x**16),x)*e**5 - 4*int((e* *(4*x)*x**8)/(e**(8*x) + 8*e**(4*x)*x**8 + 16*x**16),x) + 4*int((e**(4*x)* x**7)/(e**(8*x) + 8*e**(4*x)*x**8 + 16*x**16),x)*e**5 + 8*int((e**(4*x)*x* *7)/(e**(8*x) + 8*e**(4*x)*x**8 + 16*x**16),x) - 7*int((e**(4*x)*x**6)/(e* *(8*x) + 8*e**(4*x)*x**8 + 16*x**16),x)*e**5)