Integrand size = 71, antiderivative size = 22 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=-5+e^{\frac {11 x}{4}+\frac {x^4}{1+x^4}}+x \] Output:
x-5+exp(x^4/(x^4+1)+11/4*x)
Time = 0.34 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=\frac {1}{4} \left (4 e^{1+\frac {11 x}{4}-\frac {1}{1+x^4}}+4 x\right ) \] Input:
Integrate[(4 + 8*x^4 + 4*x^8 + E^((11*x + 4*x^4 + 11*x^5)/(4 + 4*x^4))*(11 + 16*x^3 + 22*x^4 + 11*x^8))/(4 + 8*x^4 + 4*x^8),x]
Output:
(4*E^(1 + (11*x)/4 - (1 + x^4)^(-1)) + 4*x)/4
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.
\(\displaystyle \int \frac {4 x^8+8 x^4+e^{\frac {11 x^5+4 x^4+11 x}{4 x^4+4}} \left (11 x^8+22 x^4+16 x^3+11\right )+4}{4 x^8+8 x^4+4} \, dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 4 \int \frac {4 x^8+8 x^4+e^{\frac {11 x^5+4 x^4+11 x}{4 \left (x^4+1\right )}} \left (11 x^8+22 x^4+16 x^3+11\right )+4}{16 \left (x^4+1\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {4 x^8+8 x^4+e^{\frac {11 x^5+4 x^4+11 x}{4 \left (x^4+1\right )}} \left (11 x^8+22 x^4+16 x^3+11\right )+4}{\left (x^4+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {1}{4} \int \left (\frac {e^{\frac {x \left (11 x^4+4 x^3+11\right )}{4 \left (x^4+1\right )}} \left (11 x^8+22 x^4+16 x^3+11\right )}{\left (x^4+1\right )^2}+4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (11 \int e^{\frac {x \left (11 x^4+4 x^3+11\right )}{4 \left (x^4+1\right )}}dx-4 i \int \frac {e^{\frac {x \left (11 x^4+4 x^3+11\right )}{4 \left (x^4+1\right )}} x}{\left (i-x^2\right )^2}dx+4 i \int \frac {e^{\frac {x \left (11 x^4+4 x^3+11\right )}{4 \left (x^4+1\right )}} x}{\left (x^2+i\right )^2}dx+4 x\right )\) |
Input:
Int[(4 + 8*x^4 + 4*x^8 + E^((11*x + 4*x^4 + 11*x^5)/(4 + 4*x^4))*(11 + 16* x^3 + 22*x^4 + 11*x^8))/(4 + 8*x^4 + 4*x^8),x]
Output:
$Aborted
Time = 1.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18
method | result | size |
risch | \(x +{\mathrm e}^{\frac {x \left (11 x^{4}+4 x^{3}+11\right )}{4 x^{4}+4}}\) | \(26\) |
parallelrisch | \(x +{\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}\) | \(27\) |
parts | \(x +\frac {x^{4} {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+{\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}}{x^{4}+1}\) | \(66\) |
norman | \(\frac {x +x^{5}+x^{4} {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+{\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}}{x^{4}+1}\) | \(68\) |
orering | \(\frac {\left (x +\frac {4}{11}\right ) \left (\left (11 x^{8}+22 x^{4}+16 x^{3}+11\right ) {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+4 x^{8}+8 x^{4}+4\right )}{4 x^{8}+8 x^{4}+4}-\frac {4 x \left (121 x^{8}+242 x^{4}+176 x^{3}+64 x^{2}+121\right ) \left (x^{4}+1\right )^{2} \left (\frac {\left (88 x^{7}+88 x^{3}+48 x^{2}\right ) {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+\left (11 x^{8}+22 x^{4}+16 x^{3}+11\right ) \left (\frac {55 x^{4}+16 x^{3}+11}{4 x^{4}+4}-\frac {16 \left (11 x^{5}+4 x^{4}+11 x \right ) x^{3}}{\left (4 x^{4}+4\right )^{2}}\right ) {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+32 x^{7}+32 x^{3}}{4 x^{8}+8 x^{4}+4}-\frac {\left (\left (11 x^{8}+22 x^{4}+16 x^{3}+11\right ) {\mathrm e}^{\frac {11 x^{5}+4 x^{4}+11 x}{4 x^{4}+4}}+4 x^{8}+8 x^{4}+4\right ) \left (32 x^{7}+32 x^{3}\right )}{\left (4 x^{8}+8 x^{4}+4\right )^{2}}\right )}{11 \left (121 x^{16}+484 x^{12}+352 x^{11}-320 x^{10}+726 x^{8}+704 x^{7}+128 x^{6}+484 x^{4}+352 x^{3}+192 x^{2}+121\right )}\) | \(406\) |
Input:
int(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4*x^8+8* x^4+4)/(4*x^8+8*x^4+4),x,method=_RETURNVERBOSE)
Output:
x+exp(1/4*x*(11*x^4+4*x^3+11)/(x^4+1))
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\left (\frac {11 \, x^{5} + 4 \, x^{4} + 11 \, x}{4 \, {\left (x^{4} + 1\right )}}\right )} \] Input:
integrate(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4* x^8+8*x^4+4)/(4*x^8+8*x^4+4),x, algorithm="fricas")
Output:
x + e^(1/4*(11*x^5 + 4*x^4 + 11*x)/(x^4 + 1))
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\frac {11 x^{5} + 4 x^{4} + 11 x}{4 x^{4} + 4}} \] Input:
integrate(((11*x**8+22*x**4+16*x**3+11)*exp((11*x**5+4*x**4+11*x)/(4*x**4+ 4))+4*x**8+8*x**4+4)/(4*x**8+8*x**4+4),x)
Output:
x + exp((11*x**5 + 4*x**4 + 11*x)/(4*x**4 + 4))
Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\left (\frac {11}{4} \, x - \frac {1}{x^{4} + 1} + 1\right )} \] Input:
integrate(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4* x^8+8*x^4+4)/(4*x^8+8*x^4+4),x, algorithm="maxima")
Output:
x + e^(11/4*x - 1/(x^4 + 1) + 1)
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x + e^{\left (\frac {11 \, x^{5} + 4 \, x^{4} + 11 \, x}{4 \, {\left (x^{4} + 1\right )}}\right )} \] Input:
integrate(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4* x^8+8*x^4+4)/(4*x^8+8*x^4+4),x, algorithm="giac")
Output:
x + e^(1/4*(11*x^5 + 4*x^4 + 11*x)/(x^4 + 1))
Time = 2.56 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=x+{\mathrm {e}}^{\frac {11\,x}{4\,x^4+4}}\,{\mathrm {e}}^{\frac {4\,x^4}{4\,x^4+4}}\,{\mathrm {e}}^{\frac {11\,x^5}{4\,x^4+4}} \] Input:
int((exp((11*x + 4*x^4 + 11*x^5)/(4*x^4 + 4))*(16*x^3 + 22*x^4 + 11*x^8 + 11) + 8*x^4 + 4*x^8 + 4)/(8*x^4 + 4*x^8 + 4),x)
Output:
x + exp((11*x)/(4*x^4 + 4))*exp((4*x^4)/(4*x^4 + 4))*exp((11*x^5)/(4*x^4 + 4))
Time = 0.17 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {4+8 x^4+4 x^8+e^{\frac {11 x+4 x^4+11 x^5}{4+4 x^4}} \left (11+16 x^3+22 x^4+11 x^8\right )}{4+8 x^4+4 x^8} \, dx=\frac {e^{\frac {11 x}{4}} e +e^{\frac {1}{x^{4}+1}} x}{e^{\frac {1}{x^{4}+1}}} \] Input:
int(((11*x^8+22*x^4+16*x^3+11)*exp((11*x^5+4*x^4+11*x)/(4*x^4+4))+4*x^8+8* x^4+4)/(4*x^8+8*x^4+4),x)
Output:
(e**((11*x)/4)*e + e**(1/(x**4 + 1))*x)/e**(1/(x**4 + 1))