Integrand size = 144, antiderivative size = 26 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=e^{5-x}+\frac {3}{\left (e^{2 x}+x\right ) \left (\frac {1}{x}+x\right )} \]
Time = 5.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=e^{5-x}+\frac {3 x}{\left (e^{2 x}+x\right ) \left (1+x^2\right )} \]
Integrate[(-6*x^3 + E^(5 + 3*x)*(-1 - 2*x^2 - x^4) + E^(5 - x)*(-x^2 - 2*x ^4 - x^6) + E^(2*x)*(3 - 6*x - 3*x^2 - 6*x^3 + E^(5 - x)*(-2*x - 4*x^3 - 2 *x^5)))/(x^2 + 2*x^4 + x^6 + E^(4*x)*(1 + 2*x^2 + x^4) + E^(2*x)*(2*x + 4* x^3 + 2*x^5)),x]
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.
\(\displaystyle \int \frac {-6 x^3+e^{3 x+5} \left (-x^4-2 x^2-1\right )+e^{5-x} \left (-x^6-2 x^4-x^2\right )+e^{2 x} \left (-6 x^3-3 x^2+e^{5-x} \left (-2 x^5-4 x^3-2 x\right )-6 x+3\right )}{x^6+2 x^4+x^2+e^{2 x} \left (2 x^5+4 x^3+2 x\right )+e^{4 x} \left (x^4+2 x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {-6 x^3+e^{3 x+5} \left (-x^4-2 x^2-1\right )+e^{5-x} \left (-x^6-2 x^4-x^2\right )+e^{2 x} \left (-6 x^3-3 x^2+e^{5-x} \left (-2 x^5-4 x^3-2 x\right )-6 x+3\right )}{\left (x+e^{2 x}\right )^2 \left (x^2+1\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x (2 x-1)}{\left (x+e^{2 x}\right )^2 \left (x^2+1\right )}-\frac {3 \left (2 x^3+x^2+2 x-1\right )}{\left (x+e^{2 x}\right ) \left (x^2+1\right )^2}-e^{5-x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 6 \int \frac {1}{\left (x+e^{2 x}\right ) \left (x^2+1\right )^2}dx+6 \int \frac {1}{\left (x+e^{2 x}\right )^2}dx+\left (\frac {3}{2}-3 i\right ) \int \frac {1}{(i-x) \left (x+e^{2 x}\right )^2}dx-\left (\frac {3}{2}+3 i\right ) \int \frac {1}{(x+i) \left (x+e^{2 x}\right )^2}dx+\left (3-\frac {3 i}{2}\right ) \int \frac {1}{(i-x) \left (x+e^{2 x}\right )}dx-\left (3+\frac {3 i}{2}\right ) \int \frac {1}{(x+i) \left (x+e^{2 x}\right )}dx+e^{5-x}\) |
Int[(-6*x^3 + E^(5 + 3*x)*(-1 - 2*x^2 - x^4) + E^(5 - x)*(-x^2 - 2*x^4 - x ^6) + E^(2*x)*(3 - 6*x - 3*x^2 - 6*x^3 + E^(5 - x)*(-2*x - 4*x^3 - 2*x^5)) )/(x^2 + 2*x^4 + x^6 + E^(4*x)*(1 + 2*x^2 + x^4) + E^(2*x)*(2*x + 4*x^3 + 2*x^5)),x]
3.29.68.3.1 Defintions of rubi rules used
Time = 1.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
risch | \({\mathrm e}^{5-x}+\frac {3 x}{\left (x^{2}+1\right ) \left ({\mathrm e}^{2 x}+x \right )}\) | \(26\) |
parallelrisch | \(\frac {{\mathrm e}^{2 x} {\mathrm e}^{5-x} x^{2}+{\mathrm e}^{5-x} x^{3}+{\mathrm e}^{2 x} {\mathrm e}^{5-x}+x \,{\mathrm e}^{5-x}+3 x}{{\mathrm e}^{2 x} x^{2}+x^{3}+{\mathrm e}^{2 x}+x}\) | \(68\) |
int(((-x^4-2*x^2-1)*exp(5-x)*exp(x)^4+((-2*x^5-4*x^3-2*x)*exp(5-x)-6*x^3-3 *x^2-6*x+3)*exp(x)^2+(-x^6-2*x^4-x^2)*exp(5-x)-6*x^3)/((x^4+2*x^2+1)*exp(x )^4+(2*x^5+4*x^3+2*x)*exp(x)^2+x^6+2*x^4+x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.23 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=\frac {{\left (x^{2} + 1\right )} e^{\left (-x + 15\right )} + 3 \, x e^{\left (-2 \, x + 10\right )} + {\left (x^{3} + x\right )} e^{\left (-3 \, x + 15\right )}}{{\left (x^{2} + 1\right )} e^{10} + {\left (x^{3} + x\right )} e^{\left (-2 \, x + 10\right )}} \]
integrate(((-x^4-2*x^2-1)*exp(5-x)*exp(x)^4+((-2*x^5-4*x^3-2*x)*exp(5-x)-6 *x^3-3*x^2-6*x+3)*exp(x)^2+(-x^6-2*x^4-x^2)*exp(5-x)-6*x^3)/((x^4+2*x^2+1) *exp(x)^4+(2*x^5+4*x^3+2*x)*exp(x)^2+x^6+2*x^4+x^2),x, algorithm=\
((x^2 + 1)*e^(-x + 15) + 3*x*e^(-2*x + 10) + (x^3 + x)*e^(-3*x + 15))/((x^ 2 + 1)*e^10 + (x^3 + x)*e^(-2*x + 10))
Time = 0.15 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=\frac {3 x}{x^{3} + x + \left (x^{2} + 1\right ) e^{2 x}} + \frac {e^{5}}{\sqrt {e^{2 x}}} \]
integrate(((-x**4-2*x**2-1)*exp(5-x)*exp(x)**4+((-2*x**5-4*x**3-2*x)*exp(5 -x)-6*x**3-3*x**2-6*x+3)*exp(x)**2+(-x**6-2*x**4-x**2)*exp(5-x)-6*x**3)/(( x**4+2*x**2+1)*exp(x)**4+(2*x**5+4*x**3+2*x)*exp(x)**2+x**6+2*x**4+x**2),x )
Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (24) = 48\).
Time = 0.32 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.00 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=\frac {x^{3} e^{5} + x e^{5} + {\left (x^{2} e^{5} + e^{5}\right )} e^{\left (2 \, x\right )} + 3 \, x e^{x}}{{\left (x^{2} + 1\right )} e^{\left (3 \, x\right )} + {\left (x^{3} + x\right )} e^{x}} \]
integrate(((-x^4-2*x^2-1)*exp(5-x)*exp(x)^4+((-2*x^5-4*x^3-2*x)*exp(5-x)-6 *x^3-3*x^2-6*x+3)*exp(x)^2+(-x^6-2*x^4-x^2)*exp(5-x)-6*x^3)/((x^4+2*x^2+1) *exp(x)^4+(2*x^5+4*x^3+2*x)*exp(x)^2+x^6+2*x^4+x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 927 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 927, normalized size of antiderivative = 35.65 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=\text {Too large to display} \]
integrate(((-x^4-2*x^2-1)*exp(5-x)*exp(x)^4+((-2*x^5-4*x^3-2*x)*exp(5-x)-6 *x^3-3*x^2-6*x+3)*exp(x)^2+(-x^6-2*x^4-x^2)*exp(5-x)-6*x^3)/((x^4+2*x^2+1) *exp(x)^4+(2*x^5+4*x^3+2*x)*exp(x)^2+x^6+2*x^4+x^2),x, algorithm=\
(2*(x - 5)^8*e^(-2*x + 10) + 6*(x - 5)^7*e^10 + 79*(x - 5)^7*e^(-2*x + 10) + 207*(x - 5)^6*e^10 + 6*(x - 5)^6*e^(2*x + 10) + 6*(x - 5)^6*e^(-x + 5) + 1369*(x - 5)^6*e^(-2*x + 10) + 3072*(x - 5)^5*e^10 + 2*(x - 5)^5*e^(4*x + 10) + 177*(x - 5)^5*e^(2*x + 10) + 6*(x - 5)^5*e^(x + 5) + 177*(x - 5)^5 *e^(-x + 5) + 13593*(x - 5)^5*e^(-2*x + 10) + 25419*(x - 5)^4*e^10 + 49*(x - 5)^4*e^(4*x + 10) - 6*(x - 5)^4*e^(3*x + 5) + 2187*(x - 5)^4*e^(2*x + 1 0) + 141*(x - 5)^4*e^(x + 5) + 2181*(x - 5)^4*e^(-x + 5) + 84577*(x - 5)^4 *e^(-2*x + 10) + 126636*(x - 5)^3*e^10 - 6*(x - 5)^3*e^(5*x + 5) + 484*(x - 5)^3*e^(4*x + 10) - 129*(x - 5)^3*e^(3*x + 5) + 14484*(x - 5)^3*e^(2*x + 10) + 1326*(x - 5)^3*e^(x + 5) + 14367*(x - 5)^3*e^(-x + 5) + 337664*(x - 5)^3*e^(-2*x + 10) + 379812*(x - 5)^2*e^10 - 93*(x - 5)^2*e^(5*x + 5) + 2 408*(x - 5)^2*e^(4*x + 10) - 1041*(x - 5)^2*e^(3*x + 5) + 54216*(x - 5)^2* e^(2*x + 10) + 6237*(x - 5)^2*e^(x + 5) + 53355*(x - 5)^2*e^(-x + 5) + 844 660*(x - 5)^2*e^(-2*x + 10) + 634920*(x - 5)*e^10 - 486*(x - 5)*e^(5*x + 5 ) + 6032*(x - 5)*e^(4*x + 10) - 3732*(x - 5)*e^(3*x + 5) + 108732*(x - 5)* e^(2*x + 10) + 14670*(x - 5)*e^(x + 5) + 105900*(x - 5)*e^(-x + 5) + 12103 00*(x - 5)*e^(-2*x + 10) + 456300*e^10 - 852*e^(5*x + 5) + 6084*e^(4*x + 1 0) - 5010*e^(3*x + 5) + 91260*e^(2*x + 10) + 13800*e^(x + 5) + 87750*e^(-x + 5) + 760500*e^(-2*x + 10))/(2*(x - 5)^8*e^(-x + 5) + 6*(x - 5)^7*e^(x + 5) + 79*(x - 5)^7*e^(-x + 5) + 6*(x - 5)^6*e^(3*x + 5) + 207*(x - 5)^6...
Time = 9.88 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.31 \[ \int \frac {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx=\frac {{\mathrm {e}}^{5-x}\,\left (x+3\,x\,{\mathrm {e}}^{x-5}+{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,x-10}+x^3+x^2\,{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,x-10}\right )}{\left (x^2+1\right )\,\left (x+{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,x-10}\right )} \]
int(-(exp(5 - x)*(x^2 + 2*x^4 + x^6) + 6*x^3 + exp(2*x)*(6*x + exp(5 - x)* (2*x + 4*x^3 + 2*x^5) + 3*x^2 + 6*x^3 - 3) + exp(4*x)*exp(5 - x)*(2*x^2 + x^4 + 1))/(exp(4*x)*(2*x^2 + x^4 + 1) + exp(2*x)*(2*x + 4*x^3 + 2*x^5) + x ^2 + 2*x^4 + x^6),x)