Integrand size = 102, antiderivative size = 28 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=-4-x+e^{-\frac {3 x}{5-3 e^{14} x^2}} (-4+2 x) \] Output:
-4+(2*x-4)/exp(3*x/(5-3*x^2*exp(7)^2))-x
Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=-x+e^{\frac {3 x}{-5+3 e^{14} x^2}} (-4+2 x) \] Input:
Integrate[(E^((3*x)/(-5 + 3*E^14*x^2))*(110 - 30*x + 18*E^28*x^4 + E^14*(- 24*x^2 - 18*x^3) + (-25 + 30*E^14*x^2 - 9*E^28*x^4)/E^((3*x)/(-5 + 3*E^14* x^2))))/(25 - 30*E^14*x^2 + 9*E^28*x^4),x]
Output:
-x + E^((3*x)/(-5 + 3*E^14*x^2))*(-4 + 2*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.
| \(\displaystyle \int \frac {e^{\frac {3 x}{3 e^{14} x^2-5}} \left (18 e^{28} x^4+e^{-\frac {3 x}{3 e^{14} x^2-5}} \left (-9 e^{28} x^4+30 e^{14} x^2-25\right )+e^{14} \left (-18 x^3-24 x^2\right )-30 x+110\right )}{9 e^{28} x^4-30 e^{14} x^2+25} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle 9 e^{28} \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28} \left (18 e^{28} x^4-30 x-6 e^{14} \left (3 x^3+4 x^2\right )-e^{\frac {3 x}{5-3 e^{14} x^2}} \left (9 e^{28} x^4-30 e^{14} x^2+25\right )+110\right )}{9 \left (5-3 e^{14} x^2\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e^{28} \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28} \left (18 e^{28} x^4-30 x-6 e^{14} \left (3 x^3+4 x^2\right )-e^{\frac {3 x}{5-3 e^{14} x^2}} \left (9 e^{28} x^4-30 e^{14} x^2+25\right )+110\right )}{\left (5-3 e^{14} x^2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle e^{28} \int \left (\frac {18 e^{-\frac {3 x}{5-3 e^{14} x^2}} x^4}{\left (3 e^{14} x^2-5\right )^2}-\frac {6 e^{-\frac {3 x}{5-3 e^{14} x^2}-14} (3 x+4) x^2}{\left (3 e^{14} x^2-5\right )^2}-\frac {30 e^{-\frac {3 x}{5-3 e^{14} x^2}-28} x}{\left (3 e^{14} x^2-5\right )^2}+\frac {110 e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\left (3 e^{14} x^2-5\right )^2}-\frac {1}{e^{28}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^{28} \left (2 \int e^{-\frac {3 x}{5-3 e^{14} x^2}-28}dx+\frac {\left (4+\frac {\sqrt {15}}{e^7}\right ) \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\sqrt {5}-\sqrt {3} e^7 x}dx}{\sqrt {5}}-\frac {3}{2} \sqrt {5} \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\sqrt {5}-\sqrt {3} e^7 x}dx+\frac {7 \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\sqrt {5}-\sqrt {3} e^7 x}dx}{2 \sqrt {5}}+\frac {\left (4-\frac {\sqrt {15}}{e^7}\right ) \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\sqrt {3} e^7 x+\sqrt {5}}dx}{\sqrt {5}}-\frac {3}{2} \sqrt {5} \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\sqrt {3} e^7 x+\sqrt {5}}dx+\frac {7 \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28}}{\sqrt {3} e^7 x+\sqrt {5}}dx}{2 \sqrt {5}}+18 \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-14}}{\left (\sqrt {15} e^7-3 e^{14} x\right )^2}dx+18 \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-14}}{\left (3 e^{14} x+\sqrt {15} e^7\right )^2}dx-60 \int \frac {e^{-\frac {3 x}{5-3 e^{14} x^2}-28} x}{\left (3 e^{14} x^2-5\right )^2}dx-\frac {x}{e^{28}}\right )\) |
Input:
Int[(E^((3*x)/(-5 + 3*E^14*x^2))*(110 - 30*x + 18*E^28*x^4 + E^14*(-24*x^2 - 18*x^3) + (-25 + 30*E^14*x^2 - 9*E^28*x^4)/E^((3*x)/(-5 + 3*E^14*x^2))) )/(25 - 30*E^14*x^2 + 9*E^28*x^4),x]
Output:
$Aborted
Time = 58.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
| method | result | size |
| risch | \(-x +\left (2 x -4\right ) {\mathrm e}^{\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}\) | \(26\) |
| parts | \(-x +\frac {\left (20-10 x -12 x^{2} {\mathrm e}^{14}+6 \,{\mathrm e}^{14} x^{3}\right ) {\mathrm e}^{\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}}{3 x^{2} {\mathrm e}^{14}-5}\) | \(61\) |
| norman | \(\frac {\left (20-10 x +5 x \,{\mathrm e}^{-\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}-12 x^{2} {\mathrm e}^{14}+6 \,{\mathrm e}^{14} x^{3}-3 \,{\mathrm e}^{14} x^{3} {\mathrm e}^{-\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}\right ) {\mathrm e}^{\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}}{3 x^{2} {\mathrm e}^{14}-5}\) | \(103\) |
| parallelrisch | \(\frac {\left (12500-1875 \,{\mathrm e}^{14} x^{3} {\mathrm e}^{-\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}+3750 \,{\mathrm e}^{14} x^{3}-7500 x^{2} {\mathrm e}^{14}+3125 x \,{\mathrm e}^{-\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}-6250 x \right ) {\mathrm e}^{\frac {3 x}{3 x^{2} {\mathrm e}^{14}-5}}}{1875 x^{2} {\mathrm e}^{14}-3125}\) | \(104\) |
Input:
int(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5))+18* x^4*exp(7)^4+(-18*x^3-24*x^2)*exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30*x^2*ex p(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x,method=_RETURNVERBOSE)
Output:
-x+(2*x-4)*exp(3*x/(3*x^2*exp(14)-5))
Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=2 \, {\left (x - 2\right )} e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} - x \] Input:
integrate(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5 ))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30* x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x, algorithm="fricas")
Output:
2*(x - 2)*e^(3*x/(3*x^2*e^14 - 5)) - x
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=- x + \left (2 x - 4\right ) e^{\frac {3 x}{3 x^{2} e^{14} - 5}} \] Input:
integrate(((-9*x**4*exp(7)**4+30*x**2*exp(7)**2-25)*exp(-3*x/(3*x**2*exp(7 )**2-5))+18*x**4*exp(7)**4+(-18*x**3-24*x**2)*exp(7)**2-30*x+110)/(9*x**4* exp(7)**4-30*x**2*exp(7)**2+25)/exp(-3*x/(3*x**2*exp(7)**2-5)),x)
Output:
-x + (2*x - 4)*exp(3*x/(3*x**2*exp(14) - 5))
\[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=\int { \frac {{\left (18 \, x^{4} e^{28} - 6 \, {\left (3 \, x^{3} + 4 \, x^{2}\right )} e^{14} - {\left (9 \, x^{4} e^{28} - 30 \, x^{2} e^{14} + 25\right )} e^{\left (-\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} - 30 \, x + 110\right )} e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )}}{9 \, x^{4} e^{28} - 30 \, x^{2} e^{14} + 25} \,d x } \] Input:
integrate(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5 ))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30* x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x, algorithm="maxima")
Output:
1/12*sqrt(15)*e^(-7)*log((3*x*e^14 - sqrt(15)*e^7)/(3*x*e^14 + sqrt(15)*e^ 7)) - 1/4*(sqrt(15)*e^(-35)*log((3*x*e^14 - sqrt(15)*e^7)/(3*x*e^14 + sqrt (15)*e^7)) + 4*x*e^(-28) - 10*x/(3*x^2*e^42 - 5*e^28))*e^28 + 1/6*(sqrt(15 )*e^(-21)*log((3*x*e^14 - sqrt(15)*e^7)/(3*x*e^14 + sqrt(15)*e^7)) - 30*x/ (3*x^2*e^28 - 5*e^14))*e^14 + 2*x*e^(3*x/(3*x^2*e^14 - 5)) + 5/2*x/(3*x^2* e^14 - 5) + integrate(12*(3*x^2*e^14 + 5)*e^(3*x/(3*x^2*e^14 - 5))/(9*x^4* e^28 - 30*x^2*e^14 + 25), x)
Time = 0.14 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=2 \, x e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} - x - 4 \, e^{\left (\frac {3 \, x}{3 \, x^{2} e^{14} - 5}\right )} \] Input:
integrate(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5 ))+18*x^4*exp(7)^4+(-18*x^3-24*x^2)*exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30* x^2*exp(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x, algorithm="giac")
Output:
2*x*e^(3*x/(3*x^2*e^14 - 5)) - x - 4*e^(3*x/(3*x^2*e^14 - 5))
Time = 4.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx={\mathrm {e}}^{\frac {3\,x}{3\,x^2\,{\mathrm {e}}^{14}-5}}\,\left (2\,x-4\right )-x \] Input:
int(-(exp((3*x)/(3*x^2*exp(14) - 5))*(30*x + exp(-(3*x)/(3*x^2*exp(14) - 5 ))*(9*x^4*exp(28) - 30*x^2*exp(14) + 25) + exp(14)*(24*x^2 + 18*x^3) - 18* x^4*exp(28) - 110))/(9*x^4*exp(28) - 30*x^2*exp(14) + 25),x)
Output:
exp((3*x)/(3*x^2*exp(14) - 5))*(2*x - 4) - x
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.54 \[ \int \frac {e^{\frac {3 x}{-5+3 e^{14} x^2}} \left (110-30 x+18 e^{28} x^4+e^{14} \left (-24 x^2-18 x^3\right )+e^{-\frac {3 x}{-5+3 e^{14} x^2}} \left (-25+30 e^{14} x^2-9 e^{28} x^4\right )\right )}{25-30 e^{14} x^2+9 e^{28} x^4} \, dx=2 e^{\frac {3 x}{3 e^{14} x^{2}-5}} x -4 e^{\frac {3 x}{3 e^{14} x^{2}-5}}-x \] Input:
int(((-9*x^4*exp(7)^4+30*x^2*exp(7)^2-25)*exp(-3*x/(3*x^2*exp(7)^2-5))+18* x^4*exp(7)^4+(-18*x^3-24*x^2)*exp(7)^2-30*x+110)/(9*x^4*exp(7)^4-30*x^2*ex p(7)^2+25)/exp(-3*x/(3*x^2*exp(7)^2-5)),x)
Output:
2*e**((3*x)/(3*e**14*x**2 - 5))*x - 4*e**((3*x)/(3*e**14*x**2 - 5)) - x