Integrand size = 69, antiderivative size = 25 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=3 e^{-\frac {16 x}{5}-\frac {2}{5} \left (e+e^4-x\right ) x}+x \] Output:
x+3/exp(2/5*(exp(1)+exp(4)-x)*x+16/5*x)
Time = 1.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.88 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=3 e^{\frac {2}{5} x \left (-8-e-e^4+x\right )}+x \] Input:
Integrate[(E^((-16*x - 2*E*x - 2*E^4*x + 2*x^2)/5)*(-48 - 6*E - 6*E^4 + 5* E^((16*x + 2*E*x + 2*E^4*x - 2*x^2)/5) + 12*x))/5,x]
Output:
3*E^((2*x*(-8 - E - E^4 + x))/5) + x
Time = 0.76 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {27, 7292, 7293, 2009}
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.
| \(\displaystyle \int \frac {1}{5} e^{\frac {1}{5} \left (2 x^2-2 e^4 x-2 e x-16 x\right )} \left (5 e^{\frac {1}{5} \left (-2 x^2+2 e^4 x+2 e x+16 x\right )}+12 x-6 e^4-6 e-48\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int e^{-\frac {2}{5} \left (-x^2+e^4 x+e x+8 x\right )} \left (12 x+5 e^{\frac {2}{5} \left (-x^2+e^4 x+e x+8 x\right )}-6 \left (8+e+e^4\right )\right )dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {1}{5} \int e^{-\frac {2}{5} \left (-x+e^4+e+8\right ) x} \left (12 x+5 e^{\frac {2}{5} \left (-x^2+e^4 x+e x+8 x\right )}-6 \left (8+e+e^4\right )\right )dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (12 e^{-\frac {2}{5} \left (-x+e^4+e+8\right ) x} x-6 e^{-\frac {2}{5} \left (-x+e^4+e+8\right ) x} \left (8+e+e^4\right )+5\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} \left (15 e^{\frac {2 x^2}{5}-\frac {2}{5} \left (8+e+e^4\right ) x}+5 x\right )\) |
Input:
Int[(E^((-16*x - 2*E*x - 2*E^4*x + 2*x^2)/5)*(-48 - 6*E - 6*E^4 + 5*E^((16 *x + 2*E*x + 2*E^4*x - 2*x^2)/5) + 12*x))/5,x]
Output:
(15*E^((-2*(8 + E + E^4)*x)/5 + (2*x^2)/5) + 5*x)/5
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.34 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(x +3 \,{\mathrm e}^{-\frac {2 x \left ({\mathrm e}^{4}+{\mathrm e}-x +8\right )}{5}}\) | \(18\) |
| parallelrisch | \(\frac {\left (5 \,{\mathrm e}^{\frac {2 x \left ({\mathrm e}^{4}+{\mathrm e}-x +8\right )}{5}} x +15\right ) {\mathrm e}^{-\frac {2 x \left ({\mathrm e}^{4}+{\mathrm e}-x +8\right )}{5}}}{5}\) | \(36\) |
| norman | \(\left (3+x \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}}{5}+\frac {2 x \,{\mathrm e}}{5}-\frac {2 x^{2}}{5}+\frac {16 x}{5}}\right ) {\mathrm e}^{-\frac {2 x \,{\mathrm e}^{4}}{5}-\frac {2 x \,{\mathrm e}}{5}+\frac {2 x^{2}}{5}-\frac {16 x}{5}}\) | \(48\) |
| default | \(x +\frac {12 i \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{5}+3 \,{\mathrm e}^{\frac {2 x^{2}}{5}+\left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) x}+\frac {3 i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{4}+\frac {3 i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{10}+\frac {3 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{10}\) | \(234\) |
| parts | \(x +\frac {12 i \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{5}+3 \,{\mathrm e}^{\frac {2 x^{2}}{5}+\left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) x}+\frac {3 i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{4}+\frac {3 i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{10}+\frac {3 i {\mathrm e}^{4} \sqrt {\pi }\, {\mathrm e}^{-\frac {5 \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right )^{2}}{8}} \sqrt {10}\, \operatorname {erf}\left (\frac {i \sqrt {10}\, x}{5}+\frac {i \left (-\frac {2 \,{\mathrm e}^{4}}{5}-\frac {2 \,{\mathrm e}}{5}-\frac {16}{5}\right ) \sqrt {10}}{4}\right )}{10}\) | \(234\) |
| orering | \(\frac {\left (\frac {{\mathrm e}^{4}}{2}+\frac {{\mathrm e}}{2}+x +4\right ) \left (5 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}}{5}+\frac {2 x \,{\mathrm e}}{5}-\frac {2 x^{2}}{5}+\frac {16 x}{5}}-6 \,{\mathrm e}^{4}-6 \,{\mathrm e}+12 x -48\right ) {\mathrm e}^{-\frac {2 x \,{\mathrm e}^{4}}{5}-\frac {2 x \,{\mathrm e}}{5}+\frac {2 x^{2}}{5}-\frac {16 x}{5}}}{5}+\frac {5 \left ({\mathrm e}^{8}+2 \,{\mathrm e} \,{\mathrm e}^{4}+{\mathrm e}^{2}-4 x^{2}+16 \,{\mathrm e}^{4}+16 \,{\mathrm e}+69\right ) \left (\frac {\left (5 \left (\frac {2 \,{\mathrm e}^{4}}{5}+\frac {2 \,{\mathrm e}}{5}-\frac {4 x}{5}+\frac {16}{5}\right ) {\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}}{5}+\frac {2 x \,{\mathrm e}}{5}-\frac {2 x^{2}}{5}+\frac {16 x}{5}}+12\right ) {\mathrm e}^{-\frac {2 x \,{\mathrm e}^{4}}{5}-\frac {2 x \,{\mathrm e}}{5}+\frac {2 x^{2}}{5}-\frac {16 x}{5}}}{5}-\frac {\left (5 \,{\mathrm e}^{\frac {2 x \,{\mathrm e}^{4}}{5}+\frac {2 x \,{\mathrm e}}{5}-\frac {2 x^{2}}{5}+\frac {16 x}{5}}-6 \,{\mathrm e}^{4}-6 \,{\mathrm e}+12 x -48\right ) {\mathrm e}^{-\frac {2 x \,{\mathrm e}^{4}}{5}-\frac {2 x \,{\mathrm e}}{5}+\frac {2 x^{2}}{5}-\frac {16 x}{5}} \left (\frac {2 \,{\mathrm e}^{4}}{5}+\frac {2 \,{\mathrm e}}{5}-\frac {4 x}{5}+\frac {16}{5}\right )}{5}\right )}{4 \left ({\mathrm e}^{8}+2 \,{\mathrm e} \,{\mathrm e}^{4}-4 x \,{\mathrm e}^{4}+{\mathrm e}^{2}-4 x \,{\mathrm e}+4 x^{2}+16 \,{\mathrm e}^{4}+16 \,{\mathrm e}-32 x +69\right )}\) | \(281\) |
Input:
int(1/5*(5*exp(2/5*x*exp(4)+2/5*x*exp(1)-2/5*x^2+16/5*x)-6*exp(4)-6*exp(1) +12*x-48)/exp(2/5*x*exp(4)+2/5*x*exp(1)-2/5*x^2+16/5*x),x,method=_RETURNVE RBOSE)
Output:
x+3*exp(-2/5*x*(exp(4)+exp(1)-x+8))
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.80 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx={\left (x e^{\left (-\frac {2}{5} \, x^{2} + \frac {2}{5} \, x e^{4} + \frac {2}{5} \, x e + \frac {16}{5} \, x\right )} + 3\right )} e^{\left (\frac {2}{5} \, x^{2} - \frac {2}{5} \, x e^{4} - \frac {2}{5} \, x e - \frac {16}{5} \, x\right )} \] Input:
integrate(1/5*(5*exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x)-6*exp(4)-6* exp(1)+12*x-48)/exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x),x, algorithm ="fricas")
Output:
(x*e^(-2/5*x^2 + 2/5*x*e^4 + 2/5*x*e + 16/5*x) + 3)*e^(2/5*x^2 - 2/5*x*e^4 - 2/5*x*e - 16/5*x)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.28 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=x + 3 e^{\frac {2 x^{2}}{5} - \frac {2 x e^{4}}{5} - \frac {16 x}{5} - \frac {2 e x}{5}} \] Input:
integrate(1/5*(5*exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x**2+16/5*x)-6*exp(4)-6 *exp(1)+12*x-48)/exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x**2+16/5*x),x)
Output:
x + 3*exp(2*x**2/5 - 2*x*exp(4)/5 - 16*x/5 - 2*E*x/5)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 270, normalized size of antiderivative = 10.80 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=\frac {12}{5} i \, \sqrt {5} \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{5} i \, \sqrt {5} \sqrt {2} x - \frac {1}{10} i \, \sqrt {5} \sqrt {2} {\left (e^{4} + e + 8\right )}\right ) e^{\left (-\frac {1}{10} \, {\left (e^{4} + e + 8\right )}^{2}\right )} + \frac {3}{10} i \, \sqrt {5} \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{5} i \, \sqrt {5} \sqrt {2} x - \frac {1}{10} i \, \sqrt {5} \sqrt {2} {\left (e^{4} + e + 8\right )}\right ) e^{\left (-\frac {1}{10} \, {\left (e^{4} + e + 8\right )}^{2} + 4\right )} + \frac {3}{10} i \, \sqrt {5} \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\frac {1}{5} i \, \sqrt {5} \sqrt {2} x - \frac {1}{10} i \, \sqrt {5} \sqrt {2} {\left (e^{4} + e + 8\right )}\right ) e^{\left (-\frac {1}{10} \, {\left (e^{4} + e + 8\right )}^{2} + 1\right )} + \frac {3}{20} \, \sqrt {5} \sqrt {2} {\left (\frac {\sqrt {5} \sqrt {2} \sqrt {\frac {2}{5}} \sqrt {\pi } {\left (2 \, x - e^{4} - e - 8\right )} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{10}} \sqrt {-{\left (2 \, x - e^{4} - e - 8\right )}^{2}}\right ) - 1\right )} {\left (e^{4} + e + 8\right )}}{\sqrt {-{\left (2 \, x - e^{4} - e - 8\right )}^{2}}} + 2 \, \sqrt {5} \sqrt {2} e^{\left (\frac {1}{10} \, {\left (2 \, x - e^{4} - e - 8\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{10} \, {\left (e^{4} + e + 8\right )}^{2}\right )} + x \] Input:
integrate(1/5*(5*exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x)-6*exp(4)-6* exp(1)+12*x-48)/exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x),x, algorithm ="maxima")
Output:
12/5*I*sqrt(5)*sqrt(2)*sqrt(pi)*erf(1/5*I*sqrt(5)*sqrt(2)*x - 1/10*I*sqrt( 5)*sqrt(2)*(e^4 + e + 8))*e^(-1/10*(e^4 + e + 8)^2) + 3/10*I*sqrt(5)*sqrt( 2)*sqrt(pi)*erf(1/5*I*sqrt(5)*sqrt(2)*x - 1/10*I*sqrt(5)*sqrt(2)*(e^4 + e + 8))*e^(-1/10*(e^4 + e + 8)^2 + 4) + 3/10*I*sqrt(5)*sqrt(2)*sqrt(pi)*erf( 1/5*I*sqrt(5)*sqrt(2)*x - 1/10*I*sqrt(5)*sqrt(2)*(e^4 + e + 8))*e^(-1/10*( e^4 + e + 8)^2 + 1) + 3/20*sqrt(5)*sqrt(2)*(sqrt(5)*sqrt(2)*sqrt(2/5)*sqrt (pi)*(2*x - e^4 - e - 8)*(erf(sqrt(1/10)*sqrt(-(2*x - e^4 - e - 8)^2)) - 1 )*(e^4 + e + 8)/sqrt(-(2*x - e^4 - e - 8)^2) + 2*sqrt(5)*sqrt(2)*e^(1/10*( 2*x - e^4 - e - 8)^2))*e^(-1/10*(e^4 + e + 8)^2) + x
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 179, normalized size of antiderivative = 7.16 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=\frac {3}{10} i \, \sqrt {10} \sqrt {\pi } {\left (e^{4} + e\right )} \operatorname {erf}\left (-\frac {1}{10} i \, \sqrt {10} {\left (2 \, x - e^{4} - e - 8\right )}\right ) e^{\left (-\frac {1}{10} \, e^{8} - \frac {1}{5} \, e^{5} - \frac {8}{5} \, e^{4} - \frac {1}{10} \, e^{2} - \frac {8}{5} \, e - \frac {32}{5}\right )} - \frac {3}{10} i \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{10} i \, \sqrt {10} {\left (2 \, x - e^{4} - e - 8\right )}\right ) e^{\left (-\frac {1}{10} \, e^{8} - \frac {1}{5} \, e^{5} - \frac {8}{5} \, e^{4} - \frac {1}{10} \, e^{2} - \frac {8}{5} \, e - \frac {12}{5}\right )} - \frac {3}{10} i \, \sqrt {10} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{10} i \, \sqrt {10} {\left (2 \, x - e^{4} - e - 8\right )}\right ) e^{\left (-\frac {1}{10} \, e^{8} - \frac {1}{5} \, e^{5} - \frac {8}{5} \, e^{4} - \frac {1}{10} \, e^{2} - \frac {8}{5} \, e - \frac {27}{5}\right )} + x + 3 \, e^{\left (\frac {2}{5} \, x^{2} - \frac {2}{5} \, x e^{4} - \frac {2}{5} \, x e - \frac {16}{5} \, x\right )} \] Input:
integrate(1/5*(5*exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x)-6*exp(4)-6* exp(1)+12*x-48)/exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x),x, algorithm ="giac")
Output:
3/10*I*sqrt(10)*sqrt(pi)*(e^4 + e)*erf(-1/10*I*sqrt(10)*(2*x - e^4 - e - 8 ))*e^(-1/10*e^8 - 1/5*e^5 - 8/5*e^4 - 1/10*e^2 - 8/5*e - 32/5) - 3/10*I*sq rt(10)*sqrt(pi)*erf(-1/10*I*sqrt(10)*(2*x - e^4 - e - 8))*e^(-1/10*e^8 - 1 /5*e^5 - 8/5*e^4 - 1/10*e^2 - 8/5*e - 12/5) - 3/10*I*sqrt(10)*sqrt(pi)*erf (-1/10*I*sqrt(10)*(2*x - e^4 - e - 8))*e^(-1/10*e^8 - 1/5*e^5 - 8/5*e^4 - 1/10*e^2 - 8/5*e - 27/5) + x + 3*e^(2/5*x^2 - 2/5*x*e^4 - 2/5*x*e - 16/5*x )
Time = 0.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=x+3\,{\mathrm {e}}^{\frac {2\,x^2}{5}-\frac {2\,x\,\mathrm {e}}{5}-\frac {2\,x\,{\mathrm {e}}^4}{5}-\frac {16\,x}{5}} \] Input:
int(-exp((2*x^2)/5 - (2*x*exp(1))/5 - (2*x*exp(4))/5 - (16*x)/5)*((6*exp(1 ))/5 - exp((16*x)/5 + (2*x*exp(1))/5 + (2*x*exp(4))/5 - (2*x^2)/5) - (12*x )/5 + (6*exp(4))/5 + 48/5),x)
Output:
x + 3*exp((2*x^2)/5 - (2*x*exp(1))/5 - (2*x*exp(4))/5 - (16*x)/5)
Time = 0.18 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {1}{5} e^{\frac {1}{5} \left (-16 x-2 e x-2 e^4 x+2 x^2\right )} \left (-48-6 e-6 e^4+5 e^{\frac {1}{5} \left (16 x+2 e x+2 e^4 x-2 x^2\right )}+12 x\right ) \, dx=\frac {3 e^{\frac {2 x^{2}}{5}}+e^{\frac {2}{5} e^{4} x +\frac {2}{5} e x +\frac {16}{5} x} x}{e^{\frac {2}{5} e^{4} x +\frac {2}{5} e x +\frac {16}{5} x}} \] Input:
int(1/5*(5*exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x)-6*exp(4)-6*exp(1) +12*x-48)/exp(2/5*x*exp(4)+2/5*exp(1)*x-2/5*x^2+16/5*x),x)
Output:
(3*e**((2*x**2)/5) + e**((2*e**4*x + 2*e*x + 16*x)/5)*x)/e**((2*e**4*x + 2 *e*x + 16*x)/5)