Integrand size = 38, antiderivative size = 22 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=5 e^{(2+e) x} \left (e^{76 x^2}-\frac {2 x}{3}\right ) \]
Time = 0.50 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=\frac {5}{3} e^{(2+e) x} \left (3 e^{76 x^2}-2 x\right ) \]
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {6, 27, 27, 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.
\(\displaystyle \int \frac {1}{3} e^{e x+2 x} \left (e^{76 x^2} (2280 x+15 e+30)-10 e x-20 x-10\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {1}{3} e^{e x+2 x} \left (e^{76 x^2} (2280 x+15 e+30)+(-20-10 e) x-10\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \int -5 e^{(2+e) x} \left (2 (2+e) x-3 e^{76 x^2} (152 x+e+2)+2\right )dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {5}{3} \int e^{(2+e) x} \left (2 (2+e) x-3 e^{76 x^2} (152 x+e+2)+2\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {5}{3} \int \left (2 e^{(2+e) x} (2+e) x+2 e^{(2+e) x}-3 e^{76 x^2+(2+e) x} (152 x+e+2)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5}{3} \left (2 e^{(2+e) x} x-3 e^{76 x^2+(2+e) x}\right )\) |
3.7.77.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {\left (-10 x +15 \,{\mathrm e}^{76 x^{2}}\right ) {\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}\) | \(22\) |
parallelrisch | \(-\frac {10 x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}+5 \,{\mathrm e}^{76 x^{2}} {\mathrm e}^{x \left ({\mathrm e}+2\right )}\) | \(27\) |
norman | \(-\frac {10 x \,{\mathrm e}^{x \,{\mathrm e}+2 x}}{3}+5 \,{\mathrm e}^{76 x^{2}} {\mathrm e}^{x \,{\mathrm e}+2 x}\) | \(31\) |
default | \(-\frac {10 \,{\mathrm e}^{x \,{\mathrm e}+2 x}}{3 \left ({\mathrm e}+2\right )}-\frac {20 \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )^{2}}-\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{38}-\frac {10 \,{\mathrm e} \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )^{2}}+5 \,{\mathrm e}^{76 x^{2}+x \left ({\mathrm e}+2\right )}+\frac {5 i \left ({\mathrm e}+2\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}-\frac {5 i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}\) | \(215\) |
parts | \(\frac {-\frac {20 \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )}-\frac {10 \,{\mathrm e} \left (\left ({\mathrm e}+2\right ) x \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}-{\mathrm e}^{x \left ({\mathrm e}+2\right )}\right )}{3 \left ({\mathrm e}+2\right )}-\frac {10 \,{\mathrm e}^{x \left ({\mathrm e}+2\right )}}{3}}{{\mathrm e}+2}-\frac {5 i {\mathrm e} \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}-\frac {5 i \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{38}+5 \,{\mathrm e}^{76 x^{2}+x \left ({\mathrm e}+2\right )}+\frac {5 i \left ({\mathrm e}+2\right ) \sqrt {\pi }\, {\mathrm e}^{-\frac {\left ({\mathrm e}+2\right )^{2}}{304}} \sqrt {19}\, \operatorname {erf}\left (2 i \sqrt {19}\, x +\frac {i \left ({\mathrm e}+2\right ) \sqrt {19}}{76}\right )}{76}\) | \(216\) |
int(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp(x*exp( 1)+2*x),x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-\frac {5}{3} \, {\left (2 \, x - 3 \, e^{\left (76 \, x^{2}\right )}\right )} e^{\left (x e + 2 \, x\right )} \]
integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp( x*exp(1)+2*x),x, algorithm=\
Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=\frac {\left (- 10 x + 15 e^{76 x^{2}}\right ) e^{2 x + e x}}{3} \]
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.34 (sec) , antiderivative size = 239, normalized size of antiderivative = 10.86 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-\frac {5}{38} i \, \sqrt {19} \sqrt {\pi } \operatorname {erf}\left (2 i \, \sqrt {19} x + \frac {1}{76} i \, \sqrt {19} {\left (e + 2\right )}\right ) e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2}\right )} - \frac {5}{76} i \, \sqrt {19} \sqrt {\pi } \operatorname {erf}\left (2 i \, \sqrt {19} x + \frac {1}{76} i \, \sqrt {19} {\left (e + 2\right )}\right ) e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2} + 1\right )} - \frac {5}{76} \, \sqrt {19} {\left (\frac {\sqrt {19} \sqrt {\frac {1}{19}} \sqrt {\pi } {\left (152 \, x + e + 2\right )} {\left (\operatorname {erf}\left (\frac {1}{4} \, \sqrt {\frac {1}{19}} \sqrt {-{\left (152 \, x + e + 2\right )}^{2}}\right ) - 1\right )} {\left (e + 2\right )}}{\sqrt {-{\left (152 \, x + e + 2\right )}^{2}}} - 4 \, \sqrt {19} e^{\left (\frac {1}{304} \, {\left (152 \, x + e + 2\right )}^{2}\right )}\right )} e^{\left (-\frac {1}{304} \, {\left (e + 2\right )}^{2}\right )} - \frac {10 \, {\left (x {\left (e^{2} + 2 \, e\right )} - e\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {20 \, {\left (x {\left (e + 2\right )} - 1\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {10 \, e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e + 2\right )}} \]
integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp( x*exp(1)+2*x),x, algorithm=\
-5/38*I*sqrt(19)*sqrt(pi)*erf(2*I*sqrt(19)*x + 1/76*I*sqrt(19)*(e + 2))*e^ (-1/304*(e + 2)^2) - 5/76*I*sqrt(19)*sqrt(pi)*erf(2*I*sqrt(19)*x + 1/76*I* sqrt(19)*(e + 2))*e^(-1/304*(e + 2)^2 + 1) - 5/76*sqrt(19)*(sqrt(19)*sqrt( 1/19)*sqrt(pi)*(152*x + e + 2)*(erf(1/4*sqrt(1/19)*sqrt(-(152*x + e + 2)^2 )) - 1)*(e + 2)/sqrt(-(152*x + e + 2)^2) - 4*sqrt(19)*e^(1/304*(152*x + e + 2)^2))*e^(-1/304*(e + 2)^2) - 10/3*(x*(e^2 + 2*e) - e)*e^(x*e + 2*x)/(e^ 2 + 4*e + 4) - 20/3*(x*(e + 2) - 1)*e^(x*e + 2*x)/(e^2 + 4*e + 4) - 10/3*e ^(x*e + 2*x)/(e + 2)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 3.64 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-\frac {10 \, {\left (x e + 2 \, x - 1\right )} e^{\left (x e + 2 \, x + 1\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} - \frac {10 \, {\left (2 \, x e + 4 \, x + e\right )} e^{\left (x e + 2 \, x\right )}}{3 \, {\left (e^{2} + 4 \, e + 4\right )}} + 5 \, e^{\left (76 \, x^{2} + x e + 2 \, x\right )} \]
integrate(1/3*((15*exp(1)+2280*x+30)*exp(76*x^2)-10*x*exp(1)-20*x-10)*exp( x*exp(1)+2*x),x, algorithm=\
-10/3*(x*e + 2*x - 1)*e^(x*e + 2*x + 1)/(e^2 + 4*e + 4) - 10/3*(2*x*e + 4* x + e)*e^(x*e + 2*x)/(e^2 + 4*e + 4) + 5*e^(76*x^2 + x*e + 2*x)
Time = 0.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.05 \[ \int \frac {1}{3} e^{2 x+e x} \left (-10-20 x-10 e x+e^{76 x^2} (30+15 e+2280 x)\right ) \, dx=-{\mathrm {e}}^{2\,x+x\,\mathrm {e}}\,\left (\frac {10\,x}{3}-5\,{\mathrm {e}}^{76\,x^2}\right ) \]