Integrand size = 144, antiderivative size = 36 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\frac {-3+\frac {x}{2-\frac {\log (3)}{4 e^3 \left (-e^{2 e^3}+e^x\right )}}}{x} \]
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.92 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {3}{x}-\frac {\log (3)}{16 e^{3+2 e^3}-16 e^{3+x}+\log (9)} \]
Integrate[(192*E^(6 + 4*E^3) + 192*E^(6 + 2*x) + E^(3 + x)*(-48 - 4*x^2)*L og[3] + 3*Log[3]^2 + E^(2*E^3)*(-384*E^(6 + x) + 48*E^3*Log[3]))/(64*E^(6 + 4*E^3)*x^2 + 64*E^(6 + 2*x)*x^2 - 16*E^(3 + x)*x^2*Log[3] + x^2*Log[3]^2 + E^(2*E^3)*(-128*E^(6 + x)*x^2 + 16*E^3*x^2*Log[3])),x]
Leaf count is larger than twice the leaf count of optimal. \(164\) vs. \(2(36)=72\).
Time = 1.74 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.56, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6, 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.
| \(\displaystyle \int \frac {e^{x+3} \left (-4 x^2-48\right ) \log (3)+192 e^{2 x+6}+e^{2 e^3} \left (48 e^3 \log (3)-384 e^{x+6}\right )+192 e^{6+4 e^3}+3 \log ^2(3)}{64 e^{2 x+6} x^2+64 e^{6+4 e^3} x^2+x^2 \log ^2(3)-16 e^{x+3} x^2 \log (3)+e^{2 e^3} \left (16 e^3 x^2 \log (3)-128 e^{x+6} x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {e^{x+3} \left (-4 x^2-48\right ) \log (3)+192 e^{2 x+6}+e^{2 e^3} \left (48 e^3 \log (3)-384 e^{x+6}\right )+192 e^{6+4 e^3}+3 \log ^2(3)}{64 e^{2 x+6} x^2+x^2 \left (64 e^{6+4 e^3}+\log ^2(3)\right )-16 e^{x+3} x^2 \log (3)+e^{2 e^3} \left (16 e^3 x^2 \log (3)-128 e^{x+6} x^2\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{x+3} \left (-4 x^2-48\right ) \log (3)+192 e^{2 x+6}+e^{2 e^3} \left (48 e^3 \log (3)-384 e^{x+6}\right )+192 e^{6+4 e^3} \left (1+\frac {1}{64} e^{-6-4 e^3} \log ^2(3)\right )}{x^2 \left (8 e^{x+3}-8 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {3}{x^2}+\frac {\left (-8 e^{3+2 e^3}-\log (3)\right ) \log (3)}{2 \left (8 e^{x+3}-8 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )\right )^2}+\frac {\log (3)}{16 e^{3+2 e^3} \left (1+\frac {1}{8} e^{-3-2 e^3} \log (3)\right )-16 e^{x+3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{x}+\frac {x \log (3)}{16 e^{3+2 e^3}+\log (9)}-\frac {x \log (3)}{2 \left (8 e^{3+2 e^3}+\log (3)\right )}+\frac {\log (3) \log \left (-8 e^{x+3}+8 e^{3+2 e^3}+\log (3)\right )}{2 \left (8 e^{3+2 e^3}+\log (3)\right )}-\frac {\log (3) \log \left (-16 e^{x+3}+16 e^{3+2 e^3}+\log (9)\right )}{16 e^{3+2 e^3}+\log (9)}-\frac {\log (3)}{2 \left (-8 e^{x+3}+8 e^{3+2 e^3}+\log (3)\right )}\) |
Int[(192*E^(6 + 4*E^3) + 192*E^(6 + 2*x) + E^(3 + x)*(-48 - 4*x^2)*Log[3] + 3*Log[3]^2 + E^(2*E^3)*(-384*E^(6 + x) + 48*E^3*Log[3]))/(64*E^(6 + 4*E^ 3)*x^2 + 64*E^(6 + 2*x)*x^2 - 16*E^(3 + x)*x^2*Log[3] + x^2*Log[3]^2 + E^( 2*E^3)*(-128*E^(6 + x)*x^2 + 16*E^3*x^2*Log[3])),x]
-3/x - (x*Log[3])/(2*(8*E^(3 + 2*E^3) + Log[3])) - Log[3]/(2*(8*E^(3 + 2*E ^3) - 8*E^(3 + x) + Log[3])) + (x*Log[3])/(16*E^(3 + 2*E^3) + Log[9]) + (L og[3]*Log[8*E^(3 + 2*E^3) - 8*E^(3 + x) + Log[3]])/(2*(8*E^(3 + 2*E^3) + L og[3])) - (Log[3]*Log[16*E^(3 + 2*E^3) - 16*E^(3 + x) + Log[9]])/(16*E^(3 + 2*E^3) + Log[9])
3.4.6.3.1 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]
Time = 5.68 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86
| method | result | size |
| risch | \(-\frac {3}{x}-\frac {\ln \left (3\right )}{2 \left (8 \,{\mathrm e}^{3+2 \,{\mathrm e}^{3}}-8 \,{\mathrm e}^{3+x}+\ln \left (3\right )\right )}\) | \(31\) |
| norman | \(\frac {-\frac {x \ln \left (3\right )}{2}+24 \,{\mathrm e}^{x} {\mathrm e}^{3}-24 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{3}}-3 \ln \left (3\right )}{x \left (8 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{3}}-8 \,{\mathrm e}^{x} {\mathrm e}^{3}+\ln \left (3\right )\right )}\) | \(50\) |
| parallelrisch | \(-\frac {\left (192 \,{\mathrm e}^{6} {\mathrm e}^{2 \,{\mathrm e}^{3}}+4 x \,{\mathrm e}^{3} \ln \left (3\right )-192 \,{\mathrm e}^{6} {\mathrm e}^{x}+24 \,{\mathrm e}^{3} \ln \left (3\right )\right ) {\mathrm e}^{-3}}{8 \left (8 \,{\mathrm e}^{3} {\mathrm e}^{2 \,{\mathrm e}^{3}}-8 \,{\mathrm e}^{x} {\mathrm e}^{3}+\ln \left (3\right )\right ) x}\) | \(63\) |
int((192*exp(3)^2*exp(exp(3))^4+(-384*exp(3)^2*exp(x)+48*exp(3)*ln(3))*exp (exp(3))^2+192*exp(3)^2*exp(x)^2+(-4*x^2-48)*exp(3)*ln(3)*exp(x)+3*ln(3)^2 )/(64*x^2*exp(3)^2*exp(exp(3))^4+(-128*x^2*exp(3)^2*exp(x)+16*x^2*exp(3)*l n(3))*exp(exp(3))^2+64*x^2*exp(3)^2*exp(x)^2-16*x^2*exp(3)*ln(3)*exp(x)+x^ 2*ln(3)^2),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.44 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {{\left (x + 6\right )} e^{3} \log \left (3\right ) - 48 \, e^{\left (x + 6\right )} + 48 \, e^{\left (2 \, e^{3} + 6\right )}}{2 \, {\left (x e^{3} \log \left (3\right ) - 8 \, x e^{\left (x + 6\right )} + 8 \, x e^{\left (2 \, e^{3} + 6\right )}\right )}} \]
integrate((192*exp(3)^2*exp(exp(3))^4+(-384*exp(3)^2*exp(x)+48*exp(3)*log( 3))*exp(exp(3))^2+192*exp(3)^2*exp(x)^2+(-4*x^2-48)*exp(3)*log(3)*exp(x)+3 *log(3)^2)/(64*x^2*exp(3)^2*exp(exp(3))^4+(-128*x^2*exp(3)^2*exp(x)+16*x^2 *exp(3)*log(3))*exp(exp(3))^2+64*x^2*exp(3)^2*exp(x)^2-16*x^2*exp(3)*log(3 )*exp(x)+x^2*log(3)^2),x, algorithm=\
-1/2*((x + 6)*e^3*log(3) - 48*e^(x + 6) + 48*e^(2*e^3 + 6))/(x*e^3*log(3) - 8*x*e^(x + 6) + 8*x*e^(2*e^3 + 6))
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\frac {\log {\left (3 \right )}}{16 e^{3} e^{x} - 16 e^{3} e^{2 e^{3}} - 2 \log {\left (3 \right )}} - \frac {3}{x} \]
integrate((192*exp(3)**2*exp(exp(3))**4+(-384*exp(3)**2*exp(x)+48*exp(3)*l n(3))*exp(exp(3))**2+192*exp(3)**2*exp(x)**2+(-4*x**2-48)*exp(3)*ln(3)*exp (x)+3*ln(3)**2)/(64*x**2*exp(3)**2*exp(exp(3))**4+(-128*x**2*exp(3)**2*exp (x)+16*x**2*exp(3)*ln(3))*exp(exp(3))**2+64*x**2*exp(3)**2*exp(x)**2-16*x* *2*exp(3)*ln(3)*exp(x)+x**2*ln(3)**2),x)
Time = 0.32 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.39 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=-\frac {x \log \left (3\right ) - 48 \, e^{\left (x + 3\right )} + 48 \, e^{\left (2 \, e^{3} + 3\right )} + 6 \, \log \left (3\right )}{2 \, {\left (x {\left (8 \, e^{\left (2 \, e^{3} + 3\right )} + \log \left (3\right )\right )} - 8 \, x e^{\left (x + 3\right )}\right )}} \]
integrate((192*exp(3)^2*exp(exp(3))^4+(-384*exp(3)^2*exp(x)+48*exp(3)*log( 3))*exp(exp(3))^2+192*exp(3)^2*exp(x)^2+(-4*x^2-48)*exp(3)*log(3)*exp(x)+3 *log(3)^2)/(64*x^2*exp(3)^2*exp(exp(3))^4+(-128*x^2*exp(3)^2*exp(x)+16*x^2 *exp(3)*log(3))*exp(exp(3))^2+64*x^2*exp(3)^2*exp(x)^2-16*x^2*exp(3)*log(3 )*exp(x)+x^2*log(3)^2),x, algorithm=\
-1/2*(x*log(3) - 48*e^(x + 3) + 48*e^(2*e^3 + 3) + 6*log(3))/(x*(8*e^(2*e^ 3 + 3) + log(3)) - 8*x*e^(x + 3))
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (31) = 62\).
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.17 \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\frac {{\left (x + 3\right )} \log \left (3\right ) - 48 \, e^{\left (x + 3\right )} + 48 \, e^{\left (2 \, e^{3} + 3\right )} + 3 \, \log \left (3\right )}{2 \, {\left (8 \, {\left (x + 3\right )} e^{\left (x + 3\right )} - 8 \, {\left (x + 3\right )} e^{\left (2 \, e^{3} + 3\right )} - {\left (x + 3\right )} \log \left (3\right ) - 24 \, e^{\left (x + 3\right )} + 24 \, e^{\left (2 \, e^{3} + 3\right )} + 3 \, \log \left (3\right )\right )}} \]
integrate((192*exp(3)^2*exp(exp(3))^4+(-384*exp(3)^2*exp(x)+48*exp(3)*log( 3))*exp(exp(3))^2+192*exp(3)^2*exp(x)^2+(-4*x^2-48)*exp(3)*log(3)*exp(x)+3 *log(3)^2)/(64*x^2*exp(3)^2*exp(exp(3))^4+(-128*x^2*exp(3)^2*exp(x)+16*x^2 *exp(3)*log(3))*exp(exp(3))^2+64*x^2*exp(3)^2*exp(x)^2-16*x^2*exp(3)*log(3 )*exp(x)+x^2*log(3)^2),x, algorithm=\
1/2*((x + 3)*log(3) - 48*e^(x + 3) + 48*e^(2*e^3 + 3) + 3*log(3))/(8*(x + 3)*e^(x + 3) - 8*(x + 3)*e^(2*e^3 + 3) - (x + 3)*log(3) - 24*e^(x + 3) + 2 4*e^(2*e^3 + 3) + 3*log(3))
Timed out. \[ \int \frac {192 e^{6+4 e^3}+192 e^{6+2 x}+e^{3+x} \left (-48-4 x^2\right ) \log (3)+3 \log ^2(3)+e^{2 e^3} \left (-384 e^{6+x}+48 e^3 \log (3)\right )}{64 e^{6+4 e^3} x^2+64 e^{6+2 x} x^2-16 e^{3+x} x^2 \log (3)+x^2 \log ^2(3)+e^{2 e^3} \left (-128 e^{6+x} x^2+16 e^3 x^2 \log (3)\right )} \, dx=\int \frac {192\,{\mathrm {e}}^{4\,{\mathrm {e}}^3+6}+192\,{\mathrm {e}}^{2\,x+6}-{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,\left (384\,{\mathrm {e}}^{x+6}-48\,{\mathrm {e}}^3\,\ln \left (3\right )\right )+3\,{\ln \left (3\right )}^2-{\mathrm {e}}^{x+3}\,\ln \left (3\right )\,\left (4\,x^2+48\right )}{x^2\,{\ln \left (3\right )}^2-{\mathrm {e}}^{2\,{\mathrm {e}}^3}\,\left (128\,x^2\,{\mathrm {e}}^{x+6}-16\,x^2\,{\mathrm {e}}^3\,\ln \left (3\right )\right )+64\,x^2\,{\mathrm {e}}^{4\,{\mathrm {e}}^3+6}+64\,x^2\,{\mathrm {e}}^{2\,x+6}-16\,x^2\,{\mathrm {e}}^{x+3}\,\ln \left (3\right )} \,d x \]
int((192*exp(2*x)*exp(6) + exp(2*exp(3))*(48*exp(3)*log(3) - 384*exp(6)*ex p(x)) + 3*log(3)^2 + 192*exp(4*exp(3))*exp(6) - exp(3)*exp(x)*log(3)*(4*x^ 2 + 48))/(x^2*log(3)^2 + exp(2*exp(3))*(16*x^2*exp(3)*log(3) - 128*x^2*exp (6)*exp(x)) + 64*x^2*exp(4*exp(3))*exp(6) + 64*x^2*exp(2*x)*exp(6) - 16*x^ 2*exp(3)*exp(x)*log(3)),x)
int((192*exp(4*exp(3) + 6) + 192*exp(2*x + 6) - exp(2*exp(3))*(384*exp(x + 6) - 48*exp(3)*log(3)) + 3*log(3)^2 - exp(x + 3)*log(3)*(4*x^2 + 48))/(x^ 2*log(3)^2 - exp(2*exp(3))*(128*x^2*exp(x + 6) - 16*x^2*exp(3)*log(3)) + 6 4*x^2*exp(4*exp(3) + 6) + 64*x^2*exp(2*x + 6) - 16*x^2*exp(x + 3)*log(3)), x)