Integrand size = 144, antiderivative size = 25 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=1+x+\left (-x+\frac {1}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )}\right )^2 \]
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.60 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=x+x^2+\frac {1}{\left (x+\log \left (\frac {3 e^{x+x^2}}{4}\right )\right )^2}-\frac {2 x}{x+\log \left (\frac {3 e^{x+x^2}}{4}\right )} \]
Integrate[(-4 - 4*x + 2*x^2 + 5*x^3 + 2*x^4 + (7*x^2 + 6*x^3)*Log[(3*E^(x + x^2))/4] + (-2 + 3*x + 6*x^2)*Log[(3*E^(x + x^2))/4]^2 + (1 + 2*x)*Log[( 3*E^(x + x^2))/4]^3)/(x^3 + 3*x^2*Log[(3*E^(x + x^2))/4] + 3*x*Log[(3*E^(x + x^2))/4]^2 + Log[(3*E^(x + x^2))/4]^3),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.
| \(\displaystyle \int \frac {2 x^4+5 x^3+2 x^2+(2 x+1) \log ^3\left (\frac {3 e^{x^2+x}}{4}\right )+\left (6 x^2+3 x-2\right ) \log ^2\left (\frac {3 e^{x^2+x}}{4}\right )+\left (6 x^3+7 x^2\right ) \log \left (\frac {3 e^{x^2+x}}{4}\right )-4 x-4}{x^3+\log ^3\left (\frac {3 e^{x^2+x}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x^2+x}}{4}\right )+3 x^2 \log \left (\frac {3 e^{x^2+x}}{4}\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {2 x^4+5 x^3+2 x^2+(2 x+1) \log ^3\left (\frac {3 e^{x^2+x}}{4}\right )+\left (6 x^2+3 x-2\right ) \log ^2\left (\frac {3 e^{x^2+x}}{4}\right )+\left (6 x^3+7 x^2\right ) \log \left (\frac {3 e^{x^2+x}}{4}\right )-4 x-4}{\left (\log \left (\frac {3 e^{x^2+x}}{4}\right )+x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {4 (x+1) x}{\left (\log \left (\frac {3 e^{x^2+x}}{4}\right )+x\right )^2}-\frac {2}{\log \left (\frac {3 e^{x^2+x}}{4}\right )+x}-\frac {4 (x+1)}{\left (\log \left (\frac {3 e^{x^2+x}}{4}\right )+x\right )^3}+2 x+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 \int \frac {x}{\left (x+\log \left (\frac {3 e^{x^2+x}}{4}\right )\right )^2}dx+4 \int \frac {x^2}{\left (x+\log \left (\frac {3 e^{x^2+x}}{4}\right )\right )^2}dx-2 \int \frac {1}{x+\log \left (\frac {3 e^{x^2+x}}{4}\right )}dx+x^2+\frac {1}{\left (\log \left (\frac {3 e^{x^2+x}}{4}\right )+x\right )^2}+x\) |
Int[(-4 - 4*x + 2*x^2 + 5*x^3 + 2*x^4 + (7*x^2 + 6*x^3)*Log[(3*E^(x + x^2) )/4] + (-2 + 3*x + 6*x^2)*Log[(3*E^(x + x^2))/4]^2 + (1 + 2*x)*Log[(3*E^(x + x^2))/4]^3)/(x^3 + 3*x^2*Log[(3*E^(x + x^2))/4] + 3*x*Log[(3*E^(x + x^2 ))/4]^2 + Log[(3*E^(x + x^2))/4]^3),x]
3.10.48.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(53\) vs. \(2(22)=44\).
Time = 4.83 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.16
| method | result | size |
| default | \(x^{2}+x +\frac {-2 x^{3}-4 x^{2}-2 x \left (\ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )-x^{2}-x \right )+1}{{\left (\ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )+x \right )}^{2}}\) | \(54\) |
| parallelrisch | \(\frac {1-x^{4}-2 \ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right ) x^{3}-\ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )^{2} x^{2}-x^{3}+3 x \ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )^{2}+2 \ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )^{3}+2 x \ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )+2 \ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )^{2}}{x^{2}+2 x \ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )+\ln \left (\frac {3 \,{\mathrm e}^{x} {\mathrm e}^{x^{2}}}{4}\right )^{2}}\) | \(125\) |
| risch | \(x^{2}+x -\frac {4 i \left (-i+\pi x \operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right )^{3}-\pi x \operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{2}}\right )-\pi x \operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+\pi x \,\operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-4 i x \ln \left (2\right )+2 i \ln \left (3\right ) x +2 i x^{2}+2 i x \ln \left ({\mathrm e}^{x}\right )+2 i x \ln \left ({\mathrm e}^{x^{2}}\right )\right )}{{\left (\pi \operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right )^{3}-\pi \operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x^{2}}\right )-\pi \operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{x}\right )+\pi \,\operatorname {csgn}\left (i {\mathrm e}^{\left (1+x \right ) x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x^{2}}\right ) \operatorname {csgn}\left (i {\mathrm e}^{x}\right )-4 i \ln \left (2\right )+2 i \ln \left (3\right )+2 i x +2 i \ln \left ({\mathrm e}^{x}\right )+2 i \ln \left ({\mathrm e}^{x^{2}}\right )\right )}^{2}}\) | \(249\) |
int(((1+2*x)*ln(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*ln(3/4*exp(x)*exp(x^2 ))^2+(6*x^3+7*x^2)*ln(3/4*exp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x-4)/(ln(3/ 4*exp(x)*exp(x^2))^3+3*x*ln(3/4*exp(x)*exp(x^2))^2+3*x^2*ln(3/4*exp(x)*exp (x^2))+x^3),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (22) = 44\).
Time = 0.25 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.52 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=\frac {x^{6} + 5 \, x^{5} + 8 \, x^{4} + 2 \, x^{3} + {\left (x^{2} + x\right )} \log \left (\frac {3}{4}\right )^{2} - 4 \, x^{2} + 2 \, {\left (x^{4} + 3 \, x^{3} + 2 \, x^{2} - x\right )} \log \left (\frac {3}{4}\right ) + 1}{x^{4} + 4 \, x^{3} + 4 \, x^{2} + 2 \, {\left (x^{2} + 2 \, x\right )} \log \left (\frac {3}{4}\right ) + \log \left (\frac {3}{4}\right )^{2}} \]
integrate(((1+2*x)*log(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*log(3/4*exp(x) *exp(x^2))^2+(6*x^3+7*x^2)*log(3/4*exp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x- 4)/(log(3/4*exp(x)*exp(x^2))^3+3*x*log(3/4*exp(x)*exp(x^2))^2+3*x^2*log(3/ 4*exp(x)*exp(x^2))+x^3),x, algorithm=\
(x^6 + 5*x^5 + 8*x^4 + 2*x^3 + (x^2 + x)*log(3/4)^2 - 4*x^2 + 2*(x^4 + 3*x ^3 + 2*x^2 - x)*log(3/4) + 1)/(x^4 + 4*x^3 + 4*x^2 + 2*(x^2 + 2*x)*log(3/4 ) + log(3/4)^2)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (22) = 44\).
Time = 2.01 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=x^{2} + x + \frac {- 2 x^{3} - 4 x^{2} + x \left (- 2 \log {\left (3 \right )} + 4 \log {\left (2 \right )}\right ) + 1}{x^{4} + 4 x^{3} + x^{2} \left (- 4 \log {\left (2 \right )} + 2 \log {\left (3 \right )} + 4\right ) + x \left (- 8 \log {\left (2 \right )} + 4 \log {\left (3 \right )}\right ) - 4 \log {\left (2 \right )} \log {\left (3 \right )} + \log {\left (3 \right )}^{2} + 4 \log {\left (2 \right )}^{2}} \]
integrate(((1+2*x)*ln(3/4*exp(x)*exp(x**2))**3+(6*x**2+3*x-2)*ln(3/4*exp(x )*exp(x**2))**2+(6*x**3+7*x**2)*ln(3/4*exp(x)*exp(x**2))+2*x**4+5*x**3+2*x **2-4*x-4)/(ln(3/4*exp(x)*exp(x**2))**3+3*x*ln(3/4*exp(x)*exp(x**2))**2+3* x**2*ln(3/4*exp(x)*exp(x**2))+x**3),x)
x**2 + x + (-2*x**3 - 4*x**2 + x*(-2*log(3) + 4*log(2)) + 1)/(x**4 + 4*x** 3 + x**2*(-4*log(2) + 2*log(3) + 4) + x*(-8*log(2) + 4*log(3)) - 4*log(2)* log(3) + log(3)**2 + 4*log(2)**2)
Leaf count of result is larger than twice the leaf count of optimal. 10769 vs. \(2 (22) = 44\).
Time = 2.63 (sec) , antiderivative size = 10769, normalized size of antiderivative = 430.76 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=\text {Too large to display} \]
integrate(((1+2*x)*log(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*log(3/4*exp(x) *exp(x^2))^2+(6*x^3+7*x^2)*log(3/4*exp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x- 4)/(log(3/4*exp(x)*exp(x^2))^3+3*x*log(3/4*exp(x)*exp(x^2))^2+3*x^2*log(3/ 4*exp(x)*exp(x^2))+x^3),x, algorithm=\
-1/8*(2*(3*x^3 + 9*x^2 + x*(5*log(3) - 10*log(2) + 4) + 2*log(3)^2 - 2*(4* log(3) + 1)*log(2) + 8*log(2)^2 + log(3))/((log(3)^2 - 4*(log(3) - 1)*log( 2) + 4*log(2)^2 - 2*log(3) + 1)*x^4 + 4*(log(3)^2 - 4*(log(3) - 1)*log(2) + 4*log(2)^2 - 2*log(3) + 1)*x^3 + log(3)^4 - 16*(2*log(3) - 1)*log(2)^3 + 16*log(2)^4 + 2*(log(3)^3 + 12*log(3)*log(2)^2 - 8*log(2)^3 - 6*(log(3)^2 - 1)*log(2) - 3*log(3) + 2)*x^2 - 2*log(3)^3 + 4*(6*log(3)^2 - 6*log(3) + 1)*log(2)^2 + 4*(log(3)^3 + 4*(3*log(3) - 2)*log(2)^2 - 8*log(2)^3 - 2*lo g(3)^2 - 2*(3*log(3)^2 - 4*log(3) + 1)*log(2) + log(3))*x + log(3)^2 - 4*( 2*log(3)^3 - 3*log(3)^2 + log(3))*log(2)) + 3*log((x - sqrt(-log(3) + 2*lo g(2) + 1) + 1)/(x + sqrt(-log(3) + 2*log(2) + 1) + 1))/((log(3)^2 - 4*(log (3) - 1)*log(2) + 4*log(2)^2 - 2*log(3) + 1)*sqrt(-log(3) + 2*log(2) + 1)) )*log(3/4*e^(x^2 + x))^3 + 1/16*(2*(3*x^3 + 9*x^2 + x*(5*log(3) - 10*log(2 ) + 4) + 5*log(3) - 10*log(2) - 2)/((log(3)^2 - 4*(log(3) - 1)*log(2) + 4* log(2)^2 - 2*log(3) + 1)*x^4 + 4*(log(3)^2 - 4*(log(3) - 1)*log(2) + 4*log (2)^2 - 2*log(3) + 1)*x^3 + log(3)^4 - 16*(2*log(3) - 1)*log(2)^3 + 16*log (2)^4 + 2*(log(3)^3 + 12*log(3)*log(2)^2 - 8*log(2)^3 - 6*(log(3)^2 - 1)*l og(2) - 3*log(3) + 2)*x^2 - 2*log(3)^3 + 4*(6*log(3)^2 - 6*log(3) + 1)*log (2)^2 + 4*(log(3)^3 + 4*(3*log(3) - 2)*log(2)^2 - 8*log(2)^3 - 2*log(3)^2 - 2*(3*log(3)^2 - 4*log(3) + 1)*log(2) + log(3))*x + log(3)^2 - 4*(2*log(3 )^3 - 3*log(3)^2 + log(3))*log(2)) + 3*log((x - sqrt(-log(3) + 2*log(2)...
Time = 0.27 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=x^{2} + x - \frac {2 \, x^{3} + 4 \, x^{2} + 2 \, x \log \left (3\right ) - 4 \, x \log \left (2\right ) - 1}{{\left (x^{2} + 2 \, x + \log \left (3\right ) - 2 \, \log \left (2\right )\right )}^{2}} \]
integrate(((1+2*x)*log(3/4*exp(x)*exp(x^2))^3+(6*x^2+3*x-2)*log(3/4*exp(x) *exp(x^2))^2+(6*x^3+7*x^2)*log(3/4*exp(x)*exp(x^2))+2*x^4+5*x^3+2*x^2-4*x- 4)/(log(3/4*exp(x)*exp(x^2))^3+3*x*log(3/4*exp(x)*exp(x^2))^2+3*x^2*log(3/ 4*exp(x)*exp(x^2))+x^3),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-4-4 x+2 x^2+5 x^3+2 x^4+\left (7 x^2+6 x^3\right ) \log \left (\frac {3 e^{x+x^2}}{4}\right )+\left (-2+3 x+6 x^2\right ) \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+(1+2 x) \log ^3\left (\frac {3 e^{x+x^2}}{4}\right )}{x^3+3 x^2 \log \left (\frac {3 e^{x+x^2}}{4}\right )+3 x \log ^2\left (\frac {3 e^{x+x^2}}{4}\right )+\log ^3\left (\frac {3 e^{x+x^2}}{4}\right )} \, dx=x-\frac {2\,x^3+4\,x^2+2\,\ln \left (\frac {3}{4}\right )\,x-1}{{\left (x^2+2\,x+\ln \left (\frac {3}{4}\right )\right )}^2}+x^2 \]