Integrand size = 188, antiderivative size = 26 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\left (e^{2+x} x+\frac {2 x \left (x+x^2\right )}{\log \left (e^3+x\right )}\right )^2 \] Output:
(x*exp(1)^2*exp(x)+2*x*(x^2+x)/ln(exp(3)+x))^2
Time = 0.10 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {x^2 \left (2 x (1+x)+e^{2+x} \log \left (e^3+x\right )\right )^2}{\log ^2\left (e^3+x\right )} \] Input:
Integrate[(-8*x^4 - 16*x^5 - 8*x^6 + (16*x^4 + 40*x^5 + 24*x^6 + E^(2 + x) *(-4*x^3 - 4*x^4) + E^3*(16*x^3 + 40*x^4 + 24*x^5))*Log[E^3 + x] + E^x*(E^ 5*(12*x^2 + 20*x^3 + 4*x^4) + E^2*(12*x^3 + 20*x^4 + 4*x^5))*Log[E^3 + x]^ 2 + E^(2*x)*(E^7*(2*x + 2*x^2) + E^4*(2*x^2 + 2*x^3))*Log[E^3 + x]^3)/((E^ 3 + x)*Log[E^3 + x]^3),x]
Output:
(x^2*(2*x*(1 + x) + E^(2 + x)*Log[E^3 + x])^2)/Log[E^3 + x]^2
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.
| \(\displaystyle \int \frac {-8 x^6-16 x^5-8 x^4+e^{2 x} \left (e^7 \left (2 x^2+2 x\right )+e^4 \left (2 x^3+2 x^2\right )\right ) \log ^3\left (x+e^3\right )+\left (24 x^6+40 x^5+16 x^4+e^{x+2} \left (-4 x^4-4 x^3\right )+e^3 \left (24 x^5+40 x^4+16 x^3\right )\right ) \log \left (x+e^3\right )+e^x \left (e^2 \left (4 x^5+20 x^4+12 x^3\right )+e^5 \left (4 x^4+20 x^3+12 x^2\right )\right ) \log ^2\left (x+e^3\right )}{\left (x+e^3\right ) \log ^3\left (x+e^3\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int 2 x \left (-\frac {4 (x+1)^2 x^3}{\left (x+e^3\right ) \log ^3\left (x+e^3\right )}+\frac {2 (x+1) \left (2 x (3 x+2)-e^{x+2}+e^3 (6 x+4)\right ) x^2}{\left (x+e^3\right ) \log ^2\left (x+e^3\right )}+\frac {2 e^{x+2} \left (x^2+5 x+3\right ) x}{\log \left (x+e^3\right )}+e^{2 x+4} (x+1)\right )dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int x \left (-\frac {4 (x+1)^2 x^3}{\left (x+e^3\right ) \log ^3\left (x+e^3\right )}-\frac {2 (x+1) \left (-2 x (3 x+2)-2 e^3 (3 x+2)+e^{x+2}\right ) x^2}{\left (x+e^3\right ) \log ^2\left (x+e^3\right )}+\frac {2 e^{x+2} \left (x^2+5 x+3\right ) x}{\log \left (x+e^3\right )}+e^{2 x+4} (x+1)\right )dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle 2 \int \left (\frac {4 (x+1) \left (3 \log \left (x+e^3\right ) x^2-x^2+2 \left (1+\frac {3 e^3}{2}\right ) \log \left (x+e^3\right ) x-x+2 e^3 \log \left (x+e^3\right )\right ) x^3}{\left (x+e^3\right ) \log ^3\left (x+e^3\right )}+\frac {2 e^{x+2} \left (\log \left (x+e^3\right ) x^3+5 \left (1+\frac {e^3}{5}\right ) \log \left (x+e^3\right ) x^2-x^2+3 \left (1+\frac {5 e^3}{3}\right ) \log \left (x+e^3\right ) x-x+3 e^3 \log \left (x+e^3\right )\right ) x^2}{\left (x+e^3\right ) \log ^2\left (x+e^3\right )}+e^{2 x+4} (x+1) x\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 2 \left (-\frac {12 \left (x+e^3\right )^6}{\log \left (x+e^3\right )}-\frac {20 \left (1-3 e^3\right ) \left (x+e^3\right )^5}{\log \left (x+e^3\right )}-\frac {8 \left (1-10 e^3+15 e^6\right ) \left (x+e^3\right )^4}{\log \left (x+e^3\right )}+\frac {24 e^3 \left (1-5 e^3+5 e^6\right ) \left (x+e^3\right )^3}{\log \left (x+e^3\right )}-\frac {4 e^6 \left (6-20 e^3+15 e^6\right ) \left (x+e^3\right )^2}{\log \left (x+e^3\right )}+\frac {12 x^5 \left (x+e^3\right )}{\log \left (x+e^3\right )}+\frac {10 \left (2-e^3\right ) x^4 \left (x+e^3\right )}{\log \left (x+e^3\right )}+\frac {10 e^3 x^4 \left (x+e^3\right )}{\log \left (x+e^3\right )}+\frac {8 e^3 \left (2-e^3\right ) x^3 \left (x+e^3\right )}{\log \left (x+e^3\right )}+\frac {8 \left (1-e^3\right )^2 x^3 \left (x+e^3\right )}{\log \left (x+e^3\right )}+\frac {4 e^9 \left (2-5 e^3+3 e^6\right ) \left (x+e^3\right )}{\log \left (x+e^3\right )}+\frac {2 x^5 \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}+\frac {2 \left (2-e^3\right ) x^4 \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}+\frac {2 \left (1-e^3\right )^2 x^3 \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}-\frac {2 e^3 \left (1-e^3\right )^2 x^2 \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}+\frac {2 e^6 \left (1-e^3\right )^2 x \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}-\frac {2 e^9 \left (1-e^3\right )^2 \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}+\frac {1}{2} e^{2 x+4} x^2+8 e^6 \left (6-20 e^3+15 e^6\right ) \operatorname {ExpIntegralEi}\left (2 \log \left (x+e^3\right )\right )+32 e^9 \left (2-e^3\right ) \operatorname {ExpIntegralEi}\left (2 \log \left (x+e^3\right )\right )-48 e^6 \left (1-e^3\right )^2 \operatorname {ExpIntegralEi}\left (2 \log \left (x+e^3\right )\right )-40 e^{12} \operatorname {ExpIntegralEi}\left (2 \log \left (x+e^3\right )\right )-72 e^3 \left (1-5 e^3+5 e^6\right ) \operatorname {ExpIntegralEi}\left (3 \log \left (x+e^3\right )\right )-108 e^6 \left (2-e^3\right ) \operatorname {ExpIntegralEi}\left (3 \log \left (x+e^3\right )\right )+72 e^3 \left (1-e^3\right )^2 \operatorname {ExpIntegralEi}\left (3 \log \left (x+e^3\right )\right )+180 e^9 \operatorname {ExpIntegralEi}\left (3 \log \left (x+e^3\right )\right )+32 \left (1-10 e^3+15 e^6\right ) \operatorname {ExpIntegralEi}\left (4 \log \left (x+e^3\right )\right )+128 e^3 \left (2-e^3\right ) \operatorname {ExpIntegralEi}\left (4 \log \left (x+e^3\right )\right )-32 \left (1-e^3\right )^2 \operatorname {ExpIntegralEi}\left (4 \log \left (x+e^3\right )\right )-320 e^6 \operatorname {ExpIntegralEi}\left (4 \log \left (x+e^3\right )\right )-50 \left (2-e^3\right ) \operatorname {ExpIntegralEi}\left (5 \log \left (x+e^3\right )\right )+100 \left (1-3 e^3\right ) \operatorname {ExpIntegralEi}\left (5 \log \left (x+e^3\right )\right )+250 e^3 \operatorname {ExpIntegralEi}\left (5 \log \left (x+e^3\right )\right )-4 e^9 \left (2-5 e^3+3 e^6\right ) \operatorname {LogIntegral}\left (x+e^3\right )-2 e^{12} \left (2-e^3\right ) \operatorname {LogIntegral}\left (x+e^3\right )+8 e^9 \left (1-e^3\right )^2 \operatorname {LogIntegral}\left (x+e^3\right )+2 e^{15} \operatorname {LogIntegral}\left (x+e^3\right )-2 e^6 \left (1-e^3\right ) \int \frac {e^{x+2}}{\log ^2\left (x+e^3\right )}dx-4 \left (1-e^3\right ) \int \frac {e^{x+8}}{\log ^2\left (x+e^3\right )}dx+2 \int \frac {e^{x+11}}{\log ^2\left (x+e^3\right )}dx+2 \left (1-e^3\right ) \int \frac {e^{x+11}}{\left (x+e^3\right ) \log ^2\left (x+e^3\right )}dx+6 \left (1-e^3\right ) \int \frac {e^{x+5} \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}dx-6 \int \frac {e^{x+8} \left (x+e^3\right )}{\log ^2\left (x+e^3\right )}dx-2 \left (1-e^3\right ) \int \frac {e^{x+2} \left (x+e^3\right )^2}{\log ^2\left (x+e^3\right )}dx+6 \int \frac {e^{x+5} \left (x+e^3\right )^2}{\log ^2\left (x+e^3\right )}dx-2 \int \frac {e^{x+2} \left (x+e^3\right )^3}{\log ^2\left (x+e^3\right )}dx+2 \left (3-5 e^3+e^6\right ) \int \frac {e^{x+8}}{\log \left (x+e^3\right )}dx-2 e^3 \left (6-15 e^3+4 e^6\right ) \int \frac {e^{x+2} \left (x+e^3\right )}{\log \left (x+e^3\right )}dx+6 \left (1-5 e^3+2 e^6\right ) \int \frac {e^{x+2} \left (x+e^3\right )^2}{\log \left (x+e^3\right )}dx+2 \left (5-4 e^3\right ) \int \frac {e^{x+2} \left (x+e^3\right )^3}{\log \left (x+e^3\right )}dx+2 \int \frac {e^{x+2} \left (x+e^3\right )^4}{\log \left (x+e^3\right )}dx+\frac {2 e^{12} \left (1-e^3\right )^2}{\log ^2\left (x+e^3\right )}\right )\) |
Input:
Int[(-8*x^4 - 16*x^5 - 8*x^6 + (16*x^4 + 40*x^5 + 24*x^6 + E^(2 + x)*(-4*x ^3 - 4*x^4) + E^3*(16*x^3 + 40*x^4 + 24*x^5))*Log[E^3 + x] + E^x*(E^5*(12* x^2 + 20*x^3 + 4*x^4) + E^2*(12*x^3 + 20*x^4 + 4*x^5))*Log[E^3 + x]^2 + E^ (2*x)*(E^7*(2*x + 2*x^2) + E^4*(2*x^2 + 2*x^3))*Log[E^3 + x]^3)/((E^3 + x) *Log[E^3 + x]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(54\) vs. \(2(26)=52\).
Time = 0.12 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.12
\[x^{2} {\mathrm e}^{4+2 x}+\frac {4 x^{3} \left (\ln \left ({\mathrm e}^{3}+x \right ) x \,{\mathrm e}^{2+x}+\ln \left ({\mathrm e}^{3}+x \right ) {\mathrm e}^{2+x}+x^{3}+2 x^{2}+x \right )}{\ln \left ({\mathrm e}^{3}+x \right )^{2}}\]
Input:
int((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*ln(exp( 3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp(1 )^2)*exp(x)*ln(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5+40*x^4+ 16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*ln(exp(3)+x)-8*x^6-16*x^5-8*x^4)/(exp (3)+x)/ln(exp(3)+x)^3,x)
Output:
x^2*exp(4+2*x)+4*x^3*(ln(exp(3)+x)*x*exp(2+x)+ln(exp(3)+x)*exp(2+x)+x^3+2* x^2+x)/ln(exp(3)+x)^2
Leaf count of result is larger than twice the leaf count of optimal. 59 vs. \(2 (24) = 48\).
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {4 \, x^{6} + 8 \, x^{5} + x^{2} e^{\left (2 \, x + 4\right )} \log \left (x + e^{3}\right )^{2} + 4 \, x^{4} + 4 \, {\left (x^{4} + x^{3}\right )} e^{\left (x + 2\right )} \log \left (x + e^{3}\right )}{\log \left (x + e^{3}\right )^{2}} \] Input:
integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*l og(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3 )*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5 +40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8* x^4)/(exp(3)+x)/log(exp(3)+x)^3,x, algorithm="fricas")
Output:
(4*x^6 + 8*x^5 + x^2*e^(2*x + 4)*log(x + e^3)^2 + 4*x^4 + 4*(x^4 + x^3)*e^ (x + 2)*log(x + e^3))/log(x + e^3)^2
Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.62 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {x^{2} e^{4} e^{2 x} \log {\left (x + e^{3} \right )} + \left (4 x^{4} e^{2} + 4 x^{3} e^{2}\right ) e^{x}}{\log {\left (x + e^{3} \right )}} + \frac {4 x^{6} + 8 x^{5} + 4 x^{4}}{\log {\left (x + e^{3} \right )}^{2}} \] Input:
integrate((((2*x**2+2*x)*exp(1)**4*exp(3)+(2*x**3+2*x**2)*exp(1)**4)*exp(x )**2*ln(exp(3)+x)**3+((4*x**4+20*x**3+12*x**2)*exp(1)**2*exp(3)+(4*x**5+20 *x**4+12*x**3)*exp(1)**2)*exp(x)*ln(exp(3)+x)**2+((-4*x**4-4*x**3)*exp(1)* *2*exp(x)+(24*x**5+40*x**4+16*x**3)*exp(3)+24*x**6+40*x**5+16*x**4)*ln(exp (3)+x)-8*x**6-16*x**5-8*x**4)/(exp(3)+x)/ln(exp(3)+x)**3,x)
Output:
(x**2*exp(4)*exp(2*x)*log(x + exp(3)) + (4*x**4*exp(2) + 4*x**3*exp(2))*ex p(x))/log(x + exp(3)) + (4*x**6 + 8*x**5 + 4*x**4)/log(x + exp(3))**2
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.42 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {4 \, x^{6} + 8 \, x^{5} + x^{2} e^{\left (2 \, x + 4\right )} \log \left (x + e^{3}\right )^{2} + 4 \, x^{4} + 4 \, {\left (x^{4} e^{2} + x^{3} e^{2}\right )} e^{x} \log \left (x + e^{3}\right )}{\log \left (x + e^{3}\right )^{2}} \] Input:
integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*l og(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3 )*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5 +40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8* x^4)/(exp(3)+x)/log(exp(3)+x)^3,x, algorithm="maxima")
Output:
(4*x^6 + 8*x^5 + x^2*e^(2*x + 4)*log(x + e^3)^2 + 4*x^4 + 4*(x^4*e^2 + x^3 *e^2)*e^x*log(x + e^3))/log(x + e^3)^2
Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (24) = 48\).
Time = 0.18 (sec) , antiderivative size = 349, normalized size of antiderivative = 13.42 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {4 \, {\left (x + e^{3}\right )}^{6} - 24 \, {\left (x + e^{3}\right )}^{5} e^{3} + 4 \, {\left (x + e^{3}\right )}^{4} e^{\left (x + 2\right )} \log \left (x + e^{3}\right ) + 8 \, {\left (x + e^{3}\right )}^{5} + 60 \, {\left (x + e^{3}\right )}^{4} e^{6} - 40 \, {\left (x + e^{3}\right )}^{4} e^{3} - 16 \, {\left (x + e^{3}\right )}^{3} e^{\left (x + 5\right )} \log \left (x + e^{3}\right ) + 4 \, {\left (x + e^{3}\right )}^{3} e^{\left (x + 2\right )} \log \left (x + e^{3}\right ) + {\left (x + e^{3}\right )}^{2} e^{\left (2 \, x + 4\right )} \log \left (x + e^{3}\right )^{2} + 4 \, {\left (x + e^{3}\right )}^{4} - 80 \, {\left (x + e^{3}\right )}^{3} e^{9} + 80 \, {\left (x + e^{3}\right )}^{3} e^{6} - 16 \, {\left (x + e^{3}\right )}^{3} e^{3} + 24 \, {\left (x + e^{3}\right )}^{2} e^{\left (x + 8\right )} \log \left (x + e^{3}\right ) - 12 \, {\left (x + e^{3}\right )}^{2} e^{\left (x + 5\right )} \log \left (x + e^{3}\right ) - 2 \, {\left (x + e^{3}\right )} e^{\left (2 \, x + 7\right )} \log \left (x + e^{3}\right )^{2} + 60 \, {\left (x + e^{3}\right )}^{2} e^{12} - 80 \, {\left (x + e^{3}\right )}^{2} e^{9} + 24 \, {\left (x + e^{3}\right )}^{2} e^{6} - 16 \, {\left (x + e^{3}\right )} e^{\left (x + 11\right )} \log \left (x + e^{3}\right ) + 12 \, {\left (x + e^{3}\right )} e^{\left (x + 8\right )} \log \left (x + e^{3}\right ) + e^{\left (2 \, x + 10\right )} \log \left (x + e^{3}\right )^{2} - 24 \, {\left (x + e^{3}\right )} e^{15} + 40 \, {\left (x + e^{3}\right )} e^{12} - 16 \, {\left (x + e^{3}\right )} e^{9} + 4 \, e^{\left (x + 14\right )} \log \left (x + e^{3}\right ) - 4 \, e^{\left (x + 11\right )} \log \left (x + e^{3}\right ) + 4 \, e^{18} - 8 \, e^{15} + 4 \, e^{12}}{\log \left (x + e^{3}\right )^{2}} \] Input:
integrate((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*l og(exp(3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3 )*exp(1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5 +40*x^4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8* x^4)/(exp(3)+x)/log(exp(3)+x)^3,x, algorithm="giac")
Output:
(4*(x + e^3)^6 - 24*(x + e^3)^5*e^3 + 4*(x + e^3)^4*e^(x + 2)*log(x + e^3) + 8*(x + e^3)^5 + 60*(x + e^3)^4*e^6 - 40*(x + e^3)^4*e^3 - 16*(x + e^3)^ 3*e^(x + 5)*log(x + e^3) + 4*(x + e^3)^3*e^(x + 2)*log(x + e^3) + (x + e^3 )^2*e^(2*x + 4)*log(x + e^3)^2 + 4*(x + e^3)^4 - 80*(x + e^3)^3*e^9 + 80*( x + e^3)^3*e^6 - 16*(x + e^3)^3*e^3 + 24*(x + e^3)^2*e^(x + 8)*log(x + e^3 ) - 12*(x + e^3)^2*e^(x + 5)*log(x + e^3) - 2*(x + e^3)*e^(2*x + 7)*log(x + e^3)^2 + 60*(x + e^3)^2*e^12 - 80*(x + e^3)^2*e^9 + 24*(x + e^3)^2*e^6 - 16*(x + e^3)*e^(x + 11)*log(x + e^3) + 12*(x + e^3)*e^(x + 8)*log(x + e^3 ) + e^(2*x + 10)*log(x + e^3)^2 - 24*(x + e^3)*e^15 + 40*(x + e^3)*e^12 - 16*(x + e^3)*e^9 + 4*e^(x + 14)*log(x + e^3) - 4*e^(x + 11)*log(x + e^3) + 4*e^18 - 8*e^15 + 4*e^12)/log(x + e^3)^2
Time = 4.39 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.04 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {4\,x^4}{{\ln \left (x+{\mathrm {e}}^3\right )}^2}+\frac {8\,x^5}{{\ln \left (x+{\mathrm {e}}^3\right )}^2}+\frac {4\,x^6}{{\ln \left (x+{\mathrm {e}}^3\right )}^2}+x^2\,{\mathrm {e}}^{2\,x+4}+\frac {4\,x^3\,{\mathrm {e}}^{x+2}}{\ln \left (x+{\mathrm {e}}^3\right )}+\frac {4\,x^4\,{\mathrm {e}}^{x+2}}{\ln \left (x+{\mathrm {e}}^3\right )} \] Input:
int((log(x + exp(3))*(exp(3)*(16*x^3 + 40*x^4 + 24*x^5) + 16*x^4 + 40*x^5 + 24*x^6 - exp(2)*exp(x)*(4*x^3 + 4*x^4)) - 8*x^4 - 16*x^5 - 8*x^6 + exp(x )*log(x + exp(3))^2*(exp(2)*(12*x^3 + 20*x^4 + 4*x^5) + exp(5)*(12*x^2 + 2 0*x^3 + 4*x^4)) + exp(2*x)*log(x + exp(3))^3*(exp(7)*(2*x + 2*x^2) + exp(4 )*(2*x^2 + 2*x^3)))/(log(x + exp(3))^3*(x + exp(3))),x)
Output:
(4*x^4)/log(x + exp(3))^2 + (8*x^5)/log(x + exp(3))^2 + (4*x^6)/log(x + ex p(3))^2 + x^2*exp(2*x + 4) + (4*x^3*exp(x + 2))/log(x + exp(3)) + (4*x^4*e xp(x + 2))/log(x + exp(3))
Time = 0.20 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.96 \[ \int \frac {-8 x^4-16 x^5-8 x^6+\left (16 x^4+40 x^5+24 x^6+e^{2+x} \left (-4 x^3-4 x^4\right )+e^3 \left (16 x^3+40 x^4+24 x^5\right )\right ) \log \left (e^3+x\right )+e^x \left (e^5 \left (12 x^2+20 x^3+4 x^4\right )+e^2 \left (12 x^3+20 x^4+4 x^5\right )\right ) \log ^2\left (e^3+x\right )+e^{2 x} \left (e^7 \left (2 x+2 x^2\right )+e^4 \left (2 x^2+2 x^3\right )\right ) \log ^3\left (e^3+x\right )}{\left (e^3+x\right ) \log ^3\left (e^3+x\right )} \, dx=\frac {x^{2} \left (e^{2 x} \mathrm {log}\left (e^{3}+x \right )^{2} e^{4}+4 e^{x} \mathrm {log}\left (e^{3}+x \right ) e^{2} x^{2}+4 e^{x} \mathrm {log}\left (e^{3}+x \right ) e^{2} x +4 x^{4}+8 x^{3}+4 x^{2}\right )}{\mathrm {log}\left (e^{3}+x \right )^{2}} \] Input:
int((((2*x^2+2*x)*exp(1)^4*exp(3)+(2*x^3+2*x^2)*exp(1)^4)*exp(x)^2*log(exp (3)+x)^3+((4*x^4+20*x^3+12*x^2)*exp(1)^2*exp(3)+(4*x^5+20*x^4+12*x^3)*exp( 1)^2)*exp(x)*log(exp(3)+x)^2+((-4*x^4-4*x^3)*exp(1)^2*exp(x)+(24*x^5+40*x^ 4+16*x^3)*exp(3)+24*x^6+40*x^5+16*x^4)*log(exp(3)+x)-8*x^6-16*x^5-8*x^4)/( exp(3)+x)/log(exp(3)+x)^3,x)
Output:
(x**2*(e**(2*x)*log(e**3 + x)**2*e**4 + 4*e**x*log(e**3 + x)*e**2*x**2 + 4 *e**x*log(e**3 + x)*e**2*x + 4*x**4 + 8*x**3 + 4*x**2))/log(e**3 + x)**2