Integrand size = 127, antiderivative size = 30 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=2 \left (x-\frac {-25-e^x+x+\log (2)}{\left (3+\log \left (e^{2 x} x^2\right )\right )^2}\right ) \] Output:
2*x-2*(ln(2)-exp(x)-25+x)/(3+ln(exp(x)^2*x^2))^2
Time = 0.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.20 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=2 \left (x+\frac {100+4 e^x-4 x-\log (16)}{4 \left (3+\log \left (e^{2 x} x^2\right )\right )^2}\right ) \] Input:
Integrate[(-200 + E^x*(-8 - 2*x) - 144*x + 8*x^2 + (8 + 8*x)*Log[2] + (52* x + 2*E^x*x)*Log[E^(2*x)*x^2] + 18*x*Log[E^(2*x)*x^2]^2 + 2*x*Log[E^(2*x)* x^2]^3)/(27*x + 27*x*Log[E^(2*x)*x^2] + 9*x*Log[E^(2*x)*x^2]^2 + x*Log[E^( 2*x)*x^2]^3),x]
Output:
2*(x + (100 + 4*E^x - 4*x - Log[16])/(4*(3 + Log[E^(2*x)*x^2])^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.
| \(\displaystyle \int \frac {8 x^2+2 x \log ^3\left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+\left (2 e^x x+52 x\right ) \log \left (e^{2 x} x^2\right )+e^x (-2 x-8)-144 x+(8 x+8) \log (2)-200}{x \log ^3\left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+27 x \log \left (e^{2 x} x^2\right )+27 x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {8 x^2+2 x \log ^3\left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+\left (2 e^x x+52 x\right ) \log \left (e^{2 x} x^2\right )+e^x (-2 x-8)-144 x+(8 x+8) \log (2)-200}{x \left (\log \left (e^{2 x} x^2\right )+3\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \log ^3\left (e^{2 x} x^2\right )}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}+\frac {18 \log ^2\left (e^{2 x} x^2\right )}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}+\frac {52 \log \left (e^{2 x} x^2\right )}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}+\frac {2 e^x \left (x \log \left (e^{2 x} x^2\right )-x-4\right )}{x \left (\log \left (e^{2 x} x^2\right )+3\right )^3}+\frac {8 x}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}-\frac {200}{x \left (\log \left (e^{2 x} x^2\right )+3\right )^3}+\frac {8 (x+1) \log (2)}{x \left (\log \left (e^{2 x} x^2\right )+3\right )^3}-\frac {144}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -192 \int \frac {1}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}dx-8 \int \frac {e^x}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}dx-200 \int \frac {1}{x \left (\log \left (e^{2 x} x^2\right )+3\right )^3}dx-8 \int \frac {e^x}{x \left (\log \left (e^{2 x} x^2\right )+3\right )^3}dx+8 \int \frac {x}{\left (\log \left (e^{2 x} x^2\right )+3\right )^3}dx-2 \int \frac {1}{\left (\log \left (e^{2 x} x^2\right )+3\right )^2}dx+2 \int \frac {e^x}{\left (\log \left (e^{2 x} x^2\right )+3\right )^2}dx-\frac {2 \log (2)}{\left (\log \left (e^{2 x} x^2\right )+3\right )^2}+2 x\) |
Input:
Int[(-200 + E^x*(-8 - 2*x) - 144*x + 8*x^2 + (8 + 8*x)*Log[2] + (52*x + 2* E^x*x)*Log[E^(2*x)*x^2] + 18*x*Log[E^(2*x)*x^2]^2 + 2*x*Log[E^(2*x)*x^2]^3 )/(27*x + 27*x*Log[E^(2*x)*x^2] + 9*x*Log[E^(2*x)*x^2]^2 + x*Log[E^(2*x)*x ^2]^3),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs. \(2(28)=56\).
Time = 1.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.27
| method | result | size |
| parallelrisch | \(-\frac {-100-4 x \ln \left ({\mathrm e}^{2 x} x^{2}\right )^{2}-24 x \ln \left ({\mathrm e}^{2 x} x^{2}\right )+4 \ln \left (2\right )-32 x -4 \,{\mathrm e}^{x}}{2 \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )^{2}+6 \ln \left ({\mathrm e}^{2 x} x^{2}\right )+9\right )}\) | \(68\) |
| default | \(\frac {2 \left (12+4 \ln \left ({\mathrm e}^{2 x} x^{2}\right )-8 \ln \left (x \right )-8 x \right ) x^{2}+2 \left ({\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (x \right )\right )}^{2}+4 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (x \right )\right )+4 {\left (\ln \left ({\mathrm e}^{x}\right )-x \right )}^{2}+6 \ln \left ({\mathrm e}^{2 x} x^{2}\right )-12 \ln \left (x \right )-12 x +8\right ) x +50+2 \left (12+4 \ln \left ({\mathrm e}^{2 x} x^{2}\right )-8 \ln \left (x \right )-8 x \right ) \ln \left (x \right ) x +8 x^{3}+8 x \ln \left (x \right )^{2}+16 x^{2} \ln \left (x \right )}{{\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )+3\right )}^{2}}-\frac {2 \ln \left (2\right )}{{\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )+3\right )}^{2}}+\frac {2 \,{\mathrm e}^{x}}{{\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )+3\right )}^{2}}\) | \(202\) |
| parts | \(\frac {2 \left (12+4 \ln \left ({\mathrm e}^{2 x} x^{2}\right )-8 \ln \left (x \right )-8 x \right ) x^{2}+2 \left ({\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (x \right )\right )}^{2}+4 \left (\ln \left ({\mathrm e}^{x}\right )-x \right ) \left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )-2 \ln \left ({\mathrm e}^{x}\right )-2 \ln \left (x \right )\right )+4 {\left (\ln \left ({\mathrm e}^{x}\right )-x \right )}^{2}+6 \ln \left ({\mathrm e}^{2 x} x^{2}\right )-12 \ln \left (x \right )-12 x +8\right ) x +50+2 \left (12+4 \ln \left ({\mathrm e}^{2 x} x^{2}\right )-8 \ln \left (x \right )-8 x \right ) \ln \left (x \right ) x +8 x^{3}+8 x \ln \left (x \right )^{2}+16 x^{2} \ln \left (x \right )}{{\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )+3\right )}^{2}}-\frac {2 \ln \left (2\right )}{{\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )+3\right )}^{2}}+\frac {2 \,{\mathrm e}^{x}}{{\left (\ln \left ({\mathrm e}^{2 x} x^{2}\right )+3\right )}^{2}}\) | \(202\) |
| risch | \(2 x +\frac {8 \ln \left (2\right )-8 \,{\mathrm e}^{x}-200+8 x}{\left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )-\pi \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{x}\right )^{2} \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )-2 \pi \,\operatorname {csgn}\left (i {\mathrm e}^{x}\right ) \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{2}+\pi \operatorname {csgn}\left (i {\mathrm e}^{2 x}\right )^{3}-\pi \,\operatorname {csgn}\left (i {\mathrm e}^{2 x}\right ) \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2} {\mathrm e}^{2 x}\right )^{3}+4 i \ln \left (x \right )+4 i \ln \left ({\mathrm e}^{x}\right )+6 i\right )^{2}}\) | \(215\) |
Input:
int((2*x*ln(exp(x)^2*x^2)^3+18*x*ln(exp(x)^2*x^2)^2+(2*exp(x)*x+52*x)*ln(e xp(x)^2*x^2)+(-2*x-8)*exp(x)+(8*x+8)*ln(2)+8*x^2-144*x-200)/(x*ln(exp(x)^2 *x^2)^3+9*x*ln(exp(x)^2*x^2)^2+27*x*ln(exp(x)^2*x^2)+27*x),x,method=_RETUR NVERBOSE)
Output:
-1/2*(-100-4*x*ln(exp(x)^2*x^2)^2-24*x*ln(exp(x)^2*x^2)+4*ln(2)-32*x-4*exp (x))/(ln(exp(x)^2*x^2)^2+6*ln(exp(x)^2*x^2)+9)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (28) = 56\).
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.13 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=\frac {2 \, {\left (x \log \left (x^{2} e^{\left (2 \, x\right )}\right )^{2} + 6 \, x \log \left (x^{2} e^{\left (2 \, x\right )}\right ) + 8 \, x + e^{x} - \log \left (2\right ) + 25\right )}}{\log \left (x^{2} e^{\left (2 \, x\right )}\right )^{2} + 6 \, \log \left (x^{2} e^{\left (2 \, x\right )}\right ) + 9} \] Input:
integrate((2*x*log(exp(x)^2*x^2)^3+18*x*log(exp(x)^2*x^2)^2+(2*exp(x)*x+52 *x)*log(exp(x)^2*x^2)+(-2*x-8)*exp(x)+(8*x+8)*log(2)+8*x^2-144*x-200)/(x*l og(exp(x)^2*x^2)^3+9*x*log(exp(x)^2*x^2)^2+27*x*log(exp(x)^2*x^2)+27*x),x, algorithm="fricas")
Output:
2*(x*log(x^2*e^(2*x))^2 + 6*x*log(x^2*e^(2*x)) + 8*x + e^x - log(2) + 25)/ (log(x^2*e^(2*x))^2 + 6*log(x^2*e^(2*x)) + 9)
Time = 0.10 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=2 x + \frac {- 2 x + 2 e^{x} - 2 \log {\left (2 \right )} + 50}{\log {\left (x^{2} e^{2 x} \right )}^{2} + 6 \log {\left (x^{2} e^{2 x} \right )} + 9} \] Input:
integrate((2*x*ln(exp(x)**2*x**2)**3+18*x*ln(exp(x)**2*x**2)**2+(2*exp(x)* x+52*x)*ln(exp(x)**2*x**2)+(-2*x-8)*exp(x)+(8*x+8)*ln(2)+8*x**2-144*x-200) /(x*ln(exp(x)**2*x**2)**3+9*x*ln(exp(x)**2*x**2)**2+27*x*ln(exp(x)**2*x**2 )+27*x),x)
Output:
2*x + (-2*x + 2*exp(x) - 2*log(2) + 50)/(log(x**2*exp(2*x))**2 + 6*log(x** 2*exp(2*x)) + 9)
Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (28) = 56\).
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.33 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=\frac {2 \, {\left (4 \, x^{3} + 4 \, x \log \left (x\right )^{2} + 12 \, x^{2} + 4 \, {\left (2 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 8 \, x + e^{x} - \log \left (2\right ) + 25\right )}}{4 \, x^{2} + 4 \, {\left (2 \, x + 3\right )} \log \left (x\right ) + 4 \, \log \left (x\right )^{2} + 12 \, x + 9} \] Input:
integrate((2*x*log(exp(x)^2*x^2)^3+18*x*log(exp(x)^2*x^2)^2+(2*exp(x)*x+52 *x)*log(exp(x)^2*x^2)+(-2*x-8)*exp(x)+(8*x+8)*log(2)+8*x^2-144*x-200)/(x*l og(exp(x)^2*x^2)^3+9*x*log(exp(x)^2*x^2)^2+27*x*log(exp(x)^2*x^2)+27*x),x, algorithm="maxima")
Output:
2*(4*x^3 + 4*x*log(x)^2 + 12*x^2 + 4*(2*x^2 + 3*x)*log(x) + 8*x + e^x - lo g(2) + 25)/(4*x^2 + 4*(2*x + 3)*log(x) + 4*log(x)^2 + 12*x + 9)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (28) = 56\).
Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.60 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=\frac {2 \, {\left (4 \, x^{3} + 4 \, x^{2} \log \left (x^{2}\right ) + x \log \left (x^{2}\right )^{2} + 12 \, x^{2} + 6 \, x \log \left (x^{2}\right ) + 8 \, x + e^{x} - \log \left (2\right ) + 25\right )}}{4 \, x^{2} + 4 \, x \log \left (x^{2}\right ) + \log \left (x^{2}\right )^{2} + 12 \, x + 6 \, \log \left (x^{2}\right ) + 9} \] Input:
integrate((2*x*log(exp(x)^2*x^2)^3+18*x*log(exp(x)^2*x^2)^2+(2*exp(x)*x+52 *x)*log(exp(x)^2*x^2)+(-2*x-8)*exp(x)+(8*x+8)*log(2)+8*x^2-144*x-200)/(x*l og(exp(x)^2*x^2)^3+9*x*log(exp(x)^2*x^2)^2+27*x*log(exp(x)^2*x^2)+27*x),x, algorithm="giac")
Output:
2*(4*x^3 + 4*x^2*log(x^2) + x*log(x^2)^2 + 12*x^2 + 6*x*log(x^2) + 8*x + e ^x - log(2) + 25)/(4*x^2 + 4*x*log(x^2) + log(x^2)^2 + 12*x + 6*log(x^2) + 9)
Time = 3.38 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=2\,x-\frac {2\,x+\ln \left (4\right )-2\,{\mathrm {e}}^x-50}{{\left (\ln \left (x^2\,{\mathrm {e}}^{2\,x}\right )+3\right )}^2} \] Input:
int((log(2)*(8*x + 8) - 144*x + 18*x*log(x^2*exp(2*x))^2 + 2*x*log(x^2*exp (2*x))^3 + log(x^2*exp(2*x))*(52*x + 2*x*exp(x)) - exp(x)*(2*x + 8) + 8*x^ 2 - 200)/(27*x + 9*x*log(x^2*exp(2*x))^2 + x*log(x^2*exp(2*x))^3 + 27*x*lo g(x^2*exp(2*x))),x)
Output:
2*x - (2*x + log(4) - 2*exp(x) - 50)/(log(x^2*exp(2*x)) + 3)^2
Time = 0.17 (sec) , antiderivative size = 132, normalized size of antiderivative = 4.40 \[ \int \frac {-200+e^x (-8-2 x)-144 x+8 x^2+(8+8 x) \log (2)+\left (52 x+2 e^x x\right ) \log \left (e^{2 x} x^2\right )+18 x \log ^2\left (e^{2 x} x^2\right )+2 x \log ^3\left (e^{2 x} x^2\right )}{27 x+27 x \log \left (e^{2 x} x^2\right )+9 x \log ^2\left (e^{2 x} x^2\right )+x \log ^3\left (e^{2 x} x^2\right )} \, dx=\frac {18 e^{x}+8 \mathrm {log}\left (e^{2 x} x^{2}\right )^{3}-16 \mathrm {log}\left (e^{2 x} x^{2}\right )^{2} \mathrm {log}\left (x \right )+2 \mathrm {log}\left (e^{2 x} x^{2}\right )^{2} x +36 \mathrm {log}\left (e^{2 x} x^{2}\right )^{2}-96 \,\mathrm {log}\left (e^{2 x} x^{2}\right ) \mathrm {log}\left (x \right )+12 \,\mathrm {log}\left (e^{2 x} x^{2}\right ) x -144 \,\mathrm {log}\left (x \right )-18 \,\mathrm {log}\left (2\right )+342}{9 \mathrm {log}\left (e^{2 x} x^{2}\right )^{2}+54 \,\mathrm {log}\left (e^{2 x} x^{2}\right )+81} \] Input:
int((2*x*log(exp(x)^2*x^2)^3+18*x*log(exp(x)^2*x^2)^2+(2*exp(x)*x+52*x)*lo g(exp(x)^2*x^2)+(-2*x-8)*exp(x)+(8*x+8)*log(2)+8*x^2-144*x-200)/(x*log(exp (x)^2*x^2)^3+9*x*log(exp(x)^2*x^2)^2+27*x*log(exp(x)^2*x^2)+27*x),x)
Output:
(2*(9*e**x + 4*log(e**(2*x)*x**2)**3 - 8*log(e**(2*x)*x**2)**2*log(x) + lo g(e**(2*x)*x**2)**2*x + 18*log(e**(2*x)*x**2)**2 - 48*log(e**(2*x)*x**2)*l og(x) + 6*log(e**(2*x)*x**2)*x - 72*log(x) - 9*log(2) + 171))/(9*(log(e**( 2*x)*x**2)**2 + 6*log(e**(2*x)*x**2) + 9))