Integrand size = 126, antiderivative size = 26 \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx=-2+x+\frac {e^x x^2}{5+e^3+81 \left (2+e^x\right )^2} \]
Time = 1.91 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx=x+\frac {e^x x^2}{329+e^3+324 e^x+81 e^{2 x}} \]
Integrate[(108241 + 658*E^3 + E^6 + 6561*E^(4*x) + E^(2*x)*(158274 + 162*E ^3 + 648*x) + E^(3*x)*(52488 + 162*x - 81*x^2) + E^x*(213192 + 658*x + 329 *x^2 + E^3*(648 + 2*x + x^2)))/(108241 + 658*E^3 + E^6 + 52488*E^(3*x) + 6 561*E^(4*x) + E^(2*x)*(158274 + 162*E^3) + E^x*(213192 + 648*E^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.
| \(\displaystyle \int \frac {e^{3 x} \left (-81 x^2+162 x+52488\right )+e^x \left (329 x^2+e^3 \left (x^2+2 x+648\right )+658 x+213192\right )+e^{2 x} \left (648 x+162 e^3+158274\right )+6561 e^{4 x}+e^6+658 e^3+108241}{52488 e^{3 x}+6561 e^{4 x}+\left (158274+162 e^3\right ) e^{2 x}+\left (213192+648 e^3\right ) e^x+108241+658 e^3+e^6} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{3 x} \left (-81 x^2+162 x+52488\right )+e^x \left (329 x^2+e^3 \left (x^2+2 x+648\right )+658 x+213192\right )+e^{2 x} \left (648 x+162 e^3+158274\right )+6561 e^{4 x}+108241 \left (1+\frac {e^3 \left (658+e^3\right )}{108241}\right )}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 \left (-319 \left (1-\frac {e^3}{319}\right ) e^x-658 \left (1+\frac {e^3}{329}\right )\right ) x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}+\frac {\left (-e^x x+4 x+2 e^x\right ) x}{324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )}+1\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 x^3}{3 \left (5+e^3+18 i \sqrt {5+e^3}\right )}+\frac {2 x^3}{3 \left (5+e^3-18 i \sqrt {5+e^3}\right )}-\frac {2 \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right ) x^2}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right ) x^2}{18 \sqrt {5+e^3}}-\frac {2 \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right ) x^2}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right ) x^2}{18 \sqrt {5+e^3}}-\frac {i \log \left (1+\frac {9 e^x}{18-i \sqrt {5+e^3}}\right ) x}{9 \sqrt {5+e^3}}+\frac {i \log \left (1+\frac {9 e^x}{18+i \sqrt {5+e^3}}\right ) x}{9 \sqrt {5+e^3}}-\frac {4 \operatorname {PolyLog}\left (2,-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right ) x}{5+e^3+18 i \sqrt {5+e^3}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right ) x}{9 \sqrt {5+e^3}}-\frac {4 \operatorname {PolyLog}\left (2,-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right ) x}{5+e^3-18 i \sqrt {5+e^3}}-\frac {i \operatorname {PolyLog}\left (2,-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right ) x}{9 \sqrt {5+e^3}}+x-\frac {i \operatorname {PolyLog}\left (2,-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {i \operatorname {PolyLog}\left (2,-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{5+e^3+18 i \sqrt {5+e^3}}-\frac {i \operatorname {PolyLog}\left (3,-\frac {9 e^x}{18-i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}+\frac {4 \operatorname {PolyLog}\left (3,-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{5+e^3-18 i \sqrt {5+e^3}}+\frac {i \operatorname {PolyLog}\left (3,-\frac {9 e^x}{18+i \sqrt {5+e^3}}\right )}{9 \sqrt {5+e^3}}-4 \left (329+e^3\right ) \int \frac {x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}dx-2 \left (319-e^3\right ) \int \frac {e^x x^2}{\left (324 e^x+81 e^{2 x}+329 \left (1+\frac {e^3}{329}\right )\right )^2}dx\) |
Int[(108241 + 658*E^3 + E^6 + 6561*E^(4*x) + E^(2*x)*(158274 + 162*E^3 + 6 48*x) + E^(3*x)*(52488 + 162*x - 81*x^2) + E^x*(213192 + 658*x + 329*x^2 + E^3*(648 + 2*x + x^2)))/(108241 + 658*E^3 + E^6 + 52488*E^(3*x) + 6561*E^ (4*x) + E^(2*x)*(158274 + 162*E^3) + E^x*(213192 + 648*E^3)),x]
3.4.16.3.1 Defintions of rubi rules used
Time = 0.50 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96
| method | result | size |
| risch | \(x +\frac {x^{2} {\mathrm e}^{x}}{81 \,{\mathrm e}^{2 x}+{\mathrm e}^{3}+324 \,{\mathrm e}^{x}+329}\) | \(25\) |
| norman | \(\frac {\left ({\mathrm e}^{3}+329\right ) x +{\mathrm e}^{x} x^{2}+81 x \,{\mathrm e}^{2 x}+324 \,{\mathrm e}^{x} x}{81 \,{\mathrm e}^{2 x}+{\mathrm e}^{3}+324 \,{\mathrm e}^{x}+329}\) | \(43\) |
| parallelrisch | \(\frac {81 \,{\mathrm e}^{x} x^{2}+6561 x \,{\mathrm e}^{2 x}+81 x \,{\mathrm e}^{3}+26244 \,{\mathrm e}^{x} x +26649 x}{6561 \,{\mathrm e}^{2 x}+81 \,{\mathrm e}^{3}+26244 \,{\mathrm e}^{x}+26649}\) | \(47\) |
int((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+158274 )*exp(x)^2+((x^2+2*x+648)*exp(3)+329*x^2+658*x+213192)*exp(x)+exp(3)^2+658 *exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*exp(x)^2 +(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x,method=_RETURNVE RBOSE)
Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx=\frac {x e^{3} + 81 \, x e^{\left (2 \, x\right )} + {\left (x^{2} + 324 \, x\right )} e^{x} + 329 \, x}{e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329} \]
integrate((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+ 158274)*exp(x)^2+((x^2+2*x+648)*exp(3)+329*x^2+658*x+213192)*exp(x)+exp(3) ^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*ex p(x)^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x, algorithm =\
Time = 0.10 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx=\frac {x^{2} e^{x}}{81 e^{2 x} + 324 e^{x} + e^{3} + 329} + x \]
integrate((6561*exp(x)**4+(-81*x**2+162*x+52488)*exp(x)**3+(162*exp(3)+648 *x+158274)*exp(x)**2+((x**2+2*x+648)*exp(3)+329*x**2+658*x+213192)*exp(x)+ exp(3)**2+658*exp(3)+108241)/(6561*exp(x)**4+52488*exp(x)**3+(162*exp(3)+1 58274)*exp(x)**2+(648*exp(3)+213192)*exp(x)+exp(3)**2+658*exp(3)+108241),x )
Leaf count of result is larger than twice the leaf count of optimal. 585 vs. \(2 (23) = 46\).
Time = 0.34 (sec) , antiderivative size = 585, normalized size of antiderivative = 22.50 \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx =\text {Too large to display} \]
integrate((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+ 158274)*exp(x)^2+((x^2+2*x+648)*exp(3)+329*x^2+658*x+213192)*exp(x)+exp(3) ^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*ex p(x)^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x, algorithm =\
-1/2*(54*(e^3 + 113)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/((e^9 + 663*e^6 + 1 11531*e^3 + 541205)*sqrt(e^3 + 5)) - 2*x/(e^6 + 658*e^3 + 108241) - (e^3 - 162*e^x - 319)/(81*(e^6 + 334*e^3 + 1645)*e^(2*x) + 324*(e^6 + 334*e^3 + 1645)*e^x + e^9 + 663*e^6 + 111531*e^3 + 541205) + log(e^3 + 81*e^(2*x) + 324*e^x + 329)/(e^6 + 658*e^3 + 108241))*e^6 - 329*(54*(e^3 + 113)*arctan( 9*(e^x + 2)/sqrt(e^3 + 5))/((e^9 + 663*e^6 + 111531*e^3 + 541205)*sqrt(e^3 + 5)) - 2*x/(e^6 + 658*e^3 + 108241) - (e^3 - 162*e^x - 319)/(81*(e^6 + 3 34*e^3 + 1645)*e^(2*x) + 324*(e^6 + 334*e^3 + 1645)*e^x + e^9 + 663*e^6 + 111531*e^3 + 541205) + log(e^3 + 81*e^(2*x) + 324*e^x + 329)/(e^6 + 658*e^ 3 + 108241))*e^3 - x + 9*(3*e^3 - 977)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/( e^3 + 5)^(3/2) - 2922507*(e^3 + 113)*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/((e ^9 + 663*e^6 + 111531*e^3 + 541205)*sqrt(e^3 + 5)) + 1/2*(162*x*(e^3 + 5)* e^(2*x) + 2*x*(e^6 + 334*e^3 + 1645) + 2*(x^2*(e^3 + 5) + 324*x*(e^3 + 5) + 81*e^3 - 79947)*e^x - e^6 - 10*e^3 - 321433)/(81*(e^3 + 5)*e^(2*x) + 324 *(e^3 + 5)*e^x + e^6 + 334*e^3 + 1645) + 108241*x/(e^6 + 658*e^3 + 108241) + 108241/2*(e^3 - 162*e^x - 319)/(81*(e^6 + 334*e^3 + 1645)*e^(2*x) + 324 *(e^6 + 334*e^3 + 1645)*e^x + e^9 + 663*e^6 + 111531*e^3 + 541205) + 10659 6*(e^x + 2)/(81*(e^3 + 5)*e^(2*x) + 324*(e^3 + 5)*e^x + e^6 + 334*e^3 + 16 45) - 108241/2*log(e^3 + 81*e^(2*x) + 324*e^x + 329)/(e^6 + 658*e^3 + 1082 41) + 11844*arctan(9*(e^x + 2)/sqrt(e^3 + 5))/(e^3 + 5)^(3/2) + 1/2*log...
Time = 0.35 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.69 \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx=\frac {2 \, x^{2} e^{x} + x e^{3} + 81 \, x e^{\left (2 \, x\right )} + 324 \, x e^{x} + 329 \, x}{e^{3} + 81 \, e^{\left (2 \, x\right )} + 324 \, e^{x} + 329} \]
integrate((6561*exp(x)^4+(-81*x^2+162*x+52488)*exp(x)^3+(162*exp(3)+648*x+ 158274)*exp(x)^2+((x^2+2*x+648)*exp(3)+329*x^2+658*x+213192)*exp(x)+exp(3) ^2+658*exp(3)+108241)/(6561*exp(x)^4+52488*exp(x)^3+(162*exp(3)+158274)*ex p(x)^2+(648*exp(3)+213192)*exp(x)+exp(3)^2+658*exp(3)+108241),x, algorithm =\
Timed out. \[ \int \frac {108241+658 e^3+e^6+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3+648 x\right )+e^{3 x} \left (52488+162 x-81 x^2\right )+e^x \left (213192+658 x+329 x^2+e^3 \left (648+2 x+x^2\right )\right )}{108241+658 e^3+e^6+52488 e^{3 x}+6561 e^{4 x}+e^{2 x} \left (158274+162 e^3\right )+e^x \left (213192+648 e^3\right )} \, dx=\int \frac {6561\,{\mathrm {e}}^{4\,x}+658\,{\mathrm {e}}^3+{\mathrm {e}}^6+{\mathrm {e}}^{3\,x}\,\left (-81\,x^2+162\,x+52488\right )+{\mathrm {e}}^x\,\left (658\,x+{\mathrm {e}}^3\,\left (x^2+2\,x+648\right )+329\,x^2+213192\right )+{\mathrm {e}}^{2\,x}\,\left (648\,x+162\,{\mathrm {e}}^3+158274\right )+108241}{52488\,{\mathrm {e}}^{3\,x}+6561\,{\mathrm {e}}^{4\,x}+658\,{\mathrm {e}}^3+{\mathrm {e}}^6+{\mathrm {e}}^x\,\left (648\,{\mathrm {e}}^3+213192\right )+{\mathrm {e}}^{2\,x}\,\left (162\,{\mathrm {e}}^3+158274\right )+108241} \,d x \]
int((6561*exp(4*x) + 658*exp(3) + exp(6) + exp(3*x)*(162*x - 81*x^2 + 5248 8) + exp(x)*(658*x + exp(3)*(2*x + x^2 + 648) + 329*x^2 + 213192) + exp(2* x)*(648*x + 162*exp(3) + 158274) + 108241)/(52488*exp(3*x) + 6561*exp(4*x) + 658*exp(3) + exp(6) + exp(x)*(648*exp(3) + 213192) + exp(2*x)*(162*exp( 3) + 158274) + 108241),x)
int((6561*exp(4*x) + 658*exp(3) + exp(6) + exp(3*x)*(162*x - 81*x^2 + 5248 8) + exp(x)*(658*x + exp(3)*(2*x + x^2 + 648) + 329*x^2 + 213192) + exp(2* x)*(648*x + 162*exp(3) + 158274) + 108241)/(52488*exp(3*x) + 6561*exp(4*x) + 658*exp(3) + exp(6) + exp(x)*(648*exp(3) + 213192) + exp(2*x)*(162*exp( 3) + 158274) + 108241), x)