Integrand size = 61, antiderivative size = 24 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=4-\frac {e^{-3+x}}{-7+e-\frac {26}{3} (-25+x)}-x \] Output:
4-x-exp(-3+x)/(exp(1)+629/3-26/3*x)
Time = 0.40 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=\frac {-e^3 x+\frac {3 e^x}{-629-3 e+26 x}}{e^3} \] Input:
Integrate[(-395641 - 9*E^2 + 32708*x - 676*x^2 + E^(-3 + x)*(-1965 - 9*E + 78*x) + E*(-3774 + 156*x))/(395641 + 9*E^2 + E*(3774 - 156*x) - 32708*x + 676*x^2),x]
Output:
(-(E^3*x) + (3*E^x)/(-629 - 3*E + 26*x))/E^3
Leaf count is larger than twice the leaf count of optimal. \(144\) vs. \(2(24)=48\).
Time = 1.00 (sec) , antiderivative size = 144, normalized size of antiderivative = 6.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {7292, 7277, 27, 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.
| \(\displaystyle \int \frac {-676 x^2+32708 x+e^{x-3} (78 x-9 e-1965)+e (156 x-3774)-9 e^2-395641}{676 x^2-32708 x+e (3774-156 x)+9 e^2+395641} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-676 x^2+32708 x+e^{x-3} (78 x-9 e-1965)+e (156 x-3774)-395641 \left (1+\frac {9 e^2}{395641}\right )}{676 x^2-52 (629+3 e) x+(629+3 e)^2}dx\) |
| \(\Big \downarrow \) 7277 |
| \(\displaystyle 2704 \int -\frac {676 x^2-32708 x+6 e (629-26 x)+3 e^{x-3} (-26 x+3 e+655)+9 e^2+395641}{2704 (-26 x+3 e+629)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {676 x^2-32708 x+6 e (629-26 x)+3 e^{x-3} (-26 x+3 e+655)+9 e^2+395641}{(-26 x+3 e+629)^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\int \frac {676 x^2-32708 x+6 e (629-26 x)+3 e^{x-3} (-26 x+3 e+655)+395641 \left (1+\frac {9 e^2}{395641}\right )}{(-26 x+3 e+629)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\int \left (\frac {676 x^2}{(-26 x+3 e+629)^2}-\frac {32708 x}{(-26 x+3 e+629)^2}+\frac {3 e^{x-3} (-26 x+3 e+655)}{(-26 x+3 e+629)^2}-\frac {6 e (26 x-629)}{(-26 x+3 e+629)^2}+\frac {395641+9 e^2}{(-26 x+3 e+629)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -x-\frac {3 e^{x-3}}{-26 x+3 e+629}-\frac {395641+9 e^2}{26 (-26 x+3 e+629)}-\frac {(629+3 e)^2}{26 (-26 x+3 e+629)}+\frac {629 (629+3 e)}{13 (-26 x+3 e+629)}+\frac {9 e^2}{13 (-26 x+3 e+629)}-\frac {1}{13} (629+3 e) \log (-26 x+3 e+629)+\frac {3}{13} e \log (-26 x+3 e+629)+\frac {629}{13} \log (-26 x+3 e+629)\) |
Input:
Int[(-395641 - 9*E^2 + 32708*x - 676*x^2 + E^(-3 + x)*(-1965 - 9*E + 78*x) + E*(-3774 + 156*x))/(395641 + 9*E^2 + E*(3774 - 156*x) - 32708*x + 676*x ^2),x]
Output:
(9*E^2)/(13*(629 + 3*E - 26*x)) - (3*E^(-3 + x))/(629 + 3*E - 26*x) + (629 *(629 + 3*E))/(13*(629 + 3*E - 26*x)) - (629 + 3*E)^2/(26*(629 + 3*E - 26* x)) - (395641 + 9*E^2)/(26*(629 + 3*E - 26*x)) - x + (629*Log[629 + 3*E - 26*x])/13 + (3*E*Log[629 + 3*E - 26*x])/13 - ((629 + 3*E)*Log[629 + 3*E - 26*x])/13
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> Simp[1/(4^p*c^p) Int[u*(b + 2*c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n} , x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p] && !AlgebraicFu nctionQ[u, x]
Time = 0.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.50
| method | result | size |
| norman | \(\frac {26 x^{2}-3 \,{\mathrm e}^{-3+x}-\frac {395641}{26}-\frac {9 \,{\mathrm e}^{2}}{26}-\frac {1887 \,{\mathrm e}}{13}}{3 \,{\mathrm e}-26 x +629}\) | \(36\) |
| parallelrisch | \(-\frac {9 \,{\mathrm e}^{2}+395641-676 x^{2}+3774 \,{\mathrm e}+78 \,{\mathrm e}^{-3+x}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}\) | \(37\) |
| parts | \(-x -\frac {1731 \,{\mathrm e}^{-3+x}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {1731 \,{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}+\frac {3 \,{\mathrm e}^{-3+x} \left (551+3 \,{\mathrm e}\right )}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-78 \left (\frac {577}{17576}+\frac {3 \,{\mathrm e}}{17576}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )-9 \,{\mathrm e} \left (\frac {{\mathrm e}^{-3+x}}{78 \,{\mathrm e}-676 x +16354}-\frac {{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}\right )\) | \(133\) |
| derivativedivides | \(-\frac {303601}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {\frac {303601}{13}+\frac {1653 \,{\mathrm e}}{13}}{3 \,{\mathrm e}-26 x +629}+\frac {551 \ln \left (3 \,{\mathrm e}-26 x +629\right )}{13}+\frac {26 \left (-3+x \right )^{2}-\frac {9 \,{\mathrm e}^{2}}{13}-\frac {3306 \,{\mathrm e}}{13}-\frac {303601}{13}}{3 \,{\mathrm e}-26 x +629}+\left (-\frac {3 \,{\mathrm e}}{13}-\frac {551}{13}\right ) \ln \left (3 \,{\mathrm e}-26 x +629\right )-\frac {1653 \,{\mathrm e}}{13 \left (3 \,{\mathrm e}-26 x +629\right )}-\frac {9 \,{\mathrm e}^{2}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-\frac {1731 \,{\mathrm e}^{-3+x}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {1731 \,{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}+\frac {\frac {9 \,{\mathrm e}^{2}}{13}+\frac {1653 \,{\mathrm e}}{13}}{3 \,{\mathrm e}-26 x +629}+\frac {3 \ln \left (3 \,{\mathrm e}-26 x +629\right ) {\mathrm e}}{13}+\frac {3 \,{\mathrm e}^{-3+x} \left (551+3 \,{\mathrm e}\right )}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-78 \left (\frac {577}{17576}+\frac {3 \,{\mathrm e}}{17576}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )-9 \,{\mathrm e} \left (\frac {{\mathrm e}^{-3+x}}{78 \,{\mathrm e}-676 x +16354}-\frac {{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}\right )\) | \(290\) |
| default | \(-\frac {303601}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {\frac {303601}{13}+\frac {1653 \,{\mathrm e}}{13}}{3 \,{\mathrm e}-26 x +629}+\frac {551 \ln \left (3 \,{\mathrm e}-26 x +629\right )}{13}+\frac {26 \left (-3+x \right )^{2}-\frac {9 \,{\mathrm e}^{2}}{13}-\frac {3306 \,{\mathrm e}}{13}-\frac {303601}{13}}{3 \,{\mathrm e}-26 x +629}+\left (-\frac {3 \,{\mathrm e}}{13}-\frac {551}{13}\right ) \ln \left (3 \,{\mathrm e}-26 x +629\right )-\frac {1653 \,{\mathrm e}}{13 \left (3 \,{\mathrm e}-26 x +629\right )}-\frac {9 \,{\mathrm e}^{2}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-\frac {1731 \,{\mathrm e}^{-3+x}}{26 \left (3 \,{\mathrm e}-26 x +629\right )}+\frac {1731 \,{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}+\frac {\frac {9 \,{\mathrm e}^{2}}{13}+\frac {1653 \,{\mathrm e}}{13}}{3 \,{\mathrm e}-26 x +629}+\frac {3 \ln \left (3 \,{\mathrm e}-26 x +629\right ) {\mathrm e}}{13}+\frac {3 \,{\mathrm e}^{-3+x} \left (551+3 \,{\mathrm e}\right )}{26 \left (3 \,{\mathrm e}-26 x +629\right )}-78 \left (\frac {577}{17576}+\frac {3 \,{\mathrm e}}{17576}\right ) {\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )-9 \,{\mathrm e} \left (\frac {{\mathrm e}^{-3+x}}{78 \,{\mathrm e}-676 x +16354}-\frac {{\mathrm e}^{\frac {3 \,{\mathrm e}}{26}+\frac {551}{26}} \operatorname {expIntegral}_{1}\left (\frac {629}{26}+\frac {3 \,{\mathrm e}}{26}-x \right )}{676}\right )\) | \(290\) |
Input:
int(((-9*exp(1)+78*x-1965)*exp(-3+x)-9*exp(1)^2+(156*x-3774)*exp(1)-676*x^ 2+32708*x-395641)/(9*exp(1)^2+(-156*x+3774)*exp(1)+676*x^2-32708*x+395641) ,x,method=_RETURNVERBOSE)
Output:
(26*x^2-3*exp(-3+x)-395641/26-9/26*exp(1)^2-1887/13*exp(1))/(3*exp(1)-26*x +629)
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=-\frac {26 \, x^{2} - 3 \, x e - 629 \, x - 3 \, e^{\left (x - 3\right )}}{26 \, x - 3 \, e - 629} \] Input:
integrate(((-9*exp(1)+78*x-1965)*exp(-3+x)-9*exp(1)^2+(156*x-3774)*exp(1)- 676*x^2+32708*x-395641)/(9*exp(1)^2+(-156*x+3774)*exp(1)+676*x^2-32708*x+3 95641),x, algorithm="fricas")
Output:
-(26*x^2 - 3*x*e - 629*x - 3*e^(x - 3))/(26*x - 3*e - 629)
Time = 0.08 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.71 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=- x + \frac {3 e^{x - 3}}{26 x - 629 - 3 e} \] Input:
integrate(((-9*exp(1)+78*x-1965)*exp(-3+x)-9*exp(1)**2+(156*x-3774)*exp(1) -676*x**2+32708*x-395641)/(9*exp(1)**2+(-156*x+3774)*exp(1)+676*x**2-32708 *x+395641),x)
Output:
-x + 3*exp(x - 3)/(26*x - 629 - 3*E)
\[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=\int { -\frac {676 \, x^{2} - 6 \, {\left (26 \, x - 629\right )} e - 3 \, {\left (26 \, x - 3 \, e - 655\right )} e^{\left (x - 3\right )} - 32708 \, x + 9 \, e^{2} + 395641}{676 \, x^{2} - 6 \, {\left (26 \, x - 629\right )} e - 32708 \, x + 9 \, e^{2} + 395641} \,d x } \] Input:
integrate(((-9*exp(1)+78*x-1965)*exp(-3+x)-9*exp(1)^2+(156*x-3774)*exp(1)- 676*x^2+32708*x-395641)/(9*exp(1)^2+(-156*x+3774)*exp(1)+676*x^2-32708*x+3 95641),x, algorithm="maxima")
Output:
-3/13*((3*e + 629)/(26*x - 3*e - 629) - log(26*x - 3*e - 629))*e - 1/13*(3 *e + 629)*log(26*x - 3*e - 629) - x + 78*x*e^x/(676*x^2*e^3 - 52*x*(3*e^4 + 629*e^3) + 9*e^5 + 3774*e^4 + 395641*e^3) + 1965/26*e^(3/26*e + 551/26)* exp_integral_e(2, -x + 3/26*e + 629/26)/(26*x - 3*e - 629) + 1/26*(9*e^2 + 3774*e + 395641)/(26*x - 3*e - 629) - 629/13*(3*e + 629)/(26*x - 3*e - 62 9) + 9/26*e^2/(26*x - 3*e - 629) + 1887/13*e/(26*x - 3*e - 629) + 395641/2 6/(26*x - 3*e - 629) - integrate(3*(26*x*(3*e - 26) - 9*e^2 - 1965*e - 163 54)*e^x/(17576*x^3*e^3 - 2028*x^2*(3*e^4 + 629*e^3) + 78*x*(9*e^5 + 3774*e ^4 + 395641*e^3) - 27*e^6 - 16983*e^5 - 3560769*e^4 - 248858189*e^3), x) + 629/13*log(26*x - 3*e - 629)
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=-\frac {26 \, x^{2} e^{3} - 3 \, x e^{4} - 629 \, x e^{3} - 3 \, e^{x}}{26 \, x e^{3} - 3 \, e^{4} - 629 \, e^{3}} \] Input:
integrate(((-9*exp(1)+78*x-1965)*exp(-3+x)-9*exp(1)^2+(156*x-3774)*exp(1)- 676*x^2+32708*x-395641)/(9*exp(1)^2+(-156*x+3774)*exp(1)+676*x^2-32708*x+3 95641),x, algorithm="giac")
Output:
-(26*x^2*e^3 - 3*x*e^4 - 629*x*e^3 - 3*e^x)/(26*x*e^3 - 3*e^4 - 629*e^3)
Time = 0.21 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.88 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=-x-\frac {3\,{\mathrm {e}}^{-3}\,{\mathrm {e}}^x}{3\,\mathrm {e}-26\,x+629} \] Input:
int(-(9*exp(2) - 32708*x + exp(x - 3)*(9*exp(1) - 78*x + 1965) + 676*x^2 - exp(1)*(156*x - 3774) + 395641)/(9*exp(2) - 32708*x + 676*x^2 - exp(1)*(1 56*x - 3774) + 395641),x)
Output:
- x - (3*exp(-3)*exp(x))/(3*exp(1) - 26*x + 629)
Time = 0.20 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {-395641-9 e^2+32708 x-676 x^2+e^{-3+x} (-1965-9 e+78 x)+e (-3774+156 x)}{395641+9 e^2+e (3774-156 x)-32708 x+676 x^2} \, dx=\frac {-3 e^{x}-3 e^{4} x +26 e^{3} x^{2}-629 e^{3} x}{e^{3} \left (3 e -26 x +629\right )} \] Input:
int(((-9*exp(1)+78*x-1965)*exp(-3+x)-9*exp(1)^2+(156*x-3774)*exp(1)-676*x^ 2+32708*x-395641)/(9*exp(1)^2+(-156*x+3774)*exp(1)+676*x^2-32708*x+395641) ,x)
Output:
( - 3*e**x - 3*e**4*x + 26*e**3*x**2 - 629*e**3*x)/(e**3*(3*e - 26*x + 629 ))