Integrand size = 160, antiderivative size = 36 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=3 x+\frac {(4+4 \log (2))^2}{9 \left (e^{\frac {-1+2 x}{x}}-x\right )^2 x^2} \]
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.78 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=\frac {27 e^4 x^3-54 e^{2+\frac {1}{x}} x^4+e^{2/x} \left (27 x^5+16 (1+\log (2))^2\right )}{9 x^2 \left (e^2-e^{\frac {1}{x}} x\right )^2} \]
Integrate[(64*x^2 + 27*E^((3*(-1 + 2*x))/x)*x^4 - 81*E^((2*(-1 + 2*x))/x)* x^5 - 27*x^7 + 128*x^2*Log[2] + 64*x^2*Log[2]^2 + E^((-1 + 2*x)/x)*(-32 - 32*x + 81*x^6 + (-64 - 64*x)*Log[2] + (-32 - 32*x)*Log[2]^2))/(9*E^((3*(-1 + 2*x))/x)*x^4 - 27*E^((2*(-1 + 2*x))/x)*x^5 + 27*E^((-1 + 2*x)/x)*x^6 - 9*x^7),x]
(27*E^4*x^3 - 54*E^(2 + x^(-1))*x^4 + E^(2/x)*(27*x^5 + 16*(1 + Log[2])^2) )/(9*x^2*(E^2 - E^x^(-1)*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.
| \(\displaystyle \int \frac {-27 x^7+e^{\frac {2 x-1}{x}} \left (81 x^6-32 x+(-32 x-32) \log ^2(2)+(-64 x-64) \log (2)-32\right )-81 e^{\frac {2 (2 x-1)}{x}} x^5+27 e^{\frac {3 (2 x-1)}{x}} x^4+64 x^2+64 x^2 \log ^2(2)+128 x^2 \log (2)}{-9 x^7+27 e^{\frac {2 x-1}{x}} x^6-27 e^{\frac {2 (2 x-1)}{x}} x^5+9 e^{\frac {3 (2 x-1)}{x}} x^4} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-27 x^7+e^{\frac {2 x-1}{x}} \left (81 x^6-32 x+(-32 x-32) \log ^2(2)+(-64 x-64) \log (2)-32\right )-81 e^{\frac {2 (2 x-1)}{x}} x^5+27 e^{\frac {3 (2 x-1)}{x}} x^4+64 x^2 \log ^2(2)+x^2 (64+128 \log (2))}{-9 x^7+27 e^{\frac {2 x-1}{x}} x^6-27 e^{\frac {2 (2 x-1)}{x}} x^5+9 e^{\frac {3 (2 x-1)}{x}} x^4}dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {-27 x^7+e^{\frac {2 x-1}{x}} \left (81 x^6-32 x+(-32 x-32) \log ^2(2)+(-64 x-64) \log (2)-32\right )-81 e^{\frac {2 (2 x-1)}{x}} x^5+27 e^{\frac {3 (2 x-1)}{x}} x^4+x^2 \left (64+64 \log ^2(2)+128 \log (2)\right )}{-9 x^7+27 e^{\frac {2 x-1}{x}} x^6-27 e^{\frac {2 (2 x-1)}{x}} x^5+9 e^{\frac {3 (2 x-1)}{x}} x^4}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{3/x} \left (-27 x^7+e^{\frac {2 x-1}{x}} \left (81 x^6-32 x+(-32 x-32) \log ^2(2)+(-64 x-64) \log (2)-32\right )-81 e^{\frac {2 (2 x-1)}{x}} x^5+27 e^{\frac {3 (2 x-1)}{x}} x^4+x^2 \left (64+64 \log ^2(2)+128 \log (2)\right )\right )}{9 x^4 \left (e^2-e^{\frac {1}{x}} x\right )^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{9} \int \frac {e^{3/x} \left (-27 x^7-81 e^{-\frac {2 (1-2 x)}{x}} x^5+27 e^{-\frac {3 (1-2 x)}{x}} x^4+64 (1+\log (2))^2 x^2-e^{-\frac {1-2 x}{x}} \left (-81 x^6+32 x+32 (x+1) \log ^2(2)+64 (x+1) \log (2)+32\right )\right )}{x^4 \left (e^2-e^{\frac {1}{x}} x\right )^3}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{9} \int \left (-\frac {32 e^{3/x} (1+\log (2))^2 (x-1)}{x^3 \left (e^{\frac {1}{x}} x-e^2\right )^3}-\frac {32 e^{\frac {2}{x}-4} (x+1) (1+\log (2))^2}{x^4}-\frac {32 e^{\frac {3}{x}-4} (x+1) (1+\log (2))^2}{x^3 \left (e^2-e^{\frac {1}{x}} x\right )}-\frac {32 e^{\frac {3}{x}-2} (x+1) (1+\log (2))^2}{x^3 \left (e^2-e^{\frac {1}{x}} x\right )^2}+27\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{9} \left (32 (1+\log (2))^2 \int \frac {e^{3/x}}{x^3 \left (e^{\frac {1}{x}} x-e^2\right )^3}dx-32 (1+\log (2))^2 \int \frac {e^{\frac {3}{x}-2}}{x^3 \left (e^{\frac {1}{x}} x-e^2\right )^2}dx+32 (1+\log (2))^2 \int \frac {e^{\frac {3}{x}-4}}{x^3 \left (e^{\frac {1}{x}} x-e^2\right )}dx-32 (1+\log (2))^2 \int \frac {e^{3/x}}{x^2 \left (e^{\frac {1}{x}} x-e^2\right )^3}dx-32 (1+\log (2))^2 \int \frac {e^{\frac {3}{x}-2}}{x^2 \left (e^{\frac {1}{x}} x-e^2\right )^2}dx+32 (1+\log (2))^2 \int \frac {e^{\frac {3}{x}-4}}{x^2 \left (e^{\frac {1}{x}} x-e^2\right )}dx+\frac {16 e^{\frac {2}{x}-4} (1+\log (2))^2}{x^2}+27 x\right )\) |
Int[(64*x^2 + 27*E^((3*(-1 + 2*x))/x)*x^4 - 81*E^((2*(-1 + 2*x))/x)*x^5 - 27*x^7 + 128*x^2*Log[2] + 64*x^2*Log[2]^2 + E^((-1 + 2*x)/x)*(-32 - 32*x + 81*x^6 + (-64 - 64*x)*Log[2] + (-32 - 32*x)*Log[2]^2))/(9*E^((3*(-1 + 2*x ))/x)*x^4 - 27*E^((2*(-1 + 2*x))/x)*x^5 + 27*E^((-1 + 2*x)/x)*x^6 - 9*x^7) ,x]
3.14.25.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Time = 0.38 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00
| method | result | size |
| risch | \(3 x +\frac {\frac {16 \ln \left (2\right )^{2}}{9}+\frac {32 \ln \left (2\right )}{9}+\frac {16}{9}}{x^{2} \left (x -{\mathrm e}^{\frac {-1+2 x}{x}}\right )^{2}}\) | \(36\) |
| norman | \(\frac {\left (\frac {16 \ln \left (2\right )^{2}}{9}+\frac {32 \ln \left (2\right )}{9}+\frac {16}{9}\right ) x +3 x^{6}+3 x^{4} {\mathrm e}^{\frac {4 x -2}{x}}-6 x^{5} {\mathrm e}^{\frac {-1+2 x}{x}}}{x^{3} \left (x -{\mathrm e}^{\frac {-1+2 x}{x}}\right )^{2}}\) | \(73\) |
| parallelrisch | \(\frac {27 \,{\mathrm e}^{\frac {4 x -2}{x}} x^{3}-54 \,{\mathrm e}^{\frac {-1+2 x}{x}} x^{4}+27 x^{5}+16 \ln \left (2\right )^{2}+32 \ln \left (2\right )+16}{9 x^{2} \left (x^{2}-2 x \,{\mathrm e}^{\frac {-1+2 x}{x}}+{\mathrm e}^{\frac {4 x -2}{x}}\right )}\) | \(86\) |
int((27*x^4*exp((-1+2*x)/x)^3-81*x^5*exp((-1+2*x)/x)^2+((-32*x-32)*ln(2)^2 +(-64*x-64)*ln(2)+81*x^6-32*x-32)*exp((-1+2*x)/x)+64*x^2*ln(2)^2+128*x^2*l n(2)-27*x^7+64*x^2)/(9*x^4*exp((-1+2*x)/x)^3-27*x^5*exp((-1+2*x)/x)^2+27*x ^6*exp((-1+2*x)/x)-9*x^7),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.39 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=\frac {27 \, x^{5} - 54 \, x^{4} e^{\left (\frac {2 \, x - 1}{x}\right )} + 27 \, x^{3} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} + 16 \, \log \left (2\right )^{2} + 32 \, \log \left (2\right ) + 16}{9 \, {\left (x^{4} - 2 \, x^{3} e^{\left (\frac {2 \, x - 1}{x}\right )} + x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )}} \]
integrate((27*x^4*exp((-1+2*x)/x)^3-81*x^5*exp((-1+2*x)/x)^2+((-32*x-32)*l og(2)^2+(-64*x-64)*log(2)+81*x^6-32*x-32)*exp((-1+2*x)/x)+64*x^2*log(2)^2+ 128*x^2*log(2)-27*x^7+64*x^2)/(9*x^4*exp((-1+2*x)/x)^3-27*x^5*exp((-1+2*x) /x)^2+27*x^6*exp((-1+2*x)/x)-9*x^7),x, algorithm=\
1/9*(27*x^5 - 54*x^4*e^((2*x - 1)/x) + 27*x^3*e^(2*(2*x - 1)/x) + 16*log(2 )^2 + 32*log(2) + 16)/(x^4 - 2*x^3*e^((2*x - 1)/x) + x^2*e^(2*(2*x - 1)/x) )
Time = 0.11 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.33 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=3 x + \frac {16 \log {\left (2 \right )}^{2} + 16 + 32 \log {\left (2 \right )}}{9 x^{4} - 18 x^{3} e^{\frac {2 x - 1}{x}} + 9 x^{2} e^{\frac {2 \cdot \left (2 x - 1\right )}{x}}} \]
integrate((27*x**4*exp((-1+2*x)/x)**3-81*x**5*exp((-1+2*x)/x)**2+((-32*x-3 2)*ln(2)**2+(-64*x-64)*ln(2)+81*x**6-32*x-32)*exp((-1+2*x)/x)+64*x**2*ln(2 )**2+128*x**2*ln(2)-27*x**7+64*x**2)/(9*x**4*exp((-1+2*x)/x)**3-27*x**5*ex p((-1+2*x)/x)**2+27*x**6*exp((-1+2*x)/x)-9*x**7),x)
3*x + (16*log(2)**2 + 16 + 32*log(2))/(9*x**4 - 18*x**3*exp((2*x - 1)/x) + 9*x**2*exp(2*(2*x - 1)/x))
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (31) = 62\).
Time = 0.35 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.11 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=-\frac {54 \, x^{4} e^{\left (\frac {1}{x} + 2\right )} - 27 \, x^{3} e^{4} - {\left (27 \, x^{5} + 16 \, \log \left (2\right )^{2} + 32 \, \log \left (2\right ) + 16\right )} e^{\frac {2}{x}}}{9 \, {\left (x^{4} e^{\frac {2}{x}} - 2 \, x^{3} e^{\left (\frac {1}{x} + 2\right )} + x^{2} e^{4}\right )}} \]
integrate((27*x^4*exp((-1+2*x)/x)^3-81*x^5*exp((-1+2*x)/x)^2+((-32*x-32)*l og(2)^2+(-64*x-64)*log(2)+81*x^6-32*x-32)*exp((-1+2*x)/x)+64*x^2*log(2)^2+ 128*x^2*log(2)-27*x^7+64*x^2)/(9*x^4*exp((-1+2*x)/x)^3-27*x^5*exp((-1+2*x) /x)^2+27*x^6*exp((-1+2*x)/x)-9*x^7),x, algorithm=\
-1/9*(54*x^4*e^(1/x + 2) - 27*x^3*e^4 - (27*x^5 + 16*log(2)^2 + 32*log(2) + 16)*e^(2/x))/(x^4*e^(2/x) - 2*x^3*e^(1/x + 2) + x^2*e^4)
Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (31) = 62\).
Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.28 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=-\frac {\frac {54 \, e^{\left (-\frac {1}{x} + 2\right )}}{x} - \frac {27 \, e^{\left (-\frac {2}{x} + 4\right )}}{x^{2}} - \frac {16 \, \log \left (2\right )^{2}}{x^{5}} - \frac {32 \, \log \left (2\right )}{x^{5}} - \frac {16}{x^{5}} - 27}{9 \, {\left (\frac {1}{x} - \frac {2 \, e^{\left (-\frac {1}{x} + 2\right )}}{x^{2}} + \frac {e^{\left (-\frac {2}{x} + 4\right )}}{x^{3}}\right )}} \]
integrate((27*x^4*exp((-1+2*x)/x)^3-81*x^5*exp((-1+2*x)/x)^2+((-32*x-32)*l og(2)^2+(-64*x-64)*log(2)+81*x^6-32*x-32)*exp((-1+2*x)/x)+64*x^2*log(2)^2+ 128*x^2*log(2)-27*x^7+64*x^2)/(9*x^4*exp((-1+2*x)/x)^3-27*x^5*exp((-1+2*x) /x)^2+27*x^6*exp((-1+2*x)/x)-9*x^7),x, algorithm=\
-1/9*(54*e^(-1/x + 2)/x - 27*e^(-2/x + 4)/x^2 - 16*log(2)^2/x^5 - 32*log(2 )/x^5 - 16/x^5 - 27)/(1/x - 2*e^(-1/x + 2)/x^2 + e^(-2/x + 4)/x^3)
Time = 11.00 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx=3\,x+\frac {\frac {32\,\ln \left (2\right )}{9}+\frac {16\,{\ln \left (2\right )}^2}{9}+\frac {16}{9}}{x^2\,{\left (x-{\mathrm {e}}^{2-\frac {1}{x}}\right )}^2} \]
int((64*x^2*log(2)^2 - exp((2*x - 1)/x)*(32*x + log(2)*(64*x + 64) + log(2 )^2*(32*x + 32) - 81*x^6 + 32) + 27*x^4*exp((3*(2*x - 1))/x) - 81*x^5*exp( (2*(2*x - 1))/x) + 128*x^2*log(2) + 64*x^2 - 27*x^7)/(27*x^6*exp((2*x - 1) /x) + 9*x^4*exp((3*(2*x - 1))/x) - 27*x^5*exp((2*(2*x - 1))/x) - 9*x^7),x)