Integrand size = 120, antiderivative size = 27 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=4 \left (-e^{\frac {1}{x}}+(8-x) \left (e^x-x\right )\right )^2+x \]
Leaf count is larger than twice the leaf count of optimal. \(88\) vs. \(2(27)=54\).
Time = 0.54 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.26 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=4 e^{2/x}+8 e^{\frac {1}{x}+x} (-8+x)+x+256 x^2-64 x^3+4 x^4+8 e^{2 x} \left (32-8 x+\frac {x^2}{2}\right )-8 e^{\frac {1}{x}} \left (-8 x+x^2\right )-8 e^x \left (64 x-16 x^2+x^3\right ) \]
Integrate[(-8*E^(2/x) + x^2 + 512*x^3 - 192*x^4 + 16*x^5 + E^x^(-1)*(-64*x + 72*x^2 - 16*x^3) + E^(2*x)*(448*x^2 - 120*x^3 + 8*x^4) + E^x*(-512*x^2 - 256*x^3 + 104*x^4 - 8*x^5 + E^x^(-1)*(64 - 8*x - 56*x^2 + 8*x^3)))/x^2,x ]
4*E^(2/x) + 8*E^(x^(-1) + x)*(-8 + x) + x + 256*x^2 - 64*x^3 + 4*x^4 + 8*E ^(2*x)*(32 - 8*x + x^2/2) - 8*E^x^(-1)*(-8*x + x^2) - 8*E^x*(64*x - 16*x^2 + x^3)
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.
\(\displaystyle \int \frac {16 x^5-192 x^4+512 x^3+x^2+e^{\frac {1}{x}} \left (-16 x^3+72 x^2-64 x\right )+e^{2 x} \left (8 x^4-120 x^3+448 x^2\right )+e^x \left (-8 x^5+104 x^4-256 x^3-512 x^2+e^{\frac {1}{x}} \left (8 x^3-56 x^2-8 x+64\right )\right )-8 e^{2/x}}{x^2} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {8 e^x \left (x^5-13 x^4-e^{\frac {1}{x}} x^3+32 x^3+7 e^{\frac {1}{x}} x^2+64 x^2+e^{\frac {1}{x}} x-8 e^{\frac {1}{x}}\right )}{x^2}+\frac {16 x^5-192 x^4-16 e^{\frac {1}{x}} x^3+512 x^3+72 e^{\frac {1}{x}} x^2+x^2-64 e^{\frac {1}{x}} x-8 e^{2/x}}{x^2}+8 e^{2 x} (x-8) (x-7)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 64 \int \frac {e^{x+\frac {1}{x}}}{x^2}dx-56 \int e^{x+\frac {1}{x}}dx-8 \int \frac {e^{x+\frac {1}{x}}}{x}dx+8 \int e^{x+\frac {1}{x}} xdx+4 x^4-8 e^x x^3-64 x^3+128 e^x x^2+4 e^{2 x} x^2-8 e^{\frac {1}{x}} x^2+256 x^2-512 e^x x-64 e^{2 x} x+64 e^{\frac {1}{x}} x+x+4 e^{2/x}+256 e^{2 x}\) |
Int[(-8*E^(2/x) + x^2 + 512*x^3 - 192*x^4 + 16*x^5 + E^x^(-1)*(-64*x + 72* x^2 - 16*x^3) + E^(2*x)*(448*x^2 - 120*x^3 + 8*x^4) + E^x*(-512*x^2 - 256* x^3 + 104*x^4 - 8*x^5 + E^x^(-1)*(64 - 8*x - 56*x^2 + 8*x^3)))/x^2,x]
3.8.100.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Leaf count of result is larger than twice the leaf count of optimal. \(80\) vs. \(2(25)=50\).
Time = 2.23 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.00
method | result | size |
risch | \(4 x^{4}-64 x^{3}+256 x^{2}+x +\left (4 x^{2}-64 x +256\right ) {\mathrm e}^{2 x}+\left (-8 x^{3}+128 x^{2}-512 x \right ) {\mathrm e}^{x}+4 \,{\mathrm e}^{\frac {2}{x}}+\left (-8 x^{2}+8 \,{\mathrm e}^{x} x +64 x -64 \,{\mathrm e}^{x}\right ) {\mathrm e}^{\frac {1}{x}}\) | \(81\) |
parallelrisch | \(-8 \,{\mathrm e}^{x} x^{3}+4 x^{4}+8 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x} x -8 x^{2} {\mathrm e}^{\frac {1}{x}}+128 \,{\mathrm e}^{x} x^{2}+4 \,{\mathrm e}^{2 x} x^{2}-64 x^{3}-64 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x}+64 x \,{\mathrm e}^{\frac {1}{x}}-512 \,{\mathrm e}^{x} x -64 x \,{\mathrm e}^{2 x}+256 x^{2}+256 \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{\frac {2}{x}}+x\) | \(100\) |
parts | \(x +\frac {-512 \,{\mathrm e}^{x} x^{2}+128 \,{\mathrm e}^{x} x^{3}-8 \,{\mathrm e}^{x} x^{4}-64 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x} x +8 \,{\mathrm e}^{\frac {1}{x}} {\mathrm e}^{x} x^{2}}{x}+4 \,{\mathrm e}^{2 x} x^{2}-64 x \,{\mathrm e}^{2 x}+256 \,{\mathrm e}^{2 x}+256 x^{2}-64 x^{3}+4 x^{4}+4 \,{\mathrm e}^{\frac {2}{x}}-8 x^{2} {\mathrm e}^{\frac {1}{x}}+64 x \,{\mathrm e}^{\frac {1}{x}}\) | \(110\) |
derivativedivides | \(4 x^{4}-64 x^{3}+256 x^{2}+x +4 \,{\mathrm e}^{\frac {2}{x}}+4 \,{\mathrm e}^{2 x} x^{2}-64 x \,{\mathrm e}^{2 x}+256 \,{\mathrm e}^{2 x}-8 x^{2} {\mathrm e}^{\frac {1}{x}}+64 x \,{\mathrm e}^{\frac {1}{x}}+\left (-\frac {8 \,{\mathrm e}^{x}}{x}+\frac {128 \,{\mathrm e}^{x}}{x^{2}}-\frac {512 \,{\mathrm e}^{x}}{x^{3}}+\frac {8 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{x}}}{x^{3}}-\frac {64 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{x}}}{x^{4}}\right ) x^{4}\) | \(112\) |
default | \(4 x^{4}-64 x^{3}+256 x^{2}+x +4 \,{\mathrm e}^{\frac {2}{x}}+4 \,{\mathrm e}^{2 x} x^{2}-64 x \,{\mathrm e}^{2 x}+256 \,{\mathrm e}^{2 x}-8 x^{2} {\mathrm e}^{\frac {1}{x}}+64 x \,{\mathrm e}^{\frac {1}{x}}+\left (-\frac {8 \,{\mathrm e}^{x}}{x}+\frac {128 \,{\mathrm e}^{x}}{x^{2}}-\frac {512 \,{\mathrm e}^{x}}{x^{3}}+\frac {8 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{x}}}{x^{3}}-\frac {64 \,{\mathrm e}^{x} {\mathrm e}^{\frac {1}{x}}}{x^{4}}\right ) x^{4}\) | \(112\) |
int(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x)-8*x^ 5+104*x^4-256*x^3-512*x^2)*exp(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x)*exp(1 /x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x,method=_RETURNVERBOSE)
4*x^4-64*x^3+256*x^2+x+(4*x^2-64*x+256)*exp(x)^2+(-8*x^3+128*x^2-512*x)*ex p(x)+4*exp(1/x)^2+(-8*x^2+8*exp(x)*x+64*x-64*exp(x))*exp(1/x)
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.85 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=4 \, x^{4} - 64 \, x^{3} + 256 \, x^{2} + 4 \, {\left (x^{2} - 16 \, x + 64\right )} e^{\left (2 \, x\right )} - 8 \, {\left (x^{3} - 16 \, x^{2} - {\left (x - 8\right )} e^{\frac {1}{x}} + 64 \, x\right )} e^{x} - 8 \, {\left (x^{2} - 8 \, x\right )} e^{\frac {1}{x}} + x + 4 \, e^{\frac {2}{x}} \]
integrate(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x )-8*x^5+104*x^4-256*x^3-512*x^2)*exp(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x) *exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x, algorithm=\
4*x^4 - 64*x^3 + 256*x^2 + 4*(x^2 - 16*x + 64)*e^(2*x) - 8*(x^3 - 16*x^2 - (x - 8)*e^(1/x) + 64*x)*e^x - 8*(x^2 - 8*x)*e^(1/x) + x + 4*e^(2/x)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (17) = 34\).
Time = 3.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.07 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=4 x^{4} - 64 x^{3} + 256 x^{2} + x + \left (- 8 x^{2} + 64 x\right ) e^{\frac {1}{x}} + \left (4 x^{2} - 64 x + 256\right ) e^{2 x} + \left (- 8 x^{3} + 128 x^{2} + 8 x e^{\frac {1}{x}} - 512 x - 64 e^{\frac {1}{x}}\right ) e^{x} + 4 e^{\frac {2}{x}} \]
integrate(((8*x**4-120*x**3+448*x**2)*exp(x)**2+((8*x**3-56*x**2-8*x+64)*e xp(1/x)-8*x**5+104*x**4-256*x**3-512*x**2)*exp(x)-8*exp(1/x)**2+(-16*x**3+ 72*x**2-64*x)*exp(1/x)+16*x**5-192*x**4+512*x**3+x**2)/x**2,x)
4*x**4 - 64*x**3 + 256*x**2 + x + (-8*x**2 + 64*x)*exp(1/x) + (4*x**2 - 64 *x + 256)*exp(2*x) + (-8*x**3 + 128*x**2 + 8*x*exp(1/x) - 512*x - 64*exp(1 /x))*exp(x) + 4*exp(2/x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.93 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=4 \, x^{4} - 64 \, x^{3} + 256 \, x^{2} + 2 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} - 30 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + 8 \, {\left (x - 8\right )} e^{\left (x + \frac {1}{x}\right )} - 8 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + 104 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} - 256 \, {\left (x - 1\right )} e^{x} + x + 64 \, {\rm Ei}\left (\frac {1}{x}\right ) + 224 \, e^{\left (2 \, x\right )} - 512 \, e^{x} + 4 \, e^{\frac {2}{x}} - 72 \, \Gamma \left (-1, -\frac {1}{x}\right ) - 16 \, \Gamma \left (-2, -\frac {1}{x}\right ) \]
integrate(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x )-8*x^5+104*x^4-256*x^3-512*x^2)*exp(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x) *exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x, algorithm=\
4*x^4 - 64*x^3 + 256*x^2 + 2*(2*x^2 - 2*x + 1)*e^(2*x) - 30*(2*x - 1)*e^(2 *x) + 8*(x - 8)*e^(x + 1/x) - 8*(x^3 - 3*x^2 + 6*x - 6)*e^x + 104*(x^2 - 2 *x + 2)*e^x - 256*(x - 1)*e^x + x + 64*Ei(1/x) + 224*e^(2*x) - 512*e^x + 4 *e^(2/x) - 72*gamma(-1, -1/x) - 16*gamma(-2, -1/x)
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.96 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=4 \, x^{4} - 8 \, x^{3} e^{x} - 64 \, x^{3} + 4 \, x^{2} e^{\left (2 \, x\right )} + 128 \, x^{2} e^{x} - 8 \, x^{2} e^{\frac {1}{x}} + 256 \, x^{2} - 64 \, x e^{\left (2 \, x\right )} - 512 \, x e^{x} + 8 \, x e^{\left (\frac {x^{2} + 1}{x}\right )} + 64 \, x e^{\frac {1}{x}} + x + 256 \, e^{\left (2 \, x\right )} - 64 \, e^{\left (\frac {x^{2} + 1}{x}\right )} + 4 \, e^{\frac {2}{x}} \]
integrate(((8*x^4-120*x^3+448*x^2)*exp(x)^2+((8*x^3-56*x^2-8*x+64)*exp(1/x )-8*x^5+104*x^4-256*x^3-512*x^2)*exp(x)-8*exp(1/x)^2+(-16*x^3+72*x^2-64*x) *exp(1/x)+16*x^5-192*x^4+512*x^3+x^2)/x^2,x, algorithm=\
4*x^4 - 8*x^3*e^x - 64*x^3 + 4*x^2*e^(2*x) + 128*x^2*e^x - 8*x^2*e^(1/x) + 256*x^2 - 64*x*e^(2*x) - 512*x*e^x + 8*x*e^((x^2 + 1)/x) + 64*x*e^(1/x) + x + 256*e^(2*x) - 64*e^((x^2 + 1)/x) + 4*e^(2/x)
Time = 9.91 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.44 \[ \int \frac {-8 e^{2/x}+x^2+512 x^3-192 x^4+16 x^5+e^{\frac {1}{x}} \left (-64 x+72 x^2-16 x^3\right )+e^{2 x} \left (448 x^2-120 x^3+8 x^4\right )+e^x \left (-512 x^2-256 x^3+104 x^4-8 x^5+e^{\frac {1}{x}} \left (64-8 x-56 x^2+8 x^3\right )\right )}{x^2} \, dx=x+4\,{\mathrm {e}}^{2/x}+4\,{\mathrm {e}}^{2\,x}\,{\left (x-8\right )}^2+256\,x^2-64\,x^3+4\,x^4+8\,{\mathrm {e}}^x\,\left (x-8\right )\,\left (8\,x+{\mathrm {e}}^{1/x}-x^2\right )-8\,x\,{\mathrm {e}}^{1/x}\,\left (x-8\right ) \]
int(-(8*exp(2/x) + exp(1/x)*(64*x - 72*x^2 + 16*x^3) - exp(2*x)*(448*x^2 - 120*x^3 + 8*x^4) + exp(x)*(exp(1/x)*(8*x + 56*x^2 - 8*x^3 - 64) + 512*x^2 + 256*x^3 - 104*x^4 + 8*x^5) - x^2 - 512*x^3 + 192*x^4 - 16*x^5)/x^2,x)