Integrand size = 162, antiderivative size = 23 \[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx=(1+x) \log \left (\left (3+x-\left (4-x^{\frac {1}{x}}\right )^2\right )^2\right ) \]
Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(23)=46\).
Time = 0.47 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.04 \[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx=2 \log \left (-13+x+8 x^{\frac {1}{x}}-x^{2/x}\right )+x \log \left (\left (13-x-8 x^{\frac {1}{x}}+x^{2/x}\right )^2\right ) \]
Integrate[(-2*x^2 - 2*x^3 + x^(2/x)*(4 + 4*x + (-4 - 4*x)*Log[x]) + x^x^(- 1)*(-16 - 16*x + (16 + 16*x)*Log[x]) + (13*x^2 - x^3 - 8*x^(2 + x^(-1)) + x^(2 + 2/x))*Log[169 - 26*x + x^2 + (90 - 2*x)*x^(2/x) - 16*x^(3/x) + x^(4 /x) + x^x^(-1)*(-208 + 16*x)])/(13*x^2 - x^3 - 8*x^(2 + x^(-1)) + x^(2 + 2 /x)),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.
| \(\displaystyle \int \frac {x^{2/x} (4 x+(-4 x-4) \log (x)+4)+x^{\frac {1}{x}} (-16 x+(16 x+16) \log (x)-16)-2 x^3-2 x^2+\left (-8 x^{\frac {1}{x}+2}+x^{\frac {2}{x}+2}-x^3+13 x^2\right ) \log \left ((90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+(16 x-208) x^{\frac {1}{x}}+x^2-26 x+169\right )}{-8 x^{\frac {1}{x}+2}+x^{\frac {2}{x}+2}-x^3+13 x^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {x^2 \log \left (\left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )^2\right )+4 x-4 x \log (x)-4 \log (x)+4}{x^2}-\frac {2 (x+1) \left (-8 x^{\frac {1}{x}}+8 x^{\frac {1}{x}} \log (x)+x^2-2 x+2 x \log (x)-26 \log (x)+26\right )}{x^2 \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \int \frac {1}{x^{2/x}-8 x^{\frac {1}{x}}-x+13}dx+4 \int \frac {1}{x \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}dx+16 \int \frac {x^{\frac {1}{x}-2}}{x^{2/x}-8 x^{\frac {1}{x}}-x+13}dx+4 \int \frac {\int \frac {1}{x \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}dx}{x}dx+16 \int \frac {\int \frac {x^{\frac {1}{x}-2}}{x^{2/x}-8 x^{\frac {1}{x}}-x+13}dx}{x}dx-4 \log (x) \int \frac {1}{x \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}dx-16 \log (x) \int \frac {x^{\frac {1}{x}-2}}{x^{2/x}-8 x^{\frac {1}{x}}-x+13}dx-52 \int \frac {1}{x^2 \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}dx-52 \int \frac {\int \frac {1}{x^2 \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}dx}{x}dx+52 \log (x) \int \frac {1}{x^2 \left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )}dx+x \log \left (\left (x^{2/x}-8 x^{\frac {1}{x}}-x+13\right )^2\right )+\frac {4}{x}+2 \log ^2(x)+2 (1-\log (x))^2+4 \log (x) (1-\log (x))-\frac {4 (1-\log (x))}{x}\) |
Int[(-2*x^2 - 2*x^3 + x^(2/x)*(4 + 4*x + (-4 - 4*x)*Log[x]) + x^x^(-1)*(-1 6 - 16*x + (16 + 16*x)*Log[x]) + (13*x^2 - x^3 - 8*x^(2 + x^(-1)) + x^(2 + 2/x))*Log[169 - 26*x + x^2 + (90 - 2*x)*x^(2/x) - 16*x^(3/x) + x^(4/x) + x^x^(-1)*(-208 + 16*x)])/(13*x^2 - x^3 - 8*x^(2 + x^(-1)) + x^(2 + 2/x)),x ]
3.23.8.3.1 Defintions of rubi rules used
Leaf count of result is larger than twice the leaf count of optimal. \(117\) vs. \(2(25)=50\).
Time = 2.62 (sec) , antiderivative size = 118, normalized size of antiderivative = 5.13
| method | result | size |
| parallelrisch | \(x \ln \left ({\mathrm e}^{\frac {4 \ln \left (x \right )}{x}}-16 \,{\mathrm e}^{\frac {3 \ln \left (x \right )}{x}}+\left (-2 x +90\right ) {\mathrm e}^{\frac {2 \ln \left (x \right )}{x}}+\left (16 x -208\right ) {\mathrm e}^{\frac {\ln \left (x \right )}{x}}+x^{2}-26 x +169\right )+\ln \left ({\mathrm e}^{\frac {4 \ln \left (x \right )}{x}}-16 \,{\mathrm e}^{\frac {3 \ln \left (x \right )}{x}}+\left (-2 x +90\right ) {\mathrm e}^{\frac {2 \ln \left (x \right )}{x}}+\left (16 x -208\right ) {\mathrm e}^{\frac {\ln \left (x \right )}{x}}+x^{2}-26 x +169\right )\) | \(118\) |
| risch | \(2 x \ln \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )-\frac {i \pi x {\operatorname {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )\right )}^{2} \operatorname {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )^{2}\right )}{2}+i \pi x \,\operatorname {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )\right ) {\operatorname {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )^{2}\right )}^{2}-\frac {i \pi x {\operatorname {csgn}\left (i \left (-x^{\frac {2}{x}}+8 x^{\frac {1}{x}}+x -13\right )^{2}\right )}^{3}}{2}+2 \ln \left (x^{\frac {2}{x}}-8 x^{\frac {1}{x}}-x +13\right )\) | \(189\) |
int(((x^2*exp(ln(x)/x)^2-8*x^2*exp(ln(x)/x)-x^3+13*x^2)*ln(exp(ln(x)/x)^4- 16*exp(ln(x)/x)^3+(-2*x+90)*exp(ln(x)/x)^2+(16*x-208)*exp(ln(x)/x)+x^2-26* x+169)+((-4-4*x)*ln(x)+4*x+4)*exp(ln(x)/x)^2+((16*x+16)*ln(x)-16*x-16)*exp (ln(x)/x)-2*x^3-2*x^2)/(x^2*exp(ln(x)/x)^2-8*x^2*exp(ln(x)/x)-x^3+13*x^2), x,method=_RETURNVERBOSE)
x*ln(exp(ln(x)/x)^4-16*exp(ln(x)/x)^3+(-2*x+90)*exp(ln(x)/x)^2+(16*x-208)* exp(ln(x)/x)+x^2-26*x+169)+ln(exp(ln(x)/x)^4-16*exp(ln(x)/x)^3+(-2*x+90)*e xp(ln(x)/x)^2+(16*x-208)*exp(ln(x)/x)+x^2-26*x+169)
Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (21) = 42\).
Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.22 \[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx={\left (x + 1\right )} \log \left (-2 \, {\left (x - 45\right )} x^{\frac {2}{x}} + 16 \, {\left (x - 13\right )} x^{\left (\frac {1}{x}\right )} + x^{2} + x^{\frac {4}{x}} - 16 \, x^{\frac {3}{x}} - 26 \, x + 169\right ) \]
integrate(((x^2*exp(log(x)/x)^2-8*x^2*exp(log(x)/x)-x^3+13*x^2)*log(exp(lo g(x)/x)^4-16*exp(log(x)/x)^3+(-2*x+90)*exp(log(x)/x)^2+(16*x-208)*exp(log( x)/x)+x^2-26*x+169)+((-4-4*x)*log(x)+4*x+4)*exp(log(x)/x)^2+((16*x+16)*log (x)-16*x-16)*exp(log(x)/x)-2*x^3-2*x^2)/(x^2*exp(log(x)/x)^2-8*x^2*exp(log (x)/x)-x^3+13*x^2),x, algorithm=\
(x + 1)*log(-2*(x - 45)*x^(2/x) + 16*(x - 13)*x^(1/x) + x^2 + x^(4/x) - 16 *x^(3/x) - 26*x + 169)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (19) = 38\).
Time = 1.72 (sec) , antiderivative size = 78, normalized size of antiderivative = 3.39 \[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx=x \log {\left (x^{2} - 26 x + \left (90 - 2 x\right ) e^{\frac {2 \log {\left (x \right )}}{x}} + \left (16 x - 208\right ) e^{\frac {\log {\left (x \right )}}{x}} + e^{\frac {4 \log {\left (x \right )}}{x}} - 16 e^{\frac {3 \log {\left (x \right )}}{x}} + 169 \right )} + 2 \log {\left (- x + e^{\frac {2 \log {\left (x \right )}}{x}} - 8 e^{\frac {\log {\left (x \right )}}{x}} + 13 \right )} \]
integrate(((x**2*exp(ln(x)/x)**2-8*x**2*exp(ln(x)/x)-x**3+13*x**2)*ln(exp( ln(x)/x)**4-16*exp(ln(x)/x)**3+(-2*x+90)*exp(ln(x)/x)**2+(16*x-208)*exp(ln (x)/x)+x**2-26*x+169)+((-4-4*x)*ln(x)+4*x+4)*exp(ln(x)/x)**2+((16*x+16)*ln (x)-16*x-16)*exp(ln(x)/x)-2*x**3-2*x**2)/(x**2*exp(ln(x)/x)**2-8*x**2*exp( ln(x)/x)-x**3+13*x**2),x)
x*log(x**2 - 26*x + (90 - 2*x)*exp(2*log(x)/x) + (16*x - 208)*exp(log(x)/x ) + exp(4*log(x)/x) - 16*exp(3*log(x)/x) + 169) + 2*log(-x + exp(2*log(x)/ x) - 8*exp(log(x)/x) + 13)
Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx=2 \, {\left (x + 1\right )} \log \left (x^{\frac {2}{x}} - x - 8 \, x^{\left (\frac {1}{x}\right )} + 13\right ) \]
integrate(((x^2*exp(log(x)/x)^2-8*x^2*exp(log(x)/x)-x^3+13*x^2)*log(exp(lo g(x)/x)^4-16*exp(log(x)/x)^3+(-2*x+90)*exp(log(x)/x)^2+(16*x-208)*exp(log( x)/x)+x^2-26*x+169)+((-4-4*x)*log(x)+4*x+4)*exp(log(x)/x)^2+((16*x+16)*log (x)-16*x-16)*exp(log(x)/x)-2*x^3-2*x^2)/(x^2*exp(log(x)/x)^2-8*x^2*exp(log (x)/x)-x^3+13*x^2),x, algorithm=\
\[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx=\int { -\frac {2 \, x^{3} + 4 \, {\left ({\left (x + 1\right )} \log \left (x\right ) - x - 1\right )} x^{\frac {2}{x}} - 16 \, {\left ({\left (x + 1\right )} \log \left (x\right ) - x - 1\right )} x^{\left (\frac {1}{x}\right )} + 2 \, x^{2} - {\left (x^{2} x^{\frac {2}{x}} - 8 \, x^{2} x^{\left (\frac {1}{x}\right )} - x^{3} + 13 \, x^{2}\right )} \log \left (-2 \, {\left (x - 45\right )} x^{\frac {2}{x}} + 16 \, {\left (x - 13\right )} x^{\left (\frac {1}{x}\right )} + x^{2} + x^{\frac {4}{x}} - 16 \, x^{\frac {3}{x}} - 26 \, x + 169\right )}{x^{2} x^{\frac {2}{x}} - 8 \, x^{2} x^{\left (\frac {1}{x}\right )} - x^{3} + 13 \, x^{2}} \,d x } \]
integrate(((x^2*exp(log(x)/x)^2-8*x^2*exp(log(x)/x)-x^3+13*x^2)*log(exp(lo g(x)/x)^4-16*exp(log(x)/x)^3+(-2*x+90)*exp(log(x)/x)^2+(16*x-208)*exp(log( x)/x)+x^2-26*x+169)+((-4-4*x)*log(x)+4*x+4)*exp(log(x)/x)^2+((16*x+16)*log (x)-16*x-16)*exp(log(x)/x)-2*x^3-2*x^2)/(x^2*exp(log(x)/x)^2-8*x^2*exp(log (x)/x)-x^3+13*x^2),x, algorithm=\
Time = 13.50 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.65 \[ \int \frac {-2 x^2-2 x^3+x^{2/x} (4+4 x+(-4-4 x) \log (x))+x^{\frac {1}{x}} (-16-16 x+(16+16 x) \log (x))+\left (13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}\right ) \log \left (169-26 x+x^2+(90-2 x) x^{2/x}-16 x^{3/x}+x^{4/x}+x^{\frac {1}{x}} (-208+16 x)\right )}{13 x^2-x^3-8 x^{2+\frac {1}{x}}+x^{2+\frac {2}{x}}} \, dx=2\,\ln \left (x-x^{2/x}+8\,x^{1/x}-13\right )+x\,\ln \left (90\,x^{2/x}-26\,x-16\,x^{3/x}+x^{4/x}+16\,x\,x^{1/x}-208\,x^{1/x}+x^2-2\,x\,x^{2/x}+169\right ) \]
int((log(exp((4*log(x))/x) - 16*exp((3*log(x))/x) - 26*x - exp((2*log(x))/ x)*(2*x - 90) + exp(log(x)/x)*(16*x - 208) + x^2 + 169)*(8*x^2*exp(log(x)/ x) - x^2*exp((2*log(x))/x) - 13*x^2 + x^3) - exp((2*log(x))/x)*(4*x - log( x)*(4*x + 4) + 4) + exp(log(x)/x)*(16*x - log(x)*(16*x + 16) + 16) + 2*x^2 + 2*x^3)/(8*x^2*exp(log(x)/x) - x^2*exp((2*log(x))/x) - 13*x^2 + x^3),x)