Integrand size = 100, antiderivative size = 25 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\left (20-\frac {3^{\frac {1}{x}} x \left (x^4\right )^{-1/x}}{\log (x)}\right )^2 \] Output:
(20-x/ln(x)/exp(ln(1/3*x^4)/x))^2
\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx \] Input:
Integrate[(3^(2/x)*(-2*x^2 + (-8*x + 2*x^2)*Log[x] + 2*x*Log[x]*Log[x^4/3] + ((x^4)^x^(-1)*(40*x*Log[x] + (160 - 40*x)*Log[x]^2 - 40*Log[x]^2*Log[x^ 4/3]))/3^x^(-1)))/(x*(x^4)^(2/x)*Log[x]^3),x]
Output:
Integrate[(3^(2/x)*(-2*x^2 + (-8*x + 2*x^2)*Log[x] + 2*x*Log[x]*Log[x^4/3] + ((x^4)^x^(-1)*(40*x*Log[x] + (160 - 40*x)*Log[x]^2 - 40*Log[x]^2*Log[x^ 4/3]))/3^x^(-1)))/(x*(x^4)^(2/x)*Log[x]^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.
| \(\displaystyle \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (-40 \log \left (\frac {x^4}{3}\right ) \log ^2(x)+(160-40 x) \log ^2(x)+40 x \log (x)\right )+2 x \log (x) \log \left (\frac {x^4}{3}\right )-2 x^2+\left (2 x^2-8 x\right ) \log (x)\right )}{x \log ^3(x)} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2\ 3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (\log (x) \left (\log \left (\frac {x^4}{3}\right )+x-4\right )-x\right )}{x \log ^3(x)}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int -\frac {3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (x+\log (x) \left (-x-\log \left (\frac {x^4}{3}\right )+4\right )\right )}{x \log ^3(x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -2 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-2/x} \left (3^{\frac {1}{x}} x-20 \left (x^4\right )^{\frac {1}{x}} \log (x)\right ) \left (x+\log (x) \left (-x-\log \left (\frac {x^4}{3}\right )+4\right )\right )}{x \log ^3(x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -2 \int \left (\frac {20\ 3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \left (\log (x) x-x-4 \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{x \log ^2(x)}-\frac {3^{2/x} \left (x^4\right )^{-2/x} \left (\log (x) x-x-4 \log (x)+\log (x) \log \left (\frac {x^4}{3}\right )\right )}{\log ^3(x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (\int \frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^3(x)}dx+4 \int \frac {3^{2/x} \left (x^4\right )^{-2/x}}{\log ^2(x)}dx-\int \frac {3^{2/x} x \left (x^4\right )^{-2/x}}{\log ^2(x)}dx-20 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log ^2(x)}dx-\int \frac {3^{2/x} \left (x^4\right )^{-2/x} \log \left (\frac {x^4}{3}\right )}{\log ^2(x)}dx+20 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{\log (x)}dx-80 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x}}{x \log (x)}dx+20 \int \frac {3^{\frac {1}{x}} \left (x^4\right )^{-1/x} \log \left (\frac {x^4}{3}\right )}{x \log (x)}dx\right )\) |
Input:
Int[(3^(2/x)*(-2*x^2 + (-8*x + 2*x^2)*Log[x] + 2*x*Log[x]*Log[x^4/3] + ((x ^4)^x^(-1)*(40*x*Log[x] + (160 - 40*x)*Log[x]^2 - 40*Log[x]^2*Log[x^4/3])) /3^x^(-1)))/(x*(x^4)^(2/x)*Log[x]^3),x]
Output:
$Aborted
Time = 0.67 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.68
| method | result | size |
| default | \(-\frac {40 x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )}+\frac {x^{2} {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(42\) |
| parts | \(-\frac {40 x \,{\mathrm e}^{-\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )}+\frac {x^{2} {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(42\) |
| parallelrisch | \(\frac {\left (-320 \ln \left (x \right ) {\mathrm e}^{\frac {\ln \left (\frac {x^{4}}{3}\right )}{x}} x^{2}+8 x^{3}\right ) {\mathrm e}^{-\frac {2 \ln \left (\frac {x^{4}}{3}\right )}{x}}}{8 x \ln \left (x \right )^{2}}\) | \(47\) |
| risch | \(-\frac {40 x \,x^{-\frac {4}{x}} 3^{\frac {1}{x}} {\mathrm e}^{\frac {i \pi \left (\operatorname {csgn}\left (i x^{2}\right )^{3}-2 \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )-\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{3}\right )^{3}-\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{4}\right )^{3}\right )}{2 x}}}{\ln \left (x \right )}+\frac {x^{2} x^{-\frac {8}{x}} 3^{\frac {2}{x}} {\mathrm e}^{\frac {i \pi \left (\operatorname {csgn}\left (i x^{2}\right )^{3}-2 \operatorname {csgn}\left (i x^{2}\right )^{2} \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )^{2}+\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )-\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x^{3}\right )^{2}-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right )^{2}+\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{3}\right )^{3}-\operatorname {csgn}\left (i x^{3}\right ) \operatorname {csgn}\left (i x^{4}\right )^{2}+\operatorname {csgn}\left (i x^{4}\right )^{3}\right )}{x}}}{\ln \left (x \right )^{2}}\) | \(399\) |
Input:
int(((-40*ln(x)^2*ln(1/3*x^4)+(-40*x+160)*ln(x)^2+40*x*ln(x))*exp(ln(1/3*x ^4)/x)+2*x*ln(x)*ln(1/3*x^4)+(2*x^2-8*x)*ln(x)-2*x^2)/x/ln(x)^3/exp(ln(1/3 *x^4)/x)^2,x,method=_RETURNVERBOSE)
Output:
-40*x/ln(x)/exp(ln(1/3*x^4)/x)+x^2*exp(-2*ln(1/3*x^4)/x)/ln(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=-\frac {{\left (40 \, x e^{\left (-\frac {\log \left (3\right ) - 4 \, \log \left (x\right )}{x}\right )} \log \left (x\right ) - x^{2}\right )} e^{\left (\frac {2 \, {\left (\log \left (3\right ) - 4 \, \log \left (x\right )\right )}}{x}\right )}}{\log \left (x\right )^{2}} \] Input:
integrate(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*ex p(log(1/3*x^4)/x)+2*x*log(x)*log(1/3*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log( x)^3/exp(log(1/3*x^4)/x)^2,x, algorithm="fricas")
Output:
-(40*x*e^(-(log(3) - 4*log(x))/x)*log(x) - x^2)*e^(2*(log(3) - 4*log(x))/x )/log(x)^2
Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (17) = 34\).
Time = 0.17 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.76 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {x^{2} e^{- \frac {2 \cdot \left (4 \log {\left (x \right )} - \log {\left (3 \right )}\right )}{x}} \log {\left (x \right )} - 40 x e^{- \frac {4 \log {\left (x \right )} - \log {\left (3 \right )}}{x}} \log {\left (x \right )}^{2}}{\log {\left (x \right )}^{3}} \] Input:
integrate(((-40*ln(x)**2*ln(1/3*x**4)+(-40*x+160)*ln(x)**2+40*x*ln(x))*exp (ln(1/3*x**4)/x)+2*x*ln(x)*ln(1/3*x**4)+(2*x**2-8*x)*ln(x)-2*x**2)/x/ln(x) **3/exp(ln(1/3*x**4)/x)**2,x)
Output:
(x**2*exp(-2*(4*log(x) - log(3))/x)*log(x) - 40*x*exp(-(4*log(x) - log(3)) /x)*log(x)**2)/log(x)**3
Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.84 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {x^{2} e^{\left (\frac {2 \, \log \left (3\right )}{x} - \frac {8 \, \log \left (x\right )}{x}\right )} - 40 \, x e^{\left (\frac {\log \left (3\right )}{x} - \frac {4 \, \log \left (x\right )}{x}\right )} \log \left (x\right )}{\log \left (x\right )^{2}} \] Input:
integrate(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*ex p(log(1/3*x^4)/x)+2*x*log(x)*log(1/3*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log( x)^3/exp(log(1/3*x^4)/x)^2,x, algorithm="maxima")
Output:
(x^2*e^(2*log(3)/x - 8*log(x)/x) - 40*x*e^(log(3)/x - 4*log(x)/x)*log(x))/ log(x)^2
\[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\int { \frac {2 \, {\left (x \log \left (\frac {1}{3} \, x^{4}\right ) \log \left (x\right ) - 20 \, {\left ({\left (x - 4\right )} \log \left (x\right )^{2} + \log \left (\frac {1}{3} \, x^{4}\right ) \log \left (x\right )^{2} - x \log \left (x\right )\right )} \left (\frac {1}{3} \, x^{4}\right )^{\left (\frac {1}{x}\right )} - x^{2} + {\left (x^{2} - 4 \, x\right )} \log \left (x\right )\right )}}{\left (\frac {1}{3} \, x^{4}\right )^{\frac {2}{x}} x \log \left (x\right )^{3}} \,d x } \] Input:
integrate(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*ex p(log(1/3*x^4)/x)+2*x*log(x)*log(1/3*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log( x)^3/exp(log(1/3*x^4)/x)^2,x, algorithm="giac")
Output:
integrate(2*(x*log(1/3*x^4)*log(x) - 20*((x - 4)*log(x)^2 + log(1/3*x^4)*l og(x)^2 - x*log(x))*(1/3*x^4)^(1/x) - x^2 + (x^2 - 4*x)*log(x))/((1/3*x^4) ^(2/x)*x*log(x)^3), x)
Time = 2.81 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.64 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {3^{1/x}\,x\,\left (3^{1/x}\,x-40\,\ln \left (x\right )\,{\left (x^4\right )}^{1/x}\right )}{{\ln \left (x\right )}^2\,{\left (x^4\right )}^{2/x}} \] Input:
int(-(exp(-(2*log(x^4/3))/x)*(log(x)*(8*x - 2*x^2) + 2*x^2 + exp(log(x^4/3 )/x)*(40*log(x^4/3)*log(x)^2 - 40*x*log(x) + log(x)^2*(40*x - 160)) - 2*x* log(x^4/3)*log(x)))/(x*log(x)^3),x)
Output:
(3^(1/x)*x*(3^(1/x)*x - 40*log(x)*(x^4)^(1/x)))/(log(x)^2*(x^4)^(2/x))
Time = 0.19 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {3^{2/x} \left (x^4\right )^{-2/x} \left (-2 x^2+\left (-8 x+2 x^2\right ) \log (x)+2 x \log (x) \log \left (\frac {x^4}{3}\right )+3^{-1/x} \left (x^4\right )^{\frac {1}{x}} \left (40 x \log (x)+(160-40 x) \log ^2(x)-40 \log ^2(x) \log \left (\frac {x^4}{3}\right )\right )\right )}{x \log ^3(x)} \, dx=\frac {x \left (-40 e^{\frac {\mathrm {log}\left (\frac {x^{4}}{3}\right )}{x}} \mathrm {log}\left (x \right )+x \right )}{e^{\frac {2 \,\mathrm {log}\left (\frac {x^{4}}{3}\right )}{x}} \mathrm {log}\left (x \right )^{2}} \] Input:
int(((-40*log(x)^2*log(1/3*x^4)+(-40*x+160)*log(x)^2+40*x*log(x))*exp(log( 1/3*x^4)/x)+2*x*log(x)*log(1/3*x^4)+(2*x^2-8*x)*log(x)-2*x^2)/x/log(x)^3/e xp(log(1/3*x^4)/x)^2,x)
Output:
(x*( - 40*e**(log(x**4/3)/x)*log(x) + x))/(e**((2*log(x**4/3))/x)*log(x)** 2)