Integrand size = 50, antiderivative size = 29 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {\frac {3}{4 x^2}+\frac {2}{x}+x+\frac {1+5 x}{\log (x)}}{\log ^2(x)} \] Output:
(x+2/x+3/4/x^2+(1+5*x)/ln(x))/ln(x)^2
Time = 0.13 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {1}{\log ^3(x)}+\frac {5 x}{\log ^3(x)}+\frac {3}{4 x^2 \log ^2(x)}+\frac {2}{x \log ^2(x)}+\frac {x}{\log ^2(x)} \] Input:
Integrate[(-6*x^2 - 30*x^3 + (-3 - 8*x + 6*x^3)*Log[x] + (-3 - 4*x + 2*x^3 )*Log[x]^2)/(2*x^3*Log[x]^4),x]
Output:
Log[x]^(-3) + (5*x)/Log[x]^3 + 3/(4*x^2*Log[x]^2) + 2/(x*Log[x]^2) + x/Log [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.
\(\displaystyle \int \frac {-30 x^3+\left (2 x^3-4 x-3\right ) \log ^2(x)+\left (6 x^3-8 x-3\right ) \log (x)-6 x^2}{2 x^3 \log ^4(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \int -\frac {30 x^3+6 x^2+\left (-2 x^3+4 x+3\right ) \log ^2(x)+\left (-6 x^3+8 x+3\right ) \log (x)}{x^3 \log ^4(x)}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {30 x^3+6 x^2+\left (-2 x^3+4 x+3\right ) \log ^2(x)+\left (-6 x^3+8 x+3\right ) \log (x)}{x^3 \log ^4(x)}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {6 (5 x+1)}{x \log ^4(x)}+\frac {-2 x^3+4 x+3}{x^3 \log ^2(x)}+\frac {-6 x^3+8 x+3}{x^3 \log ^3(x)}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {-6 x^3+8 x+3}{x^3 \log ^3(x)}dx-\int \frac {-2 x^3+4 x+3}{x^3 \log ^2(x)}dx-5 \operatorname {LogIntegral}(x)+\frac {10 x}{\log ^3(x)}+\frac {2}{\log ^3(x)}+\frac {5 x}{\log ^2(x)}+\frac {5 x}{\log (x)}\right )\) |
Input:
Int[(-6*x^2 - 30*x^3 + (-3 - 8*x + 6*x^3)*Log[x] + (-3 - 4*x + 2*x^3)*Log[ x]^2)/(2*x^3*Log[x]^4),x]
Output:
$Aborted
Time = 0.34 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14
method | result | size |
norman | \(\frac {x^{2}+x^{3} \ln \left (x \right )+5 x^{3}+2 x \ln \left (x \right )+\frac {3 \ln \left (x \right )}{4}}{x^{2} \ln \left (x \right )^{3}}\) | \(33\) |
default | \(\frac {x}{\ln \left (x \right )^{2}}+\frac {5 x}{\ln \left (x \right )^{3}}+\frac {2}{x \ln \left (x \right )^{2}}+\frac {1}{\ln \left (x \right )^{3}}+\frac {3}{4 x^{2} \ln \left (x \right )^{2}}\) | \(37\) |
risch | \(\frac {4 x^{3} \ln \left (x \right )+20 x^{3}+4 x^{2}+8 x \ln \left (x \right )+3 \ln \left (x \right )}{4 x^{2} \ln \left (x \right )^{3}}\) | \(37\) |
parallelrisch | \(-\frac {-4 x^{3} \ln \left (x \right )-20 x^{3}-4 x^{2}-8 x \ln \left (x \right )-3 \ln \left (x \right )}{4 x^{2} \ln \left (x \right )^{3}}\) | \(37\) |
parts | \(\frac {x}{\ln \left (x \right )^{2}}+\frac {5 x}{\ln \left (x \right )^{3}}+\frac {2}{x \ln \left (x \right )^{2}}+\frac {1}{\ln \left (x \right )^{3}}+\frac {3}{4 x^{2} \ln \left (x \right )^{2}}\) | \(37\) |
Input:
int(1/2*((2*x^3-4*x-3)*ln(x)^2+(6*x^3-8*x-3)*ln(x)-30*x^3-6*x^2)/x^3/ln(x) ^4,x,method=_RETURNVERBOSE)
Output:
(x^2+x^3*ln(x)+5*x^3+2*x*ln(x)+3/4*ln(x))/x^2/ln(x)^3
Time = 0.09 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {20 \, x^{3} + 4 \, x^{2} + {\left (4 \, x^{3} + 8 \, x + 3\right )} \log \left (x\right )}{4 \, x^{2} \log \left (x\right )^{3}} \] Input:
integrate(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x ^3/log(x)^4,x, algorithm="fricas")
Output:
1/4*(20*x^3 + 4*x^2 + (4*x^3 + 8*x + 3)*log(x))/(x^2*log(x)^3)
Time = 0.06 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {20 x^{3} + 4 x^{2} + \left (4 x^{3} + 8 x + 3\right ) \log {\left (x \right )}}{4 x^{2} \log {\left (x \right )}^{3}} \] Input:
integrate(1/2*((2*x**3-4*x-3)*ln(x)**2+(6*x**3-8*x-3)*ln(x)-30*x**3-6*x**2 )/x**3/ln(x)**4,x)
Output:
(20*x**3 + 4*x**2 + (4*x**3 + 8*x + 3)*log(x))/(4*x**2*log(x)**3)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {1}{\log \left (x\right )^{3}} + 3 \, \Gamma \left (-1, 2 \, \log \left (x\right )\right ) + \Gamma \left (-1, -\log \left (x\right )\right ) + 2 \, \Gamma \left (-1, \log \left (x\right )\right ) + 6 \, \Gamma \left (-2, 2 \, \log \left (x\right )\right ) - 3 \, \Gamma \left (-2, -\log \left (x\right )\right ) + 4 \, \Gamma \left (-2, \log \left (x\right )\right ) - 15 \, \Gamma \left (-3, -\log \left (x\right )\right ) \] Input:
integrate(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x ^3/log(x)^4,x, algorithm="maxima")
Output:
1/log(x)^3 + 3*gamma(-1, 2*log(x)) + gamma(-1, -log(x)) + 2*gamma(-1, log( x)) + 6*gamma(-2, 2*log(x)) - 3*gamma(-2, -log(x)) + 4*gamma(-2, log(x)) - 15*gamma(-3, -log(x))
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {4 \, x^{3} \log \left (x\right ) + 20 \, x^{3} + 4 \, x^{2} + 8 \, x \log \left (x\right ) + 3 \, \log \left (x\right )}{4 \, x^{2} \log \left (x\right )^{3}} \] Input:
integrate(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x ^3/log(x)^4,x, algorithm="giac")
Output:
1/4*(4*x^3*log(x) + 20*x^3 + 4*x^2 + 8*x*log(x) + 3*log(x))/(x^2*log(x)^3)
Time = 2.98 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {\ln \left (x\right )\,\left (x^3+2\,x+\frac {3}{4}\right )+x^2+5\,x^3}{x^2\,{\ln \left (x\right )}^3} \] Input:
int(-((log(x)^2*(4*x - 2*x^3 + 3))/2 + (log(x)*(8*x - 6*x^3 + 3))/2 + 3*x^ 2 + 15*x^3)/(x^3*log(x)^4),x)
Output:
(log(x)*(2*x + x^3 + 3/4) + x^2 + 5*x^3)/(x^2*log(x)^3)
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {-6 x^2-30 x^3+\left (-3-8 x+6 x^3\right ) \log (x)+\left (-3-4 x+2 x^3\right ) \log ^2(x)}{2 x^3 \log ^4(x)} \, dx=\frac {4 \,\mathrm {log}\left (x \right ) x^{3}+8 \,\mathrm {log}\left (x \right ) x +3 \,\mathrm {log}\left (x \right )+20 x^{3}+4 x^{2}}{4 \mathrm {log}\left (x \right )^{3} x^{2}} \] Input:
int(1/2*((2*x^3-4*x-3)*log(x)^2+(6*x^3-8*x-3)*log(x)-30*x^3-6*x^2)/x^3/log (x)^4,x)
Output:
(4*log(x)*x**3 + 8*log(x)*x + 3*log(x) + 20*x**3 + 4*x**2)/(4*log(x)**3*x* *2)