Integrand size = 34, antiderivative size = 28 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=\left (-\frac {1}{x}+x \log \left (5+\frac {1}{5} \left (\frac {1}{8 \log (3)}+\log (5)\right )\right )\right )^2 \] Output:
(ln(5+1/40/ln(3)+1/5*ln(5))*x-1/x)^2
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=\frac {1}{x^2}+x^2 \log ^2\left (5+\frac {1}{40 \log (3)}+\frac {\log (5)}{5}\right ) \] Input:
Integrate[(-2 + 2*x^4*Log[(1 + 200*Log[3] + 8*Log[3]*Log[5])/(40*Log[3])]^ 2)/x^3,x]
Output:
x^(-2) + x^2*Log[5 + 1/(40*Log[3]) + Log[5]/5]^2
Time = 0.17 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {802, 2009}
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.
\(\displaystyle \int \frac {2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )-2}{x^3} \, dx\) |
\(\Big \downarrow \) 802 |
\(\displaystyle \int \left (2 x \log ^2\left (5+\frac {1}{40 \log (3)}+\frac {\log (5)}{5}\right )-\frac {2}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{x^2}+x^2 \log ^2\left (5+\frac {1}{40 \log (3)}+\frac {\log (5)}{5}\right )\) |
Input:
Int[(-2 + 2*x^4*Log[(1 + 200*Log[3] + 8*Log[3]*Log[5])/(40*Log[3])]^2)/x^3 ,x]
Output:
x^(-2) + x^2*Log[5 + 1/(40*Log[3]) + Log[5]/5]^2
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[Exp andIntegrand[(c*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Time = 0.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
method | result | size |
default | \(\ln \left (\frac {8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1}{40 \ln \left (3\right )}\right )^{2} x^{2}+\frac {1}{x^{2}}\) | \(30\) |
gosper | \(\frac {x^{4} \ln \left (\frac {8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1}{40 \ln \left (3\right )}\right )^{2}+1}{x^{2}}\) | \(32\) |
parallelrisch | \(\frac {x^{4} \ln \left (\frac {8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1}{40 \ln \left (3\right )}\right )^{2}+1}{x^{2}}\) | \(32\) |
norman | \(\frac {1+\left (\ln \left (40\right )^{2}-2 \ln \left (40\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )+2 \ln \left (40\right ) \ln \left (\ln \left (3\right )\right )+\ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )^{2}-2 \ln \left (\ln \left (3\right )\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )+\ln \left (\ln \left (3\right )\right )^{2}\right ) x^{4}}{x^{2}}\) | \(78\) |
risch | \(9 x^{2} \ln \left (2\right )^{2}+6 \ln \left (2\right ) \ln \left (5\right ) x^{2}+6 \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) x^{2}-6 \ln \left (2\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right ) x^{2}+x^{2} \ln \left (5\right )^{2}+2 \ln \left (5\right ) \ln \left (\ln \left (3\right )\right ) x^{2}-2 \ln \left (5\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right ) x^{2}+\ln \left (\ln \left (3\right )\right )^{2} x^{2}-2 \ln \left (\ln \left (3\right )\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right ) x^{2}+\ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )^{2} x^{2}+\frac {1}{x^{2}}\) | \(140\) |
orering | \(\frac {\left (9 x^{4} \ln \left (2\right )^{2}+6 \ln \left (2\right ) \ln \left (5\right ) x^{4}+6 \ln \left (2\right ) \ln \left (\ln \left (3\right )\right ) x^{4}-6 \ln \left (2\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right ) x^{4}+x^{4} \ln \left (5\right )^{2}+2 \ln \left (5\right ) \ln \left (\ln \left (3\right )\right ) x^{4}-2 \ln \left (5\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right ) x^{4}+\ln \left (\ln \left (3\right )\right )^{2} x^{4}-2 \ln \left (\ln \left (3\right )\right ) \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right ) x^{4}+\ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )^{2} x^{4}+1\right ) \left (2 x^{4} \ln \left (\frac {8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1}{40 \ln \left (3\right )}\right )^{2}-2\right )}{2 x^{2} \left (3 x^{2} \ln \left (2\right )+x^{2} \ln \left (5\right )-x^{2} \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )+x^{2} \ln \left (\ln \left (3\right )\right )-1\right ) \left (3 x^{2} \ln \left (2\right )+x^{2} \ln \left (5\right )-x^{2} \ln \left (8 \ln \left (3\right ) \ln \left (5\right )+200 \ln \left (3\right )+1\right )+x^{2} \ln \left (\ln \left (3\right )\right )+1\right )}\) | \(255\) |
Input:
int((2*x^4*ln(1/40*(8*ln(3)*ln(5)+200*ln(3)+1)/ln(3))^2-2)/x^3,x,method=_R ETURNVERBOSE)
Output:
ln(1/40*(8*ln(3)*ln(5)+200*ln(3)+1)/ln(3))^2*x^2+1/x^2
Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=\frac {x^{4} \log \left (\frac {8 \, \log \left (5\right ) \log \left (3\right ) + 200 \, \log \left (3\right ) + 1}{40 \, \log \left (3\right )}\right )^{2} + 1}{x^{2}} \] Input:
integrate((2*x^4*log(1/40*(8*log(3)*log(5)+200*log(3)+1)/log(3))^2-2)/x^3, x, algorithm="fricas")
Output:
(x^4*log(1/40*(8*log(5)*log(3) + 200*log(3) + 1)/log(3))^2 + 1)/x^2
Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.14 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=x^{2} \left (- 2 \log {\left (\frac {1}{40} + \frac {\log {\left (3 \right )} \log {\left (5 \right )}}{5} + 5 \log {\left (3 \right )} \right )} \log {\left (\log {\left (3 \right )} \right )} + \log {\left (\log {\left (3 \right )} \right )}^{2} + \log {\left (\frac {1}{40} + \frac {\log {\left (3 \right )} \log {\left (5 \right )}}{5} + 5 \log {\left (3 \right )} \right )}^{2}\right ) + \frac {1}{x^{2}} \] Input:
integrate((2*x**4*ln(1/40*(8*ln(3)*ln(5)+200*ln(3)+1)/ln(3))**2-2)/x**3,x)
Output:
x**2*(-2*log(1/40 + log(3)*log(5)/5 + 5*log(3))*log(log(3)) + log(log(3))* *2 + log(1/40 + log(3)*log(5)/5 + 5*log(3))**2) + x**(-2)
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=x^{2} \log \left (\frac {8 \, \log \left (5\right ) \log \left (3\right ) + 200 \, \log \left (3\right ) + 1}{40 \, \log \left (3\right )}\right )^{2} + \frac {1}{x^{2}} \] Input:
integrate((2*x^4*log(1/40*(8*log(3)*log(5)+200*log(3)+1)/log(3))^2-2)/x^3, x, algorithm="maxima")
Output:
x^2*log(1/40*(8*log(5)*log(3) + 200*log(3) + 1)/log(3))^2 + 1/x^2
Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=x^{2} \log \left (\frac {8 \, \log \left (5\right ) \log \left (3\right ) + 200 \, \log \left (3\right ) + 1}{40 \, \log \left (3\right )}\right )^{2} + \frac {1}{x^{2}} \] Input:
integrate((2*x^4*log(1/40*(8*log(3)*log(5)+200*log(3)+1)/log(3))^2-2)/x^3, x, algorithm="giac")
Output:
x^2*log(1/40*(8*log(5)*log(3) + 200*log(3) + 1)/log(3))^2 + 1/x^2
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=x^2\,\left ({\ln \left (5\,\ln \left (3\right )+\frac {\ln \left (3\right )\,\ln \left (5\right )}{5}+\frac {1}{40}\right )}^2-2\,\ln \left (5\,\ln \left (3\right )+\frac {\ln \left (3\right )\,\ln \left (5\right )}{5}+\frac {1}{40}\right )\,\ln \left (\ln \left (3\right )\right )+{\ln \left (\ln \left (3\right )\right )}^2\right )+\frac {1}{x^2} \] Input:
int((2*x^4*log((5*log(3) + (log(3)*log(5))/5 + 1/40)/log(3))^2 - 2)/x^3,x)
Output:
x^2*(log(5*log(3) + (log(3)*log(5))/5 + 1/40)^2 - 2*log(5*log(3) + (log(3) *log(5))/5 + 1/40)*log(log(3)) + log(log(3))^2) + 1/x^2
Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {-2+2 x^4 \log ^2\left (\frac {1+200 \log (3)+8 \log (3) \log (5)}{40 \log (3)}\right )}{x^3} \, dx=\frac {\mathrm {log}\left (\frac {8 \,\mathrm {log}\left (5\right ) \mathrm {log}\left (3\right )+200 \,\mathrm {log}\left (3\right )+1}{40 \,\mathrm {log}\left (3\right )}\right )^{2} x^{4}+1}{x^{2}} \] Input:
int((2*x^4*log(1/40*(8*log(3)*log(5)+200*log(3)+1)/log(3))^2-2)/x^3,x)
Output:
(log((8*log(5)*log(3) + 200*log(3) + 1)/(40*log(3)))**2*x**4 + 1)/x**2