Integrand size = 69, antiderivative size = 15 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\left (x+\frac {(3+x) \log (2) \log (x)}{x^2}\right )^2 \] Output:
(ln(2)*(3+x)/x^2*ln(x)+x)^2
Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(15)=30\).
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.80 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=x^2+2 \log (2) \log (x)+\frac {2 \log (8) \log (x)}{x}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2} \] Input:
Integrate[(2*x^6 + (6*x^3 + 2*x^4)*Log[2] + (-6*x^3*Log[2] + (18 + 12*x + 2*x^2)*Log[2]^2)*Log[x] + (-36 - 18*x - 2*x^2)*Log[2]^2*Log[x]^2)/x^5,x]
Output:
x^2 + 2*Log[2]*Log[x] + (2*Log[8]*Log[x])/x + (9*Log[2]^2*Log[x]^2)/x^4 + (6*Log[2]^2*Log[x]^2)/x^3 + (Log[2]^2*Log[x]^2)/x^2
Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(15)=30\).
Time = 0.48 (sec) , antiderivative size = 165, normalized size of antiderivative = 11.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2010, 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.
\(\displaystyle \int \frac {2 x^6+\left (-2 x^2-18 x-36\right ) \log ^2(2) \log ^2(x)+\left (2 x^4+6 x^3\right ) \log (2)+\left (\left (2 x^2+12 x+18\right ) \log ^2(2)-6 x^3 \log (2)\right ) \log (x)}{x^5} \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (-\frac {2 (x+3) (x+6) \log ^2(2) \log ^2(x)}{x^5}+\frac {2 \left (x^3+x \log (2)+\log (8)\right )}{x^2}-\frac {2 \log (2) \left (3 x^3-x^2 \log (2)-x \log (64)-\log (512)\right ) \log (x)}{x^5}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {9 \log ^2(2)}{8 x^4}-\frac {\log (2) \log (512) \log (x)}{2 x^4}-\frac {\log (2) \log (512)}{8 x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {4 \log ^2(2) \log (x)}{x^3}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {2 \log (2) \log (64)}{9 x^3}+x^2+\frac {\log ^2(2) \log ^2(x)}{x^2}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {2 \log (8)}{x}+\frac {6 \log (2)}{x}\) |
Input:
Int[(2*x^6 + (6*x^3 + 2*x^4)*Log[2] + (-6*x^3*Log[2] + (18 + 12*x + 2*x^2) *Log[2]^2)*Log[x] + (-36 - 18*x - 2*x^2)*Log[2]^2*Log[x]^2)/x^5,x]
Output:
x^2 + (6*Log[2])/x + (9*Log[2]^2)/(8*x^4) + (4*Log[2]^2)/(3*x^3) - (2*Log[ 8])/x - (2*Log[2]*Log[64])/(9*x^3) - (Log[2]*Log[512])/(8*x^4) + 2*Log[2]* Log[x] + (6*Log[2]*Log[x])/x + (9*Log[2]^2*Log[x])/(2*x^4) + (4*Log[2]^2*L og[x])/x^3 - (2*Log[2]*Log[64]*Log[x])/(3*x^3) - (Log[2]*Log[512]*Log[x])/ (2*x^4) + (9*Log[2]^2*Log[x]^2)/x^4 + (6*Log[2]^2*Log[x]^2)/x^3 + (Log[2]^ 2*Log[x]^2)/x^2
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. \(39\) vs. \(2(15)=30\).
Time = 4.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.67
method | result | size |
risch | \(\frac {\ln \left (2\right )^{2} \left (x^{2}+6 x +9\right ) \ln \left (x \right )^{2}}{x^{4}}+\frac {6 \ln \left (2\right ) \ln \left (x \right )}{x}+x^{2}+2 \ln \left (2\right ) \ln \left (x \right )\) | \(40\) |
norman | \(\frac {x^{6}+2 x^{4} \ln \left (x \right ) \ln \left (2\right )+\ln \left (2\right )^{2} \ln \left (x \right )^{2} x^{2}+9 \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 \ln \left (2\right )^{2} x \ln \left (x \right )^{2}+6 \ln \left (x \right ) \ln \left (2\right ) x^{3}}{x^{4}}\) | \(60\) |
parallelrisch | \(\frac {x^{6}+2 x^{4} \ln \left (x \right ) \ln \left (2\right )+\ln \left (2\right )^{2} \ln \left (x \right )^{2} x^{2}+9 \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 \ln \left (2\right )^{2} x \ln \left (x \right )^{2}+6 \ln \left (x \right ) \ln \left (2\right ) x^{3}}{x^{4}}\) | \(60\) |
parts | \(x^{2}-\frac {6 \ln \left (2\right )}{x}+2 \ln \left (2\right ) \ln \left (x \right )+2 \ln \left (2\right ) \left (\ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+\frac {3 \ln \left (x \right )}{x}+\frac {3}{x}+6 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )+9 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )\right )-2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}-\frac {3 \ln \left (x \right )^{2}}{x^{3}}-\frac {2 \ln \left (x \right )}{x^{3}}-\frac {2}{3 x^{3}}-\frac {9 \ln \left (x \right )^{2}}{2 x^{4}}-\frac {9 \ln \left (x \right )}{4 x^{4}}-\frac {9}{16 x^{4}}\right )\) | \(155\) |
default | \(-2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+x^{2}-18 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{3 x^{3}}-\frac {2 \ln \left (x \right )}{9 x^{3}}-\frac {2}{27 x^{3}}\right )+2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-6 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )+2 \ln \left (2\right ) \ln \left (x \right )-36 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{4 x^{4}}-\frac {\ln \left (x \right )}{8 x^{4}}-\frac {1}{32 x^{4}}\right )+12 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )-\frac {6 \ln \left (2\right )}{x}+18 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )\) | \(176\) |
Input:
int(((-2*x^2-18*x-36)*ln(2)^2*ln(x)^2+((2*x^2+12*x+18)*ln(2)^2-6*x^3*ln(2) )*ln(x)+(2*x^4+6*x^3)*ln(2)+2*x^6)/x^5,x,method=_RETURNVERBOSE)
Output:
ln(2)^2*(x^2+6*x+9)/x^4*ln(x)^2+6*ln(2)*ln(x)/x+x^2+2*ln(2)*ln(x)
Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (15) = 30\).
Time = 0.08 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.67 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\frac {x^{6} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (2\right )^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{4} + 3 \, x^{3}\right )} \log \left (2\right ) \log \left (x\right )}{x^{4}} \] Input:
integrate(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6* x^3*log(2))*log(x)+(2*x^4+6*x^3)*log(2)+2*x^6)/x^5,x, algorithm="fricas")
Output:
(x^6 + (x^2 + 6*x + 9)*log(2)^2*log(x)^2 + 2*(x^4 + 3*x^3)*log(2)*log(x))/ x^4
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (15) = 30\).
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 3.53 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=x^{2} + 2 \log {\left (2 \right )} \log {\left (x \right )} + \frac {6 \log {\left (2 \right )} \log {\left (x \right )}}{x} + \frac {\left (x^{2} \log {\left (2 \right )}^{2} + 6 x \log {\left (2 \right )}^{2} + 9 \log {\left (2 \right )}^{2}\right ) \log {\left (x \right )}^{2}}{x^{4}} \] Input:
integrate(((-2*x**2-18*x-36)*ln(2)**2*ln(x)**2+((2*x**2+12*x+18)*ln(2)**2- 6*x**3*ln(2))*ln(x)+(2*x**4+6*x**3)*ln(2)+2*x**6)/x**5,x)
Output:
x**2 + 2*log(2)*log(x) + 6*log(2)*log(x)/x + (x**2*log(2)**2 + 6*x*log(2)* *2 + 9*log(2)**2)*log(x)**2/x**4
Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (15) = 30\).
Time = 0.03 (sec) , antiderivative size = 145, normalized size of antiderivative = 9.67 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=-\frac {1}{2} \, {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (2\right )^{2} - \frac {4}{3} \, {\left (\frac {3 \, \log \left (x\right )}{x^{3}} + \frac {1}{x^{3}}\right )} \log \left (2\right )^{2} - \frac {9}{8} \, {\left (\frac {4 \, \log \left (x\right )}{x^{4}} + \frac {1}{x^{4}}\right )} \log \left (2\right )^{2} + x^{2} + 6 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} \log \left (2\right ) + 2 \, \log \left (2\right ) \log \left (x\right ) + \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{2 \, x^{2}} - \frac {6 \, \log \left (2\right )}{x} + \frac {2 \, {\left (9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) + 2\right )} \log \left (2\right )^{2}}{3 \, x^{3}} + \frac {9 \, {\left (8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{8 \, x^{4}} \] Input:
integrate(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6* x^3*log(2))*log(x)+(2*x^4+6*x^3)*log(2)+2*x^6)/x^5,x, algorithm="maxima")
Output:
-1/2*(2*log(x)/x^2 + 1/x^2)*log(2)^2 - 4/3*(3*log(x)/x^3 + 1/x^3)*log(2)^2 - 9/8*(4*log(x)/x^4 + 1/x^4)*log(2)^2 + x^2 + 6*(log(x)/x + 1/x)*log(2) + 2*log(2)*log(x) + 1/2*(2*log(x)^2 + 2*log(x) + 1)*log(2)^2/x^2 - 6*log(2) /x + 2/3*(9*log(x)^2 + 6*log(x) + 2)*log(2)^2/x^3 + 9/8*(8*log(x)^2 + 4*lo g(x) + 1)*log(2)^2/x^4
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).
Time = 0.12 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.27 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=x^{2} + 2 \, \log \left (2\right ) \log \left (x\right ) + \frac {6 \, \log \left (2\right ) \log \left (x\right )}{x} + \frac {{\left (x^{2} \log \left (2\right )^{2} + 6 \, x \log \left (2\right )^{2} + 9 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2}}{x^{4}} \] Input:
integrate(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6* x^3*log(2))*log(x)+(2*x^4+6*x^3)*log(2)+2*x^6)/x^5,x, algorithm="giac")
Output:
x^2 + 2*log(2)*log(x) + 6*log(2)*log(x)/x + (x^2*log(2)^2 + 6*x*log(2)^2 + 9*log(2)^2)*log(x)^2/x^4
Time = 2.73 (sec) , antiderivative size = 60, normalized size of antiderivative = 4.00 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\frac {x^7+\ln \left (4\right )\,x^5\,\ln \left (x\right )+\ln \left (64\right )\,x^4\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^3\,{\ln \left (x\right )}^2+6\,{\ln \left (2\right )}^2\,x^2\,{\ln \left (x\right )}^2+9\,{\ln \left (2\right )}^2\,x\,{\ln \left (x\right )}^2}{x^5} \] Input:
int((log(2)*(6*x^3 + 2*x^4) + 2*x^6 + log(x)*(log(2)^2*(12*x + 2*x^2 + 18) - 6*x^3*log(2)) - log(2)^2*log(x)^2*(18*x + 2*x^2 + 36))/x^5,x)
Output:
(x^7 + 6*x^2*log(2)^2*log(x)^2 + x^3*log(2)^2*log(x)^2 + x^5*log(4)*log(x) + x^4*log(64)*log(x) + 9*x*log(2)^2*log(x)^2)/x^5
Time = 0.18 (sec) , antiderivative size = 59, normalized size of antiderivative = 3.93 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\frac {\mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2} x^{2}+6 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2} x +9 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2}+2 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{4}+6 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{3}+x^{6}}{x^{4}} \] Input:
int(((-2*x^2-18*x-36)*log(2)^2*log(x)^2+((2*x^2+12*x+18)*log(2)^2-6*x^3*lo g(2))*log(x)+(2*x^4+6*x^3)*log(2)+2*x^6)/x^5,x)
Output:
(log(x)**2*log(2)**2*x**2 + 6*log(x)**2*log(2)**2*x + 9*log(x)**2*log(2)** 2 + 2*log(x)*log(2)*x**4 + 6*log(x)*log(2)*x**3 + x**6)/x**4