3.55.0 ∫ 2x6+(6x3+2x4) log(2)+(−6x3 log(2)+(18+12x+2x2) log2(2)) log(x)+(−36−18x−2x2) log2(2) log2(x) x5 dx [5430]

Integrand size = 69, antiderivative size = 15 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\left (x+\frac {(3+x) \log (2) \log (x)}{x^2}\right )^2 \]

Rubi [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(165\) vs. \(2(15)=30\).

Time = 0.16 (sec) , antiderivative size = 165, normalized size of antiderivative = 11.00, number of steps used = 18, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.058, Rules used = {14, 2404, 2341, 2342} \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {9 \log ^2(2)}{8 x^4}-\frac {\log (2) \log (512) \log (x)}{2 x^4}-\frac {\log (2) \log (512)}{8 x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {4 \log ^2(2) \log (x)}{x^3}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {2 \log (2) \log (64)}{9 x^3}+x^2+\frac {\log ^2(2) \log ^2(x)}{x^2}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {2 \log (8)}{x}+\frac {6 \log (2)}{x} \]

Rubi steps \begin{align*} \text {integral}& = \int \left (\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}-\frac {2 (3+x) (6+x) \log ^2(2) \log ^2(x)}{x^5}\right ) \, dx \\ & = 2 \int \frac {x^3+x \log (2)+\log (8)}{x^2} \, dx-(2 \log (2)) \int \frac {\left (3 x^3-x^2 \log (2)-x \log (64)-\log (512)\right ) \log (x)}{x^5} \, dx-\left (2 \log ^2(2)\right ) \int \frac {(3+x) (6+x) \log ^2(x)}{x^5} \, dx \\ & = 2 \int \left (x+\frac {\log (2)}{x}+\frac {\log (8)}{x^2}\right ) \, dx-(2 \log (2)) \int \left (\frac {3 \log (x)}{x^2}-\frac {\log (2) \log (x)}{x^3}-\frac {\log (64) \log (x)}{x^4}-\frac {\log (512) \log (x)}{x^5}\right ) \, dx-\left (2 \log ^2(2)\right ) \int \left (\frac {18 \log ^2(x)}{x^5}+\frac {9 \log ^2(x)}{x^4}+\frac {\log ^2(x)}{x^3}\right ) \, dx \\ & = x^2-\frac {2 \log (8)}{x}+2 \log (2) \log (x)-(6 \log (2)) \int \frac {\log (x)}{x^2} \, dx+\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (2 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^3} \, dx-\left (18 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^4} \, dx-\left (36 \log ^2(2)\right ) \int \frac {\log ^2(x)}{x^5} \, dx+(2 \log (2) \log (64)) \int \frac {\log (x)}{x^4} \, dx+(2 \log (2) \log (512)) \int \frac {\log (x)}{x^5} \, dx \\ & = x^2+\frac {6 \log (2)}{x}-\frac {\log ^2(2)}{2 x^2}-\frac {2 \log (8)}{x}-\frac {2 \log (2) \log (64)}{9 x^3}-\frac {\log (2) \log (512)}{8 x^4}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}-\frac {\log ^2(2) \log (x)}{x^2}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {\log (2) \log (512) \log (x)}{2 x^4}+\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}-\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x^3} \, dx-\left (12 \log ^2(2)\right ) \int \frac {\log (x)}{x^4} \, dx-\left (18 \log ^2(2)\right ) \int \frac {\log (x)}{x^5} \, dx \\ & = x^2+\frac {6 \log (2)}{x}+\frac {9 \log ^2(2)}{8 x^4}+\frac {4 \log ^2(2)}{3 x^3}-\frac {2 \log (8)}{x}-\frac {2 \log (2) \log (64)}{9 x^3}-\frac {\log (2) \log (512)}{8 x^4}+2 \log (2) \log (x)+\frac {6 \log (2) \log (x)}{x}+\frac {9 \log ^2(2) \log (x)}{2 x^4}+\frac {4 \log ^2(2) \log (x)}{x^3}-\frac {2 \log (2) \log (64) \log (x)}{3 x^3}-\frac {\log (2) \log (512) \log (x)}{2 x^4}+\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} \\ \end{align*}

Mathematica [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(57\) vs. \(2(15)=30\).

Time = 0.04 (sec) , antiderivative size = 57, normalized size of antiderivative = 3.80 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=x^2+2 \log (2) \log (x)+\frac {2 \log (8) \log (x)}{x}+\frac {9 \log ^2(2) \log ^2(x)}{x^4}+\frac {6 \log ^2(2) \log ^2(x)}{x^3}+\frac {\log ^2(2) \log ^2(x)}{x^2} \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(39\) vs. \(2(15)=30\).

Time = 0.10 (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	\(\frac {x^{6}+2 x^{4} \ln \left (2\right ) \ln \left (x \right )+\ln \left (2\right )^{2} \ln \left (x \right )^{2} x^{2}+9 \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 x \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 x^{3} \ln \left (2\right ) \ln \left (x \right )}{x^{4}}\)	\(60\)
parallelrisch	\(\frac {x^{6}+2 x^{4} \ln \left (2\right ) \ln \left (x \right )+\ln \left (2\right )^{2} \ln \left (x \right )^{2} x^{2}+9 \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 x \ln \left (2\right )^{2} \ln \left (x \right )^{2}+6 x^{3} \ln \left (2\right ) \ln \left (x \right )}{x^{4}}\)	\(60\)
parts	\(x^{2}+2 \ln \left (2\right ) \ln \left (x \right )-\frac {6 \ln \left (2\right )}{x}+2 \ln \left (2\right ) \left (\ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+\frac {3 \ln \left (x \right )}{x}+\frac {3}{x}+6 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )+9 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )\right )-2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}-\frac {3 \ln \left (x \right )^{2}}{x^{3}}-\frac {2 \ln \left (x \right )}{x^{3}}-\frac {2}{3 x^{3}}-\frac {9 \ln \left (x \right )^{2}}{2 x^{4}}-\frac {9 \ln \left (x \right )}{4 x^{4}}-\frac {9}{16 x^{4}}\right )\)	\(155\)
default	\(-2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{2 x^{2}}-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+x^{2}-18 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{3 x^{3}}-\frac {2 \ln \left (x \right )}{9 x^{3}}-\frac {2}{27 x^{3}}\right )+2 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-6 \ln \left (2\right ) \left (-\frac {\ln \left (x \right )}{x}-\frac {1}{x}\right )+2 \ln \left (2\right ) \ln \left (x \right )-36 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )^{2}}{4 x^{4}}-\frac {\ln \left (x \right )}{8 x^{4}}-\frac {1}{32 x^{4}}\right )+12 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{3 x^{3}}-\frac {1}{9 x^{3}}\right )-\frac {6 \ln \left (2\right )}{x}+18 \ln \left (2\right )^{2} \left (-\frac {\ln \left (x \right )}{4 x^{4}}-\frac {1}{16 x^{4}}\right )\)	\(176\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (15) = 30\).

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 2.67 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\frac {x^{6} + {\left (x^{2} + 6 \, x + 9\right )} \log \left (2\right )^{2} \log \left (x\right )^{2} + 2 \, {\left (x^{4} + 3 \, x^{3}\right )} \log \left (2\right ) \log \left (x\right )}{x^{4}} \]

Sympy [B] (veriﬁcation not implemented)

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 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=x^{2} + 2 \log {\left (2 \right )} \log {\left (x \right )} + \frac {6 \log {\left (2 \right )} \log {\left (x \right )}}{x} + \frac {\left (x^{2} \log {\left (2 \right )}^{2} + 6 x \log {\left (2 \right )}^{2} + 9 \log {\left (2 \right )}^{2}\right ) \log {\left (x \right )}^{2}}{x^{4}} \]

Maxima [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 145 vs. \(2 (15) = 30\).

Time = 0.25 (sec) , antiderivative size = 145, normalized size of antiderivative = 9.67 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=-\frac {1}{2} \, {\left (\frac {2 \, \log \left (x\right )}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (2\right )^{2} - \frac {4}{3} \, {\left (\frac {3 \, \log \left (x\right )}{x^{3}} + \frac {1}{x^{3}}\right )} \log \left (2\right )^{2} - \frac {9}{8} \, {\left (\frac {4 \, \log \left (x\right )}{x^{4}} + \frac {1}{x^{4}}\right )} \log \left (2\right )^{2} + x^{2} + 6 \, {\left (\frac {\log \left (x\right )}{x} + \frac {1}{x}\right )} \log \left (2\right ) + 2 \, \log \left (2\right ) \log \left (x\right ) + \frac {{\left (2 \, \log \left (x\right )^{2} + 2 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{2 \, x^{2}} - \frac {6 \, \log \left (2\right )}{x} + \frac {2 \, {\left (9 \, \log \left (x\right )^{2} + 6 \, \log \left (x\right ) + 2\right )} \log \left (2\right )^{2}}{3 \, x^{3}} + \frac {9 \, {\left (8 \, \log \left (x\right )^{2} + 4 \, \log \left (x\right ) + 1\right )} \log \left (2\right )^{2}}{8 \, x^{4}} \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (15) = 30\).

Time = 0.27 (sec) , antiderivative size = 49, normalized size of antiderivative = 3.27 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=x^{2} + 2 \, \log \left (2\right ) \log \left (x\right ) + \frac {6 \, \log \left (2\right ) \log \left (x\right )}{x} + \frac {{\left (x^{2} \log \left (2\right )^{2} + 6 \, x \log \left (2\right )^{2} + 9 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2}}{x^{4}} \]

Mupad [B] (veriﬁcation not implemented)

Time = 11.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 4.00 \[ \int \frac {2 x^6+\left (6 x^3+2 x^4\right ) \log (2)+\left (-6 x^3 \log (2)+\left (18+12 x+2 x^2\right ) \log ^2(2)\right ) \log (x)+\left (-36-18 x-2 x^2\right ) \log ^2(2) \log ^2(x)}{x^5} \, dx=\frac {x^7+\ln \left (4\right )\,x^5\,\ln \left (x\right )+\ln \left (64\right )\,x^4\,\ln \left (x\right )+{\ln \left (2\right )}^2\,x^3\,{\ln \left (x\right )}^2+6\,{\ln \left (2\right )}^2\,x^2\,{\ln \left (x\right )}^2+9\,{\ln \left (2\right )}^2\,x\,{\ln \left (x\right )}^2}{x^5} \]

\(\int \frac {2 x^6+(6 x^3+2 x^4) \log (2)+(-6 x^3 \log (2)+(18+12 x+2 x^2) \log ^2(2)) \log (x)+(-36-18 x-2 x^2) \log ^2(2) \log ^2(x)}{x^5} \, dx\) [5430]

Optimal result