Integrand size = 105, antiderivative size = 33 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=\frac {3 \left (-3+\frac {-1+x}{2+x}\right ) \left (1+x+\log \left (\frac {x}{2 (x-\log (x))}\right )\right )}{x} \]
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Time = 2.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.85, number of steps used = 23, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {6873, 6874, 78, 907, 2635, 12} \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=-\frac {21}{2 x}-\frac {9}{2 (x+2)}-\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (x+2)} \]
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Rule 12
Rule 78
Rule 907
Rule 2635
Rule 6873
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {-42+9 x+36 x^2+15 x^3-\left (9 x+9 x^2\right ) \log (x)-\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{x^2 (2+x)^2 (x-\log (x))} \, dx \\ & = \int \left (\frac {36}{(2+x)^2 (x-\log (x))}-\frac {42}{x^2 (2+x)^2 (x-\log (x))}+\frac {9}{x (2+x)^2 (x-\log (x))}+\frac {15 x}{(2+x)^2 (x-\log (x))}-\frac {9 (1+x) \log (x)}{x (2+x)^2 (x-\log (x))}+\frac {6 \left (7+7 x+x^2\right ) \log \left (\frac {x}{2 (x-\log (x))}\right )}{x^2 (2+x)^2}\right ) \, dx \\ & = 6 \int \frac {\left (7+7 x+x^2\right ) \log \left (\frac {x}{2 (x-\log (x))}\right )}{x^2 (2+x)^2} \, dx+9 \int \frac {1}{x (2+x)^2 (x-\log (x))} \, dx-9 \int \frac {(1+x) \log (x)}{x (2+x)^2 (x-\log (x))} \, dx+15 \int \frac {x}{(2+x)^2 (x-\log (x))} \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-42 \int \frac {1}{x^2 (2+x)^2 (x-\log (x))} \, dx \\ & = -\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (2+x)}-6 \int \frac {(7+2 x) (-1+\log (x))}{2 x^2 (2+x) (x-\log (x))} \, dx-9 \int \left (\frac {-1-x}{x (2+x)^2}+\frac {1+x}{(2+x)^2 (x-\log (x))}\right ) \, dx+9 \int \left (\frac {1}{4 x (x-\log (x))}-\frac {1}{2 (2+x)^2 (x-\log (x))}-\frac {1}{4 (2+x) (x-\log (x))}\right ) \, dx+15 \int \left (-\frac {2}{(2+x)^2 (x-\log (x))}+\frac {1}{(2+x) (x-\log (x))}\right ) \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-42 \int \left (\frac {1}{4 x^2 (x-\log (x))}-\frac {1}{4 x (x-\log (x))}+\frac {1}{4 (2+x)^2 (x-\log (x))}+\frac {1}{4 (2+x) (x-\log (x))}\right ) \, dx \\ & = -\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (2+x)}+\frac {9}{4} \int \frac {1}{x (x-\log (x))} \, dx-\frac {9}{4} \int \frac {1}{(2+x) (x-\log (x))} \, dx-3 \int \frac {(7+2 x) (-1+\log (x))}{x^2 (2+x) (x-\log (x))} \, dx-\frac {9}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-9 \int \frac {-1-x}{x (2+x)^2} \, dx-9 \int \frac {1+x}{(2+x)^2 (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{x^2 (x-\log (x))} \, dx+\frac {21}{2} \int \frac {1}{x (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x) (x-\log (x))} \, dx+15 \int \frac {1}{(2+x) (x-\log (x))} \, dx-30 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx \\ & = -\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (2+x)}+\frac {9}{4} \int \frac {1}{x (x-\log (x))} \, dx-\frac {9}{4} \int \frac {1}{(2+x) (x-\log (x))} \, dx-3 \int \left (\frac {-7-2 x}{x^2 (2+x)}+\frac {(-1+x) (7+2 x)}{x^2 (2+x) (x-\log (x))}\right ) \, dx-\frac {9}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-9 \int \left (-\frac {1}{4 x}-\frac {1}{2 (2+x)^2}+\frac {1}{4 (2+x)}\right ) \, dx-9 \int \left (-\frac {1}{(2+x)^2 (x-\log (x))}+\frac {1}{(2+x) (x-\log (x))}\right ) \, dx-\frac {21}{2} \int \frac {1}{x^2 (x-\log (x))} \, dx+\frac {21}{2} \int \frac {1}{x (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x) (x-\log (x))} \, dx+15 \int \frac {1}{(2+x) (x-\log (x))} \, dx-30 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx \\ & = -\frac {9}{2 (2+x)}+\frac {9 \log (x)}{4}-\frac {9}{4} \log (2+x)-\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (2+x)}+\frac {9}{4} \int \frac {1}{x (x-\log (x))} \, dx-\frac {9}{4} \int \frac {1}{(2+x) (x-\log (x))} \, dx-3 \int \frac {-7-2 x}{x^2 (2+x)} \, dx-3 \int \frac {(-1+x) (7+2 x)}{x^2 (2+x) (x-\log (x))} \, dx-\frac {9}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+9 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-9 \int \frac {1}{(2+x) (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{x^2 (x-\log (x))} \, dx+\frac {21}{2} \int \frac {1}{x (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x) (x-\log (x))} \, dx+15 \int \frac {1}{(2+x) (x-\log (x))} \, dx-30 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx \\ & = -\frac {9}{2 (2+x)}+\frac {9 \log (x)}{4}-\frac {9}{4} \log (2+x)-\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (2+x)}+\frac {9}{4} \int \frac {1}{x (x-\log (x))} \, dx-\frac {9}{4} \int \frac {1}{(2+x) (x-\log (x))} \, dx-3 \int \left (-\frac {7}{2 x^2}+\frac {3}{4 x}-\frac {3}{4 (2+x)}\right ) \, dx-3 \int \left (-\frac {7}{2 x^2 (x-\log (x))}+\frac {17}{4 x (x-\log (x))}-\frac {9}{4 (2+x) (x-\log (x))}\right ) \, dx-\frac {9}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+9 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-9 \int \frac {1}{(2+x) (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{x^2 (x-\log (x))} \, dx+\frac {21}{2} \int \frac {1}{x (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x) (x-\log (x))} \, dx+15 \int \frac {1}{(2+x) (x-\log (x))} \, dx-30 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx \\ & = -\frac {21}{2 x}-\frac {9}{2 (2+x)}-\frac {21 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 x}+\frac {9 \log \left (\frac {x}{2 (x-\log (x))}\right )}{2 (2+x)}+\frac {9}{4} \int \frac {1}{x (x-\log (x))} \, dx-\frac {9}{4} \int \frac {1}{(2+x) (x-\log (x))} \, dx-\frac {9}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+\frac {27}{4} \int \frac {1}{(2+x) (x-\log (x))} \, dx+9 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-9 \int \frac {1}{(2+x) (x-\log (x))} \, dx+\frac {21}{2} \int \frac {1}{x (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx-\frac {21}{2} \int \frac {1}{(2+x) (x-\log (x))} \, dx-\frac {51}{4} \int \frac {1}{x (x-\log (x))} \, dx+15 \int \frac {1}{(2+x) (x-\log (x))} \, dx-30 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx+36 \int \frac {1}{(2+x)^2 (x-\log (x))} \, dx \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=-\frac {3 \left (7+5 x+(7+2 x) \log \left (\frac {x}{2 x-2 \log (x)}\right )\right )}{x (2+x)} \]
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Time = 2.33 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.33
method | result | size |
parallelrisch | \(\frac {-21-6 \ln \left (-\frac {x}{2 \left (\ln \left (x \right )-x \right )}\right ) x -15 x -21 \ln \left (-\frac {x}{2 \left (\ln \left (x \right )-x \right )}\right )}{x \left (2+x \right )}\) | \(44\) |
risch | \(\frac {3 \left (2 x +7\right ) \ln \left (x -\ln \left (x \right )\right )}{x \left (2+x \right )}+\frac {-3 i \pi x \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right )^{2} \operatorname {csgn}\left (i x \right )+\frac {21 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )-x}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right ) \operatorname {csgn}\left (i x \right )}{2}+\frac {21 i \pi \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )-x}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right )^{2}}{2}-3 i \pi x \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right )^{3}+3 i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )-x}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right )^{2}+3 i \pi x \,\operatorname {csgn}\left (\frac {i}{\ln \left (x \right )-x}\right ) \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right ) \operatorname {csgn}\left (i x \right )-\frac {21 i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right )^{2} \operatorname {csgn}\left (i x \right )}{2}-\frac {21 i \pi \operatorname {csgn}\left (\frac {i x}{\ln \left (x \right )-x}\right )^{3}}{2}-21+6 x \ln \left (2\right )-6 x \ln \left (x \right )+21 \ln \left (2\right )-15 x -21 \ln \left (x \right )}{x \left (2+x \right )}\) | \(277\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.03 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=-\frac {3 \, {\left ({\left (2 \, x + 7\right )} \log \left (\frac {x}{2 \, {\left (x - \log \left (x\right )\right )}}\right ) + 5 \, x + 7\right )}}{x^{2} + 2 \, x} \]
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=\frac {- 15 x - 21}{x^{2} + 2 x} + \frac {\left (- 6 x - 21\right ) \log {\left (- \frac {x}{- 2 x + 2 \log {\left (x \right )}} \right )}}{x^{2} + 2 x} \]
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Time = 0.32 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=\frac {3 \, {\left (x {\left (2 \, \log \left (2\right ) - 5\right )} + {\left (2 \, x + 7\right )} \log \left (x - \log \left (x\right )\right ) - {\left (2 \, x + 7\right )} \log \left (x\right ) + 7 \, \log \left (2\right ) - 7\right )}}{x^{2} + 2 \, x} \]
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Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=-\frac {3}{2} \, {\left (\frac {3}{x + 2} - \frac {7}{x}\right )} \log \left (2 \, x - 2 \, \log \left (x\right )\right ) + \frac {3}{2} \, {\left (\frac {3}{x + 2} - \frac {7}{x}\right )} \log \left (x\right ) - \frac {9}{2 \, {\left (x + 2\right )}} - \frac {21}{2 \, x} \]
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Time = 9.61 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {42-9 x-36 x^2-15 x^3+\left (9 x+9 x^2\right ) \log (x)+\left (-42 x-42 x^2-6 x^3+\left (42+42 x+6 x^2\right ) \log (x)\right ) \log \left (-\frac {x}{-2 x+2 \log (x)}\right )}{-4 x^3-4 x^4-x^5+\left (4 x^2+4 x^3+x^4\right ) \log (x)} \, dx=-\frac {15\,x+21}{x^2+2\,x}-\frac {\ln \left (\frac {x}{2\,x-2\,\ln \left (x\right )}\right )\,\left (6\,x+21\right )}{x^2+2\,x} \]
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