Integrand size = 66, antiderivative size = 26 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=\left (2-\log \left (1+\frac {1}{31} (-3-x)+x\right )\right ) \log (x+x (2+x)) \]
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Leaf count is larger than twice the leaf count of optimal. \(67\) vs. \(2(26)=52\).
Time = 0.27 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1608, 6860, 2580, 2439, 2438, 2441, 2440, 2465} \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=\log (14) \log (x)+\left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log \left (\frac {15 (x+3)}{31}\right ) \left (2-\log \left (\frac {2}{31} (15 x+14)\right )\right )+\log \left (\frac {15 (x+3)}{31}\right ) \log (15 x+14)-\log (x (x+3)) \log (15 x+14) \]
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Rule 1608
Rule 2438
Rule 2439
Rule 2440
Rule 2441
Rule 2465
Rule 2580
Rule 6860
Rubi steps \begin{align*} \text {integral}& = \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{x \left (42+59 x+15 x^2\right )} \, dx \\ & = \int \left (-\frac {15 \log (x (3+x))}{14+15 x}-\frac {(3+2 x) \left (-2+\log \left (\frac {2}{31} (14+15 x)\right )\right )}{x (3+x)}\right ) \, dx \\ & = -\left (15 \int \frac {\log (x (3+x))}{14+15 x} \, dx\right )-\int \frac {(3+2 x) \left (-2+\log \left (\frac {2}{31} (14+15 x)\right )\right )}{x (3+x)} \, dx \\ & = -\log (x (3+x)) \log (14+15 x)-\int \left (\frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{x}+\frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{3+x}\right ) \, dx+\int \frac {\log (14+15 x)}{x} \, dx+\int \frac {\log (14+15 x)}{3+x} \, dx \\ & = \log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)-15 \int \frac {\log \left (\frac {15 (3+x)}{31}\right )}{14+15 x} \, dx+\int \frac {\log \left (1+\frac {15 x}{14}\right )}{x} \, dx-\int \frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{x} \, dx-\int \frac {-2+\log \left (\frac {2}{31} (14+15 x)\right )}{3+x} \, dx \\ & = \left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \left (2-\log \left (\frac {2}{31} (14+15 x)\right )\right )+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)-\operatorname {PolyLog}\left (2,-\frac {15 x}{14}\right )+15 \int \frac {\log \left (\frac {15 (3+x)}{31}\right )}{14+15 x} \, dx-\int \frac {\log \left (1+\frac {15 x}{14}\right )}{x} \, dx-\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{31}\right )}{x} \, dx,x,14+15 x\right ) \\ & = \left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \left (2-\log \left (\frac {2}{31} (14+15 x)\right )\right )+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x)+\operatorname {PolyLog}\left (2,\frac {1}{31} (-14-15 x)\right )+\text {Subst}\left (\int \frac {\log \left (1+\frac {x}{31}\right )}{x} \, dx,x,14+15 x\right ) \\ & = \left (2+\log \left (\frac {31}{28}\right )\right ) \log (x)+\log (14) \log (x)+\log \left (\frac {15 (3+x)}{31}\right ) \left (2-\log \left (\frac {2}{31} (14+15 x)\right )\right )+\log \left (\frac {15 (3+x)}{31}\right ) \log (14+15 x)-\log (x (3+x)) \log (14+15 x) \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.96 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=-\log \left (\frac {961}{225}\right )-\log \left (\frac {31}{15}\right ) \log \left (\frac {31}{2}\right )+\left (2+\log \left (\frac {31}{2}\right )\right ) \log (x)+\left (2+\log \left (\frac {31}{2}\right )\right ) \log (3+x)-\log (x (3+x)) \log (14+15 x) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(19)=38\).
Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.73
method | result | size |
default | \(\ln \left (31\right ) \ln \left (\left (3+x \right ) x \right )-\ln \left (15 x +14\right ) \ln \left (x^{2}+3 x \right )+2 \ln \left (\left (3+x \right ) x \right )-\ln \left (2\right ) \ln \left (\left (3+x \right ) x \right )\) | \(45\) |
parts | \(-\ln \left (15 x +14\right ) \ln \left (x^{2}+3 x \right )+\left (\ln \left (15 x +14\right )-\ln \left (\frac {15 x}{14}+1\right )\right ) \ln \left (-\frac {15 x}{14}\right )+\ln \left (15 x +14\right ) \ln \left (\frac {15 x}{31}+\frac {45}{31}\right )+2 \ln \left (\left (3+x \right ) x \right )-\left (\ln \left (\frac {30 x}{31}+\frac {28}{31}\right )-\ln \left (\frac {15 x}{14}+1\right )\right ) \ln \left (-\frac {15 x}{14}\right )-\ln \left (\frac {30 x}{31}+\frac {28}{31}\right ) \ln \left (\frac {15 x}{31}+\frac {45}{31}\right )\) | \(94\) |
risch | \(-\ln \left (x \right ) \ln \left (\frac {15 x}{14}+1\right )-\left (\ln \left (3+x \right )-\ln \left (\frac {15 x}{31}+\frac {45}{31}\right )\right ) \ln \left (-\frac {15 x}{31}-\frac {14}{31}\right )+2 \ln \left (x \right )+\frac {i \pi \,\operatorname {csgn}\left (i x \left (3+x \right )\right ) \left (\operatorname {csgn}\left (i x \left (3+x \right )\right )^{2}-\operatorname {csgn}\left (i x \left (3+x \right )\right ) \operatorname {csgn}\left (i \left (3+x \right )\right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x \left (3+x \right )\right )+\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i \left (3+x \right )\right )\right ) \ln \left (15 x +14\right )}{2}+2 \ln \left (3+x \right )-\left (\ln \left (\frac {30 x}{31}+\frac {28}{31}\right )-\ln \left (\frac {15 x}{14}+1\right )\right ) \ln \left (-\frac {15 x}{14}\right )-\ln \left (\frac {30 x}{31}+\frac {28}{31}\right ) \ln \left (\frac {15 x}{31}+\frac {45}{31}\right )\) | \(152\) |
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Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=-{\left (\log \left (\frac {30}{31} \, x + \frac {28}{31}\right ) - 2\right )} \log \left (x^{2} + 3 \, x\right ) \]
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Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.04 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=- \log {\left (\frac {30 x}{31} + \frac {28}{31} \right )} \log {\left (x^{2} + 3 x \right )} + 2 \log {\left (x^{2} + 3 x \right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (18) = 36\).
Time = 0.33 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.50 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=-\frac {1}{31} \, {\left (31 \, \log \left (x + 3\right ) + 31 \, \log \left (x\right ) - 90\right )} \log \left (15 \, x + 14\right ) + \frac {1}{31} \, {\left (31 \, \log \left (31\right ) - 31 \, \log \left (2\right ) + 34\right )} \log \left (x + 3\right ) + {\left (\log \left (31\right ) - \log \left (2\right )\right )} \log \left (x\right ) - \frac {90}{31} \, \log \left (15 \, x + 14\right ) + \frac {28}{31} \, \log \left (x + 3\right ) + 2 \, \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (18) = 36\).
Time = 0.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.58 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=-\log \left (x^{2} + 3 \, x\right ) \log \left (15 \, x + 14\right ) + {\left (\log \left (31\right ) - \log \left (2\right ) + 2\right )} \log \left (x + 3\right ) + {\left (\log \left (31\right ) - \log \left (2\right ) + 2\right )} \log \left (x\right ) \]
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Time = 14.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.69 \[ \int \frac {84+146 x+60 x^2+\left (-42-73 x-30 x^2\right ) \log \left (\frac {1}{31} (28+30 x)\right )+\left (-45 x-15 x^2\right ) \log \left (3 x+x^2\right )}{42 x+59 x^2+15 x^3} \, dx=-\ln \left (x^2+3\,x\right )\,\left (\ln \left (\frac {30\,x}{31}+\frac {28}{31}\right )-2\right ) \]
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