3.69.0 ∫ 200x+120x2−22x3−12x4+2x5+(−60x2+22x3+18x4−4x5) log(x)+(−400x−360x2+88x3+60x4−12x5) log2(x)+(−200x−180x2+44x3+30x4−6x5) log3(x) 3 log3(x) dx [6884]

Integrand size = 114, antiderivative size = 28 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=3 \left (1-\frac {1}{9} (-5+x)^2 (2+x)^2 \left (x+\frac {x}{\log (x)}\right )^2\right ) \]

Rubi [B] (veriﬁed)

Leaf count is larger than twice the leaf count of optimal. \(122\) vs. \(2(28)=56\).

Time = 0.70 (sec) , antiderivative size = 122, normalized size of antiderivative = 4.36, number of steps used = 54, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.070, Rules used = {12, 6820, 6874, 1604, 2403, 2343, 2346, 2209} \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^6}{3 \log ^2(x)}-\frac {2 x^6}{3 \log (x)}+\frac {2 x^5}{\log ^2(x)}+\frac {4 x^5}{\log (x)}+\frac {11 x^4}{3 \log ^2(x)}+\frac {22 x^4}{3 \log (x)}-\frac {20 x^3}{\log ^2(x)}-\frac {40 x^3}{\log (x)}-\frac {1}{3} (5-x)^2 (x+2)^2 x^2-\frac {100 x^2}{3 \log ^2(x)}-\frac {200 x^2}{3 \log (x)} \]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{\log ^3(x)} \, dx \\ & = \frac {1}{3} \int \frac {2 x \left (10+3 x-x^2\right ) (1+\log (x)) \left (10+3 x-x^2+\left (-10-6 x+3 x^2\right ) \log (x)+\left (-10-6 x+3 x^2\right ) \log ^2(x)\right )}{\log ^3(x)} \, dx \\ & = \frac {2}{3} \int \frac {x \left (10+3 x-x^2\right ) (1+\log (x)) \left (10+3 x-x^2+\left (-10-6 x+3 x^2\right ) \log (x)+\left (-10-6 x+3 x^2\right ) \log ^2(x)\right )}{\log ^3(x)} \, dx \\ & = \frac {2}{3} \int \left (-\left ((-5+x) x (2+x) \left (-10-6 x+3 x^2\right )\right )+\frac {x \left (-10-3 x+x^2\right )^2}{\log ^3(x)}-\frac {x^2 \left (30-11 x-9 x^2+2 x^3\right )}{\log ^2(x)}-\frac {2 (-5+x) x (2+x) \left (-10-6 x+3 x^2\right )}{\log (x)}\right ) \, dx \\ & = -\left (\frac {2}{3} \int (-5+x) x (2+x) \left (-10-6 x+3 x^2\right ) \, dx\right )+\frac {2}{3} \int \frac {x \left (-10-3 x+x^2\right )^2}{\log ^3(x)} \, dx-\frac {2}{3} \int \frac {x^2 \left (30-11 x-9 x^2+2 x^3\right )}{\log ^2(x)} \, dx-\frac {4}{3} \int \frac {(-5+x) x (2+x) \left (-10-6 x+3 x^2\right )}{\log (x)} \, dx \\ & = -\frac {1}{3} (5-x)^2 x^2 (2+x)^2+\frac {2}{3} \int \left (\frac {100 x}{\log ^3(x)}+\frac {60 x^2}{\log ^3(x)}-\frac {11 x^3}{\log ^3(x)}-\frac {6 x^4}{\log ^3(x)}+\frac {x^5}{\log ^3(x)}\right ) \, dx-\frac {2}{3} \int \left (\frac {30 x^2}{\log ^2(x)}-\frac {11 x^3}{\log ^2(x)}-\frac {9 x^4}{\log ^2(x)}+\frac {2 x^5}{\log ^2(x)}\right ) \, dx-\frac {4}{3} \int \left (\frac {100 x}{\log (x)}+\frac {90 x^2}{\log (x)}-\frac {22 x^3}{\log (x)}-\frac {15 x^4}{\log (x)}+\frac {3 x^5}{\log (x)}\right ) \, dx \\ & = -\frac {1}{3} (5-x)^2 x^2 (2+x)^2+\frac {2}{3} \int \frac {x^5}{\log ^3(x)} \, dx-\frac {4}{3} \int \frac {x^5}{\log ^2(x)} \, dx-4 \int \frac {x^4}{\log ^3(x)} \, dx-4 \int \frac {x^5}{\log (x)} \, dx+6 \int \frac {x^4}{\log ^2(x)} \, dx-\frac {22}{3} \int \frac {x^3}{\log ^3(x)} \, dx+\frac {22}{3} \int \frac {x^3}{\log ^2(x)} \, dx-20 \int \frac {x^2}{\log ^2(x)} \, dx+20 \int \frac {x^4}{\log (x)} \, dx+\frac {88}{3} \int \frac {x^3}{\log (x)} \, dx+40 \int \frac {x^2}{\log ^3(x)} \, dx+\frac {200}{3} \int \frac {x}{\log ^3(x)} \, dx-120 \int \frac {x^2}{\log (x)} \, dx-\frac {400}{3} \int \frac {x}{\log (x)} \, dx \\ & = -\frac {1}{3} (5-x)^2 x^2 (2+x)^2-\frac {100 x^2}{3 \log ^2(x)}-\frac {20 x^3}{\log ^2(x)}+\frac {11 x^4}{3 \log ^2(x)}+\frac {2 x^5}{\log ^2(x)}-\frac {x^6}{3 \log ^2(x)}+\frac {20 x^3}{\log (x)}-\frac {22 x^4}{3 \log (x)}-\frac {6 x^5}{\log (x)}+\frac {4 x^6}{3 \log (x)}+2 \int \frac {x^5}{\log ^2(x)} \, dx-4 \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-8 \int \frac {x^5}{\log (x)} \, dx-10 \int \frac {x^4}{\log ^2(x)} \, dx-\frac {44}{3} \int \frac {x^3}{\log ^2(x)} \, dx+20 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )+\frac {88}{3} \int \frac {x^3}{\log (x)} \, dx+\frac {88}{3} \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )+30 \int \frac {x^4}{\log (x)} \, dx+60 \int \frac {x^2}{\log ^2(x)} \, dx-60 \int \frac {x^2}{\log (x)} \, dx+\frac {200}{3} \int \frac {x}{\log ^2(x)} \, dx-120 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-\frac {400}{3} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {1}{3} (5-x)^2 x^2 (2+x)^2-\frac {400}{3} \text {Ei}(2 \log (x))-120 \text {Ei}(3 \log (x))+\frac {88}{3} \text {Ei}(4 \log (x))+20 \text {Ei}(5 \log (x))-4 \text {Ei}(6 \log (x))-\frac {100 x^2}{3 \log ^2(x)}-\frac {20 x^3}{\log ^2(x)}+\frac {11 x^4}{3 \log ^2(x)}+\frac {2 x^5}{\log ^2(x)}-\frac {x^6}{3 \log ^2(x)}-\frac {200 x^2}{3 \log (x)}-\frac {40 x^3}{\log (x)}+\frac {22 x^4}{3 \log (x)}+\frac {4 x^5}{\log (x)}-\frac {2 x^6}{3 \log (x)}-8 \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )+12 \int \frac {x^5}{\log (x)} \, dx+\frac {88}{3} \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )+30 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-50 \int \frac {x^4}{\log (x)} \, dx-\frac {176}{3} \int \frac {x^3}{\log (x)} \, dx-60 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+\frac {400}{3} \int \frac {x}{\log (x)} \, dx+180 \int \frac {x^2}{\log (x)} \, dx \\ & = -\frac {1}{3} (5-x)^2 x^2 (2+x)^2-\frac {400}{3} \text {Ei}(2 \log (x))-180 \text {Ei}(3 \log (x))+\frac {176}{3} \text {Ei}(4 \log (x))+50 \text {Ei}(5 \log (x))-12 \text {Ei}(6 \log (x))-\frac {100 x^2}{3 \log ^2(x)}-\frac {20 x^3}{\log ^2(x)}+\frac {11 x^4}{3 \log ^2(x)}+\frac {2 x^5}{\log ^2(x)}-\frac {x^6}{3 \log ^2(x)}-\frac {200 x^2}{3 \log (x)}-\frac {40 x^3}{\log (x)}+\frac {22 x^4}{3 \log (x)}+\frac {4 x^5}{\log (x)}-\frac {2 x^6}{3 \log (x)}+12 \text {Subst}\left (\int \frac {e^{6 x}}{x} \, dx,x,\log (x)\right )-50 \text {Subst}\left (\int \frac {e^{5 x}}{x} \, dx,x,\log (x)\right )-\frac {176}{3} \text {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (x)\right )+\frac {400}{3} \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+180 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right ) \\ & = -\frac {1}{3} (5-x)^2 x^2 (2+x)^2-\frac {100 x^2}{3 \log ^2(x)}-\frac {20 x^3}{\log ^2(x)}+\frac {11 x^4}{3 \log ^2(x)}+\frac {2 x^5}{\log ^2(x)}-\frac {x^6}{3 \log ^2(x)}-\frac {200 x^2}{3 \log (x)}-\frac {40 x^3}{\log (x)}+\frac {22 x^4}{3 \log (x)}+\frac {4 x^5}{\log (x)}-\frac {2 x^6}{3 \log (x)} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 0.74 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^2 \left (-10-3 x+x^2\right )^2 (1+\log (x))^2}{3 \log ^2(x)} \]

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(83\) vs. \(2(24)=48\).

Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 3.00


method	result	size

risch	\(-\frac {x^{6}}{3}+2 x^{5}+\frac {11 x^{4}}{3}-20 x^{3}-\frac {100 x^{2}}{3}-\frac {x^{2} \left (2 x^{4} \ln \left (x \right )+x^{4}-12 x^{3} \ln \left (x \right )-6 x^{3}-22 x^{2} \ln \left (x \right )-11 x^{2}+120 x \ln \left (x \right )+60 x +200 \ln \left (x \right )+100\right )}{3 \ln \left (x \right )^{2}}\)	\(84\)
parallelrisch	\(-\frac {x^{6} \ln \left (x \right )^{2}+2 x^{6} \ln \left (x \right )-6 x^{5} \ln \left (x \right )^{2}+x^{6}-12 x^{5} \ln \left (x \right )-11 x^{4} \ln \left (x \right )^{2}-6 x^{5}-22 x^{4} \ln \left (x \right )+60 x^{3} \ln \left (x \right )^{2}-11 x^{4}+120 x^{3} \ln \left (x \right )+100 x^{2} \ln \left (x \right )^{2}+60 x^{3}+200 x^{2} \ln \left (x \right )+100 x^{2}}{3 \ln \left (x \right )^{2}}\)	\(110\)
norman	\(\frac {-\frac {100 x^{2}}{3}-20 x^{3}+\frac {11 x^{4}}{3}+2 x^{5}-\frac {x^{6}}{3}-\frac {200 x^{2} \ln \left (x \right )}{3}-\frac {100 x^{2} \ln \left (x \right )^{2}}{3}-40 x^{3} \ln \left (x \right )-20 x^{3} \ln \left (x \right )^{2}+\frac {22 x^{4} \ln \left (x \right )}{3}+\frac {11 x^{4} \ln \left (x \right )^{2}}{3}+4 x^{5} \ln \left (x \right )+2 x^{5} \ln \left (x \right )^{2}-\frac {2 x^{6} \ln \left (x \right )}{3}-\frac {x^{6} \ln \left (x \right )^{2}}{3}}{\ln \left (x \right )^{2}}\)	\(112\)
default	\(-\frac {2 x^{6}}{3 \ln \left (x \right )}+\frac {4 x^{5}}{\ln \left (x \right )}+\frac {2 x^{5}}{\ln \left (x \right )^{2}}-\frac {x^{6}}{3 \ln \left (x \right )^{2}}-\frac {100 x^{2}}{3 \ln \left (x \right )^{2}}+\frac {22 x^{4}}{3 \ln \left (x \right )}-\frac {40 x^{3}}{\ln \left (x \right )}+\frac {11 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {20 x^{3}}{\ln \left (x \right )^{2}}-\frac {200 x^{2}}{3 \ln \left (x \right )}-\frac {x^{6}}{3}+2 x^{5}+\frac {11 x^{4}}{3}-20 x^{3}-\frac {100 x^{2}}{3}\)	\(117\)
parts	\(-\frac {2 x^{6}}{3 \ln \left (x \right )}+\frac {4 x^{5}}{\ln \left (x \right )}+\frac {2 x^{5}}{\ln \left (x \right )^{2}}-\frac {x^{6}}{3 \ln \left (x \right )^{2}}-\frac {100 x^{2}}{3 \ln \left (x \right )^{2}}+\frac {22 x^{4}}{3 \ln \left (x \right )}-\frac {40 x^{3}}{\ln \left (x \right )}+\frac {11 x^{4}}{3 \ln \left (x \right )^{2}}-\frac {20 x^{3}}{\ln \left (x \right )^{2}}-\frac {200 x^{2}}{3 \ln \left (x \right )}-\frac {x^{6}}{3}+2 x^{5}+\frac {11 x^{4}}{3}-20 x^{3}-\frac {100 x^{2}}{3}\)	\(117\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (24) = 48\).

Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^{6} - 6 \, x^{5} - 11 \, x^{4} + 60 \, x^{3} + {\left (x^{6} - 6 \, x^{5} - 11 \, x^{4} + 60 \, x^{3} + 100 \, x^{2}\right )} \log \left (x\right )^{2} + 100 \, x^{2} + 2 \, {\left (x^{6} - 6 \, x^{5} - 11 \, x^{4} + 60 \, x^{3} + 100 \, x^{2}\right )} \log \left (x\right )}{3 \, \log \left (x\right )^{2}} \]

Sympy [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (20) = 40\).

Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.11 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=- \frac {x^{6}}{3} + 2 x^{5} + \frac {11 x^{4}}{3} - 20 x^{3} - \frac {100 x^{2}}{3} + \frac {- x^{6} + 6 x^{5} + 11 x^{4} - 60 x^{3} - 100 x^{2} + \left (- 2 x^{6} + 12 x^{5} + 22 x^{4} - 120 x^{3} - 200 x^{2}\right ) \log {\left (x \right )}}{3 \log {\left (x \right )}^{2}} \]

Maxima [C] (veriﬁcation not implemented)

Time = 0.27 (sec) , antiderivative size = 133, normalized size of antiderivative = 4.75 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {1}{3} \, x^{6} + 2 \, x^{5} + \frac {11}{3} \, x^{4} - 20 \, x^{3} - \frac {100}{3} \, x^{2} - 4 \, {\rm Ei}\left (6 \, \log \left (x\right )\right ) + 20 \, {\rm Ei}\left (5 \, \log \left (x\right )\right ) + \frac {88}{3} \, {\rm Ei}\left (4 \, \log \left (x\right )\right ) - 120 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - \frac {400}{3} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) - 60 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + \frac {88}{3} \, \Gamma \left (-1, -4 \, \log \left (x\right )\right ) + 30 \, \Gamma \left (-1, -5 \, \log \left (x\right )\right ) - 8 \, \Gamma \left (-1, -6 \, \log \left (x\right )\right ) - \frac {800}{3} \, \Gamma \left (-2, -2 \, \log \left (x\right )\right ) - 360 \, \Gamma \left (-2, -3 \, \log \left (x\right )\right ) + \frac {352}{3} \, \Gamma \left (-2, -4 \, \log \left (x\right )\right ) + 100 \, \Gamma \left (-2, -5 \, \log \left (x\right )\right ) - 24 \, \Gamma \left (-2, -6 \, \log \left (x\right )\right ) \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (24) = 48\).

Time = 0.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 4.14 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {1}{3} \, x^{6} + 2 \, x^{5} - \frac {2 \, x^{6}}{3 \, \log \left (x\right )} + \frac {11}{3} \, x^{4} - \frac {x^{6}}{3 \, \log \left (x\right )^{2}} + \frac {4 \, x^{5}}{\log \left (x\right )} - 20 \, x^{3} + \frac {2 \, x^{5}}{\log \left (x\right )^{2}} + \frac {22 \, x^{4}}{3 \, \log \left (x\right )} - \frac {100}{3} \, x^{2} + \frac {11 \, x^{4}}{3 \, \log \left (x\right )^{2}} - \frac {40 \, x^{3}}{\log \left (x\right )} - \frac {20 \, x^{3}}{\log \left (x\right )^{2}} - \frac {200 \, x^{2}}{3 \, \log \left (x\right )} - \frac {100 \, x^{2}}{3 \, \log \left (x\right )^{2}} \]

Mupad [B] (veriﬁcation not implemented)

Time = 11.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+\left (-60 x^2+22 x^3+18 x^4-4 x^5\right ) \log (x)+\left (-400 x-360 x^2+88 x^3+60 x^4-12 x^5\right ) \log ^2(x)+\left (-200 x-180 x^2+44 x^3+30 x^4-6 x^5\right ) \log ^3(x)}{3 \log ^3(x)} \, dx=-\frac {x^2\,{\left (\ln \left (x\right )+1\right )}^2\,{\left (-x^2+3\,x+10\right )}^2}{3\,{\ln \left (x\right )}^2} \]

\(\int \frac {200 x+120 x^2-22 x^3-12 x^4+2 x^5+(-60 x^2+22 x^3+18 x^4-4 x^5) \log (x)+(-400 x-360 x^2+88 x^3+60 x^4-12 x^5) \log ^2(x)+(-200 x-180 x^2+44 x^3+30 x^4-6 x^5) \log ^3(x)}{3 \log ^3(x)} \, dx\) [6884]

Optimal result