\(\int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+(-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}) \log (x)+(73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx\) [3912]

Optimal result

Integrand size = 171, antiderivative size = 32 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=3+e^5-x-9 \left (3+x-\frac {1}{x+\frac {x^4}{16}}\right )^2 \log ^2(x) \]

[Out]

3+exp(5)-9*(x-1/(1/16*x^4+x)+3)^2*ln(x)^2-x

Rubi [F]

\[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=\int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx \]

[In]

Int[(-4096*x^3 - 768*x^6 - 48*x^9 - x^12 + (-73728 + 442368*x - 516096*x^2 - 446976*x^3 - 18432*x^4 - 105984*x
^5 - 82944*x^6 - 12096*x^7 - 7200*x^8 - 5184*x^9 - 864*x^10 - 162*x^11 - 108*x^12 - 18*x^13)*Log[x] + (73728 -
221184*x - 202752*x^3 - 142848*x^4 - 13824*x^5 - 41472*x^6 - 17280*x^7 - 864*x^8 - 2592*x^9 - 864*x^10 - 54*x
^12 - 18*x^13)*Log[x]^2)/(4096*x^3 + 768*x^6 + 48*x^9 + x^12),x]

[Out]

-x + 12*Log[1 + 16/x^3]*Log[x] + ((-1)^(1/3)*Log[1 - (2*(-2)^(1/3))/x]*Log[x])/(2*2^(2/3)) - (Log[1 + (2*2^(1/
3))/x]*Log[x])/(2*2^(2/3)) - ((-1/2)^(2/3)*Log[1 + (2*(-1)^(2/3)*2^(1/3))/x]*Log[x])/2 - 63*Log[x]^2 - (9*Log[
x]^2)/x^2 + (54*Log[x]^2)/x - 54*x*Log[x]^2 - 9*x^2*Log[x]^2 - Log[x]^2/(2*2^(1/3) + x)^2 + (x*Log[x]^2)/(2^(2
/3)*(2*2^(1/3) + x)) - ((-1)^(2/3)*Log[x]^2)/(2*2^(1/3) - (-1)^(1/3)*x)^2 + (x*Log[x]^2)/(2^(2/3)*(2*2^(1/3) -
(-1)^(1/3)*x)) + ((-1)^(1/3)*Log[x]^2)/(2*2^(1/3) + (-1)^(2/3)*x)^2 + (x*Log[x]^2)/(2^(2/3)*(2*2^(1/3) + (-1)
^(2/3)*x)) + (288*Log[x]^2)/(16 + x^3) + ((-1/2)^(2/3)*Log[2*2^(1/3) - (-1)^(1/3)*x])/2 + (9*(-1)^(1/3)*Log[2*
2^(1/3) - (-1)^(1/3)*x])/(2*2^(2/3)*(1 + (-1)^(1/3))^4) - (9*(-1)^(1/3)*Log[x]*Log[1 - ((-1/2)^(1/3)*x)/2])/(2
^(2/3)*(1 + (-1)^(1/3))^4) + (81*(-1/2)^(2/3)*(2 + (-1)^(2/3))*Log[x]*Log[1 - ((-1/2)^(1/3)*x)/2])/(1 + (-1)^(
1/3))^7 - (3*(16 + 2^(1/3)*(12*(-2)^(1/3) - (-1)^(2/3)))*Log[x]*Log[1 - ((-1/2)^(1/3)*x)/2])/4 + (405*Log[x]^2
*Log[1 - ((-1/2)^(1/3)*x)/2])/(2*2^(2/3)*(1 + (-1)^(1/3))^8) - (3*(-1)^(1/3)*((-1)^(1/3) - 6*2^(1/3))*Log[x]^2
*Log[1 - ((-1/2)^(1/3)*x)/2])/(2*2^(2/3)) + (Log[x]*Log[1 + x/(2*2^(1/3))])/2^(2/3) - 2^(1/3)*Log[x]*Log[1 + x
/(2*2^(1/3))] - (3*(16 - 2^(1/3)*(1 + 12*2^(1/3)))*Log[x]*Log[1 + x/(2*2^(1/3))])/4 - (Log[x]^2*Log[1 + x/(2*2
^(1/3))])/2^(2/3) - (3*(1 + 6*2^(1/3))*Log[x]^2*Log[1 + x/(2*2^(1/3))])/(2*2^(2/3)) - (5*(2 - (-1)^(1/3))*Log[
x]^2*Log[1 + x/(2*2^(1/3))])/(2^(2/3)*Sqrt[3]*(I - Sqrt[3])) - ((-1)^(1/3)*Log[x]*Log[1 + ((-1)^(2/3)*x)/(2*2^
(1/3))])/2^(2/3) - (3*(16 + (-2)^(1/3) - 12*(-2)^(2/3))*Log[x]*Log[1 + ((-1)^(2/3)*x)/(2*2^(1/3))])/4 - (3*Log
[x]*Log[1 + ((-1)^(2/3)*x)/(2*2^(1/3))])/(2^(2/3)*(1 - (-1)^(1/3))^4) + (3*(-1)^(1/3)*(1 - 6*(-2)^(1/3))*Log[x
]^2*Log[1 + ((-1)^(2/3)*x)/(2*2^(1/3))])/(2*2^(2/3)) - (2^(1/3)*Log[x]*Log[1 - ((1 - I*Sqrt[3])*x)/(4*2^(1/3))
])/(1 - I*Sqrt[3]) + (2^(1/3)*Log[x]^2*Log[1 - ((1 - I*Sqrt[3])*x)/(4*2^(1/3))])/(1 - I*Sqrt[3]) - (2^(1/3)*Lo
g[x]*Log[1 - ((1 + I*Sqrt[3])*x)/(4*2^(1/3))])/(1 + I*Sqrt[3]) + (2^(1/3)*Log[x]^2*Log[1 - ((1 + I*Sqrt[3])*x)
/(4*2^(1/3))])/(1 + I*Sqrt[3]) - (5*(1 + I*Sqrt[3])*Log[x]^2*Log[1 - (Sqrt[3]*x)/(2^(1/3)*(3*I + Sqrt[3]))])/(
4*2^(2/3)) - 4*PolyLog[2, -16/x^3] - ((-1)^(1/3)*PolyLog[2, (2*(-2)^(1/3))/x])/(2*2^(2/3)) + PolyLog[2, (-2*2^
(1/3))/x]/(2*2^(2/3)) + ((-1/2)^(2/3)*PolyLog[2, (-2*(-1)^(2/3)*2^(1/3))/x])/2 - (9*(-1)^(1/3)*PolyLog[2, ((-1
/2)^(1/3)*x)/2])/(2^(2/3)*(1 + (-1)^(1/3))^4) + (81*(-1/2)^(2/3)*(2 + (-1)^(2/3))*PolyLog[2, ((-1/2)^(1/3)*x)/
2])/(1 + (-1)^(1/3))^7 - (3*(16 + 2^(1/3)*(12*(-2)^(1/3) - (-1)^(2/3)))*PolyLog[2, ((-1/2)^(1/3)*x)/2])/4 + (4
05*Log[x]*PolyLog[2, ((-1/2)^(1/3)*x)/2])/(2^(2/3)*(1 + (-1)^(1/3))^8) - (3*(-1)^(1/3)*((-1)^(1/3) - 6*2^(1/3)
)*Log[x]*PolyLog[2, ((-1/2)^(1/3)*x)/2])/2^(2/3) + PolyLog[2, -1/2*x/2^(1/3)]/2^(2/3) - 2^(1/3)*PolyLog[2, -1/
2*x/2^(1/3)] - (3*(16 - 2^(1/3)*(1 + 12*2^(1/3)))*PolyLog[2, -1/2*x/2^(1/3)])/4 - 2^(1/3)*Log[x]*PolyLog[2, -1
/2*x/2^(1/3)] - (3*(1 + 6*2^(1/3))*Log[x]*PolyLog[2, -1/2*x/2^(1/3)])/2^(2/3) - (5*2^(1/3)*(2 - (-1)^(1/3))*Lo
g[x]*PolyLog[2, -1/2*x/2^(1/3)])/(Sqrt[3]*(I - Sqrt[3])) - ((-1)^(1/3)*PolyLog[2, -1/2*((-1)^(2/3)*x)/2^(1/3)]
)/2^(2/3) - (3*(16 + (-2)^(1/3) - 12*(-2)^(2/3))*PolyLog[2, -1/2*((-1)^(2/3)*x)/2^(1/3)])/4 - (3*PolyLog[2, -1
/2*((-1)^(2/3)*x)/2^(1/3)])/(2^(2/3)*(1 - (-1)^(1/3))^4) + (3*(-1)^(1/3)*(1 - 6*(-2)^(1/3))*Log[x]*PolyLog[2,
-1/2*((-1)^(2/3)*x)/2^(1/3)])/2^(2/3) - (2^(1/3)*PolyLog[2, ((1 - I*Sqrt[3])*x)/(4*2^(1/3))])/(1 - I*Sqrt[3])
+ (2*2^(1/3)*Log[x]*PolyLog[2, ((1 - I*Sqrt[3])*x)/(4*2^(1/3))])/(1 - I*Sqrt[3]) - (2^(1/3)*PolyLog[2, ((1 + I
*Sqrt[3])*x)/(4*2^(1/3))])/(1 + I*Sqrt[3]) + (2*2^(1/3)*Log[x]*PolyLog[2, ((1 + I*Sqrt[3])*x)/(4*2^(1/3))])/(1
+ I*Sqrt[3]) - (5*(1 + I*Sqrt[3])*Log[x]*PolyLog[2, (Sqrt[3]*x)/(2^(1/3)*(3*I + Sqrt[3]))])/(2*2^(2/3)) - (40
5*PolyLog[3, ((-1/2)^(1/3)*x)/2])/(2^(2/3)*(1 + (-1)^(1/3))^8) + (3*(-1)^(1/3)*((-1)^(1/3) - 6*2^(1/3))*PolyLo
g[3, ((-1/2)^(1/3)*x)/2])/2^(2/3) + 2^(1/3)*PolyLog[3, -1/2*x/2^(1/3)] + (3*(1 + 6*2^(1/3))*PolyLog[3, -1/2*x/
2^(1/3)])/2^(2/3) + (5*2^(1/3)*(2 - (-1)^(1/3))*PolyLog[3, -1/2*x/2^(1/3)])/(Sqrt[3]*(I - Sqrt[3])) - (3*(-1)^
(1/3)*(1 - 6*(-2)^(1/3))*PolyLog[3, -1/2*((-1)^(2/3)*x)/2^(1/3)])/2^(2/3) - (2*2^(1/3)*PolyLog[3, ((1 - I*Sqrt
[3])*x)/(4*2^(1/3))])/(1 - I*Sqrt[3]) - (2*2^(1/3)*PolyLog[3, ((1 + I*Sqrt[3])*x)/(4*2^(1/3))])/(1 + I*Sqrt[3]
) + (5*(1 + I*Sqrt[3])*PolyLog[3, (Sqrt[3]*x)/(2^(1/3)*(3*I + Sqrt[3]))])/(2*2^(2/3)) - 2592*Defer[Int][(x*Log
[x]^2)/(16 + x^3)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-1-\frac {18 \left (-16+48 x+16 x^2+3 x^4+x^5\right )^2 \log (x)}{x^3 \left (16+x^3\right )^2}-\frac {18 \left (-4096+12288 x+11264 x^3+7936 x^4+768 x^5+2304 x^6+960 x^7+48 x^8+144 x^9+48 x^{10}+3 x^{12}+x^{13}\right ) \log ^2(x)}{x^3 \left (16+x^3\right )^3}\right ) \, dx \\ & = -x-18 \int \frac {\left (-16+48 x+16 x^2+3 x^4+x^5\right )^2 \log (x)}{x^3 \left (16+x^3\right )^2} \, dx-18 \int \frac {\left (-4096+12288 x+11264 x^3+7936 x^4+768 x^5+2304 x^6+960 x^7+48 x^8+144 x^9+48 x^{10}+3 x^{12}+x^{13}\right ) \log ^2(x)}{x^3 \left (16+x^3\right )^3} \, dx \\ & = -x-18 \int \left (6 \log (x)+\frac {\log (x)}{x^3}-\frac {6 \log (x)}{x^2}+\frac {7 \log (x)}{x}+x \log (x)-\frac {16 \log (x)}{\left (16+x^3\right )^2}+\frac {\left (-1+6 x+2 x^2\right ) \log (x)}{16+x^3}\right ) \, dx-18 \int \left (3 \log ^2(x)-\frac {\log ^2(x)}{x^3}+\frac {3 \log ^2(x)}{x^2}+x \log ^2(x)-\frac {768 \log ^2(x)}{\left (16+x^3\right )^3}+\frac {16 \left (1+9 x+3 x^2\right ) \log ^2(x)}{\left (16+x^3\right )^2}+\frac {(1-3 x) \log ^2(x)}{16+x^3}\right ) \, dx \\ & = -x-18 \int \frac {\log (x)}{x^3} \, dx-18 \int x \log (x) \, dx-18 \int \frac {\left (-1+6 x+2 x^2\right ) \log (x)}{16+x^3} \, dx+18 \int \frac {\log ^2(x)}{x^3} \, dx-18 \int x \log ^2(x) \, dx-18 \int \frac {(1-3 x) \log ^2(x)}{16+x^3} \, dx-54 \int \log ^2(x) \, dx-54 \int \frac {\log ^2(x)}{x^2} \, dx-108 \int \log (x) \, dx+108 \int \frac {\log (x)}{x^2} \, dx-126 \int \frac {\log (x)}{x} \, dx+288 \int \frac {\log (x)}{\left (16+x^3\right )^2} \, dx-288 \int \frac {\left (1+9 x+3 x^2\right ) \log ^2(x)}{\left (16+x^3\right )^2} \, dx+13824 \int \frac {\log ^2(x)}{\left (16+x^3\right )^3} \, dx \\ & = \frac {9}{2 x^2}-\frac {108}{x}+107 x+\frac {9 x^2}{2}+\frac {9 \log (x)}{x^2}-\frac {108 \log (x)}{x}-108 x \log (x)-9 x^2 \log (x)-63 \log ^2(x)-\frac {9 \log ^2(x)}{x^2}+\frac {54 \log ^2(x)}{x}-54 x \log ^2(x)-9 x^2 \log ^2(x)+18 \int \frac {\log (x)}{x^3} \, dx+18 \int x \log (x) \, dx-18 \int \left (\frac {\left (-32+2 \sqrt [3]{2}+24\ 2^{2/3}\right ) \log (x)}{48 \left (-2 \sqrt [3]{2}-x\right )}+\frac {\left (24 (-2)^{2/3}+32 \sqrt [3]{-1}+2 \sqrt [3]{2}\right ) \log (x)}{48 \left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )}+\frac {\left (-32 (-1)^{2/3}+2 \sqrt [3]{2}-24 \sqrt [3]{-1} 2^{2/3}\right ) \log (x)}{48 \left (-2 \sqrt [3]{2}-(-1)^{2/3} x\right )}\right ) \, dx-18 \int \left (\frac {\left (-2 \sqrt [3]{2}-12\ 2^{2/3}\right ) \log ^2(x)}{48 \left (-2 \sqrt [3]{2}-x\right )}+\frac {\left (-12 (-2)^{2/3}-2 \sqrt [3]{2}\right ) \log ^2(x)}{48 \left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )}+\frac {\left (-2 \sqrt [3]{2}+12 \sqrt [3]{-1} 2^{2/3}\right ) \log ^2(x)}{48 \left (-2 \sqrt [3]{2}-(-1)^{2/3} x\right )}\right ) \, dx+108 \int \log (x) \, dx-108 \int \frac {\log (x)}{x^2} \, dx+288 \int \left (\frac {(-1)^{2/3} \log (x)}{32 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^4 \left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )^2}+\frac {\log (x)}{32 \sqrt [3]{2} \left (-1+\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (2 \sqrt [3]{2}+(-1)^{2/3} x\right )^2}+\frac {\log (x)}{288 \left (4+2^{2/3} x\right )}+\frac {i \log (x)}{144 \left (8 i-2^{2/3} \left (i-\sqrt {3}\right ) x\right )}-\frac {i \log (x)}{144 \left (-8 i+2^{2/3} \left (i+\sqrt {3}\right ) x\right )}+\frac {\log (x)}{288 \left (8+4\ 2^{2/3} x+\sqrt [3]{2} x^2\right )}\right ) \, dx-288 \int \left (\frac {\log ^2(x)}{\left (16+x^3\right )^2}+\frac {9 x \log ^2(x)}{\left (16+x^3\right )^2}+\frac {3 x^2 \log ^2(x)}{\left (16+x^3\right )^2}\right ) \, dx+13824 \int \left (\frac {\log ^2(x)}{6912 \left (2 \sqrt [3]{2}+x\right )^3}-\frac {3 \left (-2+\sqrt [3]{-1}\right ) \log ^2(x)}{512 \sqrt [3]{2} \left (-1+\sqrt [3]{-1}\right )^4 \left (1+\sqrt [3]{-1}\right )^7 \left (2 \sqrt [3]{2}+x\right )^2}-\frac {5 \left (-2+\sqrt [3]{-1}\right ) \log ^2(x)}{13824\ 2^{2/3} \sqrt {3} \left (-i+\sqrt {3}\right ) \left (2 \sqrt [3]{2}+x\right )}+\frac {\log ^2(x)}{6912 \left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )^3}-\frac {3 \left (2+(-1)^{2/3}\right ) \log ^2(x)}{512 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^7 \left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )^2}+\frac {15 \sqrt [3]{-1} \log ^2(x)}{1024\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^8 \left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )}+\frac {\log ^2(x)}{6912 \left (2 \sqrt [3]{2}+(-1)^{2/3} x\right )^3}+\frac {(-1)^{2/3} \log ^2(x)}{4608 \sqrt [3]{2} \left (-1+\sqrt [3]{-1}\right )^4 \left (2 \sqrt [3]{2}+(-1)^{2/3} x\right )^2}-\frac {5 \left (3 i+\sqrt {3}\right ) \log ^2(x)}{55296 \left (-6 i-2 \sqrt {3}+2^{2/3} \sqrt {3} x\right )}\right ) \, dx \\ & = -x-63 \log ^2(x)-\frac {9 \log ^2(x)}{x^2}+\frac {54 \log ^2(x)}{x}-54 x \log ^2(x)-9 x^2 \log ^2(x)+2 i \int \frac {\log (x)}{8 i-2^{2/3} \left (i-\sqrt {3}\right ) x} \, dx-2 i \int \frac {\log (x)}{-8 i+2^{2/3} \left (i+\sqrt {3}\right ) x} \, dx+2 \int \frac {\log ^2(x)}{\left (2 \sqrt [3]{2}+x\right )^3} \, dx+2 \int \frac {\log ^2(x)}{\left (2 \sqrt [3]{2}-\sqrt [3]{-1} x\right )^3} \, dx+2 \int \frac {\log ^2(x)}{\left (2 \sqrt [3]{2}+(-1)^{2/3} x\right )^3} \, dx-288 \int \frac {\log ^2(x)}{\left (16+x^3\right )^2} \, dx-864 \int \frac {x^2 \log ^2(x)}{\left (16+x^3\right )^2} \, dx-2592 \int \frac {x \log ^2(x)}{\left (16+x^3\right )^2} \, dx+\frac {\int \frac {\log (x)}{\left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )^2} \, dx}{\sqrt [3]{2}}+\frac {\int \frac {\log (x)}{\left (2 \sqrt [3]{2}+(-1)^{2/3} x\right )^2} \, dx}{\sqrt [3]{2}}+\frac {3 \int \frac {\log ^2(x)}{\left (2 \sqrt [3]{2}+x\right )^2} \, dx}{\sqrt [3]{2}}+\frac {3 \int \frac {\log ^2(x)}{\left (-2 \sqrt [3]{2}+\sqrt [3]{-1} x\right )^2} \, dx}{\sqrt [3]{2}}+\frac {3 \int \frac {\log ^2(x)}{\left (2 \sqrt [3]{2}+(-1)^{2/3} x\right )^2} \, dx}{\sqrt [3]{2}}+\frac {\left (3 \left (1-6 \sqrt [3]{-2}\right )\right ) \int \frac {\log ^2(x)}{-2 \sqrt [3]{2}-(-1)^{2/3} x} \, dx}{2\ 2^{2/3}}-\frac {\left (3 \left (1-12 \sqrt [3]{-2}-8 (-2)^{2/3}\right )\right ) \int \frac {\log (x)}{-2 \sqrt [3]{2}-(-1)^{2/3} x} \, dx}{2\ 2^{2/3}}+\frac {\left (405 \sqrt [3]{-1}\right ) \int \frac {\log ^2(x)}{-2 \sqrt [3]{2}+\sqrt [3]{-1} x} \, dx}{2\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^8}+\frac {1}{4} \left (3 \left (6 (-2)^{2/3}+\sqrt [3]{2}\right )\right ) \int \frac {\log ^2(x)}{-2 \sqrt [3]{2}+\sqrt [3]{-1} x} \, dx-\frac {1}{4} \left (3 \left (12 (-2)^{2/3}+16 \sqrt [3]{-1}+\sqrt [3]{2}\right )\right ) \int \frac {\log (x)}{-2 \sqrt [3]{2}+\sqrt [3]{-1} x} \, dx+\frac {\left (3 \left (1+6 \sqrt [3]{2}\right )\right ) \int \frac {\log ^2(x)}{-2 \sqrt [3]{2}-x} \, dx}{2\ 2^{2/3}}-\frac {1}{8} \left (3 \left (-32+2 \sqrt [3]{2}+24\ 2^{2/3}\right )\right ) \int \frac {\log (x)}{-2 \sqrt [3]{2}-x} \, dx-\frac {\left (5 \left (2-\sqrt [3]{-1}\right )\right ) \int \frac {\log ^2(x)}{2 \sqrt [3]{2}+x} \, dx}{2^{2/3} \sqrt {3} \left (i-\sqrt {3}\right )}-\frac {1}{4} \left (5 \left (3 i+\sqrt {3}\right )\right ) \int \frac {\log ^2(x)}{-6 i-2 \sqrt {3}+2^{2/3} \sqrt {3} x} \, dx+\int \frac {\log (x)}{4+2^{2/3} x} \, dx+\int \frac {\log (x)}{8+4\ 2^{2/3} x+\sqrt [3]{2} x^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.95 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.56 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=-\frac {x^3 \left (16+x^3\right )^2+9 \left (-16+48 x+16 x^2+3 x^4+x^5\right )^2 \log ^2(x)}{x^2 \left (16+x^3\right )^2} \]

[In]

Integrate[(-4096*x^3 - 768*x^6 - 48*x^9 - x^12 + (-73728 + 442368*x - 516096*x^2 - 446976*x^3 - 18432*x^4 - 10
5984*x^5 - 82944*x^6 - 12096*x^7 - 7200*x^8 - 5184*x^9 - 864*x^10 - 162*x^11 - 108*x^12 - 18*x^13)*Log[x] + (7
3728 - 221184*x - 202752*x^3 - 142848*x^4 - 13824*x^5 - 41472*x^6 - 17280*x^7 - 864*x^8 - 2592*x^9 - 864*x^10
- 54*x^12 - 18*x^13)*Log[x]^2)/(4096*x^3 + 768*x^6 + 48*x^9 + x^12),x]

[Out]

-((x^3*(16 + x^3)^2 + 9*(-16 + 48*x + 16*x^2 + 3*x^4 + x^5)^2*Log[x]^2)/(x^2*(16 + x^3)^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(29)=58\).

Time = 0.91 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31

method	result	size
risch	\(-\frac {9 \left (x^{10}+6 x^{9}+9 x^{8}+32 x^{7}+192 x^{6}+256 x^{5}+160 x^{4}+1536 x^{3}+1792 x^{2}-1536 x +256\right ) \ln \left (x \right )^{2}}{x^{2} \left (x^{6}+32 x^{3}+256\right )}-x\)	\(74\)
parallelrisch	\(-\frac {288 x^{10} \ln \left (x \right )^{2}+1728 x^{9} \ln \left (x \right )^{2}+2592 x^{8} \ln \left (x \right )^{2}+32 x^{9}+9216 x^{7} \ln \left (x \right )^{2}+55296 x^{6} \ln \left (x \right )^{2}+73728 x^{5} \ln \left (x \right )^{2}+1024 x^{6}+46080 x^{4} \ln \left (x \right )^{2}+442368 x^{3} \ln \left (x \right )^{2}+516096 x^{2} \ln \left (x \right )^{2}+8192 x^{3}-442368 x \ln \left (x \right )^{2}+73728 \ln \left (x \right )^{2}}{32 x^{2} \left (x^{6}+32 x^{3}+256\right )}\)	\(128\)

[In]

int(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6-13824*x^5-142848*x^4-202752*x^3-221184*x+
73728)*ln(x)^2+(-18*x^13-108*x^12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-18432*x^4
-446976*x^3-516096*x^2+442368*x-73728)*ln(x)-x^12-48*x^9-768*x^6-4096*x^3)/(x^12+48*x^9+768*x^6+4096*x^3),x,me
thod=_RETURNVERBOSE)

[Out]

-9*(x^10+6*x^9+9*x^8+32*x^7+192*x^6+256*x^5+160*x^4+1536*x^3+1792*x^2-1536*x+256)/x^2/(x^6+32*x^3+256)*ln(x)^2
-x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (29) = 58\).

Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=-\frac {x^{9} + 32 \, x^{6} + 256 \, x^{3} + 9 \, {\left (x^{10} + 6 \, x^{9} + 9 \, x^{8} + 32 \, x^{7} + 192 \, x^{6} + 256 \, x^{5} + 160 \, x^{4} + 1536 \, x^{3} + 1792 \, x^{2} - 1536 \, x + 256\right )} \log \left (x\right )^{2}}{x^{8} + 32 \, x^{5} + 256 \, x^{2}} \]

[In]

integrate(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6-13824*x^5-142848*x^4-202752*x^3-221
184*x+73728)*log(x)^2+(-18*x^13-108*x^12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-18
432*x^4-446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-4096*x^3)/(x^12+48*x^9+768*x^6+4096*x
^3),x, algorithm="fricas")

[Out]

-(x^9 + 32*x^6 + 256*x^3 + 9*(x^10 + 6*x^9 + 9*x^8 + 32*x^7 + 192*x^6 + 256*x^5 + 160*x^4 + 1536*x^3 + 1792*x^
2 - 1536*x + 256)*log(x)^2)/(x^8 + 32*x^5 + 256*x^2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (26) = 52\).

Time = 0.20 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.19 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=- x + \frac {\left (- 9 x^{10} - 54 x^{9} - 81 x^{8} - 288 x^{7} - 1728 x^{6} - 2304 x^{5} - 1440 x^{4} - 13824 x^{3} - 16128 x^{2} + 13824 x - 2304\right ) \log {\left (x \right )}^{2}}{x^{8} + 32 x^{5} + 256 x^{2}} \]

[In]

integrate(((-18*x**13-54*x**12-864*x**10-2592*x**9-864*x**8-17280*x**7-41472*x**6-13824*x**5-142848*x**4-20275
2*x**3-221184*x+73728)*ln(x)**2+(-18*x**13-108*x**12-162*x**11-864*x**10-5184*x**9-7200*x**8-12096*x**7-82944*
x**6-105984*x**5-18432*x**4-446976*x**3-516096*x**2+442368*x-73728)*ln(x)-x**12-48*x**9-768*x**6-4096*x**3)/(x
**12+48*x**9+768*x**6+4096*x**3),x)

[Out]

-x + (-9*x**10 - 54*x**9 - 81*x**8 - 288*x**7 - 1728*x**6 - 2304*x**5 - 1440*x**4 - 13824*x**3 - 16128*x**2 +
13824*x - 2304)*log(x)**2/(x**8 + 32*x**5 + 256*x**2)

Maxima [F(-2)]

Exception generated. \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6-13824*x^5-142848*x^4-202752*x^3-221
184*x+73728)*log(x)^2+(-18*x^13-108*x^12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-18
432*x^4-446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-4096*x^3)/(x^12+48*x^9+768*x^6+4096*x
^3),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (29) = 58\).

Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.06 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=-9 \, {\left (x^{2} + 6 \, x + \frac {6 \, x^{5} - x^{4} - 32 \, x^{3} + 96 \, x^{2} - 32 \, x - 512}{x^{6} + 32 \, x^{3} + 256} - \frac {6 \, x - 1}{x^{2}} + 9\right )} \log \left (x\right )^{2} - x \]

[In]

integrate(((-18*x^13-54*x^12-864*x^10-2592*x^9-864*x^8-17280*x^7-41472*x^6-13824*x^5-142848*x^4-202752*x^3-221
184*x+73728)*log(x)^2+(-18*x^13-108*x^12-162*x^11-864*x^10-5184*x^9-7200*x^8-12096*x^7-82944*x^6-105984*x^5-18
432*x^4-446976*x^3-516096*x^2+442368*x-73728)*log(x)-x^12-48*x^9-768*x^6-4096*x^3)/(x^12+48*x^9+768*x^6+4096*x
^3),x, algorithm="giac")

[Out]

-9*(x^2 + 6*x + (6*x^5 - x^4 - 32*x^3 + 96*x^2 - 32*x - 512)/(x^6 + 32*x^3 + 256) - (6*x - 1)/x^2 + 9)*log(x)^
2 - x

Mupad [B] (verification not implemented)

Time = 9.96 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.31 \[ \int \frac {-4096 x^3-768 x^6-48 x^9-x^{12}+\left (-73728+442368 x-516096 x^2-446976 x^3-18432 x^4-105984 x^5-82944 x^6-12096 x^7-7200 x^8-5184 x^9-864 x^{10}-162 x^{11}-108 x^{12}-18 x^{13}\right ) \log (x)+\left (73728-221184 x-202752 x^3-142848 x^4-13824 x^5-41472 x^6-17280 x^7-864 x^8-2592 x^9-864 x^{10}-54 x^{12}-18 x^{13}\right ) \log ^2(x)}{4096 x^3+768 x^6+48 x^9+x^{12}} \, dx=\left (-\frac {9\,x^{10}+54\,x^9+288\,x^7+1728\,x^6-288\,x^5+1440\,x^4+13824\,x^3-4608\,x^2-13824\,x+2304}{x^8+32\,x^5+256\,x^2}-81\right )\,{\ln \left (x\right )}^2-x \]

[In]

int(-(log(x)*(516096*x^2 - 442368*x + 446976*x^3 + 18432*x^4 + 105984*x^5 + 82944*x^6 + 12096*x^7 + 7200*x^8 +
5184*x^9 + 864*x^10 + 162*x^11 + 108*x^12 + 18*x^13 + 73728) + 4096*x^3 + 768*x^6 + 48*x^9 + x^12 + log(x)^2*
(221184*x + 202752*x^3 + 142848*x^4 + 13824*x^5 + 41472*x^6 + 17280*x^7 + 864*x^8 + 2592*x^9 + 864*x^10 + 54*x
^12 + 18*x^13 - 73728))/(4096*x^3 + 768*x^6 + 48*x^9 + x^12),x)

[Out]

- x - log(x)^2*((13824*x^3 - 4608*x^2 - 13824*x + 1440*x^4 - 288*x^5 + 1728*x^6 + 288*x^7 + 54*x^9 + 9*x^10 +
2304)/(256*x^2 + 32*x^5 + x^8) + 81)