Integrand size = 91, antiderivative size = 30 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=\frac {\log \left (\frac {4}{\frac {1}{x}+x}\right )}{6 x (3+x-x (4+2 x))} \] Output:
1/6*ln(4/(1/x+x))/(3+x-x*(4+2*x))/x
Time = 0.81 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {\log \left (\frac {4 x}{1+x^2}\right )}{6 x \left (-3+3 x+2 x^2\right )} \] Input:
Integrate[(3 - 3*x - 5*x^2 + 3*x^3 + 2*x^4 + (-3 + 6*x + 3*x^2 + 6*x^3 + 6 *x^4)*Log[(4*x)/(1 + x^2)])/(54*x^2 - 108*x^3 + 36*x^4 - 36*x^5 + 6*x^6 + 72*x^7 + 24*x^8),x]
Output:
-1/6*Log[(4*x)/(1 + x^2)]/(x*(-3 + 3*x + 2*x^2))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 7.93 (sec) , antiderivative size = 2527, normalized size of antiderivative = 84.23, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2026, 2463, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {2 x^4+3 x^3-5 x^2+\left (6 x^4+6 x^3+3 x^2+6 x-3\right ) \log \left (\frac {4 x}{x^2+1}\right )-3 x+3}{24 x^8+72 x^7+6 x^6-36 x^5+36 x^4-108 x^3+54 x^2} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {2 x^4+3 x^3-5 x^2+\left (6 x^4+6 x^3+3 x^2+6 x-3\right ) \log \left (\frac {4 x}{x^2+1}\right )-3 x+3}{x^2 \left (24 x^6+72 x^5+6 x^4-36 x^3+36 x^2-108 x+54\right )}dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {(15 x+8) \left (2 x^4+3 x^3-5 x^2+\left (6 x^4+6 x^3+3 x^2+6 x-3\right ) \log \left (\frac {4 x}{x^2+1}\right )-3 x+3\right )}{3468 x^2 \left (x^2+1\right )}+\frac {(-30 x-61) \left (2 x^4+3 x^3-5 x^2+\left (6 x^4+6 x^3+3 x^2+6 x-3\right ) \log \left (\frac {4 x}{x^2+1}\right )-3 x+3\right )}{3468 x^2 \left (2 x^2+3 x-3\right )}+\frac {(6 x+19) \left (2 x^4+3 x^3-5 x^2+\left (6 x^4+6 x^3+3 x^2+6 x-3\right ) \log \left (\frac {4 x}{x^2+1}\right )-3 x+3\right )}{204 x^2 \left (2 x^2+3 x-3\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x (2 x+29)}{1122 \left (-2 x^2-3 x+3\right )}+\frac {2 x+29}{748 \left (-2 x^2-3 x+3\right )}+\frac {4 \left (3+\sqrt {33}\right ) \arctan (x)}{99 \left (29+3 \sqrt {33}\right )}-\frac {56 \arctan (x)}{99 \left (29+3 \sqrt {33}\right )}+\frac {4 \left (3-\sqrt {33}\right ) \arctan (x)}{99 \left (29-3 \sqrt {33}\right )}-\frac {56 \arctan (x)}{99 \left (29-3 \sqrt {33}\right )}+\frac {19 \arctan (x)}{306}-\frac {\log \left (-4 x+\sqrt {33}-3\right ) \log \left (-\frac {4 (i-x)}{(-3-4 i)+\sqrt {33}}\right )}{9 \sqrt {33}}+\frac {\left (33-5 \sqrt {33}\right ) \log \left (3+\sqrt {33}\right ) \log (x)}{1224}-\frac {\left (1089-301 \sqrt {33}\right ) \log \left (3+\sqrt {33}\right ) \log (x)}{40392}-\frac {\log \left (3+\sqrt {33}\right ) \log (x)}{9 \sqrt {33}}+\frac {\log \left (-3+\sqrt {33}\right ) \log (x)}{9 \sqrt {33}}+\frac {14 \log (x)}{99 \left (3+\sqrt {33}\right )}+\frac {14 \log (x)}{99 \left (3-\sqrt {33}\right )}+\frac {7 \log (x)}{198}-\frac {\log \left (-4 x+\sqrt {33}-3\right ) \log \left (\frac {4 (x+i)}{(-3+4 i)+\sqrt {33}}\right )}{9 \sqrt {33}}+\frac {\left (1089+301 \sqrt {33}\right ) \log \left (-\frac {4 (i-x)}{(-3-4 i)+\sqrt {33}}\right ) \log \left (4 x-\sqrt {33}+3\right )}{40392}-\frac {\left (33+5 \sqrt {33}\right ) \log \left (-\frac {4 (i-x)}{(-3-4 i)+\sqrt {33}}\right ) \log \left (4 x-\sqrt {33}+3\right )}{1224}-\frac {\left (1089+301 \sqrt {33}\right ) \log \left (-\frac {4 x}{3-\sqrt {33}}\right ) \log \left (4 x-\sqrt {33}+3\right )}{40392}+\frac {\left (33+5 \sqrt {33}\right ) \log \left (-\frac {4 x}{3-\sqrt {33}}\right ) \log \left (4 x-\sqrt {33}+3\right )}{1224}+\frac {\left (1089+301 \sqrt {33}\right ) \log \left (\frac {4 (x+i)}{(-3+4 i)+\sqrt {33}}\right ) \log \left (4 x-\sqrt {33}+3\right )}{40392}-\frac {\left (33+5 \sqrt {33}\right ) \log \left (\frac {4 (x+i)}{(-3+4 i)+\sqrt {33}}\right ) \log \left (4 x-\sqrt {33}+3\right )}{1224}+\frac {\left (2299+333 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{148104}-\frac {\left (1331+241 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{37026}+\frac {\left (1089+17 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{148104}-\frac {\left (7-\sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{33 \left (29-3 \sqrt {33}\right )}+\frac {7 \left (3-\sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right )}{99 \left (29-3 \sqrt {33}\right )}+\frac {569 \log \left (4 x-\sqrt {33}+3\right )}{6732 \sqrt {33}}-\frac {\left (33-5 \sqrt {33}\right ) \log \left (\frac {4 (i-x)}{(3+4 i)+\sqrt {33}}\right ) \log \left (4 x+\sqrt {33}+3\right )}{1224}+\frac {\left (1089-301 \sqrt {33}\right ) \log \left (\frac {4 (i-x)}{(3+4 i)+\sqrt {33}}\right ) \log \left (4 x+\sqrt {33}+3\right )}{40392}+\frac {\log \left (\frac {4 (i-x)}{(3+4 i)+\sqrt {33}}\right ) \log \left (4 x+\sqrt {33}+3\right )}{9 \sqrt {33}}-\frac {\left (33-5 \sqrt {33}\right ) \log \left (-\frac {4 (x+i)}{(3-4 i)+\sqrt {33}}\right ) \log \left (4 x+\sqrt {33}+3\right )}{1224}+\frac {\left (1089-301 \sqrt {33}\right ) \log \left (-\frac {4 (x+i)}{(3-4 i)+\sqrt {33}}\right ) \log \left (4 x+\sqrt {33}+3\right )}{40392}+\frac {\log \left (-\frac {4 (x+i)}{(3-4 i)+\sqrt {33}}\right ) \log \left (4 x+\sqrt {33}+3\right )}{9 \sqrt {33}}-\frac {\left (7+\sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{33 \left (29+3 \sqrt {33}\right )}+\frac {7 \left (3+\sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{99 \left (29+3 \sqrt {33}\right )}+\frac {\left (1089-17 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{148104}-\frac {\left (1331-241 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{37026}+\frac {\left (2299-333 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right )}{148104}-\frac {569 \log \left (4 x+\sqrt {33}+3\right )}{6732 \sqrt {33}}+\frac {4 \left (3-\sqrt {33}\right ) \log \left (2 \left (93-19 \sqrt {33}\right ) x+3 \left (151-25 \sqrt {33}\right )\right )}{99 \left (93-19 \sqrt {33}\right )}-\frac {56 \log \left (2 \left (93-19 \sqrt {33}\right ) x+3 \left (151-25 \sqrt {33}\right )\right )}{99 \left (93-19 \sqrt {33}\right )}+\frac {4 \left (3+\sqrt {33}\right ) \log \left (2 \left (93+19 \sqrt {33}\right ) x+3 \left (151+25 \sqrt {33}\right )\right )}{99 \left (93+19 \sqrt {33}\right )}-\frac {56 \log \left (2 \left (93+19 \sqrt {33}\right ) x+3 \left (151+25 \sqrt {33}\right )\right )}{99 \left (93+19 \sqrt {33}\right )}-\frac {\log \left (-4 x+\sqrt {33}-3\right ) \log \left (\frac {4 x}{x^2+1}\right )}{9 \sqrt {33}}+\frac {\left (1089+301 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right ) \log \left (\frac {4 x}{x^2+1}\right )}{40392}-\frac {\left (33+5 \sqrt {33}\right ) \log \left (4 x-\sqrt {33}+3\right ) \log \left (\frac {4 x}{x^2+1}\right )}{1224}-\frac {\left (33-5 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right ) \log \left (\frac {4 x}{x^2+1}\right )}{1224}+\frac {\left (1089-301 \sqrt {33}\right ) \log \left (4 x+\sqrt {33}+3\right ) \log \left (\frac {4 x}{x^2+1}\right )}{40392}+\frac {\log \left (4 x+\sqrt {33}+3\right ) \log \left (\frac {4 x}{x^2+1}\right )}{9 \sqrt {33}}+\frac {\log \left (\frac {4 x}{x^2+1}\right )}{18 x}+\frac {\left (3-\sqrt {33}\right ) \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x-\sqrt {33}+3\right )}-\frac {14 \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x-\sqrt {33}+3\right )}+\frac {\left (3+\sqrt {33}\right ) \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x+\sqrt {33}+3\right )}-\frac {14 \log \left (\frac {4 x}{x^2+1}\right )}{99 \left (4 x+\sqrt {33}+3\right )}+\frac {\left (7+\sqrt {33}\right ) \log \left (x^2+1\right )}{33 \left (29+3 \sqrt {33}\right )}-\frac {7 \left (3+\sqrt {33}\right ) \log \left (x^2+1\right )}{99 \left (29+3 \sqrt {33}\right )}+\frac {\left (7-\sqrt {33}\right ) \log \left (x^2+1\right )}{33 \left (29-3 \sqrt {33}\right )}-\frac {7 \left (3-\sqrt {33}\right ) \log \left (x^2+1\right )}{99 \left (29-3 \sqrt {33}\right )}-\frac {1}{68} \log \left (x^2+1\right )-\frac {\operatorname {PolyLog}\left (2,-\frac {4 x}{3-\sqrt {33}}\right )}{9 \sqrt {33}}-\frac {\left (33-5 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,-\frac {4 x}{3+\sqrt {33}}\right )}{1224}+\frac {\left (1089-301 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,-\frac {4 x}{3+\sqrt {33}}\right )}{40392}+\frac {\operatorname {PolyLog}\left (2,-\frac {4 x}{3+\sqrt {33}}\right )}{9 \sqrt {33}}+\frac {\left (1089+301 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {33}+3}{(-3-4 i)+\sqrt {33}}\right )}{40392}-\frac {\left (33+5 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {33}+3}{(-3-4 i)+\sqrt {33}}\right )}{1224}-\frac {\operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {33}+3}{(-3-4 i)+\sqrt {33}}\right )}{9 \sqrt {33}}+\frac {\left (1089+301 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {33}+3}{(-3+4 i)+\sqrt {33}}\right )}{40392}-\frac {\left (33+5 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {33}+3}{(-3+4 i)+\sqrt {33}}\right )}{1224}-\frac {\operatorname {PolyLog}\left (2,-\frac {4 x-\sqrt {33}+3}{(-3+4 i)+\sqrt {33}}\right )}{9 \sqrt {33}}-\frac {\left (33-5 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {33}+3}{(3-4 i)+\sqrt {33}}\right )}{1224}+\frac {\left (1089-301 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {33}+3}{(3-4 i)+\sqrt {33}}\right )}{40392}+\frac {\operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {33}+3}{(3-4 i)+\sqrt {33}}\right )}{9 \sqrt {33}}-\frac {\left (33-5 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {33}+3}{(3+4 i)+\sqrt {33}}\right )}{1224}+\frac {\left (1089-301 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {33}+3}{(3+4 i)+\sqrt {33}}\right )}{40392}+\frac {\operatorname {PolyLog}\left (2,\frac {4 x+\sqrt {33}+3}{(3+4 i)+\sqrt {33}}\right )}{9 \sqrt {33}}-\frac {\left (1089+301 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,\frac {4 x}{3-\sqrt {33}}+1\right )}{40392}+\frac {\left (33+5 \sqrt {33}\right ) \operatorname {PolyLog}\left (2,\frac {4 x}{3-\sqrt {33}}+1\right )}{1224}-\frac {151}{6732 x}-\frac {5 (58 x+93)}{6732 \left (-2 x^2-3 x+3\right )}+\frac {62 x+151}{2244 x \left (-2 x^2-3 x+3\right )}-\frac {62 x+151}{2244 \left (-2 x^2-3 x+3\right )}\) |
Input:
Int[(3 - 3*x - 5*x^2 + 3*x^3 + 2*x^4 + (-3 + 6*x + 3*x^2 + 6*x^3 + 6*x^4)* Log[(4*x)/(1 + x^2)])/(54*x^2 - 108*x^3 + 36*x^4 - 36*x^5 + 6*x^6 + 72*x^7 + 24*x^8),x]
Output:
-151/(6732*x) + (29 + 2*x)/(748*(3 - 3*x - 2*x^2)) + (x*(29 + 2*x))/(1122* (3 - 3*x - 2*x^2)) - (5*(93 + 58*x))/(6732*(3 - 3*x - 2*x^2)) - (151 + 62* x)/(2244*(3 - 3*x - 2*x^2)) + (151 + 62*x)/(2244*x*(3 - 3*x - 2*x^2)) + (1 9*ArcTan[x])/306 - (56*ArcTan[x])/(99*(29 - 3*Sqrt[33])) + (4*(3 - Sqrt[33 ])*ArcTan[x])/(99*(29 - 3*Sqrt[33])) - (56*ArcTan[x])/(99*(29 + 3*Sqrt[33] )) + (4*(3 + Sqrt[33])*ArcTan[x])/(99*(29 + 3*Sqrt[33])) - (Log[-3 + Sqrt[ 33] - 4*x]*Log[(-4*(I - x))/((-3 - 4*I) + Sqrt[33])])/(9*Sqrt[33]) + (7*Lo g[x])/198 + (14*Log[x])/(99*(3 - Sqrt[33])) + (14*Log[x])/(99*(3 + Sqrt[33 ])) + (Log[-3 + Sqrt[33]]*Log[x])/(9*Sqrt[33]) - (Log[3 + Sqrt[33]]*Log[x] )/(9*Sqrt[33]) - ((1089 - 301*Sqrt[33])*Log[3 + Sqrt[33]]*Log[x])/40392 + ((33 - 5*Sqrt[33])*Log[3 + Sqrt[33]]*Log[x])/1224 - (Log[-3 + Sqrt[33] - 4 *x]*Log[(4*(I + x))/((-3 + 4*I) + Sqrt[33])])/(9*Sqrt[33]) + (569*Log[3 - Sqrt[33] + 4*x])/(6732*Sqrt[33]) + (7*(3 - Sqrt[33])*Log[3 - Sqrt[33] + 4* x])/(99*(29 - 3*Sqrt[33])) - ((7 - Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/(33* (29 - 3*Sqrt[33])) + ((1089 + 17*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/148104 - ((1331 + 241*Sqrt[33])*Log[3 - Sqrt[33] + 4*x])/37026 + ((2299 + 333*Sq rt[33])*Log[3 - Sqrt[33] + 4*x])/148104 - ((33 + 5*Sqrt[33])*Log[(-4*(I - x))/((-3 - 4*I) + Sqrt[33])]*Log[3 - Sqrt[33] + 4*x])/1224 + ((1089 + 301* Sqrt[33])*Log[(-4*(I - x))/((-3 - 4*I) + Sqrt[33])]*Log[3 - Sqrt[33] + 4*x ])/40392 + ((33 + 5*Sqrt[33])*Log[(-4*x)/(3 - Sqrt[33])]*Log[3 - Sqrt[3...
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 1.19 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97
| method | result | size |
| norman | \(-\frac {\ln \left (\frac {4 x}{x^{2}+1}\right )}{6 x \left (2 x^{2}+3 x -3\right )}\) | \(29\) |
| risch | \(-\frac {\ln \left (\frac {4 x}{x^{2}+1}\right )}{6 x \left (2 x^{2}+3 x -3\right )}\) | \(29\) |
| parallelrisch | \(-\frac {\ln \left (\frac {4 x}{x^{2}+1}\right )}{6 x \left (2 x^{2}+3 x -3\right )}\) | \(29\) |
| orering | \(-\frac {x \left (14 x^{6}+15 x^{5}-17 x^{4}-12 x^{3}-2 x^{2}-15 x +9\right ) \left (\left (6 x^{4}+6 x^{3}+3 x^{2}+6 x -3\right ) \ln \left (\frac {4 x}{x^{2}+1}\right )+2 x^{4}+3 x^{3}-5 x^{2}-3 x +3\right )}{3 \left (6 x^{6}+4 x^{5}-9 x^{4}-8 x^{3}-2 x^{2}-4 x +1\right ) \left (24 x^{8}+72 x^{7}+6 x^{6}-36 x^{5}+36 x^{4}-108 x^{3}+54 x^{2}\right )}-\frac {\left (x^{2}-1\right ) x^{2} \left (2 x^{2}+3 x -3\right ) \left (x^{2}+1\right ) \left (\frac {\left (24 x^{3}+18 x^{2}+6 x +6\right ) \ln \left (\frac {4 x}{x^{2}+1}\right )+\frac {\left (6 x^{4}+6 x^{3}+3 x^{2}+6 x -3\right ) \left (\frac {4}{x^{2}+1}-\frac {8 x^{2}}{\left (x^{2}+1\right )^{2}}\right ) \left (x^{2}+1\right )}{4 x}+8 x^{3}+9 x^{2}-10 x -3}{24 x^{8}+72 x^{7}+6 x^{6}-36 x^{5}+36 x^{4}-108 x^{3}+54 x^{2}}-\frac {\left (\left (6 x^{4}+6 x^{3}+3 x^{2}+6 x -3\right ) \ln \left (\frac {4 x}{x^{2}+1}\right )+2 x^{4}+3 x^{3}-5 x^{2}-3 x +3\right ) \left (192 x^{7}+504 x^{6}+36 x^{5}-180 x^{4}+144 x^{3}-324 x^{2}+108 x \right )}{\left (24 x^{8}+72 x^{7}+6 x^{6}-36 x^{5}+36 x^{4}-108 x^{3}+54 x^{2}\right )^{2}}\right )}{3 \left (6 x^{6}+4 x^{5}-9 x^{4}-8 x^{3}-2 x^{2}-4 x +1\right )}\) | \(474\) |
Input:
int(((6*x^4+6*x^3+3*x^2+6*x-3)*ln(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3*x+3)/(2 4*x^8+72*x^7+6*x^6-36*x^5+36*x^4-108*x^3+54*x^2),x,method=_RETURNVERBOSE)
Output:
-1/6*ln(4*x/(x^2+1))/x/(2*x^2+3*x-3)
Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {\log \left (\frac {4 \, x}{x^{2} + 1}\right )}{6 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} \] Input:
integrate(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3* x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*x^4-108*x^3+54*x^2),x, algorithm="fric as")
Output:
-1/6*log(4*x/(x^2 + 1))/(2*x^3 + 3*x^2 - 3*x)
Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=- \frac {\log {\left (\frac {4 x}{x^{2} + 1} \right )}}{12 x^{3} + 18 x^{2} - 18 x} \] Input:
integrate(((6*x**4+6*x**3+3*x**2+6*x-3)*ln(4*x/(x**2+1))+2*x**4+3*x**3-5*x **2-3*x+3)/(24*x**8+72*x**7+6*x**6-36*x**5+36*x**4-108*x**3+54*x**2),x)
Output:
-log(4*x/(x**2 + 1))/(12*x**3 + 18*x**2 - 18*x)
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (28) = 56\).
Time = 1.01 (sec) , antiderivative size = 191, normalized size of antiderivative = 6.37 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {350 \, x^{2} + 587 \, x - 374}{2244 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} + \frac {68 \, x^{2} + 3 \, {\left (6 \, x^{3} + 9 \, x^{2} - 9 \, x + 34\right )} \log \left (x^{2} + 1\right ) - 34 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x + 3\right )} \log \left (x\right ) + 102 \, x - 204 \, \log \left (2\right ) - 102}{612 \, {\left (2 \, x^{3} + 3 \, x^{2} - 3 \, x\right )}} + \frac {62 \, x + 151}{2244 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} + \frac {5 \, {\left (58 \, x + 93\right )}}{6732 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {26 \, x + 3}{1122 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {2 \, x + 29}{748 \, {\left (2 \, x^{2} + 3 \, x - 3\right )}} - \frac {1}{68} \, \log \left (x^{2} + 1\right ) + \frac {1}{18} \, \log \left (x\right ) \] Input:
integrate(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3* x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*x^4-108*x^3+54*x^2),x, algorithm="maxi ma")
Output:
-1/2244*(350*x^2 + 587*x - 374)/(2*x^3 + 3*x^2 - 3*x) + 1/612*(68*x^2 + 3* (6*x^3 + 9*x^2 - 9*x + 34)*log(x^2 + 1) - 34*(2*x^3 + 3*x^2 - 3*x + 3)*log (x) + 102*x - 204*log(2) - 102)/(2*x^3 + 3*x^2 - 3*x) + 1/2244*(62*x + 151 )/(2*x^2 + 3*x - 3) + 5/6732*(58*x + 93)/(2*x^2 + 3*x - 3) - 1/1122*(26*x + 3)/(2*x^2 + 3*x - 3) - 1/748*(2*x + 29)/(2*x^2 + 3*x - 3) - 1/68*log(x^2 + 1) + 1/18*log(x)
Time = 0.12 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {1}{18} \, {\left (\frac {2 \, x + 3}{2 \, x^{2} + 3 \, x - 3} - \frac {1}{x}\right )} \log \left (\frac {4 \, x}{x^{2} + 1}\right ) \] Input:
integrate(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3* x+3)/(24*x^8+72*x^7+6*x^6-36*x^5+36*x^4-108*x^3+54*x^2),x, algorithm="giac ")
Output:
-1/18*((2*x + 3)/(2*x^2 + 3*x - 3) - 1/x)*log(4*x/(x^2 + 1))
Time = 1.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.10 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=-\frac {2\,\ln \left (2\right )-\ln \left (x^2+1\right )+\ln \left (x\right )}{12\,\left (x^3+\frac {3\,x^2}{2}-\frac {3\,x}{2}\right )} \] Input:
int((log((4*x)/(x^2 + 1))*(6*x + 3*x^2 + 6*x^3 + 6*x^4 - 3) - 3*x - 5*x^2 + 3*x^3 + 2*x^4 + 3)/(54*x^2 - 108*x^3 + 36*x^4 - 36*x^5 + 6*x^6 + 72*x^7 + 24*x^8),x)
Output:
-(2*log(2) - log(x^2 + 1) + log(x))/(12*((3*x^2)/2 - (3*x)/2 + x^3))
Time = 0.21 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {3-3 x-5 x^2+3 x^3+2 x^4+\left (-3+6 x+3 x^2+6 x^3+6 x^4\right ) \log \left (\frac {4 x}{1+x^2}\right )}{54 x^2-108 x^3+36 x^4-36 x^5+6 x^6+72 x^7+24 x^8} \, dx=\frac {-9 \,\mathrm {log}\left (\frac {4 x}{x^{2}+1}\right )-4 x^{3}-6 x^{2}+6 x}{54 x \left (2 x^{2}+3 x -3\right )} \] Input:
int(((6*x^4+6*x^3+3*x^2+6*x-3)*log(4*x/(x^2+1))+2*x^4+3*x^3-5*x^2-3*x+3)/( 24*x^8+72*x^7+6*x^6-36*x^5+36*x^4-108*x^3+54*x^2),x)
Output:
( - 9*log((4*x)/(x**2 + 1)) - 4*x**3 - 6*x**2 + 6*x)/(54*x*(2*x**2 + 3*x - 3))