Integrand size = 106, antiderivative size = 31 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=3+e^3+\frac {x^2}{-x+x^2+\log (2)}-x \left (4+\log \left (\frac {5}{x^2}\right )\right ) \] Output:
x^2/(ln(2)+x^2-x)-(4+ln(5/x^2))*x+exp(3)+3
Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(31)=62\).
Time = 5.07 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.23 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=-4 x-\frac {x-4 \log ^2(2)+8 x \log ^2(2)-x (1+\log (4)) \log (16)-\log (2) (1+\log (16))+\log (4) \log (64)}{\left (-x+x^2+\log (2)\right ) (-1+\log (16))}-x \log \left (\frac {5}{x^2}\right ) \] Input:
Integrate[(-3*x^2 + 4*x^3 - 2*x^4 + (6*x - 4*x^2)*Log[2] - 2*Log[2]^2 + (- x^2 + 2*x^3 - x^4 + (2*x - 2*x^2)*Log[2] - Log[2]^2)*Log[5/x^2])/(x^2 - 2* x^3 + x^4 + (-2*x + 2*x^2)*Log[2] + Log[2]^2),x]
Output:
-4*x - (x - 4*Log[2]^2 + 8*x*Log[2]^2 - x*(1 + Log[4])*Log[16] - Log[2]*(1 + Log[16]) + Log[4]*Log[64])/((-x + x^2 + Log[2])*(-1 + Log[16])) - x*Log [5/x^2]
Result contains complex when optimal does not.
Time = 6.84 (sec) , antiderivative size = 2439, normalized size of antiderivative = 78.68, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {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.
| \(\displaystyle \int \frac {-2 x^4+4 x^3-3 x^2+\left (6 x-4 x^2\right ) \log (2)+\left (-x^4+2 x^3-x^2+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )-2 \log ^2(2)}{x^4-2 x^3+x^2+\left (2 x^2-2 x\right ) \log (2)+\log ^2(2)} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {4 i \left (-2 x^4+4 x^3-3 x^2+\left (6 x-4 x^2\right ) \log (2)+\left (-x^4+2 x^3-x^2+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )-2 \log ^2(2)\right )}{(4 \log (2)-1)^{3/2} \left (-2 x+1+i \sqrt {4 \log (2)-1}\right )}+\frac {4 i \left (-2 x^4+4 x^3-3 x^2+\left (6 x-4 x^2\right ) \log (2)+\left (-x^4+2 x^3-x^2+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )-2 \log ^2(2)\right )}{(4 \log (2)-1)^{3/2} \left (2 x-1+i \sqrt {4 \log (2)-1}\right )}-\frac {4 \left (-2 x^4+4 x^3-3 x^2+\left (6 x-4 x^2\right ) \log (2)+\left (-x^4+2 x^3-x^2+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )-2 \log ^2(2)\right )}{(4 \log (2)-1) \left (-2 x+1+i \sqrt {4 \log (2)-1}\right )^2}-\frac {4 \left (-2 x^4+4 x^3-3 x^2+\left (6 x-4 x^2\right ) \log (2)+\left (-x^4+2 x^3-x^2+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )-2 \log ^2(2)\right )}{(4 \log (2)-1) \left (2 x-1+i \sqrt {4 \log (2)-1}\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \log \left (\frac {5}{x^2}\right ) \left (i x+\frac {\log (4)}{i-\sqrt {-1+\log (16)}}\right )^4}{2 (-1+\log (16))^{3/2}}-\frac {\left (2 \sqrt {-1+\log (16)}-i (2-\log (256))\right ) \log \left (\frac {5}{x^2}\right ) \left (i x+\frac {\log (4)}{i-\sqrt {-1+\log (16)}}\right )^3}{3 \left (i-\sqrt {-1+\log (16)}\right ) (-1+\log (16))^{3/2}}-\frac {i \log \left (\frac {5}{x^2}\right ) \left (i x+\frac {\log (4)}{i-\sqrt {-1+\log (16)}}\right )^3}{3 (1-\log (16))}-\frac {i \left (i x+\frac {\log (4)}{i+\sqrt {-1+\log (16)}}\right )^4 \log \left (\frac {5}{x^2}\right )}{2 (-1+\log (16))^{3/2}}-\frac {i \left (i x+\frac {\log (4)}{i+\sqrt {-1+\log (16)}}\right )^3 \log \left (\frac {5}{x^2}\right )}{3 (1-\log (16))}+\frac {2 \left (i x+\frac {\log (4)}{i+\sqrt {-1+\log (16)}}\right )^3 \left (1-i \sqrt {-1+\log (16)}\right ) \log \left (\frac {5}{x^2}\right )}{3 \left (i+\sqrt {-1+\log (16)}\right ) (1-\log (16))}-\frac {2 \log ^3(4) \left (2 \sqrt {-1+\log (16)}-i (2-\log (256))\right ) \log (x)}{3 \left (i-\sqrt {-1+\log (16)}\right )^4 (-1+\log (16))^{3/2}}-\frac {i \log ^4(4) \log (x)}{\left (i+\sqrt {-1+\log (16)}\right )^4 (-1+\log (16))^{3/2}}+\frac {i \log ^4(4) \log (x)}{\left (i-\sqrt {-1+\log (16)}\right )^4 (-1+\log (16))^{3/2}}+\frac {2 \log ^3(4) \log (x)}{3 \left (1+i \sqrt {-1+\log (16)}\right )^3 (1-\log (16))}+\frac {2 \log ^3(4) \log (x)}{\left (1-i \sqrt {-1+\log (16)}\right )^3 (1-\log (16))}+\frac {\left (16 i \log ^2(2)-4 (1-\log (4)) \sqrt {-1+\log (16)}+i \left (4-\log ^2(16)-\log (65536)\right )\right ) \log \left (-2 i x-\sqrt {-1+\log (16)}+i\right )}{4 (-1+\log (16))^{3/2}}-\frac {\left (4+i (4-\log (16)) \sqrt {-1+\log (16)}-\log (4096)\right ) \log \left (-2 i x-\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}+\frac {3 \left (2+2 i \sqrt {-1+\log (16)}-\log (16)\right ) \log \left (-2 i x-\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}+\frac {2 \log (2) \left (1-2 i \sqrt {-1+\log (16)}\right ) \log \left (-2 i x-\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}-\frac {3 \left (1+i \sqrt {-1+\log (16)}\right ) \log \left (-2 i x-\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}-\frac {\left (16 i \log ^2(2)+4 (1-\log (4)) \sqrt {-1+\log (16)}+i \left (4-\log ^2(16)-\log (65536)\right )\right ) \log \left (-2 i x+\sqrt {-1+\log (16)}+i\right )}{4 (-1+\log (16))^{3/2}}-\frac {\left (4-i (4-\log (16)) \sqrt {-1+\log (16)}-\log (4096)\right ) \log \left (-2 i x+\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}+\frac {3 \left (2-2 i \sqrt {-1+\log (16)}-\log (16)\right ) \log \left (-2 i x+\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}+\frac {2 \log (2) \left (1+2 i \sqrt {-1+\log (16)}\right ) \log \left (-2 i x+\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}-\frac {3 \left (1-i \sqrt {-1+\log (16)}\right ) \log \left (-2 i x+\sqrt {-1+\log (16)}+i\right )}{1-\log (16)}-\frac {x \left (24 i \log (2)+\sqrt {-1+\log (16)} (2+\log (16))-i (2+\log (4096))\right )}{2 (-1+\log (16))^{3/2}}+\frac {x \left (24 i \log (2)-\sqrt {-1+\log (16)} (2+\log (16))-i (2+\log (4096))\right )}{2 (-1+\log (16))^{3/2}}+\frac {(4-\log (16)) \sqrt {-1+\log (16)}+i (4-\log (4096))}{\left (-2 i x+\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}-\frac {(4-\log (16)) \sqrt {-1+\log (16)}-i (4-\log (4096))}{\left (-2 i x-\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}+\frac {2 x^3 \left (2+2 i \sqrt {-1+\log (16)}-\log (256)\right )}{9 \left (i-\sqrt {-1+\log (16)}\right ) (-1+\log (16))^{3/2}}-\frac {2 x \log ^2(4) \left (2+2 i \sqrt {-1+\log (16)}-\log (256)\right )}{\left (i-\sqrt {-1+\log (16)}\right )^3 (-1+\log (16))^{3/2}}+\frac {x^2 \log (4) \left (2 \sqrt {-1+\log (16)}-i (2-\log (256))\right )}{\left (i-\sqrt {-1+\log (16)}\right )^2 (-1+\log (16))^{3/2}}-\frac {4 (2-\log (16)) \sqrt {-1+\log (16)}+i \left (8-8 \log (16)+\log ^2(16)\right )}{4 \left (-2 i x+\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}+\frac {4 (2-\log (16)) \sqrt {-1+\log (16)}-i \left (8-8 \log (16)+\log ^2(16)\right )}{4 \left (-2 i x-\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}+\frac {x^2 \left (2 \sqrt {-1+\log (16)}+i (4+\log (16))\right )}{2 (-1+\log (16))^{3/2}}+\frac {x^2 \left (2 \sqrt {-1+\log (16)}-i (4+\log (16))\right )}{2 (-1+\log (16))^{3/2}}+\frac {2 \log (2) \left (\sqrt {-1+\log (16)}+i (1+\log (16))\right )}{\left (-2 i x+\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}-\frac {2 \log (2) \left (\sqrt {-1+\log (16)}-i (1+\log (16))\right )}{\left (-2 i x-\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}-\frac {2 x^3 \left (3 i+\sqrt {-1+\log (16)}\right )}{3 (-1+\log (16))^{3/2}}-\frac {4 x^3 \log (4)}{3 \left (i+\sqrt {-1+\log (16)}\right ) (-1+\log (16))^{3/2}}+\frac {3 i x^2 \log ^2(4)}{\left (i+\sqrt {-1+\log (16)}\right )^2 (-1+\log (16))^{3/2}}+\frac {4 x \log ^3(4)}{\left (i+\sqrt {-1+\log (16)}\right )^3 (-1+\log (16))^{3/2}}+\frac {2 x^3 \left (3 i-\sqrt {-1+\log (16)}\right )}{3 (-1+\log (16))^{3/2}}+\frac {4 x^3 \log (4)}{3 \left (i-\sqrt {-1+\log (16)}\right ) (-1+\log (16))^{3/2}}-\frac {3 i x^2 \log ^2(4)}{\left (i-\sqrt {-1+\log (16)}\right )^2 (-1+\log (16))^{3/2}}-\frac {4 x \log ^3(4)}{\left (i-\sqrt {-1+\log (16)}\right )^3 (-1+\log (16))^{3/2}}-\frac {20 x^3}{9 (1-\log (16))}+\frac {4 x^2}{1-\log (16)}-\frac {6 x}{1-\log (16)}-\frac {3 \left (i (2-\log (16))-2 \sqrt {-1+\log (16)}\right )}{2 \left (-2 i x-\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}-\frac {4 i \log ^2(2)}{\left (-2 i x-\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}-\frac {4 i \log ^2(2)}{\left (-2 i x+\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}-\frac {3 \left (i (2-\log (16))+2 \sqrt {-1+\log (16)}\right )}{2 \left (-2 i x+\sqrt {-1+\log (16)}+i\right ) (1-\log (16))}+\frac {3 x \left (i+\sqrt {-1+\log (16)}\right )^2}{2 (1-\log (16))}+\frac {6 x \log ^2(4)}{\left (i+\sqrt {-1+\log (16)}\right )^2 (1-\log (16))}-\frac {x^2 \left (1+i \sqrt {-1+\log (16)}\right )}{1-\log (16)}+\frac {4 x \left (1+i \sqrt {-1+\log (16)}\right )}{1-\log (16)}+\frac {x^2 \log (4)}{\left (1+i \sqrt {-1+\log (16)}\right ) (1-\log (16))}-\frac {x^2 \left (1-i \sqrt {-1+\log (16)}\right )}{1-\log (16)}+\frac {4 x \left (1-i \sqrt {-1+\log (16)}\right )}{1-\log (16)}+\frac {3 x^2 \log (4)}{\left (1-i \sqrt {-1+\log (16)}\right ) (1-\log (16))}+\frac {3 x \left (i-\sqrt {-1+\log (16)}\right )^2}{2 (1-\log (16))}+\frac {2 x \log ^2(4)}{\left (i-\sqrt {-1+\log (16)}\right )^2 (1-\log (16))}-\frac {8 x \log (2)}{1-\log (16)}\) |
Input:
Int[(-3*x^2 + 4*x^3 - 2*x^4 + (6*x - 4*x^2)*Log[2] - 2*Log[2]^2 + (-x^2 + 2*x^3 - x^4 + (2*x - 2*x^2)*Log[2] - Log[2]^2)*Log[5/x^2])/(x^2 - 2*x^3 + x^4 + (-2*x + 2*x^2)*Log[2] + Log[2]^2),x]
Output:
(-6*x)/(1 - Log[16]) + (4*x^2)/(1 - Log[16]) - (20*x^3)/(9*(1 - Log[16])) - (8*x*Log[2])/(1 - Log[16]) + (2*x*Log[4]^2)/((I - Sqrt[-1 + Log[16]])^2* (1 - Log[16])) + (3*x*(I - Sqrt[-1 + Log[16]])^2)/(2*(1 - Log[16])) - ((4* I)*Log[2]^2)/((I - (2*I)*x - Sqrt[-1 + Log[16]])*(1 - Log[16])) - (3*(I*(2 - Log[16]) - 2*Sqrt[-1 + Log[16]]))/(2*(I - (2*I)*x - Sqrt[-1 + Log[16]]) *(1 - Log[16])) + (3*x^2*Log[4])/((1 - I*Sqrt[-1 + Log[16]])*(1 - Log[16]) ) + (4*x*(1 - I*Sqrt[-1 + Log[16]]))/(1 - Log[16]) - (x^2*(1 - I*Sqrt[-1 + Log[16]]))/(1 - Log[16]) + (x^2*Log[4])/((1 + I*Sqrt[-1 + Log[16]])*(1 - Log[16])) + (4*x*(1 + I*Sqrt[-1 + Log[16]]))/(1 - Log[16]) - (x^2*(1 + I*S qrt[-1 + Log[16]]))/(1 - Log[16]) + (6*x*Log[4]^2)/((I + Sqrt[-1 + Log[16] ])^2*(1 - Log[16])) + (3*x*(I + Sqrt[-1 + Log[16]])^2)/(2*(1 - Log[16])) - ((4*I)*Log[2]^2)/((I - (2*I)*x + Sqrt[-1 + Log[16]])*(1 - Log[16])) - (3* (I*(2 - Log[16]) + 2*Sqrt[-1 + Log[16]]))/(2*(I - (2*I)*x + Sqrt[-1 + Log[ 16]])*(1 - Log[16])) - (4*x*Log[4]^3)/((I - Sqrt[-1 + Log[16]])^3*(-1 + Lo g[16])^(3/2)) - ((3*I)*x^2*Log[4]^2)/((I - Sqrt[-1 + Log[16]])^2*(-1 + Log [16])^(3/2)) + (4*x^3*Log[4])/(3*(I - Sqrt[-1 + Log[16]])*(-1 + Log[16])^( 3/2)) + (2*x^3*(3*I - Sqrt[-1 + Log[16]]))/(3*(-1 + Log[16])^(3/2)) + (4*x *Log[4]^3)/((I + Sqrt[-1 + Log[16]])^3*(-1 + Log[16])^(3/2)) + ((3*I)*x^2* Log[4]^2)/((I + Sqrt[-1 + Log[16]])^2*(-1 + Log[16])^(3/2)) - (4*x^3*Log[4 ])/(3*(I + Sqrt[-1 + Log[16]])*(-1 + Log[16])^(3/2)) - (2*x^3*(3*I + Sq...
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 = 1.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03
| method | result | size |
| derivativedivides | \(-x \ln \left (5\right )-x \ln \left (\frac {1}{x^{2}}\right )-4 x +\frac {1}{\frac {\ln \left (2\right )}{x^{2}}-\frac {1}{x}+1}\) | \(32\) |
| default | \(-x \ln \left (5\right )-x \ln \left (\frac {1}{x^{2}}\right )-4 x +\frac {1}{\frac {\ln \left (2\right )}{x^{2}}-\frac {1}{x}+1}\) | \(32\) |
| parts | \(-4 x +\frac {x -\ln \left (2\right )}{\ln \left (2\right )+x^{2}-x}-x \ln \left (5\right )-x \ln \left (\frac {1}{x^{2}}\right )\) | \(35\) |
| risch | \(-x \ln \left (\frac {5}{x^{2}}\right )-\frac {4 x^{3}+4 x \ln \left (2\right )-4 x^{2}+\ln \left (2\right )-x}{\ln \left (2\right )+x^{2}-x}\) | \(45\) |
| norman | \(\frac {x^{2} \ln \left (\frac {5}{x^{2}}\right )+\left (5-4 \ln \left (2\right )\right ) x -4 x^{3}-\ln \left (\frac {5}{x^{2}}\right ) x^{3}-\ln \left (2\right ) \ln \left (\frac {5}{x^{2}}\right ) x -5 \ln \left (2\right )}{\ln \left (2\right )+x^{2}-x}\) | \(63\) |
| parallelrisch | \(\frac {-2 \ln \left (\frac {5}{x^{2}}\right ) x^{3}-2 \ln \left (2\right ) \ln \left (\frac {5}{x^{2}}\right ) x -8 x^{3}+2 x^{2} \ln \left (\frac {5}{x^{2}}\right )-8 x \ln \left (2\right )-10 \ln \left (2\right )+10 x}{2 \ln \left (2\right )+2 x^{2}-2 x}\) | \(65\) |
Input:
int(((-ln(2)^2+(-2*x^2+2*x)*ln(2)-x^4+2*x^3-x^2)*ln(5/x^2)-2*ln(2)^2+(-4*x ^2+6*x)*ln(2)-2*x^4+4*x^3-3*x^2)/(ln(2)^2+(2*x^2-2*x)*ln(2)+x^4-2*x^3+x^2) ,x,method=_RETURNVERBOSE)
Output:
-x*ln(5)-x*ln(1/x^2)-4*x+1/(ln(2)/x^2-1/x+1)
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.77 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=-\frac {4 \, x^{3} - 4 \, x^{2} + {\left (4 \, x + 1\right )} \log \left (2\right ) + {\left (x^{3} - x^{2} + x \log \left (2\right )\right )} \log \left (\frac {5}{x^{2}}\right ) - x}{x^{2} - x + \log \left (2\right )} \] Input:
integrate(((-log(2)^2+(-2*x^2+2*x)*log(2)-x^4+2*x^3-x^2)*log(5/x^2)-2*log( 2)^2+(-4*x^2+6*x)*log(2)-2*x^4+4*x^3-3*x^2)/(log(2)^2+(2*x^2-2*x)*log(2)+x ^4-2*x^3+x^2),x, algorithm="fricas")
Output:
-(4*x^3 - 4*x^2 + (4*x + 1)*log(2) + (x^3 - x^2 + x*log(2))*log(5/x^2) - x )/(x^2 - x + log(2))
Time = 0.22 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=- x \log {\left (\frac {5}{x^{2}} \right )} - 4 x - \frac {- x + \log {\left (2 \right )}}{x^{2} - x + \log {\left (2 \right )}} \] Input:
integrate(((-ln(2)**2+(-2*x**2+2*x)*ln(2)-x**4+2*x**3-x**2)*ln(5/x**2)-2*l n(2)**2+(-4*x**2+6*x)*ln(2)-2*x**4+4*x**3-3*x**2)/(ln(2)**2+(2*x**2-2*x)*l n(2)+x**4-2*x**3+x**2),x)
Output:
-x*log(5/x**2) - 4*x - (-x + log(2))/(x**2 - x + log(2))
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (30) = 60\).
Time = 0.37 (sec) , antiderivative size = 243, normalized size of antiderivative = 7.84 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=-2 \, {\left (\frac {2 \, x - 1}{x^{2} {\left (4 \, \log \left (2\right ) - 1\right )} - x {\left (4 \, \log \left (2\right ) - 1\right )} + 4 \, \log \left (2\right )^{2} - \log \left (2\right )} + \frac {4 \, \arctan \left (\frac {2 \, x - 1}{\sqrt {4 \, \log \left (2\right ) - 1}}\right )}{{\left (4 \, \log \left (2\right ) - 1\right )}^{\frac {3}{2}}}\right )} \log \left (2\right )^{2} + \frac {8 \, \arctan \left (\frac {2 \, x - 1}{\sqrt {4 \, \log \left (2\right ) - 1}}\right ) \log \left (2\right )^{2}}{{\left (4 \, \log \left (2\right ) - 1\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, {\left (\log \left (5\right ) + 4\right )} \log \left (2\right ) - \log \left (5\right ) - 4\right )} x^{3} - {\left (4 \, {\left (\log \left (5\right ) + 4\right )} \log \left (2\right ) - \log \left (5\right ) - 4\right )} x^{2} + {\left (4 \, {\left (\log \left (5\right ) + 3\right )} \log \left (2\right )^{2} - {\left (\log \left (5\right ) + 8\right )} \log \left (2\right ) + 1\right )} x + 6 \, \log \left (2\right )^{2} - 2 \, {\left (x^{3} {\left (4 \, \log \left (2\right ) - 1\right )} - x^{2} {\left (4 \, \log \left (2\right ) - 1\right )} + {\left (4 \, \log \left (2\right )^{2} - \log \left (2\right )\right )} x\right )} \log \left (x\right ) - \log \left (2\right )}{x^{2} {\left (4 \, \log \left (2\right ) - 1\right )} - x {\left (4 \, \log \left (2\right ) - 1\right )} + 4 \, \log \left (2\right )^{2} - \log \left (2\right )} \] Input:
integrate(((-log(2)^2+(-2*x^2+2*x)*log(2)-x^4+2*x^3-x^2)*log(5/x^2)-2*log( 2)^2+(-4*x^2+6*x)*log(2)-2*x^4+4*x^3-3*x^2)/(log(2)^2+(2*x^2-2*x)*log(2)+x ^4-2*x^3+x^2),x, algorithm="maxima")
Output:
-2*((2*x - 1)/(x^2*(4*log(2) - 1) - x*(4*log(2) - 1) + 4*log(2)^2 - log(2) ) + 4*arctan((2*x - 1)/sqrt(4*log(2) - 1))/(4*log(2) - 1)^(3/2))*log(2)^2 + 8*arctan((2*x - 1)/sqrt(4*log(2) - 1))*log(2)^2/(4*log(2) - 1)^(3/2) - ( (4*(log(5) + 4)*log(2) - log(5) - 4)*x^3 - (4*(log(5) + 4)*log(2) - log(5) - 4)*x^2 + (4*(log(5) + 3)*log(2)^2 - (log(5) + 8)*log(2) + 1)*x + 6*log( 2)^2 - 2*(x^3*(4*log(2) - 1) - x^2*(4*log(2) - 1) + (4*log(2)^2 - log(2))* x)*log(x) - log(2))/(x^2*(4*log(2) - 1) - x*(4*log(2) - 1) + 4*log(2)^2 - log(2))
Time = 0.12 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=-x {\left (\log \left (5\right ) + 4\right )} + x \log \left (x^{2}\right ) + \frac {x - \log \left (2\right )}{x^{2} - x + \log \left (2\right )} \] Input:
integrate(((-log(2)^2+(-2*x^2+2*x)*log(2)-x^4+2*x^3-x^2)*log(5/x^2)-2*log( 2)^2+(-4*x^2+6*x)*log(2)-2*x^4+4*x^3-3*x^2)/(log(2)^2+(2*x^2-2*x)*log(2)+x ^4-2*x^3+x^2),x, algorithm="giac")
Output:
-x*(log(5) + 4) + x*log(x^2) + (x - log(2))/(x^2 - x + log(2))
Time = 0.75 (sec) , antiderivative size = 239, normalized size of antiderivative = 7.71 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=\left (\sum _{k=1}^4\ln \left (\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,\ln \left (2\right )\,2-\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,x\,2-\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,{\ln \left (2\right )}^2\,14+\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,{\ln \left (2\right )}^3\,24-4\,x\,{\ln \left (2\right )}^2+8\,x\,{\ln \left (2\right )}^3+{\ln \left (2\right )}^2+\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,x\,\ln \left (2\right )\,16-\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,x\,{\ln \left (2\right )}^2\,36+\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\,x\,{\ln \left (2\right )}^3\,16\right )\,\mathrm {root}\left ({\ln \left (2\right )}^2+16\,{\ln \left (2\right )}^4-8\,{\ln \left (2\right )}^3,z,k\right )\right )-x\,\ln \left (\frac {1}{x^2}\right )-4\,x-x\,\ln \left (5\right ) \] Input:
int(-(log(5/x^2)*(log(2)^2 - log(2)*(2*x - 2*x^2) + x^2 - 2*x^3 + x^4) - l og(2)*(6*x - 4*x^2) + 2*log(2)^2 + 3*x^2 - 4*x^3 + 2*x^4)/(log(2)^2 - log( 2)*(2*x - 2*x^2) + x^2 - 2*x^3 + x^4),x)
Output:
symsum(log(2*root(log(2)^2 + 16*log(2)^4 - 8*log(2)^3, z, k)*log(2) - 2*ro ot(log(2)^2 + 16*log(2)^4 - 8*log(2)^3, z, k)*x - 14*root(log(2)^2 + 16*lo g(2)^4 - 8*log(2)^3, z, k)*log(2)^2 + 24*root(log(2)^2 + 16*log(2)^4 - 8*l og(2)^3, z, k)*log(2)^3 - 4*x*log(2)^2 + 8*x*log(2)^3 + log(2)^2 + 16*root (log(2)^2 + 16*log(2)^4 - 8*log(2)^3, z, k)*x*log(2) - 36*root(log(2)^2 + 16*log(2)^4 - 8*log(2)^3, z, k)*x*log(2)^2 + 16*root(log(2)^2 + 16*log(2)^ 4 - 8*log(2)^3, z, k)*x*log(2)^3)*root(log(2)^2 + 16*log(2)^4 - 8*log(2)^3 , z, k), k, 1, 4) - x*log(1/x^2) - 4*x - x*log(5)
Time = 0.21 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.35 \[ \int \frac {-3 x^2+4 x^3-2 x^4+\left (6 x-4 x^2\right ) \log (2)-2 \log ^2(2)+\left (-x^2+2 x^3-x^4+\left (2 x-2 x^2\right ) \log (2)-\log ^2(2)\right ) \log \left (\frac {5}{x^2}\right )}{x^2-2 x^3+x^4+\left (-2 x+2 x^2\right ) \log (2)+\log ^2(2)} \, dx=\frac {-\mathrm {log}\left (\frac {5}{x^{2}}\right ) \mathrm {log}\left (2\right ) x -\mathrm {log}\left (\frac {5}{x^{2}}\right ) x^{3}+\mathrm {log}\left (\frac {5}{x^{2}}\right ) x^{2}-2 \mathrm {log}\left (2\right )^{2}-2 \,\mathrm {log}\left (2\right ) x^{2}-2 \,\mathrm {log}\left (2\right ) x -4 x^{3}+5 x^{2}}{\mathrm {log}\left (2\right )+x^{2}-x} \] Input:
int(((-log(2)^2+(-2*x^2+2*x)*log(2)-x^4+2*x^3-x^2)*log(5/x^2)-2*log(2)^2+( -4*x^2+6*x)*log(2)-2*x^4+4*x^3-3*x^2)/(log(2)^2+(2*x^2-2*x)*log(2)+x^4-2*x ^3+x^2),x)
Output:
( - log(5/x**2)*log(2)*x - log(5/x**2)*x**3 + log(5/x**2)*x**2 - 2*log(2)* *2 - 2*log(2)*x**2 - 2*log(2)*x - 4*x**3 + 5*x**2)/(log(2) + x**2 - x)