Integrand size = 43, antiderivative size = 16 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=\frac {x}{\left (3-\frac {1}{x^2}+x\right ) \log ^2(4)} \]
Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.50 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=\frac {1-3 x^2}{\left (-1+3 x^2+x^3\right ) \log ^2(4)} \]
Result contains complex when optimal does not.
Time = 9.04 (sec) , antiderivative size = 1767, normalized size of antiderivative = 110.44, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {27, 27, 2027, 2462, 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.
\(\displaystyle \int \frac {3 x^4-3 x^2}{\left (x^6+6 x^5+9 x^4-2 x^3-6 x^2+1\right ) \log ^2(4)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {3 \left (x^2-x^4\right )}{x^6+6 x^5+9 x^4-2 x^3-6 x^2+1}dx}{\log ^2(4)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {3 \int \frac {x^2-x^4}{x^6+6 x^5+9 x^4-2 x^3-6 x^2+1}dx}{\log ^2(4)}\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle -\frac {3 \int \frac {x^2 \left (1-x^2\right )}{x^6+6 x^5+9 x^4-2 x^3-6 x^2+1}dx}{\log ^2(4)}\) |
\(\Big \downarrow \) 2462 |
\(\displaystyle -\frac {3 \int \left (\frac {3-x}{x^3+3 x^2-1}+\frac {-8 x^2-x+3}{\left (x^3+3 x^2-1\right )^2}\right )dx}{\log ^2(4)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \left (\frac {i 2^{2/3} \sqrt [3]{1-i \sqrt {3}} \left (\frac {\sqrt {3} \left (2\ 2^{2/3} \left (2 i+\sqrt {3}\right )-\sqrt [3]{2-2 i \sqrt {3}} \left (5 i+\sqrt {3}\right )\right ) (x+1)}{\left (1-i \sqrt {3}\right )^{2/3}}+6\right )}{\sqrt {3} \left (2-\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}\right ) \left (x+\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}+\frac {1}{\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )}}+1\right ) \left (-2 (x+1)^2+\frac {\left (2 \sqrt [3]{2}+\left (2-2 i \sqrt {3}\right )^{2/3}\right ) (x+1)}{\sqrt [3]{1-i \sqrt {3}}}-\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}-\frac {2}{\left (\frac {1}{2} \left (1-i \sqrt {3}\right )\right )^{2/3}}+2\right )}-\frac {64 \left (i+\sqrt {3}\right ) \left (\left (2-2 i \sqrt {3}\right )^{2/3} \left (i+2 \sqrt {3}\right )+\sqrt [3]{2} \left (5 i+3 \sqrt {3}\right )\right ) \arctan \left (\frac {-4 i \left (1-i \sqrt {3}\right )^{2/3} (x+1)+2^{2/3} \left (i+\sqrt {3}\right )+2 i \sqrt [3]{2-2 i \sqrt {3}}}{2 \sqrt {3 \left (2 \left (1-i \sqrt {3}\right )^{4/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )-\left (2-2 i \sqrt {3}\right )^{2/3}\right )}}\right )}{\left (i-\sqrt {3}\right ) \left (2\ 2^{2/3}-4 \left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{4/3}\right ) \left (2\ 2^{2/3}+2 \left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{4/3}\right )^3 \sqrt {\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )^{4/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )-\left (2-2 i \sqrt {3}\right )^{2/3}\right )}}+\frac {2 \left (\left (1-2 i \sqrt {3}\right ) \left (2-2 i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (5-3 i \sqrt {3}\right )\right ) \text {arctanh}\left (\frac {-4 \left (1-i \sqrt {3}\right )^{2/3} (x+1)+2 \sqrt [3]{2-2 i \sqrt {3}}+2^{2/3} \left (1-i \sqrt {3}\right )}{2 \sqrt {3 \left (2 \left (1-i \sqrt {3}\right )^{4/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )-\left (2-2 i \sqrt {3}\right )^{2/3}\right )}}\right )}{\left (2\ 2^{2/3}+2 \left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{4/3}\right ) \sqrt {3 \left (2 \left (1-i \sqrt {3}\right )^{4/3}+\sqrt [3]{2} \left (1+i \sqrt {3}\right )-\left (2-2 i \sqrt {3}\right )^{2/3}\right )}}+\frac {\sqrt [3]{1-i \sqrt {3}} \left (2 \sqrt [3]{2}+8 \sqrt [3]{1-i \sqrt {3}}+\left (2-2 i \sqrt {3}\right )^{2/3}\right ) \log \left (2 \sqrt [3]{1-i \sqrt {3}} (x+1)+\left (2-2 i \sqrt {3}\right )^{2/3}+2 \sqrt [3]{2}\right )}{3 \left (2\ 2^{2/3}+2 \left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{4/3}\right )}-\frac {\left (16 i+2 i 2^{2/3} \sqrt [3]{1-i \sqrt {3}}+\sqrt [3]{2} \left (i-\sqrt {3}\right ) \left (1-i \sqrt {3}\right )^{2/3}\right ) \log \left (2 \sqrt [3]{1-i \sqrt {3}} (x+1)+\left (2-2 i \sqrt {3}\right )^{2/3}+2 \sqrt [3]{2}\right )}{3 \left (4 i+2^{2/3} \left (i-\sqrt {3}\right ) \sqrt [3]{1-i \sqrt {3}}+2 i \sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3}\right )}-\frac {\left (1-i \sqrt {3}\right )^{4/3} \left (8 \left (1-i \sqrt {3}\right )^{2/3}+2^{2/3} \left (1-i \sqrt {3}\right )+2 \sqrt [3]{2-2 i \sqrt {3}}\right ) \log \left (2 i \left (1-i \sqrt {3}\right )^{2/3} (x+1)^2-2^{2/3} \left (i+\sqrt {3}+i 2^{2/3} \sqrt [3]{1-i \sqrt {3}}\right ) (x+1)+\sqrt [3]{2-2 i \sqrt {3}} \left (i+\sqrt {3}\right )-2 i \left (1-i \sqrt {3}\right )^{2/3}+2 i 2^{2/3}\right )}{12 \left (2+2 i \sqrt {3}-2^{2/3} \left (1-i \sqrt {3}\right )^{4/3}+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{2/3} \left (1+i \sqrt {3}\right )\right )}-\frac {\sqrt [3]{\frac {1}{2} \left (1-i \sqrt {3}\right )} \left (2^{2/3}+\left (1-i \sqrt {3}\right )^{2/3}+4 \sqrt [3]{2-2 i \sqrt {3}}\right ) \log \left (2 \left (1-i \sqrt {3}\right )^{2/3} (x+1)^2-2^{2/3} \left (1-i \sqrt {3}+2^{2/3} \sqrt [3]{1-i \sqrt {3}}\right ) (x+1)+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{4/3}-2 \left (1-i \sqrt {3}\right )^{2/3}+2\ 2^{2/3}\right )}{3 \left (2\ 2^{2/3}+2 \left (1-i \sqrt {3}\right )^{2/3}+\sqrt [3]{2} \left (1-i \sqrt {3}\right )^{4/3}\right )}-\frac {8}{3 \left (-x^3-3 x^2+1\right )}+\frac {2 \sqrt [3]{1-i \sqrt {3}}}{2 \sqrt [3]{1-i \sqrt {3}} (x+1)+\left (2-2 i \sqrt {3}\right )^{2/3}+2 \sqrt [3]{2}}\right )}{\log ^2(4)}\) |
(-3*(-8/(3*(1 - 3*x^2 - x^3)) + (2*(1 - I*Sqrt[3])^(1/3))/(2*2^(1/3) + (2 - (2*I)*Sqrt[3])^(2/3) + 2*(1 - I*Sqrt[3])^(1/3)*(1 + x)) + (I*2^(2/3)*(1 - I*Sqrt[3])^(1/3)*(6 + (Sqrt[3]*(2*2^(2/3)*(2*I + Sqrt[3]) - (2 - (2*I)*S qrt[3])^(1/3)*(5*I + Sqrt[3]))*(1 + x))/(1 - I*Sqrt[3])^(2/3)))/(Sqrt[3]*( 2 - 2^(1/3)*(1 - I*Sqrt[3])^(2/3))*(1 + ((1 - I*Sqrt[3])/2)^(-1/3) + ((1 - I*Sqrt[3])/2)^(1/3) + x)*(2 - 2/((1 - I*Sqrt[3])/2)^(2/3) - 2^(1/3)*(1 - I*Sqrt[3])^(2/3) + ((2*2^(1/3) + (2 - (2*I)*Sqrt[3])^(2/3))*(1 + x))/(1 - I*Sqrt[3])^(1/3) - 2*(1 + x)^2)) - (64*(I + Sqrt[3])*((2 - (2*I)*Sqrt[3])^ (2/3)*(I + 2*Sqrt[3]) + 2^(1/3)*(5*I + 3*Sqrt[3]))*ArcTan[((2*I)*(2 - (2*I )*Sqrt[3])^(1/3) + 2^(2/3)*(I + Sqrt[3]) - (4*I)*(1 - I*Sqrt[3])^(2/3)*(1 + x))/(2*Sqrt[3*(2*(1 - I*Sqrt[3])^(4/3) + 2^(1/3)*(1 + I*Sqrt[3]) - (2 - (2*I)*Sqrt[3])^(2/3))])])/((I - Sqrt[3])*(2*2^(2/3) - 4*(1 - I*Sqrt[3])^(2 /3) + 2^(1/3)*(1 - I*Sqrt[3])^(4/3))*(2*2^(2/3) + 2*(1 - I*Sqrt[3])^(2/3) + 2^(1/3)*(1 - I*Sqrt[3])^(4/3))^3*Sqrt[(2*(1 - I*Sqrt[3])^(4/3) + 2^(1/3) *(1 + I*Sqrt[3]) - (2 - (2*I)*Sqrt[3])^(2/3))/3]) + (2*((1 - (2*I)*Sqrt[3] )*(2 - (2*I)*Sqrt[3])^(2/3) + 2^(1/3)*(5 - (3*I)*Sqrt[3]))*ArcTanh[(2^(2/3 )*(1 - I*Sqrt[3]) + 2*(2 - (2*I)*Sqrt[3])^(1/3) - 4*(1 - I*Sqrt[3])^(2/3)* (1 + x))/(2*Sqrt[3*(2*(1 - I*Sqrt[3])^(4/3) + 2^(1/3)*(1 + I*Sqrt[3]) - (2 - (2*I)*Sqrt[3])^(2/3))])])/((2*2^(2/3) + 2*(1 - I*Sqrt[3])^(2/3) + 2^(1/ 3)*(1 - I*Sqrt[3])^(4/3))*Sqrt[3*(2*(1 - I*Sqrt[3])^(4/3) + 2^(1/3)*(1 ...
3.5.59.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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] && GtQ [Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0 ] && RationalFunctionQ[u, x]
Time = 0.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.62
method | result | size |
gosper | \(-\frac {3 x^{2}-1}{4 \ln \left (2\right )^{2} \left (x^{3}+3 x^{2}-1\right )}\) | \(26\) |
default | \(\frac {-\frac {3 x^{2}}{4}+\frac {1}{4}}{\ln \left (2\right )^{2} \left (x^{3}+3 x^{2}-1\right )}\) | \(26\) |
risch | \(\frac {-3 x^{2}+1}{4 \ln \left (2\right )^{2} \left (x^{3}+3 x^{2}-1\right )}\) | \(26\) |
parallelrisch | \(\frac {-3 x^{2}+1}{4 \ln \left (2\right )^{2} \left (x^{3}+3 x^{2}-1\right )}\) | \(26\) |
norman | \(\frac {-\frac {3 x^{2}}{4 \ln \left (2\right )}+\frac {1}{4 \ln \left (2\right )}}{\left (x^{3}+3 x^{2}-1\right ) \ln \left (2\right )}\) | \(34\) |
Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {3 \, x^{2} - 1}{4 \, {\left (x^{3} + 3 \, x^{2} - 1\right )} \log \left (2\right )^{2}} \]
Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.17 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.00 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=\frac {1 - 3 x^{2}}{4 x^{3} \log {\left (2 \right )}^{2} + 12 x^{2} \log {\left (2 \right )}^{2} - 4 \log {\left (2 \right )}^{2}} \]
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {3 \, x^{2} - 1}{4 \, {\left (x^{3} + 3 \, x^{2} - 1\right )} \log \left (2\right )^{2}} \]
Time = 0.25 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {3 \, x^{2} - 1}{4 \, {\left (x^{3} + 3 \, x^{2} - 1\right )} \log \left (2\right )^{2}} \]
Time = 8.97 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {-3 x^2+3 x^4}{\left (1-6 x^2-2 x^3+9 x^4+6 x^5+x^6\right ) \log ^2(4)} \, dx=-\frac {3\,x^2-1}{4\,{\ln \left (2\right )}^2\,\left (x^3+3\,x^2-1\right )} \]