Integrand size = 118, antiderivative size = 38 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=x-\log (4)+\frac {2}{\frac {1}{2}-25 x^2+5 \left (1+i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )} \]
Time = 0.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=x-\frac {4}{-11-10 i \pi +50 x^2-10 \log (-\log (\log (2)))} \]
Integrate[(121 + 400*x - 1100*x^2 + 2500*x^4 + (220 - 1000*x^2)*(I*Pi + Lo g[-Log[Log[4]/2]]) + 100*(I*Pi + Log[-Log[Log[4]/2]])^2)/(121 - 1100*x^2 + 2500*x^4 + (220 - 1000*x^2)*(I*Pi + Log[-Log[Log[4]/2]]) + 100*(I*Pi + Lo g[-Log[Log[4]/2]])^2),x]
Time = 0.44 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.68, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {2454, 1380, 27, 2345, 24}
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.
\(\displaystyle \int \frac {2500 x^4-1100 x^2+\left (220-1000 x^2\right ) \left (\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )+i \pi \right )+400 x+121+100 \left (\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )+i \pi \right )^2}{2500 x^4-1100 x^2+\left (220-1000 x^2\right ) \left (\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )+i \pi \right )+121+100 \left (\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )+i \pi \right )^2} \, dx\) |
\(\Big \downarrow \) 2454 |
\(\displaystyle \int \frac {2500 x^4-1100 x^2+\left (220-1000 x^2\right ) \left (\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )+i \pi \right )+400 x+121+100 \left (\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )+i \pi \right )^2}{2500 x^4-100 x^2 (11+10 i \pi +10 \log (-\log (\log (2))))-(11 i-10 \pi +10 i \log (-\log (\log (2))))^2}dx\) |
\(\Big \downarrow \) 1380 |
\(\displaystyle 2500 \int \frac {2500 x^4-1100 x^2+400 x+100 (i \pi +\log (-\log (\log (2))))^2+20 \left (11-50 x^2\right ) (i \pi +\log (-\log (\log (2))))+121}{2500 \left (-50 x^2+10 \log (-\log (\log (2)))+10 i \pi +11\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {2500 x^4-1100 x^2+20 \left (11-50 x^2\right ) (\log (-\log (\log (2)))+i \pi )+400 x+121+100 (\log (-\log (\log (2)))+i \pi )^2}{\left (-50 x^2+10 i \pi +11+10 \log (-\log (\log (2)))\right )^2}dx\) |
\(\Big \downarrow \) 2345 |
\(\displaystyle \frac {4}{-50 x^2+10 i \pi +11+10 \log (-\log (\log (2)))}-\frac {\int -2 (11+10 i \pi +10 \log (-\log (\log (2))))dx}{2 (11+10 i \pi +10 \log (-\log (\log (2))))}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle x+\frac {4}{-50 x^2+10 i \pi +11+10 \log (-\log (\log (2)))}\) |
Int[(121 + 400*x - 1100*x^2 + 2500*x^4 + (220 - 1000*x^2)*(I*Pi + Log[-Log [Log[4]/2]]) + 100*(I*Pi + Log[-Log[Log[4]/2]])^2)/(121 - 1100*x^2 + 2500* x^4 + (220 - 1000*x^2)*(I*Pi + Log[-Log[Log[4]/2]]) + 100*(I*Pi + Log[-Log [Log[4]/2]])^2),x]
3.29.10.3.1 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[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Int[(Pq_)*(u_)^(p_.), x_Symbol] :> Int[Pq*ExpandToSum[u, x]^p, x] /; FreeQ[ p, x] && PolyQ[Pq, x] && TrinomialQ[u, x] && !TrinomialMatchQ[u, x]
Time = 0.70 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.47
method | result | size |
risch | \(x +\frac {2}{5 \left (-5 x^{2}+\ln \left (\ln \left (\ln \left (2\right )\right )\right )+\frac {11}{10}\right )}\) | \(18\) |
gosper | \(\frac {-50 x^{3}+10 x \ln \left (\ln \left (\ln \left (2\right )\right )\right )+11 x +4}{-50 x^{2}+10 \ln \left (\ln \left (\ln \left (2\right )\right )\right )+11}\) | \(34\) |
norman | \(\frac {-50 x^{3}+4+\left (10 \ln \left (\ln \left (\ln \left (2\right )\right )\right )+11\right ) x}{-50 x^{2}+10 \ln \left (\ln \left (\ln \left (2\right )\right )\right )+11}\) | \(34\) |
parallelrisch | \(\frac {-2500 x^{3}+200+500 x \ln \left (\ln \left (\ln \left (2\right )\right )\right )+550 x}{-2500 x^{2}+500 \ln \left (\ln \left (\ln \left (2\right )\right )\right )+550}\) | \(35\) |
default | \(x +\frac {8800+8000 \ln \left (\ln \left (\ln \left (2\right )\right )\right )}{\left (-2000 \ln \left (\ln \left (\ln \left (2\right )\right )\right )-2200\right ) \left (50 x^{2}-10 \ln \left (\ln \left (\ln \left (2\right )\right )\right )-11\right )}\) | \(38\) |
int((100*ln(ln(ln(2)))^2+(-1000*x^2+220)*ln(ln(ln(2)))+2500*x^4-1100*x^2+4 00*x+121)/(100*ln(ln(ln(2)))^2+(-1000*x^2+220)*ln(ln(ln(2)))+2500*x^4-1100 *x^2+121),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.87 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=\frac {50 \, x^{3} - 10 \, x \log \left (\log \left (\log \left (2\right )\right )\right ) - 11 \, x - 4}{50 \, x^{2} - 10 \, \log \left (\log \left (\log \left (2\right )\right )\right ) - 11} \]
integrate((100*log(log(log(2)))^2+(-1000*x^2+220)*log(log(log(2)))+2500*x^ 4-1100*x^2+400*x+121)/(100*log(log(log(2)))^2+(-1000*x^2+220)*log(log(log( 2)))+2500*x^4-1100*x^2+121),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.63 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=x - \frac {4}{50 x^{2} - 11 - 10 \log {\left (- \log {\left (\log {\left (2 \right )} \right )} \right )} - 10 i \pi } \]
integrate((100*ln(ln(ln(2)))**2+(-1000*x**2+220)*ln(ln(ln(2)))+2500*x**4-1 100*x**2+400*x+121)/(100*ln(ln(ln(2)))**2+(-1000*x**2+220)*ln(ln(ln(2)))+2 500*x**4-1100*x**2+121),x)
Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=x - \frac {4}{50 \, x^{2} - 10 \, \log \left (\log \left (\log \left (2\right )\right )\right ) - 11} \]
integrate((100*log(log(log(2)))^2+(-1000*x^2+220)*log(log(log(2)))+2500*x^ 4-1100*x^2+400*x+121)/(100*log(log(log(2)))^2+(-1000*x^2+220)*log(log(log( 2)))+2500*x^4-1100*x^2+121),x, algorithm=\
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=x - \frac {4}{50 \, x^{2} - 10 \, \log \left (\log \left (\log \left (2\right )\right )\right ) - 11} \]
integrate((100*log(log(log(2)))^2+(-1000*x^2+220)*log(log(log(2)))+2500*x^ 4-1100*x^2+400*x+121)/(100*log(log(log(2)))^2+(-1000*x^2+220)*log(log(log( 2)))+2500*x^4-1100*x^2+121),x, algorithm=\
Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.50 \[ \int \frac {121+400 x-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2}{121-1100 x^2+2500 x^4+\left (220-1000 x^2\right ) \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )+100 \left (i \pi +\log \left (-\log \left (\frac {\log (4)}{2}\right )\right )\right )^2} \, dx=x+\frac {4}{-50\,x^2+10\,\ln \left (\ln \left (\ln \left (2\right )\right )\right )+11} \]