Integrand size = 17, antiderivative size = 91 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=\frac {x}{32 \left (1-x^2\right )}+\frac {x \left (99-17 x^2\right )}{128 \left (1-6 x^2+x^4\right )}+\frac {5 \text {arctanh}(x)}{32}+\frac {1}{512} \left (-4+3 \sqrt {2}\right ) \text {arctanh}\left (\left (-1+\sqrt {2}\right ) x\right )+\frac {1}{512} \left (4+3 \sqrt {2}\right ) \text {arctanh}\left (\left (1+\sqrt {2}\right ) x\right ) \]
1/32*x/(-x^2+1)+1/128*x*(-17*x^2+99)/(x^4-6*x^2+1)+5/32*arctanh(x)+1/512*a rctanh(x*(2^(1/2)-1))*(-4+3*2^(1/2))+1/512*arctanh(x*(1+2^(1/2)))*(4+3*2^( 1/2))
Time = 0.06 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=\frac {-\frac {8 x \left (103-140 x^2+21 x^4\right )}{-1+7 x^2-7 x^4+x^6}-80 \log (1-x)-\left (4+3 \sqrt {2}\right ) \log \left (-1+\sqrt {2}-x\right )+\left (4-3 \sqrt {2}\right ) \log \left (1+\sqrt {2}-x\right )+80 \log (1+x)+\left (4+3 \sqrt {2}\right ) \log \left (-1+\sqrt {2}+x\right )+\left (-4+3 \sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right )}{1024} \]
((-8*x*(103 - 140*x^2 + 21*x^4))/(-1 + 7*x^2 - 7*x^4 + x^6) - 80*Log[1 - x ] - (4 + 3*Sqrt[2])*Log[-1 + Sqrt[2] - x] + (4 - 3*Sqrt[2])*Log[1 + Sqrt[2 ] - x] + 80*Log[1 + x] + (4 + 3*Sqrt[2])*Log[-1 + Sqrt[2] + x] + (-4 + 3*S qrt[2])*Log[1 + Sqrt[2] + x])/1024
Leaf count is larger than twice the leaf count of optimal. \(245\) vs. \(2(91)=182\).
Time = 0.44 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.69, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2460, 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.
| \(\displaystyle \int \frac {1}{\left (x^6-7 x^4+7 x^2-1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {29-12 x}{64 \left (x^2-2 x-1\right )^2}-\frac {5}{32 \left (x^2-1\right )}+\frac {x+6}{128 \left (x^2-2 x-1\right )}+\frac {6-x}{128 \left (x^2+2 x-1\right )}+\frac {12 x+29}{64 \left (x^2+2 x-1\right )^2}+\frac {1}{64 (x-1)^2}+\frac {1}{64 (x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5 \text {arctanh}(x)}{32}-\frac {41-17 x}{256 \left (-x^2+2 x+1\right )}+\frac {17 x+41}{256 \left (-x^2-2 x+1\right )}+\frac {1}{64 (1-x)}-\frac {1}{64 (x+1)}+\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (-x-\sqrt {2}+1\right )+\frac {17 \log \left (-x-\sqrt {2}+1\right )}{512 \sqrt {2}}+\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (-x+\sqrt {2}+1\right )-\frac {17 \log \left (-x+\sqrt {2}+1\right )}{512 \sqrt {2}}-\frac {1}{512} \left (2-7 \sqrt {2}\right ) \log \left (x-\sqrt {2}+1\right )-\frac {17 \log \left (x-\sqrt {2}+1\right )}{512 \sqrt {2}}-\frac {1}{512} \left (2+7 \sqrt {2}\right ) \log \left (x+\sqrt {2}+1\right )+\frac {17 \log \left (x+\sqrt {2}+1\right )}{512 \sqrt {2}}\) |
1/(64*(1 - x)) - 1/(64*(1 + x)) + (41 + 17*x)/(256*(1 - 2*x - x^2)) - (41 - 17*x)/(256*(1 + 2*x - x^2)) + (5*ArcTanh[x])/32 + (17*Log[1 - Sqrt[2] - x])/(512*Sqrt[2]) + ((2 - 7*Sqrt[2])*Log[1 - Sqrt[2] - x])/512 - (17*Log[1 + Sqrt[2] - x])/(512*Sqrt[2]) + ((2 + 7*Sqrt[2])*Log[1 + Sqrt[2] - x])/51 2 - (17*Log[1 - Sqrt[2] + x])/(512*Sqrt[2]) - ((2 - 7*Sqrt[2])*Log[1 - Sqr t[2] + x])/512 + (17*Log[1 + Sqrt[2] + x])/(512*Sqrt[2]) - ((2 + 7*Sqrt[2] )*Log[1 + Sqrt[2] + x])/512
3.1.77.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px /. x -> Sqrt[x]]}, Int[ExpandIntegrand[u*(Qx /. x -> x^2)^p, x], x] /; !SumQ[NonfreeFactors[Q x, x]]] /; PolyQ[Px, x^2] && GtQ[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0] && RationalFunctionQ[u, x]
Time = 0.13 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.27
| method | result | size |
| default | \(-\frac {1}{64 \left (x +1\right )}+\frac {5 \ln \left (x +1\right )}{64}-\frac {\frac {17 x}{2}+\frac {41}{2}}{128 \left (x^{2}+2 x -1\right )}-\frac {\ln \left (x^{2}+2 x -1\right )}{256}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4}\right )}{512}-\frac {1}{64 \left (x -1\right )}-\frac {5 \ln \left (x -1\right )}{64}+\frac {-\frac {17 x}{2}+\frac {41}{2}}{128 x^{2}-256 x -128}+\frac {\ln \left (x^{2}-2 x -1\right )}{256}+\frac {3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x -2\right ) \sqrt {2}}{4}\right )}{512}\) | \(116\) |
| risch | \(\frac {-\frac {21}{128} x^{5}+\frac {35}{32} x^{3}-\frac {103}{128} x}{x^{6}-7 x^{4}+7 x^{2}-1}+\frac {\ln \left (x -1+\sqrt {2}\right )}{256}+\frac {3 \ln \left (x -1+\sqrt {2}\right ) \sqrt {2}}{1024}+\frac {\ln \left (x -1-\sqrt {2}\right )}{256}-\frac {3 \ln \left (x -1-\sqrt {2}\right ) \sqrt {2}}{1024}-\frac {\ln \left (1+x +\sqrt {2}\right )}{256}+\frac {3 \ln \left (1+x +\sqrt {2}\right ) \sqrt {2}}{1024}-\frac {\ln \left (1+x -\sqrt {2}\right )}{256}-\frac {3 \ln \left (1+x -\sqrt {2}\right ) \sqrt {2}}{1024}+\frac {5 \ln \left (x +1\right )}{64}-\frac {5 \ln \left (x -1\right )}{64}\) | \(138\) |
-1/64/(x+1)+5/64*ln(x+1)-1/128*(17/2*x+41/2)/(x^2+2*x-1)-1/256*ln(x^2+2*x- 1)+3/512*2^(1/2)*arctanh(1/4*(2*x+2)*2^(1/2))-1/64/(x-1)-5/64*ln(x-1)+1/12 8*(-17/2*x+41/2)/(x^2-2*x-1)+1/256*ln(x^2-2*x-1)+3/512*2^(1/2)*arctanh(1/4 *(2*x-2)*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (71) = 142\).
Time = 0.28 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.45 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {168 \, x^{5} - 1120 \, x^{3} - 3 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} + 2 \, \sqrt {2} {\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) - 3 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} + 2 \, \sqrt {2} {\left (x - 1\right )} - 2 \, x + 3}{x^{2} - 2 \, x - 1}\right ) + 4 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) - 4 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 80 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 80 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 824 \, x}{1024 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \]
-1/1024*(168*x^5 - 1120*x^3 - 3*sqrt(2)*(x^6 - 7*x^4 + 7*x^2 - 1)*log((x^2 + 2*sqrt(2)*(x + 1) + 2*x + 3)/(x^2 + 2*x - 1)) - 3*sqrt(2)*(x^6 - 7*x^4 + 7*x^2 - 1)*log((x^2 + 2*sqrt(2)*(x - 1) - 2*x + 3)/(x^2 - 2*x - 1)) + 4* (x^6 - 7*x^4 + 7*x^2 - 1)*log(x^2 + 2*x - 1) - 4*(x^6 - 7*x^4 + 7*x^2 - 1) *log(x^2 - 2*x - 1) - 80*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x + 1) + 80*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x - 1) + 824*x)/(x^6 - 7*x^4 + 7*x^2 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 296 vs. \(2 (75) = 150\).
Time = 0.81 (sec) , antiderivative size = 296, normalized size of antiderivative = 3.25 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=\frac {- 21 x^{5} + 140 x^{3} - 103 x}{128 x^{6} - 896 x^{4} + 896 x^{2} - 128} - \frac {5 \log {\left (x - 1 \right )}}{64} + \frac {5 \log {\left (x + 1 \right )}}{64} + \left (- \frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right ) \log {\left (x - \frac {8071264001}{202624020} - \frac {471550901878784 \left (- \frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{3}}{2979765} + \frac {1299552375287054336 \left (- \frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{5}}{50656005} + \frac {8071264001 \sqrt {2}}{270165360} \right )} + \left (- \frac {3 \sqrt {2}}{1024} - \frac {1}{256}\right ) \log {\left (x - \frac {8071264001 \sqrt {2}}{270165360} - \frac {8071264001}{202624020} + \frac {1299552375287054336 \left (- \frac {3 \sqrt {2}}{1024} - \frac {1}{256}\right )^{5}}{50656005} - \frac {471550901878784 \left (- \frac {3 \sqrt {2}}{1024} - \frac {1}{256}\right )^{3}}{2979765} \right )} + \left (\frac {1}{256} - \frac {3 \sqrt {2}}{1024}\right ) \log {\left (x - \frac {8071264001 \sqrt {2}}{270165360} + \frac {1299552375287054336 \left (\frac {1}{256} - \frac {3 \sqrt {2}}{1024}\right )^{5}}{50656005} - \frac {471550901878784 \left (\frac {1}{256} - \frac {3 \sqrt {2}}{1024}\right )^{3}}{2979765} + \frac {8071264001}{202624020} \right )} + \left (\frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right ) \log {\left (x - \frac {471550901878784 \left (\frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{3}}{2979765} + \frac {1299552375287054336 \left (\frac {1}{256} + \frac {3 \sqrt {2}}{1024}\right )^{5}}{50656005} + \frac {8071264001}{202624020} + \frac {8071264001 \sqrt {2}}{270165360} \right )} \]
(-21*x**5 + 140*x**3 - 103*x)/(128*x**6 - 896*x**4 + 896*x**2 - 128) - 5*l og(x - 1)/64 + 5*log(x + 1)/64 + (-1/256 + 3*sqrt(2)/1024)*log(x - 8071264 001/202624020 - 471550901878784*(-1/256 + 3*sqrt(2)/1024)**3/2979765 + 129 9552375287054336*(-1/256 + 3*sqrt(2)/1024)**5/50656005 + 8071264001*sqrt(2 )/270165360) + (-3*sqrt(2)/1024 - 1/256)*log(x - 8071264001*sqrt(2)/270165 360 - 8071264001/202624020 + 1299552375287054336*(-3*sqrt(2)/1024 - 1/256) **5/50656005 - 471550901878784*(-3*sqrt(2)/1024 - 1/256)**3/2979765) + (1/ 256 - 3*sqrt(2)/1024)*log(x - 8071264001*sqrt(2)/270165360 + 1299552375287 054336*(1/256 - 3*sqrt(2)/1024)**5/50656005 - 471550901878784*(1/256 - 3*s qrt(2)/1024)**3/2979765 + 8071264001/202624020) + (1/256 + 3*sqrt(2)/1024) *log(x - 471550901878784*(1/256 + 3*sqrt(2)/1024)**3/2979765 + 12995523752 87054336*(1/256 + 3*sqrt(2)/1024)**5/50656005 + 8071264001/202624020 + 807 1264001*sqrt(2)/270165360)
Time = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {3}{1024} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} + 1}{x + \sqrt {2} + 1}\right ) - \frac {3}{1024} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} - 1}{x + \sqrt {2} - 1}\right ) - \frac {21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac {1}{256} \, \log \left (x^{2} + 2 \, x - 1\right ) + \frac {1}{256} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac {5}{64} \, \log \left (x + 1\right ) - \frac {5}{64} \, \log \left (x - 1\right ) \]
-3/1024*sqrt(2)*log((x - sqrt(2) + 1)/(x + sqrt(2) + 1)) - 3/1024*sqrt(2)* log((x - sqrt(2) - 1)/(x + sqrt(2) - 1)) - 1/128*(21*x^5 - 140*x^3 + 103*x )/(x^6 - 7*x^4 + 7*x^2 - 1) - 1/256*log(x^2 + 2*x - 1) + 1/256*log(x^2 - 2 *x - 1) + 5/64*log(x + 1) - 5/64*log(x - 1)
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {3}{1024} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + 2 \right |}}\right ) - \frac {3}{1024} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} - 2 \right |}}\right ) - \frac {21 \, x^{5} - 140 \, x^{3} + 103 \, x}{128 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} - \frac {1}{256} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) + \frac {1}{256} \, \log \left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac {5}{64} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {5}{64} \, \log \left ({\left | x - 1 \right |}\right ) \]
-3/1024*sqrt(2)*log(abs(2*x - 2*sqrt(2) + 2)/abs(2*x + 2*sqrt(2) + 2)) - 3 /1024*sqrt(2)*log(abs(2*x - 2*sqrt(2) - 2)/abs(2*x + 2*sqrt(2) - 2)) - 1/1 28*(21*x^5 - 140*x^3 + 103*x)/(x^6 - 7*x^4 + 7*x^2 - 1) - 1/256*log(abs(x^ 2 + 2*x - 1)) + 1/256*log(abs(x^2 - 2*x - 1)) + 5/64*log(abs(x + 1)) - 5/6 4*log(abs(x - 1))
Time = 10.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.38 \[ \int \frac {1}{\left (-1+7 x^2-7 x^4+x^6\right )^2} \, dx=-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,5{}\mathrm {i}}{32}-\frac {\frac {21\,x^5}{128}-\frac {35\,x^3}{32}+\frac {103\,x}{128}}{x^6-7\,x^4+7\,x^2-1}-\mathrm {atan}\left (\frac {x\,940311{}\mathrm {i}}{134217728\,\left (\frac {275445\,\sqrt {2}}{134217728}-\frac {389421}{134217728}\right )}-\frac {\sqrt {2}\,x\,332433{}\mathrm {i}}{67108864\,\left (\frac {275445\,\sqrt {2}}{134217728}-\frac {389421}{134217728}\right )}\right )\,\left (\frac {\sqrt {2}\,3{}\mathrm {i}}{512}-\frac {1}{128}{}\mathrm {i}\right )-\mathrm {atan}\left (\frac {x\,940311{}\mathrm {i}}{134217728\,\left (\frac {275445\,\sqrt {2}}{134217728}+\frac {389421}{134217728}\right )}+\frac {\sqrt {2}\,x\,332433{}\mathrm {i}}{67108864\,\left (\frac {275445\,\sqrt {2}}{134217728}+\frac {389421}{134217728}\right )}\right )\,\left (\frac {\sqrt {2}\,3{}\mathrm {i}}{512}+\frac {1}{128}{}\mathrm {i}\right ) \]
- (atan(x*1i)*5i)/32 - ((103*x)/128 - (35*x^3)/32 + (21*x^5)/128)/(7*x^2 - 7*x^4 + x^6 - 1) - atan((x*940311i)/(134217728*((275445*2^(1/2))/13421772 8 - 389421/134217728)) - (2^(1/2)*x*332433i)/(67108864*((275445*2^(1/2))/1 34217728 - 389421/134217728)))*((2^(1/2)*3i)/512 - 1i/128) - atan((x*94031 1i)/(134217728*((275445*2^(1/2))/134217728 + 389421/134217728)) + (2^(1/2) *x*332433i)/(67108864*((275445*2^(1/2))/134217728 + 389421/134217728)))*(( 2^(1/2)*3i)/512 + 1i/128)