Integrand size = 25, antiderivative size = 111 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {x}{16 \left (1-x^2\right )}+\frac {x \left (29-5 x^2\right )}{32 \left (1-6 x^2+x^4\right )}+\frac {\text {arctanh}(x)}{4}+\frac {\text {arctanh}\left (\sqrt {3-2 \sqrt {2}} x\right )}{64 \sqrt {17+12 \sqrt {2}}}-\frac {\text {arctanh}\left (\sqrt {3+2 \sqrt {2}} x\right )}{64 \sqrt {17-12 \sqrt {2}}} \] Output:
x/(-16*x^2+16)+x*(-5*x^2+29)/(32*x^4-192*x^2+32)+1/4*arctanh(x)+1/64*arcta nh((2^(1/2)-1)*x)/(3+2*2^(1/2))-1/64*arctanh((1+2^(1/2))*x)/(3-2*2^(1/2))
Time = 0.10 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.19 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {1}{128} \left (-\frac {4 x \left (31-46 x^2+7 x^4\right )}{-1+7 x^2-7 x^4+x^6}-16 \log (1-x)+\left (3+2 \sqrt {2}\right ) \log \left (-1+\sqrt {2}-x\right )+\left (-3+2 \sqrt {2}\right ) \log \left (1+\sqrt {2}-x\right )+16 \log (1+x)-\left (3+2 \sqrt {2}\right ) \log \left (-1+\sqrt {2}+x\right )+\left (3-2 \sqrt {2}\right ) \log \left (1+\sqrt {2}+x\right )\right ) \] Input:
Integrate[(1 + x^2)/(1 - 7*x^2 + 7*x^4 - x^6)^2,x]
Output:
((-4*x*(31 - 46*x^2 + 7*x^4))/(-1 + 7*x^2 - 7*x^4 + x^6) - 16*Log[1 - x] + (3 + 2*Sqrt[2])*Log[-1 + Sqrt[2] - x] + (-3 + 2*Sqrt[2])*Log[1 + Sqrt[2] - x] + 16*Log[1 + x] - (3 + 2*Sqrt[2])*Log[-1 + Sqrt[2] + x] + (3 - 2*Sqrt [2])*Log[1 + Sqrt[2] + x])/128
Leaf count is larger than twice the leaf count of optimal. \(245\) vs. \(2(111)=222\).
Time = 0.72 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.
| \(\displaystyle \int \frac {x^2+1}{\left (-x^6+7 x^4-7 x^2+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {17-7 x}{32 \left (x^2-2 x-1\right )^2}-\frac {1}{4 \left (x^2-1\right )}-\frac {3 (x-4)}{64 \left (x^2-2 x-1\right )}+\frac {3 (x+4)}{64 \left (x^2+2 x-1\right )}+\frac {7 x+17}{32 \left (x^2+2 x-1\right )^2}+\frac {1}{32 (x-1)^2}+\frac {1}{32 (x+1)^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\text {arctanh}(x)}{4}-\frac {12-5 x}{64 \left (-x^2+2 x+1\right )}+\frac {5 x+12}{64 \left (-x^2-2 x+1\right )}+\frac {1}{32 (1-x)}-\frac {1}{32 (x+1)}-\frac {3}{256} \left (2+3 \sqrt {2}\right ) \log \left (-x-\sqrt {2}+1\right )+\frac {5 \log \left (-x-\sqrt {2}+1\right )}{128 \sqrt {2}}-\frac {3}{256} \left (2-3 \sqrt {2}\right ) \log \left (-x+\sqrt {2}+1\right )-\frac {5 \log \left (-x+\sqrt {2}+1\right )}{128 \sqrt {2}}+\frac {3}{256} \left (2+3 \sqrt {2}\right ) \log \left (x-\sqrt {2}+1\right )-\frac {5 \log \left (x-\sqrt {2}+1\right )}{128 \sqrt {2}}+\frac {3}{256} \left (2-3 \sqrt {2}\right ) \log \left (x+\sqrt {2}+1\right )+\frac {5 \log \left (x+\sqrt {2}+1\right )}{128 \sqrt {2}}\) |
Input:
Int[(1 + x^2)/(1 - 7*x^2 + 7*x^4 - x^6)^2,x]
Output:
1/(32*(1 - x)) - 1/(32*(1 + x)) + (12 + 5*x)/(64*(1 - 2*x - x^2)) - (12 - 5*x)/(64*(1 + 2*x - x^2)) + ArcTanh[x]/4 + (5*Log[1 - Sqrt[2] - x])/(128*S qrt[2]) - (3*(2 + 3*Sqrt[2])*Log[1 - Sqrt[2] - x])/256 - (5*Log[1 + Sqrt[2 ] - x])/(128*Sqrt[2]) - (3*(2 - 3*Sqrt[2])*Log[1 + Sqrt[2] - x])/256 - (5* Log[1 - Sqrt[2] + x])/(128*Sqrt[2]) + (3*(2 + 3*Sqrt[2])*Log[1 - Sqrt[2] + x])/256 + (5*Log[1 + Sqrt[2] + x])/(128*Sqrt[2]) + (3*(2 - 3*Sqrt[2])*Log [1 + Sqrt[2] + x])/256
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 = 0.08 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(-\frac {1}{32 \left (x -1\right )}-\frac {\ln \left (x -1\right )}{8}+\frac {-5 x -12}{64 x^{2}+128 x -64}+\frac {3 \ln \left (x^{2}+2 x -1\right )}{128}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x +2\right ) \sqrt {2}}{4}\right )}{32}-\frac {1}{32 \left (x +1\right )}+\frac {\ln \left (x +1\right )}{8}-\frac {5 x -12}{64 \left (x^{2}-2 x -1\right )}-\frac {3 \ln \left (x^{2}-2 x -1\right )}{128}-\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (2 x -2\right ) \sqrt {2}}{4}\right )}{32}\) | \(116\) |
| risch | \(\frac {-\frac {7}{32} x^{5}+\frac {23}{16} x^{3}-\frac {31}{32} x}{x^{6}-7 x^{4}+7 x^{2}-1}+\frac {3 \ln \left (1-\sqrt {2}+x \right )}{128}+\frac {\ln \left (1-\sqrt {2}+x \right ) \sqrt {2}}{64}+\frac {3 \ln \left (1+\sqrt {2}+x \right )}{128}-\frac {\ln \left (1+\sqrt {2}+x \right ) \sqrt {2}}{64}+\frac {\ln \left (x +1\right )}{8}-\frac {3 \ln \left (2 x -2-2 \sqrt {2}\right )}{128}+\frac {\ln \left (2 x -2-2 \sqrt {2}\right ) \sqrt {2}}{64}-\frac {3 \ln \left (2 x -2+2 \sqrt {2}\right )}{128}-\frac {\ln \left (2 x -2+2 \sqrt {2}\right ) \sqrt {2}}{64}-\frac {\ln \left (x -1\right )}{8}\) | \(150\) |
Input:
int((x^2+1)/(-x^6+7*x^4-7*x^2+1)^2,x,method=_RETURNVERBOSE)
Output:
-1/32/(x-1)-1/8*ln(x-1)+1/64*(-5*x-12)/(x^2+2*x-1)+3/128*ln(x^2+2*x-1)-1/3 2*2^(1/2)*arctanh(1/4*(2*x+2)*2^(1/2))-1/32/(x+1)+1/8*ln(x+1)-1/64*(5*x-12 )/(x^2-2*x-1)-3/128*ln(x^2-2*x-1)-1/32*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. \(2 (75) = 150\).
Time = 0.08 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.01 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=-\frac {28 \, x^{5} - 184 \, x^{3} - 2 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} - 2 \, \sqrt {2} {\left (x + 1\right )} + 2 \, x + 3}{x^{2} + 2 \, x - 1}\right ) - 2 \, \sqrt {2} {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (\frac {x^{2} - 2 \, \sqrt {2} {\left (x - 1\right )} - 2 \, x + 3}{x^{2} - 2 \, x - 1}\right ) - 3 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} + 2 \, x - 1\right ) + 3 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x^{2} - 2 \, x - 1\right ) - 16 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x + 1\right ) + 16 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )} \log \left (x - 1\right ) + 124 \, x}{128 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} \] Input:
integrate((x^2+1)/(-x^6+7*x^4-7*x^2+1)^2,x, algorithm="fricas")
Output:
-1/128*(28*x^5 - 184*x^3 - 2*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)) - 2*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*(x^ 6 - 7*x^4 + 7*x^2 - 1)*log(x^2 + 2*x - 1) + 3*(x^6 - 7*x^4 + 7*x^2 - 1)*lo g(x^2 - 2*x - 1) - 16*(x^6 - 7*x^4 + 7*x^2 - 1)*log(x + 1) + 16*(x^6 - 7*x ^4 + 7*x^2 - 1)*log(x - 1) + 124*x)/(x^6 - 7*x^4 + 7*x^2 - 1)
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (70) = 140\).
Time = 0.93 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.45 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {- 7 x^{5} + 46 x^{3} - 31 x}{32 x^{6} - 224 x^{4} + 224 x^{2} - 32} - \frac {\log {\left (x - 1 \right )}}{8} + \frac {\log {\left (x + 1 \right )}}{8} + \left (- \frac {3}{128} - \frac {\sqrt {2}}{64}\right ) \log {\left (x - \frac {38423555}{909328} - \frac {38423555 \sqrt {2}}{1363992} + \frac {9549859782656 \left (- \frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{5}}{170499} - \frac {56267374592 \left (- \frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{3}}{56833} \right )} + \left (- \frac {3}{128} + \frac {\sqrt {2}}{64}\right ) \log {\left (x - \frac {38423555}{909328} + \frac {9549859782656 \left (- \frac {3}{128} + \frac {\sqrt {2}}{64}\right )^{5}}{170499} - \frac {56267374592 \left (- \frac {3}{128} + \frac {\sqrt {2}}{64}\right )^{3}}{56833} + \frac {38423555 \sqrt {2}}{1363992} \right )} + \left (\frac {3}{128} - \frac {\sqrt {2}}{64}\right ) \log {\left (x - \frac {38423555 \sqrt {2}}{1363992} - \frac {56267374592 \left (\frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{3}}{56833} + \frac {9549859782656 \left (\frac {3}{128} - \frac {\sqrt {2}}{64}\right )^{5}}{170499} + \frac {38423555}{909328} \right )} + \left (\frac {\sqrt {2}}{64} + \frac {3}{128}\right ) \log {\left (x - \frac {56267374592 \left (\frac {\sqrt {2}}{64} + \frac {3}{128}\right )^{3}}{56833} + \frac {9549859782656 \left (\frac {\sqrt {2}}{64} + \frac {3}{128}\right )^{5}}{170499} + \frac {38423555 \sqrt {2}}{1363992} + \frac {38423555}{909328} \right )} \] Input:
integrate((x**2+1)/(-x**6+7*x**4-7*x**2+1)**2,x)
Output:
(-7*x**5 + 46*x**3 - 31*x)/(32*x**6 - 224*x**4 + 224*x**2 - 32) - log(x - 1)/8 + log(x + 1)/8 + (-3/128 - sqrt(2)/64)*log(x - 38423555/909328 - 3842 3555*sqrt(2)/1363992 + 9549859782656*(-3/128 - sqrt(2)/64)**5/170499 - 562 67374592*(-3/128 - sqrt(2)/64)**3/56833) + (-3/128 + sqrt(2)/64)*log(x - 3 8423555/909328 + 9549859782656*(-3/128 + sqrt(2)/64)**5/170499 - 562673745 92*(-3/128 + sqrt(2)/64)**3/56833 + 38423555*sqrt(2)/1363992) + (3/128 - s qrt(2)/64)*log(x - 38423555*sqrt(2)/1363992 - 56267374592*(3/128 - sqrt(2) /64)**3/56833 + 9549859782656*(3/128 - sqrt(2)/64)**5/170499 + 38423555/90 9328) + (sqrt(2)/64 + 3/128)*log(x - 56267374592*(sqrt(2)/64 + 3/128)**3/5 6833 + 9549859782656*(sqrt(2)/64 + 3/128)**5/170499 + 38423555*sqrt(2)/136 3992 + 38423555/909328)
Time = 0.11 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.03 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {1}{64} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} + 1}{x + \sqrt {2} + 1}\right ) + \frac {1}{64} \, \sqrt {2} \log \left (\frac {x - \sqrt {2} - 1}{x + \sqrt {2} - 1}\right ) - \frac {7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac {3}{128} \, \log \left (x^{2} + 2 \, x - 1\right ) - \frac {3}{128} \, \log \left (x^{2} - 2 \, x - 1\right ) + \frac {1}{8} \, \log \left (x + 1\right ) - \frac {1}{8} \, \log \left (x - 1\right ) \] Input:
integrate((x^2+1)/(-x^6+7*x^4-7*x^2+1)^2,x, algorithm="maxima")
Output:
1/64*sqrt(2)*log((x - sqrt(2) + 1)/(x + sqrt(2) + 1)) + 1/64*sqrt(2)*log(( x - sqrt(2) - 1)/(x + sqrt(2) - 1)) - 1/32*(7*x^5 - 46*x^3 + 31*x)/(x^6 - 7*x^4 + 7*x^2 - 1) + 3/128*log(x^2 + 2*x - 1) - 3/128*log(x^2 - 2*x - 1) + 1/8*log(x + 1) - 1/8*log(x - 1)
Time = 0.14 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=\frac {1}{64} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} + 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} + 2 \right |}}\right ) + \frac {1}{64} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} - 2 \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} - 2 \right |}}\right ) - \frac {7 \, x^{5} - 46 \, x^{3} + 31 \, x}{32 \, {\left (x^{6} - 7 \, x^{4} + 7 \, x^{2} - 1\right )}} + \frac {3}{128} \, \log \left ({\left | x^{2} + 2 \, x - 1 \right |}\right ) - \frac {3}{128} \, \log \left ({\left | x^{2} - 2 \, x - 1 \right |}\right ) + \frac {1}{8} \, \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{8} \, \log \left ({\left | x - 1 \right |}\right ) \] Input:
integrate((x^2+1)/(-x^6+7*x^4-7*x^2+1)^2,x, algorithm="giac")
Output:
1/64*sqrt(2)*log(abs(2*x - 2*sqrt(2) + 2)/abs(2*x + 2*sqrt(2) + 2)) + 1/64 *sqrt(2)*log(abs(2*x - 2*sqrt(2) - 2)/abs(2*x + 2*sqrt(2) - 2)) - 1/32*(7* x^5 - 46*x^3 + 31*x)/(x^6 - 7*x^4 + 7*x^2 - 1) + 3/128*log(abs(x^2 + 2*x - 1)) - 3/128*log(abs(x^2 - 2*x - 1)) + 1/8*log(abs(x + 1)) - 1/8*log(abs(x - 1))
Time = 0.13 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.12 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx=-\frac {\mathrm {atan}\left (x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4}-\frac {\frac {7\,x^5}{32}-\frac {23\,x^3}{16}+\frac {31\,x}{32}}{x^6-7\,x^4+7\,x^2-1}+\mathrm {atan}\left (\frac {x\,23313{}\mathrm {i}}{8192\,\left (\frac {27309\,\sqrt {2}}{32768}-\frac {19317}{16384}\right )}-\frac {\sqrt {2}\,x\,65943{}\mathrm {i}}{32768\,\left (\frac {27309\,\sqrt {2}}{32768}-\frac {19317}{16384}\right )}\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{32}-\frac {3}{64}{}\mathrm {i}\right )+\mathrm {atan}\left (\frac {x\,23313{}\mathrm {i}}{8192\,\left (\frac {27309\,\sqrt {2}}{32768}+\frac {19317}{16384}\right )}+\frac {\sqrt {2}\,x\,65943{}\mathrm {i}}{32768\,\left (\frac {27309\,\sqrt {2}}{32768}+\frac {19317}{16384}\right )}\right )\,\left (\frac {\sqrt {2}\,1{}\mathrm {i}}{32}+\frac {3}{64}{}\mathrm {i}\right ) \] Input:
int((x^2 + 1)/(7*x^2 - 7*x^4 + x^6 - 1)^2,x)
Output:
atan((x*23313i)/(8192*((27309*2^(1/2))/32768 - 19317/16384)) - (2^(1/2)*x* 65943i)/(32768*((27309*2^(1/2))/32768 - 19317/16384)))*((2^(1/2)*1i)/32 - 3i/64) - ((31*x)/32 - (23*x^3)/16 + (7*x^5)/32)/(7*x^2 - 7*x^4 + x^6 - 1) - (atan(x*1i)*1i)/4 + atan((x*23313i)/(8192*((27309*2^(1/2))/32768 + 19317 /16384)) + (2^(1/2)*x*65943i)/(32768*((27309*2^(1/2))/32768 + 19317/16384) ))*((2^(1/2)*1i)/32 + 3i/64)
Time = 0.23 (sec) , antiderivative size = 492, normalized size of antiderivative = 4.43 \[ \int \frac {1+x^2}{\left (1-7 x^2+7 x^4-x^6\right )^2} \, dx =\text {Too large to display} \] Input:
int((x^2+1)/(-x^6+7*x^4-7*x^2+1)^2,x)
Output:
(2*sqrt(2)*log( - sqrt(2) + x - 1)*x**6 - 14*sqrt(2)*log( - sqrt(2) + x - 1)*x**4 + 14*sqrt(2)*log( - sqrt(2) + x - 1)*x**2 - 2*sqrt(2)*log( - sqrt( 2) + x - 1) + 2*sqrt(2)*log( - sqrt(2) + x + 1)*x**6 - 14*sqrt(2)*log( - s qrt(2) + x + 1)*x**4 + 14*sqrt(2)*log( - sqrt(2) + x + 1)*x**2 - 2*sqrt(2) *log( - sqrt(2) + x + 1) - 2*sqrt(2)*log(sqrt(2) + x - 1)*x**6 + 14*sqrt(2 )*log(sqrt(2) + x - 1)*x**4 - 14*sqrt(2)*log(sqrt(2) + x - 1)*x**2 + 2*sqr t(2)*log(sqrt(2) + x - 1) - 2*sqrt(2)*log(sqrt(2) + x + 1)*x**6 + 14*sqrt( 2)*log(sqrt(2) + x + 1)*x**4 - 14*sqrt(2)*log(sqrt(2) + x + 1)*x**2 + 2*sq rt(2)*log(sqrt(2) + x + 1) - 3*log( - sqrt(2) + x - 1)*x**6 + 21*log( - sq rt(2) + x - 1)*x**4 - 21*log( - sqrt(2) + x - 1)*x**2 + 3*log( - sqrt(2) + x - 1) + 3*log( - sqrt(2) + x + 1)*x**6 - 21*log( - sqrt(2) + x + 1)*x**4 + 21*log( - sqrt(2) + x + 1)*x**2 - 3*log( - sqrt(2) + x + 1) - 3*log(sqr t(2) + x - 1)*x**6 + 21*log(sqrt(2) + x - 1)*x**4 - 21*log(sqrt(2) + x - 1 )*x**2 + 3*log(sqrt(2) + x - 1) + 3*log(sqrt(2) + x + 1)*x**6 - 21*log(sqr t(2) + x + 1)*x**4 + 21*log(sqrt(2) + x + 1)*x**2 - 3*log(sqrt(2) + x + 1) - 16*log(x - 1)*x**6 + 112*log(x - 1)*x**4 - 112*log(x - 1)*x**2 + 16*log (x - 1) + 16*log(x + 1)*x**6 - 112*log(x + 1)*x**4 + 112*log(x + 1)*x**2 - 16*log(x + 1) - 28*x**5 + 184*x**3 - 124*x)/(128*(x**6 - 7*x**4 + 7*x**2 - 1))