Integrand size = 57, antiderivative size = 186 \[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=2 \arctan (1-x)+2 \arctan (1+x)-\sqrt {2} \arctan \left (1-\sqrt {2}-\sqrt {2} x\right )+\sqrt {2} \arctan \left (1+\sqrt {2}-\sqrt {2} x\right )-\sqrt {2} \arctan \left (1-\sqrt {2}+\sqrt {2} x\right )+\sqrt {2} \arctan \left (1+\sqrt {2}+\sqrt {2} x\right )-2 \log (x)+\log \left (-4+x^2\right )+\frac {\log \left (6+4 \sqrt {2}-2 x^2-2 \sqrt {2} x^2+x^4\right )}{\sqrt {2}}-\frac {\log \left (6-4 \sqrt {2}-2 x^2+2 \sqrt {2} x^2+x^4\right )}{\sqrt {2}} \] Output:
-2*arctan(-1+x)+2*arctan(1+x)+2^(1/2)*arctan(-1+2^(1/2)+x*2^(1/2))-arctan( -1-2^(1/2)+x*2^(1/2))*2^(1/2)-arctan(1-2^(1/2)+x*2^(1/2))*2^(1/2)+2^(1/2)* arctan(1+2^(1/2)+x*2^(1/2))-2*ln(x)+ln(x^2-4)+1/2*ln(6+4*2^(1/2)-2*x^2-2*x ^2*2^(1/2)+x^4)*2^(1/2)-1/2*ln(6-4*2^(1/2)-2*x^2+2*x^2*2^(1/2)+x^4)*2^(1/2 )
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.05 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.72 \[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=2 \arctan (1-x)+2 \arctan (1+x)+\log (2-x)-2 \log (x)+\log (2+x)-\text {RootSum}\left [2-4 \text {$\#$1}+6 \text {$\#$1}^2-4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{-1+3 \text {$\#$1}-3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]+\text {RootSum}\left [2+4 \text {$\#$1}+6 \text {$\#$1}^2+4 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1})}{1+3 \text {$\#$1}+3 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ] \] Input:
Integrate[(128 + 896*x^2 + 896*x^4 + 128*x^6)/(-64*x - 112*x^3 - 112*x^5 + 68*x^7 - 56*x^9 + 28*x^11 - 8*x^13 + x^15),x]
Output:
2*ArcTan[1 - x] + 2*ArcTan[1 + x] + Log[2 - x] - 2*Log[x] + Log[2 + x] - R ootSum[2 - 4*#1 + 6*#1^2 - 4*#1^3 + #1^4 & , Log[x - #1]/(-1 + 3*#1 - 3*#1 ^2 + #1^3) & ] + RootSum[2 + 4*#1 + 6*#1^2 + 4*#1^3 + #1^4 & , Log[x - #1] /(1 + 3*#1 + 3*#1^2 + #1^3) & ]
Time = 1.08 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2026, 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.
| \(\displaystyle \int \frac {128 x^6+896 x^4+896 x^2+128}{x^{15}-8 x^{13}+28 x^{11}-56 x^9+68 x^7-112 x^5-112 x^3-64 x} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {128 x^6+896 x^4+896 x^2+128}{x \left (x^{14}-8 x^{12}+28 x^{10}-56 x^8+68 x^6-112 x^4-112 x^2-64\right )}dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (-\frac {2}{x^2-2 x+2}+\frac {2}{x^2+2 x+2}-\frac {4}{x^4-4 x^3+6 x^2-4 x+2}+\frac {4}{x^4+4 x^3+6 x^2+4 x+2}-\frac {2}{x}+\frac {1}{x+2}+\frac {1}{x-2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\sqrt {2} \arctan \left (1-\sqrt {2} (1-x)\right )+\sqrt {2} \arctan \left (\sqrt {2} (1-x)+1\right )+2 \arctan (1-x)+2 \arctan (x+1)-\sqrt {2} \arctan \left (1-\sqrt {2} (x+1)\right )+\sqrt {2} \arctan \left (\sqrt {2} (x+1)+1\right )-\frac {\log \left ((x-1)^2-\sqrt {2} (1-x)+1\right )}{\sqrt {2}}+\frac {\log \left ((x-1)^2+\sqrt {2} (1-x)+1\right )}{\sqrt {2}}+\log (2-x)-2 \log (x)+\log (x+2)-\frac {\log \left ((x+1)^2-\sqrt {2} (x+1)+1\right )}{\sqrt {2}}+\frac {\log \left ((x+1)^2+\sqrt {2} (x+1)+1\right )}{\sqrt {2}}\) |
Input:
Int[(128 + 896*x^2 + 896*x^4 + 128*x^6)/(-64*x - 112*x^3 - 112*x^5 + 68*x^ 7 - 56*x^9 + 28*x^11 - 8*x^13 + x^15),x]
Output:
-(Sqrt[2]*ArcTan[1 - Sqrt[2]*(1 - x)]) + Sqrt[2]*ArcTan[1 + Sqrt[2]*(1 - x )] + 2*ArcTan[1 - x] + 2*ArcTan[1 + x] - Sqrt[2]*ArcTan[1 - Sqrt[2]*(1 + x )] + Sqrt[2]*ArcTan[1 + Sqrt[2]*(1 + x)] - Log[1 - Sqrt[2]*(1 - x) + (-1 + x)^2]/Sqrt[2] + Log[1 + Sqrt[2]*(1 - x) + (-1 + x)^2]/Sqrt[2] + Log[2 - x ] - 2*Log[x] + Log[2 + x] - Log[1 - Sqrt[2]*(1 + x) + (1 + x)^2]/Sqrt[2] + Log[1 + Sqrt[2]*(1 + x) + (1 + x)^2]/Sqrt[2]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.24
| method | result | size |
| risch | \(-2 \ln \left (x \right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\textit {\_R} \ln \left (-\textit {\_R}^{2}+x^{2}-2 \textit {\_R} -1\right )\right )+\ln \left (x^{2}-4\right )-2 \arctan \left (\frac {x^{2}}{2}\right )\) | \(45\) |
| default | \(\ln \left (x -2\right )+2 \arctan \left (1+x \right )-2 \ln \left (x \right )+\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+4 \textit {\_Z}^{3}+6 \textit {\_Z}^{2}+4 \textit {\_Z} +2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}+3 \textit {\_R}^{2}+3 \textit {\_R} +1}\right )+\ln \left (x +2\right )-2 \arctan \left (-1+x \right )-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}-4 \textit {\_Z}^{3}+6 \textit {\_Z}^{2}-4 \textit {\_Z} +2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}-3 \textit {\_R}^{2}+3 \textit {\_R} -1}\right )\) | \(116\) |
Input:
int((128*x^6+896*x^4+896*x^2+128)/(x^15-8*x^13+28*x^11-56*x^9+68*x^7-112*x ^5-112*x^3-64*x),x,method=_RETURNVERBOSE)
Output:
-2*ln(x)+sum(_R*ln(-_R^2+x^2-2*_R-1),_R=RootOf(_Z^4+1))+ln(x^2-4)-2*arctan (1/2*x^2)
Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.59 \[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=-\sqrt {2} \arctan \left (\sqrt {2} x^{2} + x^{2} + 1\right ) - \sqrt {2} \arctan \left (\sqrt {2} x^{2} - x^{2} - 1\right ) - \frac {1}{2} \, \sqrt {2} \log \left (x^{4} - 2 \, x^{2} + 2 \, \sqrt {2} {\left (x^{2} - 2\right )} + 6\right ) + \frac {1}{2} \, \sqrt {2} \log \left (x^{4} - 2 \, x^{2} - 2 \, \sqrt {2} {\left (x^{2} - 2\right )} + 6\right ) - 2 \, \arctan \left (\frac {1}{2} \, x^{2}\right ) + \log \left (x^{2} - 4\right ) - 2 \, \log \left (x\right ) \] Input:
integrate((128*x^6+896*x^4+896*x^2+128)/(x^15-8*x^13+28*x^11-56*x^9+68*x^7 -112*x^5-112*x^3-64*x),x, algorithm="fricas")
Output:
-sqrt(2)*arctan(sqrt(2)*x^2 + x^2 + 1) - sqrt(2)*arctan(sqrt(2)*x^2 - x^2 - 1) - 1/2*sqrt(2)*log(x^4 - 2*x^2 + 2*sqrt(2)*(x^2 - 2) + 6) + 1/2*sqrt(2 )*log(x^4 - 2*x^2 - 2*sqrt(2)*(x^2 - 2) + 6) - 2*arctan(1/2*x^2) + log(x^2 - 4) - 2*log(x)
Result contains complex when optimal does not.
Time = 1.35 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.27 \[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=- 2 \log {\left (x \right )} + \log {\left (x^{2} - 4 \right )} + i \log {\left (x^{2} - 2 i \right )} - i \log {\left (x^{2} + 2 i \right )} + \operatorname {RootSum} {\left (t^{4} + 1, \left ( t \mapsto t \log {\left (- t^{2} - 2 t + x^{2} - 1 \right )} \right )\right )} \] Input:
integrate((128*x**6+896*x**4+896*x**2+128)/(x**15-8*x**13+28*x**11-56*x**9 +68*x**7-112*x**5-112*x**3-64*x),x)
Output:
-2*log(x) + log(x**2 - 4) + I*log(x**2 - 2*I) - I*log(x**2 + 2*I) + RootSu m(_t**4 + 1, Lambda(_t, _t*log(-_t**2 - 2*_t + x**2 - 1)))
\[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=\int { \frac {128 \, {\left (x^{6} + 7 \, x^{4} + 7 \, x^{2} + 1\right )}}{x^{15} - 8 \, x^{13} + 28 \, x^{11} - 56 \, x^{9} + 68 \, x^{7} - 112 \, x^{5} - 112 \, x^{3} - 64 \, x} \,d x } \] Input:
integrate((128*x^6+896*x^4+896*x^2+128)/(x^15-8*x^13+28*x^11-56*x^9+68*x^7 -112*x^5-112*x^3-64*x),x, algorithm="maxima")
Output:
2*arctan(x + 1) - 2*arctan(x - 1) + 4*integrate(1/(x^4 + 4*x^3 + 6*x^2 + 4 *x + 2), x) - 4*integrate(1/(x^4 - 4*x^3 + 6*x^2 - 4*x + 2), x) + log(x + 2) + log(x - 2) - 2*log(x)
Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.12 \[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=-2 \, \arctan \left (\frac {1}{2} \, x^{2}\right ) - \log \left (x^{2}\right ) + \log \left ({\left | x^{2} - 4 \right |}\right ) \] Input:
integrate((128*x^6+896*x^4+896*x^2+128)/(x^15-8*x^13+28*x^11-56*x^9+68*x^7 -112*x^5-112*x^3-64*x),x, algorithm="giac")
Output:
-2*arctan(1/2*x^2) - log(x^2) + log(abs(x^2 - 4))
Time = 10.47 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.70 \[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=2\,\mathrm {atan}\left (\frac {187197827062294009568444137017003579801600000}{1861\,\left (2669509464018326614341299852165706153984\,x^2+22550971673203714510510249852712018509824\right )}-\frac {15721}{3722}\right )+\ln \left (x^2-4\right )-2\,\ln \left (x\right )+\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (33557403127924312826424756750204083896320000-67818241846069012311426833408864802570240000{}\mathrm {i}\right )}{x^2\,\left (173059045693899167869323889017475741777920000-58486988538473719350589843481476831641600000{}\mathrm {i}\right )+95909550540234862597060065681222968279040000+47457250899576822865884532035590742343680000{}\mathrm {i}}+\frac {\sqrt {2}\,x^2\,\left (-41356569179401369608088805152146472304640000-122371223206902505300398093937941298544640000{}\mathrm {i}\right )}{x^2\,\left (173059045693899167869323889017475741777920000-58486988538473719350589843481476831641600000{}\mathrm {i}\right )+95909550540234862597060065681222968279040000+47457250899576822865884532035590742343680000{}\mathrm {i}}\right )\,\left (1-\mathrm {i}\right )+\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\left (33557403127924312826424756750204083896320000+67818241846069012311426833408864802570240000{}\mathrm {i}\right )}{x^2\,\left (173059045693899167869323889017475741777920000+58486988538473719350589843481476831641600000{}\mathrm {i}\right )+95909550540234862597060065681222968279040000-47457250899576822865884532035590742343680000{}\mathrm {i}}+\frac {\sqrt {2}\,x^2\,\left (-41356569179401369608088805152146472304640000+122371223206902505300398093937941298544640000{}\mathrm {i}\right )}{x^2\,\left (173059045693899167869323889017475741777920000+58486988538473719350589843481476831641600000{}\mathrm {i}\right )+95909550540234862597060065681222968279040000-47457250899576822865884532035590742343680000{}\mathrm {i}}\right )\,\left (1+1{}\mathrm {i}\right ) \] Input:
int(-(896*x^2 + 896*x^4 + 128*x^6 + 128)/(64*x + 112*x^3 + 112*x^5 - 68*x^ 7 + 56*x^9 - 28*x^11 + 8*x^13 - x^15),x)
Output:
2*atan(187197827062294009568444137017003579801600000/(1861*(26695094640183 26614341299852165706153984*x^2 + 22550971673203714510510249852712018509824 )) - 15721/3722) + log(x^2 - 4) - 2*log(x) + 2^(1/2)*atan((2^(1/2)*(335574 03127924312826424756750204083896320000 - 678182418460690123114268334088648 02570240000i))/(x^2*(173059045693899167869323889017475741777920000 - 58486 988538473719350589843481476831641600000i) + (95909550540234862597060065681 222968279040000 + 47457250899576822865884532035590742343680000i)) - (2^(1/ 2)*x^2*(41356569179401369608088805152146472304640000 + 1223712232069025053 00398093937941298544640000i))/(x^2*(17305904569389916786932388901747574177 7920000 - 58486988538473719350589843481476831641600000i) + (95909550540234 862597060065681222968279040000 + 47457250899576822865884532035590742343680 000i)))*(1 - 1i) + 2^(1/2)*atan((2^(1/2)*(33557403127924312826424756750204 083896320000 + 67818241846069012311426833408864802570240000i))/(x^2*(17305 9045693899167869323889017475741777920000 + 5848698853847371935058984348147 6831641600000i) + (95909550540234862597060065681222968279040000 - 47457250 899576822865884532035590742343680000i)) - (2^(1/2)*x^2*(413565691794013696 08088805152146472304640000 - 122371223206902505300398093937941298544640000 i))/(x^2*(173059045693899167869323889017475741777920000 + 5848698853847371 9350589843481476831641600000i) + (9590955054023486259706006568122296827904 0000 - 47457250899576822865884532035590742343680000i)))*(1 + 1i)
\[ \int \frac {128+896 x^2+896 x^4+128 x^6}{-64 x-112 x^3-112 x^5+68 x^7-56 x^9+28 x^{11}-8 x^{13}+x^{15}} \, dx=\frac {12 \mathit {atan} \left (x -1\right )}{7}-\frac {12 \mathit {atan} \left (x +1\right )}{7}+\frac {11520 \left (\int \frac {x^{3}}{x^{14}-8 x^{12}+28 x^{10}-56 x^{8}+68 x^{6}-112 x^{4}-112 x^{2}-64}d x \right )}{7}+\frac {11904 \left (\int \frac {x}{x^{14}-8 x^{12}+28 x^{10}-56 x^{8}+68 x^{6}-112 x^{4}-112 x^{2}-64}d x \right )}{7}+\frac {4224 \left (\int \frac {1}{x^{15}-8 x^{13}+28 x^{11}-56 x^{9}+68 x^{7}-112 x^{5}-112 x^{3}-64 x}d x \right )}{7}-\frac {4 \,\mathrm {log}\left (x^{8}-4 x^{6}+8 x^{4}+8 x^{2}+4\right )}{7}-\frac {4 \,\mathrm {log}\left (x^{2}-2 x +2\right )}{7}-\frac {4 \,\mathrm {log}\left (x^{2}+2 x +2\right )}{7}-\frac {2 \,\mathrm {log}\left (x -2\right )}{7}-\frac {2 \,\mathrm {log}\left (x +2\right )}{7}+\frac {52 \,\mathrm {log}\left (x \right )}{7} \] Input:
int((128*x^6+896*x^4+896*x^2+128)/(x^15-8*x^13+28*x^11-56*x^9+68*x^7-112*x ^5-112*x^3-64*x),x)
Output:
(2*(6*atan(x - 1) - 6*atan(x + 1) + 5760*int(x**3/(x**14 - 8*x**12 + 28*x* *10 - 56*x**8 + 68*x**6 - 112*x**4 - 112*x**2 - 64),x) + 5952*int(x/(x**14 - 8*x**12 + 28*x**10 - 56*x**8 + 68*x**6 - 112*x**4 - 112*x**2 - 64),x) + 2112*int(1/(x**15 - 8*x**13 + 28*x**11 - 56*x**9 + 68*x**7 - 112*x**5 - 1 12*x**3 - 64*x),x) - 2*log(x**8 - 4*x**6 + 8*x**4 + 8*x**2 + 4) - 2*log(x* *2 - 2*x + 2) - 2*log(x**2 + 2*x + 2) - log(x - 2) - log(x + 2) + 26*log(x )))/7