Integrand size = 74, antiderivative size = 141 \[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\left (-1-\sqrt {5}\right ) \arctan \left (\frac {1}{2} \left (-1+\sqrt {5}\right ) x^4\right )+\left (1-\sqrt {5}\right ) \arctan \left (\frac {1}{2} \left (1+\sqrt {5}\right ) x^4\right )-\sqrt {5} \text {arctanh}\left (\frac {1}{5} \left (3 \sqrt {5}+2 \sqrt {5} x^4\right )\right )-\sqrt {5} \text {arctanh}\left (\frac {1}{15} \left (7 \sqrt {5}+2 \sqrt {5} x^4\right )\right )+\frac {1}{2} \log \left (1+3 x^4+x^8\right )-\frac {1}{2} \log \left (1+7 x^4+x^8\right ) \] Output:
(-5^(1/2)-1)*arctan(1/2*(5^(1/2)-1)*x^4)+(-5^(1/2)+1)*arctan(1/2*(5^(1/2)+ 1)*x^4)-5^(1/2)*arctanh(3/5*5^(1/2)+2/5*5^(1/2)*x^4)-5^(1/2)*arctanh(7/15* 5^(1/2)+2/15*5^(1/2)*x^4)+1/2*ln(x^8+3*x^4+1)-1/2*ln(x^8+7*x^4+1)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.07 \[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {5}\right ) \log \left (-3+\sqrt {5}-2 x^4\right )+\left (-1+\sqrt {5}\right ) \log \left (-7+3 \sqrt {5}-2 x^4\right )-\left (-1+\sqrt {5}\right ) \log \left (3+\sqrt {5}+2 x^4\right )-\left (1+\sqrt {5}\right ) \log \left (7+3 \sqrt {5}+2 x^4\right )-2 \text {RootSum}\left [1+3 \text {$\#$1}^8+\text {$\#$1}^{16}\&,\frac {2 \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1}) \text {$\#$1}^8}{3 \text {$\#$1}^4+2 \text {$\#$1}^{12}}\&\right ]\right ) \] Input:
Integrate[(16*x^3 - 248*x^11 + 80*x^15 - 392*x^19 - 80*x^23 + 24*x^27)/(1 + 10*x^4 + 26*x^8 + 40*x^12 + 71*x^16 + 40*x^20 + 26*x^24 + 10*x^28 + x^32 ),x]
Output:
((1 + Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4] + (-1 + Sqrt[5])*Log[-7 + 3*Sqrt[ 5] - 2*x^4] - (-1 + Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4] - (1 + Sqrt[5])*Log[ 7 + 3*Sqrt[5] + 2*x^4] - 2*RootSum[1 + 3*#1^8 + #1^16 & , (2*Log[x - #1] + 3*Log[x - #1]*#1^8)/(3*#1^4 + 2*#1^12) & ])/2
Time = 1.00 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, 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.
| \(\displaystyle \int \frac {24 x^{27}-80 x^{23}-392 x^{19}+80 x^{15}-248 x^{11}+16 x^3}{x^{32}+10 x^{28}+26 x^{24}+40 x^{20}+71 x^{16}+40 x^{12}+26 x^8+10 x^4+1} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (-\frac {8 \left (3 x^8+2\right ) x^3}{x^{16}+3 x^8+1}+\frac {4 \left (x^4+4\right ) x^3}{x^8+3 x^4+1}-\frac {4 \left (x^4-4\right ) x^3}{x^8+7 x^4+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\sqrt {2 \left (3+\sqrt {5}\right )} \arctan \left (\sqrt {\frac {2}{3+\sqrt {5}}} x^4\right )-\sqrt {2 \left (3-\sqrt {5}\right )} \arctan \left (\sqrt {\frac {1}{2} \left (3+\sqrt {5}\right )} x^4\right )-\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (2 x^4-3 \sqrt {5}+7\right )+\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (2 x^4-\sqrt {5}+3\right )+\frac {1}{2} \left (1-\sqrt {5}\right ) \log \left (2 x^4+\sqrt {5}+3\right )-\frac {1}{2} \left (1+\sqrt {5}\right ) \log \left (2 x^4+3 \sqrt {5}+7\right )\) |
Input:
Int[(16*x^3 - 248*x^11 + 80*x^15 - 392*x^19 - 80*x^23 + 24*x^27)/(1 + 10*x ^4 + 26*x^8 + 40*x^12 + 71*x^16 + 40*x^20 + 26*x^24 + 10*x^28 + x^32),x]
Output:
-(Sqrt[2*(3 + Sqrt[5])]*ArcTan[Sqrt[2/(3 + Sqrt[5])]*x^4]) - Sqrt[2*(3 - S qrt[5])]*ArcTan[Sqrt[(3 + Sqrt[5])/2]*x^4] - ((1 - Sqrt[5])*Log[7 - 3*Sqrt [5] + 2*x^4])/2 + ((1 + Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/2 + ((1 - Sqrt[ 5])*Log[3 + Sqrt[5] + 2*x^4])/2 - ((1 + Sqrt[5])*Log[7 + 3*Sqrt[5] + 2*x^4 ])/2
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 = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.96
| method | result | size |
| default | \(\frac {\ln \left (x^{8}+3 x^{4}+1\right )}{2}-\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{4}+3\right ) \sqrt {5}}{5}\right )-\frac {\ln \left (x^{8}+7 x^{4}+1\right )}{2}-\sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{4}+7\right ) \sqrt {5}}{15}\right )-\frac {4 \sqrt {5}\, \left (5+3 \sqrt {5}\right ) \arctan \left (\frac {4 x^{4}}{2+2 \sqrt {5}}\right )}{5 \left (2+2 \sqrt {5}\right )}-\frac {4 \left (3 \sqrt {5}-5\right ) \sqrt {5}\, \arctan \left (\frac {4 x^{4}}{-2+2 \sqrt {5}}\right )}{5 \left (-2+2 \sqrt {5}\right )}\) | \(136\) |
| risch | \(\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+3 \textit {\_Z}^{2}+1\right )}{\sum }\textit {\_R} \ln \left (x^{4}-\textit {\_R} \right )\right )+\frac {\ln \left (2 x^{4}-\sqrt {5}+3\right )}{2}+\frac {\ln \left (2 x^{4}-\sqrt {5}+3\right ) \sqrt {5}}{2}+\frac {\ln \left (2 x^{4}+\sqrt {5}+3\right )}{2}-\frac {\ln \left (2 x^{4}+\sqrt {5}+3\right ) \sqrt {5}}{2}+\frac {\ln \left (2 x^{4}-3 \sqrt {5}+7\right ) \sqrt {5}}{2}-\frac {\ln \left (2 x^{4}-3 \sqrt {5}+7\right )}{2}-\frac {\ln \left (2 x^{4}+3 \sqrt {5}+7\right )}{2}-\frac {\ln \left (2 x^{4}+3 \sqrt {5}+7\right ) \sqrt {5}}{2}\) | \(154\) |
Input:
int((24*x^27-80*x^23-392*x^19+80*x^15-248*x^11+16*x^3)/(x^32+10*x^28+26*x^ 24+40*x^20+71*x^16+40*x^12+26*x^8+10*x^4+1),x,method=_RETURNVERBOSE)
Output:
1/2*ln(x^8+3*x^4+1)-5^(1/2)*arctanh(1/5*(2*x^4+3)*5^(1/2))-1/2*ln(x^8+7*x^ 4+1)-5^(1/2)*arctanh(1/15*(2*x^4+7)*5^(1/2))-4/5*5^(1/2)*(5+3*5^(1/2))/(2+ 2*5^(1/2))*arctan(4*x^4/(2+2*5^(1/2)))-4/5*(3*5^(1/2)-5)*5^(1/2)/(-2+2*5^( 1/2))*arctan(4*x^4/(-2+2*5^(1/2)))
Time = 0.13 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.11 \[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=-{\left (\sqrt {5} - 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {5} x^{4} + \frac {1}{2} \, x^{4}\right ) - {\left (\sqrt {5} + 1\right )} \arctan \left (\frac {1}{2} \, \sqrt {5} x^{4} - \frac {1}{2} \, x^{4}\right ) + \frac {1}{2} \, \sqrt {5} \log \left (\frac {2 \, x^{8} + 14 \, x^{4} - 3 \, \sqrt {5} {\left (2 \, x^{4} + 7\right )} + 47}{x^{8} + 7 \, x^{4} + 1}\right ) + \frac {1}{2} \, \sqrt {5} \log \left (\frac {2 \, x^{8} + 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) - \frac {1}{2} \, \log \left (x^{8} + 7 \, x^{4} + 1\right ) + \frac {1}{2} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) \] Input:
integrate((24*x^27-80*x^23-392*x^19+80*x^15-248*x^11+16*x^3)/(x^32+10*x^28 +26*x^24+40*x^20+71*x^16+40*x^12+26*x^8+10*x^4+1),x, algorithm="fricas")
Output:
-(sqrt(5) - 1)*arctan(1/2*sqrt(5)*x^4 + 1/2*x^4) - (sqrt(5) + 1)*arctan(1/ 2*sqrt(5)*x^4 - 1/2*x^4) + 1/2*sqrt(5)*log((2*x^8 + 14*x^4 - 3*sqrt(5)*(2* x^4 + 7) + 47)/(x^8 + 7*x^4 + 1)) + 1/2*sqrt(5)*log((2*x^8 + 6*x^4 - sqrt( 5)*(2*x^4 + 3) + 7)/(x^8 + 3*x^4 + 1)) - 1/2*log(x^8 + 7*x^4 + 1) + 1/2*lo g(x^8 + 3*x^4 + 1)
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (126) = 252\).
Time = 0.39 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.24 \[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right ) \log {\left (x^{4} - \frac {1}{12} - \frac {\left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right )^{6}}{6} + \frac {\left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right )^{7}}{6} + \frac {\left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right )^{4}}{6} - \frac {4 \left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right )^{3}}{3} + \frac {11 \left (\frac {1}{2} - \frac {\sqrt {5}}{2}\right )^{2}}{6} + \frac {3 \sqrt {5}}{4} \right )} + \left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right ) \log {\left (x^{4} - \frac {3 \sqrt {5}}{4} - \frac {4 \left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{3}}{3} - \frac {\left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{6}}{6} + \frac {\left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{7}}{6} + \frac {\left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{4}}{6} + \frac {11 \left (- \frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{2}}{6} + \frac {17}{12} \right )} + \left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right ) \log {\left (x^{4} - \frac {4 \left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{3}}{3} - \frac {\left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{6}}{6} - \frac {3 \sqrt {5}}{4} - \frac {1}{12} + \frac {\left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{4}}{6} + \frac {11 \left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{2}}{6} + \frac {\left (\frac {1}{2} + \frac {\sqrt {5}}{2}\right )^{7}}{6} \right )} + \left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right ) \log {\left (x^{4} + \frac {\left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right )^{7}}{6} - \frac {\left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right )^{6}}{6} + \frac {\left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right )^{4}}{6} + \frac {17}{12} + \frac {3 \sqrt {5}}{4} + \frac {11 \left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right )^{2}}{6} - \frac {4 \left (- \frac {\sqrt {5}}{2} - \frac {1}{2}\right )^{3}}{3} \right )} + \operatorname {RootSum} {\left (t^{4} + 3 t^{2} + 1, \left ( t \mapsto t \log {\left (\frac {t^{7}}{6} - \frac {t^{6}}{6} + \frac {t^{4}}{6} - \frac {4 t^{3}}{3} + \frac {11 t^{2}}{6} - \frac {3 t}{2} + x^{4} + \frac {2}{3} \right )} \right )\right )} \] Input:
integrate((24*x**27-80*x**23-392*x**19+80*x**15-248*x**11+16*x**3)/(x**32+ 10*x**28+26*x**24+40*x**20+71*x**16+40*x**12+26*x**8+10*x**4+1),x)
Output:
(1/2 - sqrt(5)/2)*log(x**4 - 1/12 - (1/2 - sqrt(5)/2)**6/6 + (1/2 - sqrt(5 )/2)**7/6 + (1/2 - sqrt(5)/2)**4/6 - 4*(1/2 - sqrt(5)/2)**3/3 + 11*(1/2 - sqrt(5)/2)**2/6 + 3*sqrt(5)/4) + (-1/2 + sqrt(5)/2)*log(x**4 - 3*sqrt(5)/4 - 4*(-1/2 + sqrt(5)/2)**3/3 - (-1/2 + sqrt(5)/2)**6/6 + (-1/2 + sqrt(5)/2 )**7/6 + (-1/2 + sqrt(5)/2)**4/6 + 11*(-1/2 + sqrt(5)/2)**2/6 + 17/12) + ( 1/2 + sqrt(5)/2)*log(x**4 - 4*(1/2 + sqrt(5)/2)**3/3 - (1/2 + sqrt(5)/2)** 6/6 - 3*sqrt(5)/4 - 1/12 + (1/2 + sqrt(5)/2)**4/6 + 11*(1/2 + sqrt(5)/2)** 2/6 + (1/2 + sqrt(5)/2)**7/6) + (-sqrt(5)/2 - 1/2)*log(x**4 + (-sqrt(5)/2 - 1/2)**7/6 - (-sqrt(5)/2 - 1/2)**6/6 + (-sqrt(5)/2 - 1/2)**4/6 + 17/12 + 3*sqrt(5)/4 + 11*(-sqrt(5)/2 - 1/2)**2/6 - 4*(-sqrt(5)/2 - 1/2)**3/3) + Ro otSum(_t**4 + 3*_t**2 + 1, Lambda(_t, _t*log(_t**7/6 - _t**6/6 + _t**4/6 - 4*_t**3/3 + 11*_t**2/6 - 3*_t/2 + x**4 + 2/3)))
\[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\int { \frac {8 \, {\left (3 \, x^{27} - 10 \, x^{23} - 49 \, x^{19} + 10 \, x^{15} - 31 \, x^{11} + 2 \, x^{3}\right )}}{x^{32} + 10 \, x^{28} + 26 \, x^{24} + 40 \, x^{20} + 71 \, x^{16} + 40 \, x^{12} + 26 \, x^{8} + 10 \, x^{4} + 1} \,d x } \] Input:
integrate((24*x^27-80*x^23-392*x^19+80*x^15-248*x^11+16*x^3)/(x^32+10*x^28 +26*x^24+40*x^20+71*x^16+40*x^12+26*x^8+10*x^4+1),x, algorithm="maxima")
Output:
8*integrate((3*x^27 - 10*x^23 - 49*x^19 + 10*x^15 - 31*x^11 + 2*x^3)/(x^32 + 10*x^28 + 26*x^24 + 40*x^20 + 71*x^16 + 40*x^12 + 26*x^8 + 10*x^4 + 1), x)
Exception generated. \[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((24*x^27-80*x^23-392*x^19+80*x^15-248*x^11+16*x^3)/(x^32+10*x^28 +26*x^24+40*x^20+71*x^16+40*x^12+26*x^8+10*x^4+1),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:int(sage0,sageVARx) Error: Bad Arg ument Type
Time = 11.16 (sec) , antiderivative size = 2153, normalized size of antiderivative = 15.27 \[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\text {Too large to display} \] Input:
int((16*x^3 - 248*x^11 + 80*x^15 - 392*x^19 - 80*x^23 + 24*x^27)/(10*x^4 + 26*x^8 + 40*x^12 + 71*x^16 + 40*x^20 + 26*x^24 + 10*x^28 + x^32 + 1),x)
Output:
atan(((- 5^(1/2)/2 - 3/2)^(1/2)*117779131845107704926873936971012505600000 00000000000000000000000000000000000000000i)/(51058337120893890751916603222 89090560000000000000000000000000000000000000000000000*5^(1/2) - 5267248564 920505865498027973043814400000000000000000000000000000000000000000000000*5 ^(1/2)*x^4 + 1177791318451077049268739369710125056000000000000000000000000 0000000000000000000000*x^4 - 114170233933457692052694155217233510400000000 00000000000000000000000000000000000000) - (5^(1/2)*(- 5^(1/2)/2 - 3/2)^(1/ 2)*52672485649205058654980279730438144000000000000000000000000000000000000 00000000000i)/(51058337120893890751916603222890905600000000000000000000000 00000000000000000000000*5^(1/2) - 5267248564920505865498027973043814400000 000000000000000000000000000000000000000000*5^(1/2)*x^4 + 11777913184510770 492687393697101250560000000000000000000000000000000000000000000000*x^4 - 1 14170233933457692052694155217233510400000000000000000000000000000000000000 00000000) + (x^4*(- 5^(1/2)/2 - 3/2)^(1/2)*2989011937024212649588327408830 7752960000000000000000000000000000000000000000000000i)/(510583371208938907 5191660322289090560000000000000000000000000000000000000000000000*5^(1/2) - 5267248564920505865498027973043814400000000000000000000000000000000000000 000000000*5^(1/2)*x^4 + 11777913184510770492687393697101250560000000000000 000000000000000000000000000000000*x^4 - 1141702339334576920526941552172335 1040000000000000000000000000000000000000000000000) - (5^(1/2)*x^4*(- 5^...
\[ \int \frac {16 x^3-248 x^{11}+80 x^{15}-392 x^{19}-80 x^{23}+24 x^{27}}{1+10 x^4+26 x^8+40 x^{12}+71 x^{16}+40 x^{20}+26 x^{24}+10 x^{28}+x^{32}} \, dx=\int \frac {24 x^{27}-80 x^{23}-392 x^{19}+80 x^{15}-248 x^{11}+16 x^{3}}{x^{32}+10 x^{28}+26 x^{24}+40 x^{20}+71 x^{16}+40 x^{12}+26 x^{8}+10 x^{4}+1}d x \] Input:
int((24*x^27-80*x^23-392*x^19+80*x^15-248*x^11+16*x^3)/(x^32+10*x^28+26*x^ 24+40*x^20+71*x^16+40*x^12+26*x^8+10*x^4+1),x)
Output:
int((24*x^27-80*x^23-392*x^19+80*x^15-248*x^11+16*x^3)/(x^32+10*x^28+26*x^ 24+40*x^20+71*x^16+40*x^12+26*x^8+10*x^4+1),x)