Integrand size = 73, antiderivative size = 147 \[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\frac {1}{24} \left (\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+3 \left (-1+\sqrt {5}\right ) x^2-2 x^6\right )+\left (-1+\sqrt {5}\right ) \log \left (-1-\sqrt {5}+3 \left (1+\sqrt {5}\right ) x^2-2 x^6\right )-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+3 \left (-1+\sqrt {5}\right ) x^2+2 x^6\right )-\left (-1+\sqrt {5}\right ) \log \left (-1-\sqrt {5}+3 \left (1+\sqrt {5}\right ) x^2+2 x^6\right )\right ) \] Output:
1/24*(5^(1/2)+1)*ln(1-5^(1/2)+3*(5^(1/2)-1)*x^2-2*x^6)+1/24*(5^(1/2)-1)*ln (-1-5^(1/2)+3*(5^(1/2)+1)*x^2-2*x^6)-1/24*(5^(1/2)+1)*ln(1-5^(1/2)+3*(5^(1 /2)-1)*x^2+2*x^6)-1/24*(5^(1/2)-1)*ln(-1-5^(1/2)+3*(5^(1/2)+1)*x^2+2*x^6)
Time = 0.44 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00 \[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\frac {1}{24} \left (\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+3 \left (-1+\sqrt {5}\right ) x^2-2 x^6\right )+\left (-1+\sqrt {5}\right ) \log \left (-1-\sqrt {5}+3 \left (1+\sqrt {5}\right ) x^2-2 x^6\right )-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+3 \left (-1+\sqrt {5}\right ) x^2+2 x^6\right )-\left (-1+\sqrt {5}\right ) \log \left (-1-\sqrt {5}+3 \left (1+\sqrt {5}\right ) x^2+2 x^6\right )\right ) \] Input:
Integrate[(x^5*(3 - 24*x^2 + 63*x^4 - 54*x^6 - 2*x^12 + 4*x^14))/(1 - 12*x ^2 + 54*x^4 - 108*x^6 + 81*x^8 - 3*x^12 + 18*x^14 - 27*x^16 + x^24),x]
Output:
((1 + Sqrt[5])*Log[1 - Sqrt[5] + 3*(-1 + Sqrt[5])*x^2 - 2*x^6] + (-1 + Sqr t[5])*Log[-1 - Sqrt[5] + 3*(1 + Sqrt[5])*x^2 - 2*x^6] - (1 + Sqrt[5])*Log[ 1 - Sqrt[5] + 3*(-1 + Sqrt[5])*x^2 + 2*x^6] - (-1 + Sqrt[5])*Log[-1 - Sqrt [5] + 3*(1 + Sqrt[5])*x^2 + 2*x^6])/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.
| \(\displaystyle \int \frac {x^5 \left (4 x^{14}-2 x^{12}-54 x^6+63 x^4-24 x^2+3\right )}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^8-108 x^6+54 x^4-12 x^2+1} \, dx\) |
| \(\Big \downarrow \) 2460 |
| \(\displaystyle \int \left (\frac {x \left (-x^{10}-7 x^6+3 x^4+3 x^2-1\right )}{2 \left (-x^{12}-3 x^8+x^6+9 x^4-6 x^2+1\right )}-\frac {x \left (x^{10}-7 x^6+3 x^4-3 x^2+1\right )}{2 \left (x^{12}-3 x^8+x^6-9 x^4+6 x^2-1\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{8} \text {Subst}\left (\int \frac {x^2}{x^6-3 x^4+x^3-9 x^2+6 x-1}dx,x,x^2\right )+\frac {5}{4} \text {Subst}\left (\int \frac {x^3}{x^6-3 x^4+x^3-9 x^2+6 x-1}dx,x,x^2\right )-\frac {5}{8} \text {Subst}\left (\int \frac {x^2}{x^6+3 x^4-x^3-9 x^2+6 x-1}dx,x,x^2\right )+\frac {5}{4} \text {Subst}\left (\int \frac {x^3}{x^6+3 x^4-x^3-9 x^2+6 x-1}dx,x,x^2\right )+\frac {1}{24} \log \left (-x^{12}-3 x^8+x^6+9 x^4-6 x^2+1\right )-\frac {1}{24} \log \left (x^{12}-3 x^8+x^6-9 x^4+6 x^2-1\right )\) |
Input:
Int[(x^5*(3 - 24*x^2 + 63*x^4 - 54*x^6 - 2*x^12 + 4*x^14))/(1 - 12*x^2 + 5 4*x^4 - 108*x^6 + 81*x^8 - 3*x^12 + 18*x^14 - 27*x^16 + x^24),x]
Output:
$Aborted
Time = 0.14 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\left (\frac {\sqrt {5}}{6}-\frac {1}{6}\right ) \ln \left (2 x^{6}+\left (-3-3 \sqrt {5}\right ) x^{2}+\sqrt {5}+1\right )}{4}+\frac {\left (-\frac {1}{6}-\frac {\sqrt {5}}{6}\right ) \ln \left (2 x^{6}+\left (3 \sqrt {5}-3\right ) x^{2}-\sqrt {5}+1\right )}{4}+\frac {\left (\frac {1}{6}+\frac {\sqrt {5}}{6}\right ) \ln \left (2 x^{6}+\left (3-3 \sqrt {5}\right ) x^{2}+\sqrt {5}-1\right )}{4}+\frac {\left (\frac {1}{6}-\frac {\sqrt {5}}{6}\right ) \ln \left (2 x^{6}+\left (3 \sqrt {5}+3\right ) x^{2}-\sqrt {5}-1\right )}{4}\) | \(130\) |
| risch | \(\frac {\ln \left (2 x^{6}+\left (3-3 \sqrt {5}\right ) x^{2}+\sqrt {5}-1\right )}{24}+\frac {\ln \left (2 x^{6}+\left (3-3 \sqrt {5}\right ) x^{2}+\sqrt {5}-1\right ) \sqrt {5}}{24}+\frac {\ln \left (2 x^{6}+\left (3 \sqrt {5}+3\right ) x^{2}-\sqrt {5}-1\right )}{24}-\frac {\ln \left (2 x^{6}+\left (3 \sqrt {5}+3\right ) x^{2}-\sqrt {5}-1\right ) \sqrt {5}}{24}+\frac {\ln \left (2 x^{6}+\left (-3-3 \sqrt {5}\right ) x^{2}+\sqrt {5}+1\right ) \sqrt {5}}{24}-\frac {\ln \left (2 x^{6}+\left (-3-3 \sqrt {5}\right ) x^{2}+\sqrt {5}+1\right )}{24}-\frac {\ln \left (2 x^{6}+\left (3 \sqrt {5}-3\right ) x^{2}-\sqrt {5}+1\right )}{24}-\frac {\ln \left (2 x^{6}+\left (3 \sqrt {5}-3\right ) x^{2}-\sqrt {5}+1\right ) \sqrt {5}}{24}\) | \(214\) |
Input:
int(x^5*(4*x^14-2*x^12-54*x^6+63*x^4-24*x^2+3)/(x^24-27*x^16+18*x^14-3*x^1 2+81*x^8-108*x^6+54*x^4-12*x^2+1),x,method=_RETURNVERBOSE)
Output:
1/4*(1/6*5^(1/2)-1/6)*ln(2*x^6+(-3-3*5^(1/2))*x^2+5^(1/2)+1)+1/4*(-1/6-1/6 *5^(1/2))*ln(2*x^6+(3*5^(1/2)-3)*x^2-5^(1/2)+1)+1/4*(1/6+1/6*5^(1/2))*ln(2 *x^6+(3-3*5^(1/2))*x^2+5^(1/2)-1)+1/4*(1/6-1/6*5^(1/2))*ln(2*x^6+(3*5^(1/2 )+3)*x^2-5^(1/2)-1)
Time = 0.12 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.56 \[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\frac {1}{24} \, \sqrt {5} \log \left (\frac {2 \, x^{12} + 6 \, x^{8} - 2 \, x^{6} + 27 \, x^{4} - 18 \, x^{2} - \sqrt {5} {\left (6 \, x^{8} - 2 \, x^{6} + 9 \, x^{4} - 6 \, x^{2} + 1\right )} + 3}{x^{12} + 3 \, x^{8} - x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1}\right ) + \frac {1}{24} \, \sqrt {5} \log \left (\frac {2 \, x^{12} - 6 \, x^{8} + 2 \, x^{6} + 27 \, x^{4} - 18 \, x^{2} - \sqrt {5} {\left (6 \, x^{8} - 2 \, x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1\right )} + 3}{x^{12} - 3 \, x^{8} + x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1}\right ) + \frac {1}{24} \, \log \left (x^{12} + 3 \, x^{8} - x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1\right ) - \frac {1}{24} \, \log \left (x^{12} - 3 \, x^{8} + x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1\right ) \] Input:
integrate(x^5*(4*x^14-2*x^12-54*x^6+63*x^4-24*x^2+3)/(x^24-27*x^16+18*x^14 -3*x^12+81*x^8-108*x^6+54*x^4-12*x^2+1),x, algorithm="fricas")
Output:
1/24*sqrt(5)*log((2*x^12 + 6*x^8 - 2*x^6 + 27*x^4 - 18*x^2 - sqrt(5)*(6*x^ 8 - 2*x^6 + 9*x^4 - 6*x^2 + 1) + 3)/(x^12 + 3*x^8 - x^6 - 9*x^4 + 6*x^2 - 1)) + 1/24*sqrt(5)*log((2*x^12 - 6*x^8 + 2*x^6 + 27*x^4 - 18*x^2 - sqrt(5) *(6*x^8 - 2*x^6 - 9*x^4 + 6*x^2 - 1) + 3)/(x^12 - 3*x^8 + x^6 - 9*x^4 + 6* x^2 - 1)) + 1/24*log(x^12 + 3*x^8 - x^6 - 9*x^4 + 6*x^2 - 1) - 1/24*log(x^ 12 - 3*x^8 + x^6 - 9*x^4 + 6*x^2 - 1)
Time = 0.24 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.92 \[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\left (\frac {1}{24} - \frac {\sqrt {5}}{24}\right ) \log {\left (x^{6} + x^{2} \left (- \frac {9}{2} + 5184 \left (\frac {1}{24} - \frac {\sqrt {5}}{24}\right )^{3} + \frac {9 \sqrt {5}}{2}\right ) - \frac {3 \sqrt {5}}{2} - 1728 \left (\frac {1}{24} - \frac {\sqrt {5}}{24}\right )^{3} + \frac {3}{2} \right )} + \left (\frac {1}{24} + \frac {\sqrt {5}}{24}\right ) \log {\left (x^{6} + x^{2} \left (- \frac {9 \sqrt {5}}{2} - \frac {9}{2} + 5184 \left (\frac {1}{24} + \frac {\sqrt {5}}{24}\right )^{3}\right ) - 1728 \left (\frac {1}{24} + \frac {\sqrt {5}}{24}\right )^{3} + \frac {3}{2} + \frac {3 \sqrt {5}}{2} \right )} + \left (- \frac {1}{24} + \frac {\sqrt {5}}{24}\right ) \log {\left (x^{6} + x^{2} \left (- \frac {9 \sqrt {5}}{2} + 5184 \left (- \frac {1}{24} + \frac {\sqrt {5}}{24}\right )^{3} + \frac {9}{2}\right ) - \frac {3}{2} - 1728 \left (- \frac {1}{24} + \frac {\sqrt {5}}{24}\right )^{3} + \frac {3 \sqrt {5}}{2} \right )} + \left (- \frac {\sqrt {5}}{24} - \frac {1}{24}\right ) \log {\left (x^{6} + x^{2} \cdot \left (5184 \left (- \frac {\sqrt {5}}{24} - \frac {1}{24}\right )^{3} + \frac {9}{2} + \frac {9 \sqrt {5}}{2}\right ) - \frac {3 \sqrt {5}}{2} - \frac {3}{2} - 1728 \left (- \frac {\sqrt {5}}{24} - \frac {1}{24}\right )^{3} \right )} \] Input:
integrate(x**5*(4*x**14-2*x**12-54*x**6+63*x**4-24*x**2+3)/(x**24-27*x**16 +18*x**14-3*x**12+81*x**8-108*x**6+54*x**4-12*x**2+1),x)
Output:
(1/24 - sqrt(5)/24)*log(x**6 + x**2*(-9/2 + 5184*(1/24 - sqrt(5)/24)**3 + 9*sqrt(5)/2) - 3*sqrt(5)/2 - 1728*(1/24 - sqrt(5)/24)**3 + 3/2) + (1/24 + sqrt(5)/24)*log(x**6 + x**2*(-9*sqrt(5)/2 - 9/2 + 5184*(1/24 + sqrt(5)/24) **3) - 1728*(1/24 + sqrt(5)/24)**3 + 3/2 + 3*sqrt(5)/2) + (-1/24 + sqrt(5) /24)*log(x**6 + x**2*(-9*sqrt(5)/2 + 5184*(-1/24 + sqrt(5)/24)**3 + 9/2) - 3/2 - 1728*(-1/24 + sqrt(5)/24)**3 + 3*sqrt(5)/2) + (-sqrt(5)/24 - 1/24)* log(x**6 + x**2*(5184*(-sqrt(5)/24 - 1/24)**3 + 9/2 + 9*sqrt(5)/2) - 3*sqr t(5)/2 - 3/2 - 1728*(-sqrt(5)/24 - 1/24)**3)
\[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\int { \frac {{\left (4 \, x^{14} - 2 \, x^{12} - 54 \, x^{6} + 63 \, x^{4} - 24 \, x^{2} + 3\right )} x^{5}}{x^{24} - 27 \, x^{16} + 18 \, x^{14} - 3 \, x^{12} + 81 \, x^{8} - 108 \, x^{6} + 54 \, x^{4} - 12 \, x^{2} + 1} \,d x } \] Input:
integrate(x^5*(4*x^14-2*x^12-54*x^6+63*x^4-24*x^2+3)/(x^24-27*x^16+18*x^14 -3*x^12+81*x^8-108*x^6+54*x^4-12*x^2+1),x, algorithm="maxima")
Output:
integrate((4*x^14 - 2*x^12 - 54*x^6 + 63*x^4 - 24*x^2 + 3)*x^5/(x^24 - 27* x^16 + 18*x^14 - 3*x^12 + 81*x^8 - 108*x^6 + 54*x^4 - 12*x^2 + 1), x)
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.39 \[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\frac {1}{24} \, \log \left ({\left | x^{12} + 3 \, x^{8} - x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1 \right |}\right ) - \frac {1}{24} \, \log \left ({\left | x^{12} - 3 \, x^{8} + x^{6} - 9 \, x^{4} + 6 \, x^{2} - 1 \right |}\right ) \] Input:
integrate(x^5*(4*x^14-2*x^12-54*x^6+63*x^4-24*x^2+3)/(x^24-27*x^16+18*x^14 -3*x^12+81*x^8-108*x^6+54*x^4-12*x^2+1),x, algorithm="giac")
Output:
1/24*log(abs(x^12 + 3*x^8 - x^6 - 9*x^4 + 6*x^2 - 1)) - 1/24*log(abs(x^12 - 3*x^8 + x^6 - 9*x^4 + 6*x^2 - 1))
Time = 12.32 (sec) , antiderivative size = 1389, normalized size of antiderivative = 9.45 \[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=\text {Too large to display} \] Input:
int(-(x^5*(24*x^2 - 63*x^4 + 54*x^6 + 2*x^12 - 4*x^14 - 3))/(54*x^4 - 12*x ^2 - 108*x^6 + 81*x^8 - 3*x^12 + 18*x^14 - 27*x^16 + x^24 + 1),x)
Output:
(atan((x^2*263824405954560000000i)/(2011661095403520000000*x^4 - 134110739 6935680000000*x^2 + 293138228838400000000*x^6 - 879414686515200000000*x^8 + 46902116614144000000*x^12 + 223517899489280000000) - (x^4*39573660893184 0000000i)/(2011661095403520000000*x^4 - 1341107396935680000000*x^2 + 29313 8228838400000000*x^6 - 879414686515200000000*x^8 + 46902116614144000000*x^ 12 + 223517899489280000000) - (x^6*88307891437568000000i)/(201166109540352 0000000*x^4 - 1341107396935680000000*x^2 + 293138228838400000000*x^6 - 879 414686515200000000*x^8 + 46902116614144000000*x^12 + 223517899489280000000 ) + (x^8*264923674312704000000i)/(2011661095403520000000*x^4 - 13411073969 35680000000*x^2 + 293138228838400000000*x^6 - 879414686515200000000*x^8 + 46902116614144000000*x^12 + 223517899489280000000) + (x^12*439707343257600 00000i)/(2011661095403520000000*x^4 - 1341107396935680000000*x^2 + 2931382 28838400000000*x^6 - 879414686515200000000*x^8 + 46902116614144000000*x^12 + 223517899489280000000) + (5^(1/2)*49833498902528000000i)/(2011661095403 520000000*x^4 - 1341107396935680000000*x^2 + 293138228838400000000*x^6 - 8 79414686515200000000*x^8 + 46902116614144000000*x^12 + 2235178994892800000 00) - 43970734325760000000i/(2011661095403520000000*x^4 - 1341107396935680 000000*x^2 + 293138228838400000000*x^6 - 879414686515200000000*x^8 + 46902 116614144000000*x^12 + 223517899489280000000) - (5^(1/2)*x^2*2990009934151 68000000i)/(2011661095403520000000*x^4 - 1341107396935680000000*x^2 + 2...
\[ \int \frac {x^5 \left (3-24 x^2+63 x^4-54 x^6-2 x^{12}+4 x^{14}\right )}{1-12 x^2+54 x^4-108 x^6+81 x^8-3 x^{12}+18 x^{14}-27 x^{16}+x^{24}} \, dx=4 \left (\int \frac {x^{19}}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-2 \left (\int \frac {x^{17}}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-54 \left (\int \frac {x^{11}}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+63 \left (\int \frac {x^{9}}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )-24 \left (\int \frac {x^{7}}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right )+3 \left (\int \frac {x^{5}}{x^{24}-27 x^{16}+18 x^{14}-3 x^{12}+81 x^{8}-108 x^{6}+54 x^{4}-12 x^{2}+1}d x \right ) \] Input:
int(x^5*(4*x^14-2*x^12-54*x^6+63*x^4-24*x^2+3)/(x^24-27*x^16+18*x^14-3*x^1 2+81*x^8-108*x^6+54*x^4-12*x^2+1),x)
Output:
4*int(x**19/(x**24 - 27*x**16 + 18*x**14 - 3*x**12 + 81*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) - 2*int(x**17/(x**24 - 27*x**16 + 18*x**14 - 3*x **12 + 81*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) - 54*int(x**11/(x**2 4 - 27*x**16 + 18*x**14 - 3*x**12 + 81*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) + 63*int(x**9/(x**24 - 27*x**16 + 18*x**14 - 3*x**12 + 81*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) - 24*int(x**7/(x**24 - 27*x**16 + 18* x**14 - 3*x**12 + 81*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x) + 3*int(x **5/(x**24 - 27*x**16 + 18*x**14 - 3*x**12 + 81*x**8 - 108*x**6 + 54*x**4 - 12*x**2 + 1),x)