Integrand size = 71, antiderivative size = 71 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (1+x+\sqrt {2} x+\sqrt {2} x^2-x^7\right )-\left (-1+\sqrt {2}\right ) \log \left (-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right )\right ) \]
-1/2*ln(-1+x^7+x*(2^(1/2)-1)+x^2*2^(1/2))*(2^(1/2)-1)+1/2*ln(1+x-x^7+x*2^( 1/2)+x^2*2^(1/2))*(1+2^(1/2))
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \left (\left (1+\sqrt {2}\right ) \log \left (1+x+\sqrt {2} x+\sqrt {2} x^2-x^7\right )-\left (-1+\sqrt {2}\right ) \log \left (-1+\left (-1+\sqrt {2}\right ) x+\sqrt {2} x^2+x^7\right )\right ) \]
Integrate[(3 + 3*x - 4*x^2 - 4*x^3 - 7*x^6 + 4*x^7 + 10*x^8 + 7*x^13)/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 + x^14),x]
((1 + Sqrt[2])*Log[1 + x + Sqrt[2]*x + Sqrt[2]*x^2 - x^7] - (-1 + Sqrt[2]) *Log[-1 + (-1 + Sqrt[2])*x + Sqrt[2]*x^2 + x^7])/2
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.
\(\displaystyle \int \frac {7 x^{13}+10 x^8+4 x^7-7 x^6-4 x^3-4 x^2+3 x+3}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1} \, dx\) |
\(\Big \downarrow \) 2525 |
\(\displaystyle \frac {1}{14} \int \frac {28 \left (5 x^8+6 x^7+x^2+2 x+1\right )}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx+\frac {1}{2} \log \left (x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \int \frac {5 x^8+6 x^7+x^2+2 x+1}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx+\frac {1}{2} \log \left (x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1\right )\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 2 \int \left (\frac {5 x^8}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}+\frac {6 x^7}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}+\frac {x^2}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}+\frac {2 x}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}+\frac {1}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}\right )dx+\frac {1}{2} \log \left (x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\int \frac {1}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx+2 \int \frac {x}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx+\int \frac {x^2}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx+6 \int \frac {x^7}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx+5 \int \frac {x^8}{x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1}dx\right )+\frac {1}{2} \log \left (x^{14}-2 x^8-2 x^7-2 x^4-4 x^3-x^2+2 x+1\right )\) |
Int[(3 + 3*x - 4*x^2 - 4*x^3 - 7*x^6 + 4*x^7 + 10*x^8 + 7*x^13)/(1 + 2*x - x^2 - 4*x^3 - 2*x^4 - 2*x^7 - 2*x^8 + x^14),x]
3.3.84.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Si mp[Coeff[Pm, x, m]*(Log[Qn]/(n*Coeff[Qn, x, n])), x] + Simp[1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x], x ]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.86
method | result | size |
default | \(\left (\frac {1}{2}+\frac {\sqrt {2}}{2}\right ) \ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )+\left (-\frac {\sqrt {2}}{2}+\frac {1}{2}\right ) \ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )\) | \(61\) |
risch | \(\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right )}{2}+\frac {\ln \left (x^{7}-x^{2} \sqrt {2}+\left (-1-\sqrt {2}\right ) x -1\right ) \sqrt {2}}{2}-\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right ) \sqrt {2}}{2}+\frac {\ln \left (-1+x^{7}+x \left (\sqrt {2}-1\right )+x^{2} \sqrt {2}\right )}{2}\) | \(102\) |
int((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7-2*x^4- 4*x^3-x^2+2*x+1),x,method=_RETURNVERBOSE)
(1/2+1/2*2^(1/2))*ln(x^7-x^2*2^(1/2)+(-1-2^(1/2))*x-1)+(-1/2*2^(1/2)+1/2)* ln(-1+x^7+x*(2^(1/2)-1)+x^2*2^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (56) = 112\).
Time = 0.26 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.93 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{14} - 2 \, x^{8} - 2 \, x^{7} + 2 \, x^{4} + 4 \, x^{3} + 3 \, x^{2} - 2 \, \sqrt {2} {\left (x^{9} + x^{8} - x^{3} - 2 \, x^{2} - x\right )} + 2 \, x + 1}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1}\right ) + \frac {1}{2} \, \log \left (x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1\right ) \]
integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7- 2*x^4-4*x^3-x^2+2*x+1),x, algorithm="fricas")
1/2*sqrt(2)*log((x^14 - 2*x^8 - 2*x^7 + 2*x^4 + 4*x^3 + 3*x^2 - 2*sqrt(2)* (x^9 + x^8 - x^3 - 2*x^2 - x) + 2*x + 1)/(x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4 *x^3 - x^2 + 2*x + 1)) + 1/2*log(x^14 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 - x^ 2 + 2*x + 1)
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.07 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} - \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} + \frac {\sqrt {2}}{2}\right ) - 1 \right )} + \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) \log {\left (x^{7} + \sqrt {2} x^{2} - 2 x \left (\frac {1}{2} - \frac {\sqrt {2}}{2}\right ) - 1 \right )} \]
integrate((7*x**13+10*x**8+4*x**7-7*x**6-4*x**3-4*x**2+3*x+3)/(x**14-2*x** 8-2*x**7-2*x**4-4*x**3-x**2+2*x+1),x)
(1/2 + sqrt(2)/2)*log(x**7 - sqrt(2)*x**2 - 2*x*(1/2 + sqrt(2)/2) - 1) + ( 1/2 - sqrt(2)/2)*log(x**7 + sqrt(2)*x**2 - 2*x*(1/2 - sqrt(2)/2) - 1)
\[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\int { \frac {7 \, x^{13} + 10 \, x^{8} + 4 \, x^{7} - 7 \, x^{6} - 4 \, x^{3} - 4 \, x^{2} + 3 \, x + 3}{x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1} \,d x } \]
integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7- 2*x^4-4*x^3-x^2+2*x+1),x, algorithm="maxima")
integrate((7*x^13 + 10*x^8 + 4*x^7 - 7*x^6 - 4*x^3 - 4*x^2 + 3*x + 3)/(x^1 4 - 2*x^8 - 2*x^7 - 2*x^4 - 4*x^3 - x^2 + 2*x + 1), x)
Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} + \sqrt {2} x^{2} + \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \sqrt {2} \log \left ({\left | x^{7} - \sqrt {2} x^{2} - \sqrt {2} x - x - 1 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | x^{14} - 2 \, x^{8} - 2 \, x^{7} - 2 \, x^{4} - 4 \, x^{3} - x^{2} + 2 \, x + 1 \right |}\right ) \]
integrate((7*x^13+10*x^8+4*x^7-7*x^6-4*x^3-4*x^2+3*x+3)/(x^14-2*x^8-2*x^7- 2*x^4-4*x^3-x^2+2*x+1),x, algorithm="giac")
-1/2*sqrt(2)*log(abs(x^7 + sqrt(2)*x^2 + sqrt(2)*x - x - 1)) + 1/2*sqrt(2) *log(abs(x^7 - sqrt(2)*x^2 - sqrt(2)*x - x - 1)) + 1/2*log(abs(x^14 - 2*x^ 8 - 2*x^7 - 2*x^4 - 4*x^3 - x^2 + 2*x + 1))
Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.45 \[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=\frac {\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2}-\frac {\sqrt {2}\,\ln \left (\sqrt {2}\,x-x+\sqrt {2}\,x^2+x^7-1\right )}{2}+\frac {\sqrt {2}\,\ln \left (x^7-\sqrt {2}\,x-\sqrt {2}\,x^2-x-1\right )}{2} \]
int(-(3*x - 4*x^2 - 4*x^3 - 7*x^6 + 4*x^7 + 10*x^8 + 7*x^13 + 3)/(x^2 - 2* x + 4*x^3 + 2*x^4 + 2*x^7 + 2*x^8 - x^14 - 1),x)
log(2^(1/2)*x - x + 2^(1/2)*x^2 + x^7 - 1)/2 + log(x^7 - 2^(1/2)*x - 2^(1/ 2)*x^2 - x - 1)/2 - (2^(1/2)*log(2^(1/2)*x - x + 2^(1/2)*x^2 + x^7 - 1))/2 + (2^(1/2)*log(x^7 - 2^(1/2)*x - 2^(1/2)*x^2 - x - 1))/2
\[ \int \frac {3+3 x-4 x^2-4 x^3-7 x^6+4 x^7+10 x^8+7 x^{13}}{1+2 x-x^2-4 x^3-2 x^4-2 x^7-2 x^8+x^{14}} \, dx=7 \left (\int \frac {x^{13}}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )+10 \left (\int \frac {x^{8}}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )+4 \left (\int \frac {x^{7}}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )-7 \left (\int \frac {x^{6}}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )-4 \left (\int \frac {x^{3}}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )-4 \left (\int \frac {x^{2}}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )+3 \left (\int \frac {x}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right )+3 \left (\int \frac {1}{x^{14}-2 x^{8}-2 x^{7}-2 x^{4}-4 x^{3}-x^{2}+2 x +1}d x \right ) \]
int((7*x**13 + 10*x**8 + 4*x**7 - 7*x**6 - 4*x**3 - 4*x**2 + 3*x + 3)/(x** 14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x**2 + 2*x + 1),x)
7*int(x**13/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x**2 + 2*x + 1),x ) + 10*int(x**8/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x**2 + 2*x + 1),x) + 4*int(x**7/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x**2 + 2*x + 1),x) - 7*int(x**6/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x**2 + 2*x + 1),x) - 4*int(x**3/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x**2 + 2*x + 1),x) - 4*int(x**2/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x **2 + 2*x + 1),x) + 3*int(x/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x **2 + 2*x + 1),x) + 3*int(1/(x**14 - 2*x**8 - 2*x**7 - 2*x**4 - 4*x**3 - x **2 + 2*x + 1),x)