Integrand size = 52, antiderivative size = 58 \[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=\frac {1}{2} \left (\left (2+\sqrt {7}\right ) \log \left (1+\left (2+\sqrt {7}\right ) x^2-x^3\right )-\left (-2+\sqrt {7}\right ) \log \left (-1+\left (-2+\sqrt {7}\right ) x^2+x^3\right )\right ) \] Output:
1/2*(2+7^(1/2))*ln(1+(2+7^(1/2))*x^2-x^3)-1/2*(-2+7^(1/2))*ln(-1+(-2+7^(1/ 2))*x^2+x^3)
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=\frac {1}{2} \left (\left (2+\sqrt {7}\right ) \log \left (1+\left (2+\sqrt {7}\right ) x^2-x^3\right )-\left (-2+\sqrt {7}\right ) \log \left (-1+\left (-2+\sqrt {7}\right ) x^2+x^3\right )\right ) \] Input:
Integrate[(22*x - 6*x^2 - 12*x^3 - 13*x^4 + 6*x^5)/(1 + 4*x^2 - 2*x^3 - 3* x^4 - 4*x^5 + x^6),x]
Output:
((2 + Sqrt[7])*Log[1 + (2 + Sqrt[7])*x^2 - x^3] - (-2 + Sqrt[7])*Log[-1 + (-2 + Sqrt[7])*x^2 + x^3])/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.
\(\displaystyle \int \frac {6 x^5-13 x^4-12 x^3-6 x^2+22 x}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1} \, dx\) |
\(\Big \downarrow \) 2525 |
\(\displaystyle \frac {1}{6} \int \frac {42 \left (x^4+2 x\right )}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}dx+\log \left (x^6-4 x^5-3 x^4-2 x^3+4 x^2+1\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 7 \int \frac {x^4+2 x}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}dx+\log \left (x^6-4 x^5-3 x^4-2 x^3+4 x^2+1\right )\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle 7 \int \frac {x \left (x^3+2\right )}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}dx+\log \left (x^6-4 x^5-3 x^4-2 x^3+4 x^2+1\right )\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle 7 \int \left (\frac {x^4}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}+\frac {2 x}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}\right )dx+\log \left (x^6-4 x^5-3 x^4-2 x^3+4 x^2+1\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 7 \left (2 \int \frac {x}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}dx+\int \frac {x^4}{x^6-4 x^5-3 x^4-2 x^3+4 x^2+1}dx\right )+\log \left (x^6-4 x^5-3 x^4-2 x^3+4 x^2+1\right )\) |
Input:
Int[(22*x - 6*x^2 - 12*x^3 - 13*x^4 + 6*x^5)/(1 + 4*x^2 - 2*x^3 - 3*x^4 - 4*x^5 + x^6),x]
Output:
$Aborted
Time = 0.05 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.86
method | result | size |
default | \(\left (1+\frac {\sqrt {7}}{2}\right ) \ln \left (x^{3}+\left (-\sqrt {7}-2\right ) x^{2}-1\right )+\left (1-\frac {\sqrt {7}}{2}\right ) \ln \left (-1+\left (-2+\sqrt {7}\right ) x^{2}+x^{3}\right )\) | \(50\) |
risch | \(\ln \left (x^{3}+\left (-\sqrt {7}-2\right ) x^{2}-1\right )+\frac {\ln \left (x^{3}+\left (-\sqrt {7}-2\right ) x^{2}-1\right ) \sqrt {7}}{2}+\ln \left (-1+\left (-2+\sqrt {7}\right ) x^{2}+x^{3}\right )-\frac {\ln \left (-1+\left (-2+\sqrt {7}\right ) x^{2}+x^{3}\right ) \sqrt {7}}{2}\) | \(76\) |
Input:
int((6*x^5-13*x^4-12*x^3-6*x^2+22*x)/(x^6-4*x^5-3*x^4-2*x^3+4*x^2+1),x,met hod=_RETURNVERBOSE)
Output:
(1+1/2*7^(1/2))*ln(x^3+(-7^(1/2)-2)*x^2-1)+(1-1/2*7^(1/2))*ln(-1+(-2+7^(1/ 2))*x^2+x^3)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (47) = 94\).
Time = 0.07 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.81 \[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=\frac {1}{2} \, \sqrt {7} \log \left (\frac {x^{6} - 4 \, x^{5} + 11 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} - 2 \, \sqrt {7} {\left (x^{5} - 2 \, x^{4} - x^{2}\right )} + 1}{x^{6} - 4 \, x^{5} - 3 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1}\right ) + \log \left (x^{6} - 4 \, x^{5} - 3 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1\right ) \] Input:
integrate((6*x^5-13*x^4-12*x^3-6*x^2+22*x)/(x^6-4*x^5-3*x^4-2*x^3+4*x^2+1) ,x, algorithm="fricas")
Output:
1/2*sqrt(7)*log((x^6 - 4*x^5 + 11*x^4 - 2*x^3 + 4*x^2 - 2*sqrt(7)*(x^5 - 2 *x^4 - x^2) + 1)/(x^6 - 4*x^5 - 3*x^4 - 2*x^3 + 4*x^2 + 1)) + log(x^6 - 4* x^5 - 3*x^4 - 2*x^3 + 4*x^2 + 1)
Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.97 \[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=\left (1 - \frac {\sqrt {7}}{2}\right ) \log {\left (x^{3} - 2 x^{2} \cdot \left (1 - \frac {\sqrt {7}}{2}\right ) - 1 \right )} + \left (1 + \frac {\sqrt {7}}{2}\right ) \log {\left (x^{3} - 2 x^{2} \cdot \left (1 + \frac {\sqrt {7}}{2}\right ) - 1 \right )} \] Input:
integrate((6*x**5-13*x**4-12*x**3-6*x**2+22*x)/(x**6-4*x**5-3*x**4-2*x**3+ 4*x**2+1),x)
Output:
(1 - sqrt(7)/2)*log(x**3 - 2*x**2*(1 - sqrt(7)/2) - 1) + (1 + sqrt(7)/2)*l og(x**3 - 2*x**2*(1 + sqrt(7)/2) - 1)
\[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=\int { \frac {6 \, x^{5} - 13 \, x^{4} - 12 \, x^{3} - 6 \, x^{2} + 22 \, x}{x^{6} - 4 \, x^{5} - 3 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1} \,d x } \] Input:
integrate((6*x^5-13*x^4-12*x^3-6*x^2+22*x)/(x^6-4*x^5-3*x^4-2*x^3+4*x^2+1) ,x, algorithm="maxima")
Output:
integrate((6*x^5 - 13*x^4 - 12*x^3 - 6*x^2 + 22*x)/(x^6 - 4*x^5 - 3*x^4 - 2*x^3 + 4*x^2 + 1), x)
Time = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.33 \[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=-\frac {1}{2} \, \sqrt {7} \log \left ({\left | x^{3} + \sqrt {7} x^{2} - 2 \, x^{2} - 1 \right |}\right ) + \frac {1}{2} \, \sqrt {7} \log \left ({\left | x^{3} - \sqrt {7} x^{2} - 2 \, x^{2} - 1 \right |}\right ) + \log \left ({\left | x^{6} - 4 \, x^{5} - 3 \, x^{4} - 2 \, x^{3} + 4 \, x^{2} + 1 \right |}\right ) \] Input:
integrate((6*x^5-13*x^4-12*x^3-6*x^2+22*x)/(x^6-4*x^5-3*x^4-2*x^3+4*x^2+1) ,x, algorithm="giac")
Output:
-1/2*sqrt(7)*log(abs(x^3 + sqrt(7)*x^2 - 2*x^2 - 1)) + 1/2*sqrt(7)*log(abs (x^3 - sqrt(7)*x^2 - 2*x^2 - 1)) + log(abs(x^6 - 4*x^5 - 3*x^4 - 2*x^3 + 4 *x^2 + 1))
Time = 0.22 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.47 \[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=\ln \left (\sqrt {7}\,x^2-2\,x^2+x^3-1\right )+\ln \left (x^3-2\,x^2-\sqrt {7}\,x^2-1\right )-\frac {\sqrt {7}\,\ln \left (\sqrt {7}\,x^2-2\,x^2+x^3-1\right )}{2}+\frac {\sqrt {7}\,\ln \left (x^3-2\,x^2-\sqrt {7}\,x^2-1\right )}{2} \] Input:
int(-(6*x^2 - 22*x + 12*x^3 + 13*x^4 - 6*x^5)/(4*x^2 - 2*x^3 - 3*x^4 - 4*x ^5 + x^6 + 1),x)
Output:
log(7^(1/2)*x^2 - 2*x^2 + x^3 - 1) + log(x^3 - 2*x^2 - 7^(1/2)*x^2 - 1) - (7^(1/2)*log(7^(1/2)*x^2 - 2*x^2 + x^3 - 1))/2 + (7^(1/2)*log(x^3 - 2*x^2 - 7^(1/2)*x^2 - 1))/2
\[ \int \frac {22 x-6 x^2-12 x^3-13 x^4+6 x^5}{1+4 x^2-2 x^3-3 x^4-4 x^5+x^6} \, dx=7 \left (\int \frac {x^{4}}{x^{6}-4 x^{5}-3 x^{4}-2 x^{3}+4 x^{2}+1}d x \right )+14 \left (\int \frac {x}{x^{6}-4 x^{5}-3 x^{4}-2 x^{3}+4 x^{2}+1}d x \right )+\mathrm {log}\left (x^{6}-4 x^{5}-3 x^{4}-2 x^{3}+4 x^{2}+1\right ) \] Input:
int((6*x^5-13*x^4-12*x^3-6*x^2+22*x)/(x^6-4*x^5-3*x^4-2*x^3+4*x^2+1),x)
Output:
7*int(x**4/(x**6 - 4*x**5 - 3*x**4 - 2*x**3 + 4*x**2 + 1),x) + 14*int(x/(x **6 - 4*x**5 - 3*x**4 - 2*x**3 + 4*x**2 + 1),x) + log(x**6 - 4*x**5 - 3*x* *4 - 2*x**3 + 4*x**2 + 1)