Integrand size = 39, antiderivative size = 45 \[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^5}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^5}}\right ) \]
Time = 1.37 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=2 \text {arctanh}\left (\frac {x}{\sqrt {1+x^2+x^5}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {1+x^2+x^5}}\right ) \]
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.
\(\displaystyle \int \frac {\sqrt {x^5+x^2+1} \left (3 x^5-2\right )}{\left (x^5+1\right ) \left (x^5-x^2+1\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt {x^5+x^2+1}}{x+1}+\frac {\left (5 x^3-2\right ) \sqrt {x^5+x^2+1}}{x^5-x^2+1}+\frac {\sqrt {x^5+x^2+1} \left (-x^3-3 x^2+2 x-1\right )}{x^4-x^3+x^2-x+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \int \frac {\sqrt {x^5+x^2+1}}{x+1}dx-2 \int \frac {\sqrt {x^5+x^2+1}}{x^5-x^2+1}dx+5 \int \frac {x^3 \sqrt {x^5+x^2+1}}{x^5-x^2+1}dx+\int \frac {\sqrt {x^5+x^2+1}}{-x^4+x^3-x^2+x-1}dx+2 \int \frac {x \sqrt {x^5+x^2+1}}{x^4-x^3+x^2-x+1}dx-3 \int \frac {x^2 \sqrt {x^5+x^2+1}}{x^4-x^3+x^2-x+1}dx-\int \frac {x^3 \sqrt {x^5+x^2+1}}{x^4-x^3+x^2-x+1}dx\) |
3.6.82.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.89 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.42
method | result | size |
pseudoelliptic | \(-\ln \left (\frac {\sqrt {x^{5}+x^{2}+1}-x}{x}\right )-2 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{5}+x^{2}+1}\, \sqrt {2}}{2 x}\right )+\ln \left (\frac {\sqrt {x^{5}+x^{2}+1}+x}{x}\right )\) | \(64\) |
trager | \(-\ln \left (-\frac {-x^{5}+2 \sqrt {x^{5}+x^{2}+1}\, x -2 x^{2}-1}{\left (1+x \right ) \left (x^{4}-x^{3}+x^{2}-x +1\right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{5}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}+4 \sqrt {x^{5}+x^{2}+1}\, x -\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{5}-x^{2}+1}\right )\) | \(121\) |
-ln(((x^5+x^2+1)^(1/2)-x)/x)-2*2^(1/2)*arctanh(1/2*(x^5+x^2+1)^(1/2)/x*2^( 1/2))+ln(((x^5+x^2+1)^(1/2)+x)/x)
Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (37) = 74\).
Time = 0.28 (sec) , antiderivative size = 115, normalized size of antiderivative = 2.56 \[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (\frac {x^{10} + 14 \, x^{7} + 2 \, x^{5} + 17 \, x^{4} - 4 \, \sqrt {2} {\left (x^{6} + 3 \, x^{3} + x\right )} \sqrt {x^{5} + x^{2} + 1} + 14 \, x^{2} + 1}{x^{10} - 2 \, x^{7} + 2 \, x^{5} + x^{4} - 2 \, x^{2} + 1}\right ) + \log \left (\frac {x^{5} + 2 \, x^{2} + 2 \, \sqrt {x^{5} + x^{2} + 1} x + 1}{x^{5} + 1}\right ) \]
1/2*sqrt(2)*log((x^10 + 14*x^7 + 2*x^5 + 17*x^4 - 4*sqrt(2)*(x^6 + 3*x^3 + x)*sqrt(x^5 + x^2 + 1) + 14*x^2 + 1)/(x^10 - 2*x^7 + 2*x^5 + x^4 - 2*x^2 + 1)) + log((x^5 + 2*x^2 + 2*sqrt(x^5 + x^2 + 1)*x + 1)/(x^5 + 1))
Timed out. \[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=\int { \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + x^{2} + 1}}{{\left (x^{5} - x^{2} + 1\right )} {\left (x^{5} + 1\right )}} \,d x } \]
\[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=\int { \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + x^{2} + 1}}{{\left (x^{5} - x^{2} + 1\right )} {\left (x^{5} + 1\right )}} \,d x } \]
Time = 8.87 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.71 \[ \int \frac {\sqrt {1+x^2+x^5} \left (-2+3 x^5\right )}{\left (1+x^5\right ) \left (1-x^2+x^5\right )} \, dx=\ln \left (\frac {2\,x\,\sqrt {x^5+x^2+1}+2\,x^2+x^5+1}{x^5+1}\right )+\sqrt {2}\,\ln \left (\frac {3\,x^2+x^5-2\,\sqrt {2}\,x\,\sqrt {x^5+x^2+1}+1}{x^5-x^2+1}\right ) \]