Integrand size = 32, antiderivative size = 71 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {1+x^5}}{1-x^2+x^5}\right )}{\sqrt {2}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} x \sqrt {1+x^5}}{1+x^2+x^5}\right )}{\sqrt {2}} \]
-1/2*arctan(2^(1/2)*x*(x^5+1)^(1/2)/(x^5-x^2+1))*2^(1/2)-1/2*arctanh(2^(1/ 2)*x*(x^5+1)^(1/2)/(x^5+x^2+1))*2^(1/2)
Time = 3.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=-\frac {\arctan \left (\frac {\sqrt {2} x \sqrt {1+x^5}}{1-x^2+x^5}\right )+\text {arctanh}\left (\frac {1+x^2+x^5}{\sqrt {2} x \sqrt {1+x^5}}\right )}{\sqrt {2}} \]
-((ArcTan[(Sqrt[2]*x*Sqrt[1 + x^5])/(1 - x^2 + x^5)] + ArcTanh[(1 + x^2 + x^5)/(Sqrt[2]*x*Sqrt[1 + x^5])])/Sqrt[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.
\(\displaystyle \int \frac {\sqrt {x^5+1} \left (3 x^5-2\right )}{x^{10}+2 x^5+x^4+1} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x^5 \sqrt {x^5+1}}{x^{10}+2 x^5+x^4+1}-\frac {2 \sqrt {x^5+1}}{x^{10}+2 x^5+x^4+1}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 3 \int \frac {x^5 \sqrt {x^5+1}}{x^{10}+2 x^5+x^4+1}dx-2 \int \frac {\sqrt {x^5+1}}{x^{10}+2 x^5+x^4+1}dx\) |
3.10.41.3.1 Defintions of rubi rules used
Time = 5.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.32
method | result | size |
pseudoelliptic | \(\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{5}-\sqrt {x^{5}+1}\, \sqrt {2}\, x +x^{2}+1}{x^{5}+\sqrt {x^{5}+1}\, \sqrt {2}\, x +x^{2}+1}\right )+2 \arctan \left (\frac {\sqrt {x^{5}+1}\, \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\sqrt {x^{5}+1}\, \sqrt {2}-x}{x}\right )\right )}{4}\) | \(94\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 x \sqrt {x^{5}+1}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{5} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}+2 x \sqrt {x^{5}+1}}{-x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{2}-1}\right )}{2}\) | \(153\) |
1/4*2^(1/2)*(ln((x^5-(x^5+1)^(1/2)*2^(1/2)*x+x^2+1)/(x^5+(x^5+1)^(1/2)*2^( 1/2)*x+x^2+1))+2*arctan(((x^5+1)^(1/2)*2^(1/2)+x)/x)+2*arctan(((x^5+1)^(1/ 2)*2^(1/2)-x)/x))
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 289, normalized size of antiderivative = 4.07 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {x^{10} + 2 i \, x^{7} + 2 \, x^{5} - x^{4} + \sqrt {2} {\left (\left (i + 1\right ) \, x^{6} + \left (i - 1\right ) \, x^{3} + \left (i + 1\right ) \, x\right )} \sqrt {x^{5} + 1} + 2 i \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}\right ) - \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {x^{10} + 2 i \, x^{7} + 2 \, x^{5} - x^{4} + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{6} - \left (i - 1\right ) \, x^{3} - \left (i + 1\right ) \, x\right )} \sqrt {x^{5} + 1} + 2 i \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {x^{10} - 2 i \, x^{7} + 2 \, x^{5} - x^{4} + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{6} - \left (i + 1\right ) \, x^{3} - \left (i - 1\right ) \, x\right )} \sqrt {x^{5} + 1} - 2 i \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \log \left (\frac {x^{10} - 2 i \, x^{7} + 2 \, x^{5} - x^{4} + \sqrt {2} {\left (\left (i - 1\right ) \, x^{6} + \left (i + 1\right ) \, x^{3} + \left (i - 1\right ) \, x\right )} \sqrt {x^{5} + 1} - 2 i \, x^{2} + 1}{x^{10} + 2 \, x^{5} + x^{4} + 1}\right ) \]
(1/8*I - 1/8)*sqrt(2)*log((x^10 + 2*I*x^7 + 2*x^5 - x^4 + sqrt(2)*((I + 1) *x^6 + (I - 1)*x^3 + (I + 1)*x)*sqrt(x^5 + 1) + 2*I*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 1)) - (1/8*I - 1/8)*sqrt(2)*log((x^10 + 2*I*x^7 + 2*x^5 - x^4 + s qrt(2)*(-(I + 1)*x^6 - (I - 1)*x^3 - (I + 1)*x)*sqrt(x^5 + 1) + 2*I*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 1)) - (1/8*I + 1/8)*sqrt(2)*log((x^10 - 2*I*x^7 + 2*x^5 - x^4 + sqrt(2)*(-(I - 1)*x^6 - (I + 1)*x^3 - (I - 1)*x)*sqrt(x^5 + 1) - 2*I*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 1)) + (1/8*I + 1/8)*sqrt(2)*log(( x^10 - 2*I*x^7 + 2*x^5 - x^4 + sqrt(2)*((I - 1)*x^6 + (I + 1)*x^3 + (I - 1 )*x)*sqrt(x^5 + 1) - 2*I*x^2 + 1)/(x^10 + 2*x^5 + x^4 + 1))
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=\int \frac {\sqrt {\left (x + 1\right ) \left (x^{4} - x^{3} + x^{2} - x + 1\right )} \left (3 x^{5} - 2\right )}{x^{10} + 2 x^{5} + x^{4} + 1}\, dx \]
Integral(sqrt((x + 1)*(x**4 - x**3 + x**2 - x + 1))*(3*x**5 - 2)/(x**10 + 2*x**5 + x**4 + 1), x)
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{x^{10} + 2 \, x^{5} + x^{4} + 1} \,d x } \]
\[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=\int { \frac {{\left (3 \, x^{5} - 2\right )} \sqrt {x^{5} + 1}}{x^{10} + 2 \, x^{5} + x^{4} + 1} \,d x } \]
Time = 19.12 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.44 \[ \int \frac {\sqrt {1+x^5} \left (-2+3 x^5\right )}{1+x^4+2 x^5+x^{10}} \, dx=\frac {{\left (-1\right )}^{1/4}\,\ln \left (2\,x^5-x^4+x^{10}+1-x^2\,2{}\mathrm {i}-x^7\,2{}\mathrm {i}+\sqrt {2}\,x^3\,\sqrt {x^5+1}\,\left (1+1{}\mathrm {i}\right )+\sqrt {2}\,x\,{\left (x^5+1\right )}^{3/2}\,\left (-1+1{}\mathrm {i}\right )\right )}{2}-\frac {{\left (-1\right )}^{1/4}\,\ln \left (x^{10}+2\,x^5+x^4+1\right )}{2}+\sqrt {2}\,\ln \left (2\,x^5-x^4+x^{10}+1+x^2\,2{}\mathrm {i}+x^7\,2{}\mathrm {i}+\sqrt {2}\,x^3\,\sqrt {x^5+1}\,\left (1-\mathrm {i}\right )+\sqrt {2}\,x\,{\left (x^5+1\right )}^{3/2}\,\left (-1-\mathrm {i}\right )\right )\,\left (\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )+\sqrt {2}\,\ln \left (x^{10}+2\,x^5+x^4+1\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right ) \]
2^(1/2)*log(x^2*2i - x^4 + 2*x^5 + x^7*2i + x^10 + 2^(1/2)*x^3*(x^5 + 1)^( 1/2)*(1 - 1i) - 2^(1/2)*x*(x^5 + 1)^(3/2)*(1 + 1i) + 1)*(1/4 - 1i/4) + ((- 1)^(1/4)*log(2*x^5 - x^4 - x^2*2i - x^7*2i + x^10 + 2^(1/2)*x^3*(x^5 + 1)^ (1/2)*(1 + 1i) - 2^(1/2)*x*(x^5 + 1)^(3/2)*(1 - 1i) + 1))/2 - 2^(1/2)*log( x^4 + 2*x^5 + x^10 + 1)*(1/4 - 1i/4) - ((-1)^(1/4)*log(x^4 + 2*x^5 + x^10 + 1))/2