Integrand size = 37, antiderivative size = 49 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {2-x^2+x^3}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {2-x^2+x^3}}\right ) \]
Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt {2-x^2+x^3}}\right )-2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x}{\sqrt {2-x^2+x^3}}\right ) \]
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.
\(\displaystyle \int \frac {\left (x^3-4\right ) \sqrt {x^3-x^2+2}}{\left (x^3+2\right ) \left (x^3+x^2+2\right )} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {\sqrt {x^3-x^2+2} (-3 x-2)}{x^3+x^2+2}+\frac {3 x \sqrt {x^3-x^2+2}}{x^3+2}\right )dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {3 x \sqrt {x^3-x^2+2}}{x^3+2}-\frac {(3 x+2) \sqrt {x^3-x^2+2}}{x^3+x^2+2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {3 x \sqrt {x^3-x^2+2}}{x^3+2}-\frac {(3 x+2) \sqrt {x^3-x^2+2}}{x^3+x^2+2}\right )dx\) |
3.7.20.3.1 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 = 3.78 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.96
method | result | size |
default | \(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{3}-x^{2}+2}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{3}-x^{2}+2}}{x}\right )\) | \(47\) |
pseudoelliptic | \(2 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {x^{3}-x^{2}+2}}{2 x}\right )-2 \arctan \left (\frac {\sqrt {x^{3}-x^{2}+2}}{x}\right )\) | \(47\) |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{2}+4 x \sqrt {x^{3}-x^{2}+2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x^{3}+x^{2}+2}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 x \sqrt {x^{3}-x^{2}+2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{3}+2}\right )\) | \(130\) |
elliptic | \(\text {Expression too large to display}\) | \(1019\) |
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (41) = 82\).
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.80 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 3 \, x^{2} + 2\right )}}{4 \, {\left (x^{4} - x^{3} + 2 \, x\right )}}\right ) - \arctan \left (\frac {\sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 2 \, x^{2} + 2\right )}}{2 \, {\left (x^{4} - x^{3} + 2 \, x\right )}}\right ) \]
sqrt(2)*arctan(1/4*sqrt(2)*sqrt(x^3 - x^2 + 2)*(x^3 - 3*x^2 + 2)/(x^4 - x^ 3 + 2*x)) - arctan(1/2*sqrt(x^3 - x^2 + 2)*(x^3 - 2*x^2 + 2)/(x^4 - x^3 + 2*x))
\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int \frac {\sqrt {\left (x + 1\right ) \left (x^{2} - 2 x + 2\right )} \left (x^{3} - 4\right )}{\left (x^{3} + 2\right ) \left (x^{3} + x^{2} + 2\right )}\, dx \]
\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 4\right )}}{{\left (x^{3} + x^{2} + 2\right )} {\left (x^{3} + 2\right )}} \,d x } \]
\[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\int { \frac {\sqrt {x^{3} - x^{2} + 2} {\left (x^{3} - 4\right )}}{{\left (x^{3} + x^{2} + 2\right )} {\left (x^{3} + 2\right )}} \,d x } \]
Time = 6.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 4.80 \[ \int \frac {\left (-4+x^3\right ) \sqrt {2-x^2+x^3}}{\left (2+x^3\right ) \left (2+x^2+x^3\right )} \, dx=\left (\sum _{_{\mathrm {X494}}\in \left \{-2^{1/3},2^{1/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right ),-2^{1/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\right \}\cup \mathrm {root}\left (z^3+z^2+2,z\right )}\frac {\sqrt {5}\,\sqrt {x\,\left (2-\mathrm {i}\right )+2-\mathrm {i}}\,\sqrt {3+x\,\left (-2+1{}\mathrm {i}\right )+1{}\mathrm {i}}\,\sqrt {3+x\,\left (-2-\mathrm {i}\right )-\mathrm {i}}\,\Pi \left (\frac {2+1{}\mathrm {i}}{_{\mathrm {X494}}+1};\mathrm {asin}\left (\frac {\sqrt {5}\,\sqrt {x\,\left (2-\mathrm {i}\right )+2-\mathrm {i}}}{5}\right )\middle |\frac {3}{5}+\frac {4}{5}{}\mathrm {i}\right )\,\left (2\,{_{\mathrm {X494}}}^5+6\,{_{\mathrm {X494}}}^3-2\,{_{\mathrm {X494}}}^2+12\right )\,\left (\frac {4}{25}+\frac {2}{25}{}\mathrm {i}\right )}{_{\mathrm {X494}}\,\left (_{\mathrm {X494}}+1\right )\,\sqrt {x^3-x^2+2}\,\left (6\,{_{\mathrm {X494}}}^4+5\,{_{\mathrm {X494}}}^3+12\,_{\mathrm {X494}}+4\right )}\right )+\frac {\sqrt {x\,\left (\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )+\frac {2}{5}-\frac {1}{5}{}\mathrm {i}}\,\sqrt {\frac {3}{5}+x\,\left (-\frac {2}{5}+\frac {1}{5}{}\mathrm {i}\right )+\frac {1}{5}{}\mathrm {i}}\,\sqrt {\frac {3}{5}+x\,\left (-\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )-\frac {1}{5}{}\mathrm {i}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {x\,\left (\frac {2}{5}-\frac {1}{5}{}\mathrm {i}\right )+\frac {2}{5}-\frac {1}{5}{}\mathrm {i}}\right )\middle |\frac {3}{5}+\frac {4}{5}{}\mathrm {i}\right )\,\left (4+2{}\mathrm {i}\right )}{\sqrt {x^3-x^2+2}} \]
symsum((5^(1/2)*(x*(2 - 1i) + (2 - 1i))^(1/2)*((3 + 1i) - x*(2 - 1i))^(1/2 )*((3 - 1i) - x*(2 + 1i))^(1/2)*ellipticPi((2 + 1i)/(_X494 + 1), asin((5^( 1/2)*(x*(2 - 1i) + (2 - 1i))^(1/2))/5), 3/5 + 4i/5)*(6*_X494^3 - 2*_X494^2 + 2*_X494^5 + 12)*(4/25 + 2i/25))/(_X494*(_X494 + 1)*(x^3 - x^2 + 2)^(1/2 )*(12*_X494 + 5*_X494^3 + 6*_X494^4 + 4)), _X494 in {-2^(1/3), 2^(1/3)*((3 ^(1/2)*1i)/2 + 1/2), -2^(1/3)*((3^(1/2)*1i)/2 - 1/2)} union root(z^3 + z^2 + 2, z)) + ((x*(2/5 - 1i/5) + (2/5 - 1i/5))^(1/2)*((3/5 + 1i/5) - x*(2/5 - 1i/5))^(1/2)*((3/5 - 1i/5) - x*(2/5 + 1i/5))^(1/2)*ellipticF(asin((x*(2/ 5 - 1i/5) + (2/5 - 1i/5))^(1/2)), 3/5 + 4i/5)*(4 + 2i))/(x^3 - x^2 + 2)^(1 /2)